enVision Math Common Core Grade 2 Answer Key

enVision Math Common Core Grade 2 Answer Key | enVision Math Common Core 2nd Grade Answers

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enVision Math Common Core Grade 2 Volume 1 Answer Key | enVision Math Common Core 2nd Grade Volume 1 Answers

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enVision Math Common Core Grade 1 Answer Key

enVision Math Common Core Grade 1 Answer Key | enVision Math Common Core 1st Grade Answers

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enVision Math Common Core Grade 1 Answers | enVision Math Common Core 1st Grade Textbook Answer Key

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enVision Math Common Core Grade 1 Volume 1 Answer Key | enVision Math Common Core 1st Grade Volume 1 Answers

enVision Math Common Core 1st Grade Volume 2 Answer Key | enVision Math Common Core Grade 1 Volume 2 Answers

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Envision Math Common Core Grade 3 Answer Key

Envision Math Common Core Grade 3 Answer Key | Envision Math Common Core 3rd Grade Answers

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Envision Math Common Core Grade 3 Volume 1 Answer Key | Envision Math Common Core 3rd Grade Volume 1 Answers

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enVision Math Common Core Grade K Answer Key

enVision Math Common Core Kindergarten Answer Key | enVision Math Common Core Grade K Answers

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enVision Math Common Core Grade Kindergarten Answers | enVision Math Common Core Grade K Textbook Answer Key

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Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations

enVision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations

Go through the enVision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations regularly and improve your accuracy in solving questions.

enVision Math Common Core 8th Grade Answers Key Topic 2 Analyze And Solve Linear Equations

Topic Essential Question
How can we analyze connections between linear equations, and use them to solve problems?
Answer:
One of the more obvious “connections” between linear equations is the presence of the same two variables (Generally, in most cases x and y) in these equations.
Assuming that your two equations are distinct (neither is merely a multiple of the other), we can use the “elimination by addition and subtraction” method or “Substitution method” to eliminate one variable, leaving us with an equation in one variable,
solve this 1-variable (x) equation, and then use the resulting value in the other equation to find the value of the other variable (y).
By doing this we find a unique solution (x, y) that satisfies both original equations.
Not only that but also this solution (x, y) will also satisfy both of the original linear equations.

3-ACT MATH
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 1

Powering Down
Do you know that feeling when you realize you left your charger at home? Uh-oh. It’s only a matter of time before your device runs out of power. Your battery percentage is dropping, but you still have so much left to do. Think about this during the 3-Act Mathematical Modeling lesson.
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 2

Topic 2 enVision STEM Project

Did You Know?
Demography is the study of changes, such as the number of births, deaths, or net migration, occurring in the human population over time.
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 3

Deaths Worldwide in 2015 (estimated)
Emigration is the act of leaving one’s country to settle elsewhere. In 2015, 244 million people, or 3.3% of the world’s population, lived outside their country of origin.
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 4
Immigration is the act of entering and settling in a foreign country. The United States has the largest immigrant population in the world.

Your Task: Modeling Population Growth
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 5
Human population numbers are in constant flux. Suppose a country has a population of 20 million people at the start of one year and during the year there are 600,000 births, 350,000 deaths, 100,000 immigrants, and 5,000 emigrants. You and your classmates will determine the total population at the end of the year and then model the expected change over a longer period.
Answer:
It is given that the population at the start of the year is 20 million people and during that year, there are 600,000 births, 350,000 deaths, 100,000 immigrants, and 5,000 emigrants
So,
The total population at the end of the year = (Total population at the start of the year) – ( Births + Deaths + Immigrants + Emigrants at that year)
= 20 million – (600,000 + 350,000 + 100,000 + 5,000)
= 20 million – 10.5 million
= 9.5 million
Change in Population = (Births + Immigration) – (Deaths + Emigration)
= (600,000 + 100,000) – (350,000 + 5,000)
= 700,000 – 355,000
= 345,000
Hence, from the above,
We can conclude that
The total population at the end of the year is: 9.5 million
The change in population at that year is: 345,000

Analyze And Solve Linear Equations 1

Topic 2 GET READY!

Review What You Know!

Vocabulary
Choose the best term from the box to complete each definition.
inverse operations
like terms
proportion
variables

Question 1.
In an algebraic expression, __________ are terms that have the same variables raised to the same exponents.
Answer:
We know that,
In an algebraic expression, “Like terms” are terms that have the same variables raised to the same exponents.
Hence, from the above,
We can conclude that the best term that fits the given definition is: Like terms

Question 2.
Quantities that represent an unknown value are _________.
Answer:
We know that,
Quantities that represent an unknown value are “Variables”
Hence, from the above,
We can conclude that the best term that fits the given definition is: Variables

Question 3.
A _________ is a statement that two ratios are equal.
Answer:
We know that,
A “Proportion” is a statement that two ratios are equal.
Hence, from the above,
We can conclude that the best term that fits the given definition is: Proportion

Analyze And Solve Linear Equations 2

Question 4.
Operations that “undo” each other are __________.
Answer:
We know that,
Operations that “undo” each other are ” Inverse Operations”
Hence, from the above,
We can conclude that the best term that fits the given definition is: Inverse Operations

Identify Like Terms

Complete the statements to identify the like terms in each expression.
Question 5.
4x + 7y – 62 + 6y – 9x
4x and ______ are like terms.
7y and _______ are like terms.
Answer:
The given expression is:
4x + 7y – 62 + 6y – 9x
We know that,
The “Like terms” are terms that have the same variables raised to the same exponents.
Hence, from the above,
We can conclude that
4x and 9x are like terms
7y and 6y are like terms

Question 6.
\(\frac{1}{2}\)s – (6u – 9u) + \(\frac{1}{10}\)s + 25
\(\frac{1}{2}\)s and _______ are like terms.
6u and _______ are like terms.
Answer:
The given expression is:
\(\frac{1}{2}\)s – (6u – 9u) + \(\frac{1}{10}\)s + 25
= \(\frac{1}{2}\)s + 9u – 6u + \(\frac{1}{10}\)s + 25
We know that,
The “Like terms” are terms that have the same variables raised to the same exponents.
Hence, from the above,
We can conclude that
\(\frac{1}{2}\)s and \(\frac{1}{10}\)s are like terms
6u and 9u are like terms

Solve One-Step Equations

Simplify each equation.
Question 7.
2x = 10
Answer:
The given expression is:
2x = 10
Divide by 2 into both sides
\(\frac{2}{2}\)x = \(\frac{10}{2}\)
x = 5
Hence, from the above,
We can conclude that the value of x is: 5

Question 8.
x + 3 = 12
Answer:
The given expression is:
x + 3 = 12
Subtract with 3 on both sides
x + 3 – 3 = 12 – 3
x = 9
Hence, from the above,
We can conclude that the vaue of x is: 9

Question 9.
x – 7 = 1
Answer:
The given expression is:
x – 7 = 1
Add with 7 on both sides
x – 7 + 7 = 1 + 7
x = 8
Hence, from the above,
We can conclude that the value of x is: 8

Simplify Fractions

Question 10.
Explain how to simplify the fraction \(\frac{12}{36}\).
Answer:
The given fraction is:
\(\frac{12}{36}\)
From the given fraction,
We can observe that the numerator and the denominator are the multiples of 12
So,
Divide the numerator by 12 and the denominator by 12
So,
\(\frac{12}{36}\) = \(\frac{1}{3}\)
Hence,
The simplified form of the given fraction is: \(\frac{1}{3}\)

Language Development
Fill in the Venn diagram to compare and contrast linear equations of the form y = mx and y = x + b.
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 6

In the box below, draw graphs to represent each form of the linear equations.

Topic 2 PICK A PROJECT

PROJECT 2A
If you had to escape from a locked room, how would you start?
PROJECT: DESIGN AN ESCAPE-ROOM ADVENTURE
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 7

PROJECT 2B
What animal would you most like to play with for an hour? Why?
PROJECT: PLAN A PET CAFÉ
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 8

PROJECT 2C
If you wrote a play, what would it be about?
PROJECT: WRITE A PLAY
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 9

PROJECT 2D
How many tiny steps does it take to cross a slackline?
PROJECT: GRAPH A WALKING PATTERN
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 10

Lesson 2.1 Combine Like Terms to solve Equations

Explore It!
A superintendent orders the new laptops shown below for two schools in her district. She receives a bill for $7,500.
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 11
I can… solve equations that have like terms on one side.

A. Draw a representation to show the relationship between the number of laptops and the total cost.
Answer:
It is given that she receives a bill for $7,500
So,
The total cost of the laptops that are given in the above figure = $7,500
Now,
Let the cost of a laptop be $x
So,
$3x + $4x + $3x = $7,500
$10x = $7,500
Hence, from the above,
We can conclude that
The representation to show the relationship between the number of laptops and the total cost is:
$10x = $7,500

B. Use the representation to write an equation that can be used to determine the cost of one laptop.
Answer:
From part (a),
The representation to show the relationship between the number of laptops and the total cost is:
$10x = $7,500
Divide with 10 into both sides
So,
\(\frac{$10x}{10}\) = \(\frac{$7,500}{10}\)
$x = $750
Hence, from the above,
We can conclude that
The representation to write an equation that can be used to determine the cost of one laptop is:
$x = $750

Analyze And Solve Linear Equations 3

Focus on math practices
Reasoning Why is it important to know that each laptop costs the same amount?
Answer:
From the given figure,
We can observe that all the laptops are of the same type
So,
Each laptop will cost the same amount since all the laptops are the same

Essential Question
How do you solve equations that contain like terms?
Answer:
We will solve the equations that contain like terms by rearranging the like terms on either the left side or the right side

Try It!

Selena spends $53.94 to buy a necklace and bracelet set for each of her friends. Each necklace costs $9.99, and each bracelet costs $7.99. How many necklace and bracelet sets, s, did Selena buy?
Selena buys necklace and bracelet sets for _________ friends.
_____ s + ______ s = 53.94
______ s = 53.94
s = ______
Answer:
Let each necklace and each bracelet be s
It is given that
The cost of each necklace is: $9.99
The cost of each bracelet is: $7.99
The total cost of a necklace and a bracelet is: $53.94
So,
$9.99s + $7.99s = $53.94
$17.98s = $53.94
$1798s = $5394
Divide by 1798 on both sides
\(\frac{$1798}{1798}\)s = \(\frac{$5394}{1798}\)
s = 3
Hence, from the above,
We can conclude that the number of necklace and bracelet sets that Selena buy is: 3

Convince Me!
Suppose the equation is 9.99s + 7.99s + 4.6 = 53.94. Can you combine the s terms and 4.6? Explain.
Answer:
The given equation is:
9.99s + 7.99s + 4.6 = 53.94
We know that,
We can only combine the terms only when they are the “Like terms”
So,
In the given equation,
9.99s and 7.99s are the like terms
53.94 and 4.6 are the like terms
Hence, from the above,
We can conclude that we can not combine the s terms and 4.6

Analyze And Solve Linear Equations 4

Try It!

Nat’s grocery bill was $150, which included a 5% club discount. What was Nat’s bill before the discount? Write and solve an equation.
Answer:
It is given that Nat’s grocery bill was $150 which included a 5% club discount
Now,
Let x be Nat’s bill before the discount
So,
To find Nat’s bill before discount, we have to find the value of 5% of 150 and add its value from 150
We know that,
The value of the bill will always be less after discount when compared to the value of the bill before discount
Now,
Nat’s bill before the discount = (Nat’s bill which included a 5% club discount) + (Value of 5% of 150)
x = $150 + (\(\frac{5}{100}\) × 150)
x = $150 + \(\frac{5 × 150}{100}\)
x = $150 + \(\frac{750}{100}\)
x = $150 + $7.5
x = $157.5
Hence, from the above,
We can conclude that Nat’s bill before the discount is: $157.5

Try It!

Solve for d.
a. –\(\frac{1}{4}\)d – \(\frac{2}{5}\)d = 39
Answer:
The given expression is:
–\(\frac{1}{4}\)d – \(\frac{2}{5}\)d = 39
-d (\(\frac{1}{4}\) + \(\frac{2}{5}\)) = 39
-d (0.25 + 0.40) = 39
-d (0.65) = 39
-d = \(\frac{39}{0.65}\)
-d = \(\frac{39 × 100}{65}\)
-d = 60
d = -60
Hence, from the above,
We can conclude that the value of d is: -60

b. -9.760 – (-12.81d) = 8.54
Answer:
The given expression is:
-9.760 – (-12.81d) = 8.54
-9.760 + 12.81d = 8.54
Rearrange the like terms in the above equation
So,
12.81d = 8.54 + 9.760
12.81d = 18.3
Divide by 12.81 on both sides
So,
\(\frac{12.81d}{12.81}\) = \(\frac{18.3}{12.81}\)
d = 1.428
Hence, from the above,
We can conclude that the value of d is: 1.428

KEY CONCEPT

In an equation with variable terms on one side, you can combine like terms before using inverse operations and properties of equality to solve the equation.
0.8n + 0.6n = 42
1.4n = 42 → Combine like terms.
\(\frac{1.4 n}{1.4}=\frac{42}{1.4}\)
n = 30

Do You Understand?
Question 1.
Essential Question How do you solve equations that contain like terms?
Answer:
In the equations that contain “Like terms”,
First, arrange the like terms at one side i.e., either the left side or the right side and combine them and then solve the equation for the desired result

Question 2.
Look for Relationships How do you recognize when an equation has like terms?
Answer:
We know that,
“Like terms” are terms that have the same variables raised to the same exponents.
Hence,
When there are the same variables in the given equation, we can call that terms “Like terms” in the given equation

Question 3.
Make Sense and Persevere in the equation 0.755 – \(\frac{5}{8}\)s = 44, how do you combine the like terms?
Answer:
The given equation is:
0.755 – \(\frac{5}{8}\)s = 44
We know that,
“Like terms” are terms that have the same variables raised to the same exponents.
So,
In the given equation,
0.755 and 44 are the like terms
So,
\(\frac{5}{8}\)s = 0.755 + 44
\(\frac{5}{8}\)s = 44.755
Multiply with \(\frac{8}{5}\) on both sides
So,
\(\frac{5}{8}\)s × \(\frac{8}{5}\) = 44.755 × \(\frac{8}{5}\)
s = 71.608
Hence, from the above,
We can conclude that the value of s is: 71.608

Do You Know How?
Question 4.
Henry is following the recipe card to make a cake. He has 95 cups of flour. How many cakes can Henry make?
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 12
Answer:
It is given that Henry is following the recipe card to make a cake and he has 95 cups of flour
It is also given that
We need
2\(\frac{2}{3}\) cups of flour for the batter
\(\frac{1}{2}\) cup of flour for the topping
Now,
Let the number of cakes be x
So,
By using the flour for the batter and the topping, Henry can make x cakes
Now,
(2\(\frac{2}{3}\) + \(\frac{1}{2}\))x = 95
We know that,
2\(\frac{2}{3}\) = \(\frac{8}{3}\)
So,
(\(\frac{8}{3}\) + \(\frac{1}{2}\))x = 95
\(\frac{19}{6}\)x = 95
Multiply with \(\frac{6}{19}\) on both sides
So,
\(\frac{19}{6}\)x × \(\frac{6}{19}\) = 95 × \(\frac{6}{19}\)
x = \(\frac{95 × 6}{19}\)
x = 30
Hence, from the above,
We can conclude that the number of cakes made by Henry is: 30

Question 5.
A city has a population of 350,000. The population has decreased by 30% in the past ten years. What was the population of the city ten years ago?
Answer:
It is given that a city has a population of 350,000 and it has decreased by 30% in the past ten years
Now,
Let the population of the city ten years ago be: x
To find the population of the city ten years ago,
We have to find the value of 30% of 350,000 and add it to the 350,000
The reason is it is given that the population i.e., 350,000 decreased in the past ten years. So, the population will be more than 350,000 ten years ago
So,
The population of the city ten years ago = (The population of the city in the past ten years) + (The value of 30% of 350,000)
x = 350,000 + \(\frac{30}{100}\) × 350,000
x = 350,000 + \(\frac{30 × 350,000}{100}\)
x = 350,000 + 105,000
x = 455,000
Hence, from the above,
We can conclude that the population of the city ten years ago is: 455,000

Question 6.
Solve the equation –12.2z – 13.4z = -179.2.
Answer:
The given equation is:
-12.2z – 13.4z = -179.2
From the given equation,
We can observe that 12.2 and 13.4 are the like terms
So,
-z(12.2 + 13.4) = -179.2
z(12.2 + 13.4) = 179.2
z(25.6) = 179.2
Divide by 25.6 into both sides
So,
\(\frac{25.6}{25.6}\)z = \(\frac{179.2}{25.6}\)
z = 7
Hence, from the above,
We can conclude that the value of z is: 7

Practice & Problem Solving

Leveled Practice In 7 and 8, complete the steps to solve for x.
Question 7.
\(\frac{4}{5}\)x – \(\frac{1}{4}\)x = 11
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 13
Answer:
The given equation is:
\(\frac{4}{5}\)x – \(\frac{1}{4}\)x = 11
x (\(\frac{4}{5}\) – \(\frac{1}{4}\)) = 11
x (\(\frac{16 – 5}{20}\)) = 11
\(\frac{11}{20}\)x = 11
Multiply with \(\frac{20}{11}\) on both sides
So,
\(\frac{20}{11}\) (\(\frac{11}{20}\)x) = 11 × \(\frac{20}{11}\)
x = \(\frac{11 × 20}{11}\)
x = 20
Hence, from the above,
We can conclude that the value of x is: 20

Question 8.
-0.65x + 0.45x = 5.4
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 14
Answer:
The given equation is:
-0.65x + 0.45x = 5.4
So,
x (0.45 – 0.65) = 5.4
x (-0.20) =5.4
Divide by -0.20 into both sides
So,
\(\frac{-0.20}{-0.20}\)x = \(\frac{5.4}{-0.20}\)
x = -27
Hence, from the above,
We can conclude that the avlue of x is: -27

In 9-12, solve for x.
Question 9.
\(\frac{4}{9}\)x + \(\frac{1}{5}\)x = 87
Answer:
The given equation is:
\(\frac{4}{9}\)x + \(\frac{1}{5}\)x = 87
So,
x (\(\frac{4}{9}\) + \(\frac{1}{5}\)) = 87
x (\(\frac{20 + 9}{45}\)) = 87
\(\frac{29}{45}\)x = 87
Multiply with \(\frac{45}{29}\) on both sides
So,
\(\frac{45}{29}\) (\(\frac{29}{45}\)x) = 87 × \(\frac{45}{29}\)
x = \(\frac{87 × 45}{29}\)
x = 135
Hence, from the above,
We can conclude that the value of x is: 135

Analyze And Solve Linear Equations 5

Question 10.
-3.8x – 5.9x = 223.1
Answer:
The given equation is:
-3.8x – 5.9x = 223.1
So,
-x (3.8 + 5.9) = 223.1
-x (9.7) =223.1
Divide by -9.7 into both sides
So,
\(\frac{-9.7}{-9.7}\)x = \(\frac{223.1}{-9.7}\)
x = -23
Hence, from the above,
We can conclude that the avlue of x is: -23

Question 11.
x + 0.15x = 3.45
Answer:
The givene quation is:
x + 0.15x = 3.45
So,
x (1 + 0.15) = 3.45
x (1.15) = 3.45
Divide be 1.15 into both sides
So,
\(\frac{1.15}{1.15}\)x = \(\frac{3.45}{1.15}\)
x = 3
Hence, from the above,
We can conclude that the value of x is: 3

Question 12.
–\(\frac{3}{5}\)x – \(\frac{7}{10}\) + \(\frac{1}{2}\)x = 56
Answer:
The given equation is:
–\(\frac{3}{5}\)x – \(\frac{7}{10}\) + \(\frac{1}{2}\)x = 56
x (\(\frac{1}{2}\) – \(\frac{3}{5}\)) – \(\frac{7}{10}\) = 56
x (\(\frac{5 – 6}{10}\)) – \(\frac{7}{10}\) = 56
–\(\frac{1}{10}\)x = 56 + \(\frac{7}{10}\)
–\(\frac{1}{10}\)x = \(\frac{560 + 7}{10}\)
Multiply with 10 on both sides
So,
–\(\frac{10}{10}\)x = \(\frac{567 × 10}{10}\)
-x = 567
x = -567
Hence, from the above,
We can conclude that the value of x is: -567

Question 13.
A contractor buys 8.2 square feet of sheet metal. She used 2.1 square feet so far and has $183 worth of sheet metal remaining. Write and solve an equation to find out how many sheets of metal costs per square foot.
Answer:
It is given that a contractor buys 8.2 square feet of sheet metal. She used 2.1 square feet so far and has $183 worth of sheet metal remaining.
So,
The remaining square feet of sheet metal = (Total square feet of sheet metal) – (The total square feet of sheet metal used so far)
The remaining square feet of sheet metal = 8.2 – 2.1
The remaining square feet of sheet metal = 6.1 square feet
Now,
It is given that there is$183 worth of sheet metal remaining
Now,
Let x be the number of sheet metals per square foot
So,
6.1x = $183
Divide by 6.1 into both sides
So,
\(\frac{6.1}{6.1}\)x = \(\frac{$183}{6.1}\)
x = 30
Hence, from the above,
We can conclude that the number of metal sheets per square foot is: 30

Question 14.
Make Sense and Persevere Clint prepares and sells trail mixes at his store. This week, he uses \(\frac{3}{8}\) his supply of raisins to make regular trail mix and \(\frac{1}{4}\) of his supply to make spicy trail mix. If Clint uses 20 pounds of raisins this week, how many pounds of raisins did he have at the beginning of the week?
Answer:
It is given that Clint prepares and sells trail mixes at his store and this week, he uses \(\frac{3}{8}\) his supply of raisins to make regular trail mix and \(\frac{1}{4}\) of his supply to make spicy trail mix.
So,
The total amount of raisins to make trail mix = (The supply of raisins to make regular mix) + (The supply of raisins to make spicy mix)
The total amount of raisins to make trail mix = \(\frac{3}{8}\) + \(\frac{1}{4}\)
The total amount of raisins to make trail mix = \(\frac{5}{8}\)
Now,
Let the number of pounds of raisins at the beginning of the week be x
So,
\(\frac{5}{8}\)x = 20
Multiply with \(\frac{8}{5}\) on both sides
So,
x = 20 × \(\frac{8}{5}\)
x = \(\frac{20 × 8}{5}\)
x = 32 pounds
Hence, from the above,
We can conclude that the number of pounds of raisins at the beginning of the week is: 32 pounds

Question 15.
Make Sense and Persevere A submarine descends to \(\frac{1}{6}\) of its maximum depth. Then it descends another \(\frac{2}{3}\) of its maximum depth. If it is now at 650 feet below sea level, what is its maximum depth?
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 15
Answer:
It is given that a submarine descends to \(\frac{1}{6}\) of its maximum depth and then it descends another \(\frac{2}{3}\) of its maximum depth and it is now at 650 feet below sea level
Now,
Let x be the maximum depth
So,
\(\frac{1}{6}\)x + \(\frac{2}{3}\)x = 650
\(\frac{1 + 4}{6}\)x = 650
\(\frac{5}{6}\)x = 650
Multiply with \(\frac{6}{5}\) on both sides
So,
x = 650 × \(\frac{6}{5}\)
x = \(\frac{650 × 6}{5}\)
x = 780 feet
Hence, from the above,
We can conclude that the maximum depth is: 780 feet

Question 16.
Model with Math Write an equation that can be represented by the bar diagram, then solve.
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 16
Answer:
The given bar diagram is:
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 16
So,
From the bar diagram,
The representation of the equation is:
-1.2y + (-4.2y) = -3.78
-1.2y – 4.2y = -3.78
– (1.2y + 4.2y) = -3.78
1.2y + 4.2y = 3.78
5.4y = 3.78
Divide by 5.4 into both sides
So,
\(\frac{5.4}{5.4}\)y = \(\frac{3.78}{5.4}\)
y = 0.7
Hence, from the above,
We can conclude that the value of y is: 0.7

Question 17.
Higher Order Thinking Solve \(\frac{2}{3}\)h – 156 = 3\(\frac{13}{24}\).
Answer:
The given equation is:
\(\frac{2}{3}\)h – 156 = 3\(\frac{13}{24}\)
We know that,
3\(\frac{13}{24}\) = \(\frac{85}{24}\)
So,
\(\frac{2}{3}\)h – 156 = \(\frac{85}{24}\)
\(\frac{2}{3}\)h = \(\frac{85}{24}\) + 156
0.666h = 3.541 + 156
0.666h = 159.541
Divide by 0.666 into both sides
So,
h = \(\frac{159.541}{0.666}\)
h = 239.552
Hence, from the above,
We can conclude that the value of ‘h’ is: 239.552

Question 18.
Model with Math Nathan bought one notebook and one binder for each of his college classes. The total cost of the notebooks and binders was $27.08. Draw a bar diagram to represent the situation. How many classes is Nathan taking?
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 17
Answer:
It is given that Nathan bought one notebook and one binder for each of his college classes. The total cost of the notebooks and binders was $27.08.
Now,
Let the number of notebooks and binders that Nathan bought be x
From the figure,
It is given that
The cost of 1 notebook is: $0.95
The cost of 1 binder is: $5.82
So,
The representation of the cost of total notebooks and binders in the form of the equation is:
$0.95x + $5.82x = $27.08
Hence,
The representation of the above equation in the form of a bar diagram is:

Assessment Practice
Question 19.
Construct Arguments Your friend incorrectly says the solution to the equation –\(\frac{3}{5}\)y – \(\frac{1}{7}\)y = 910 is y = 676. What error did your friend make?
A. Added –\(\frac{1}{7}\) to –\(\frac{3}{5}\)
B. Subtracted \(\frac{1}{7}\) from –\(\frac{3}{5}\)
C. Multiplied 910 by \(\frac{26}{35}\)
D. Multiplied 910 by \(\frac{35}{26}\)
Answer:
The given equation is:
–\(\frac{3}{5}\)y – \(\frac{1}{7}\)y = 910
-y (\(\frac{3}{5}\) + \(\frac{1}{7}\)) = 910
–\(\frac{26}{35}\)y = 910
Multiply with –\(\frac{35}{26}\) on both sides
So,
y = -910 × \(\frac{35}{26}\)
y = -1,225
Hence from the above,
We can conclude that the error your friend makes is:
Multiplied 910 by \(\frac{26}{35}\)

Question 20.
A 132-inch board is cut into two pieces. One piece is three times the length of the other. Find the length of the shorter piece.
PART A
Draw a bar diagram to represent the situation.
Answer:
It is given that a 132-inch board is cut into two pieces and one piece is 3 times the length of the other
Now,
Let the length of 1 piece be x inches
So,
The length of the other piece is: 3x inches
So,
The representation of the given situation in the form of an equation is:
3x + x = 132
Hence,
The representation of the above equation in the form of a bar diagram is:

PART B
Write and solve an equation to find the length of the shorter piece.
Answer:
From part (a),
The equation that represents the given situation is:
3x + x = 132
4x = 132
Divide by 4 into both sides
So,
x = \(\frac{132}{4}\)
x = 33 inches
Hence,from the above,
We can conclude that the length of the shorter piece is: 33 inches

Lesson 2.2 Solve Equations with Variables on Both Sides

Solve & Discuss It!
Jaxson and Bryon collected an equal amount of money during a car wash. They collected cash and checks as shown below. If each check is written for the same amount, x, what is the total amount of money collected by both boys? Explain.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 18
I can… solve equations with variables on both sides of the equal sign.
Answer:
It is given that Jaxson and Bryon collected an equal amount of money during the car wash.
It is also given that they collected cash and checks and each check is written for the same amount x
So,
The amount earned by Jaxson = The amount earned by Bryon
Now,
From the given figure,
The amount earned by Jaxson = The amount earned by cash + The amount earned by checks
= 15 + 14x
The amount earned by Bryon = The amount earned by cash + The amount earned by checks
= 50 + 7x
So,
Now,
15 + 14x = 50 + 7x
Subtract with 7x on both sides
15 + 14x – 7x = 50 + 7x – 7x
15 + 7x = 50
Subtract with 15 on both sides
15 + 7x – 15 = 50 – 15
7x = 35
Divide by 7 on both sides
\(\frac{7}{7}\)x = \(\frac{35}{7}\)
x = 5
So,
The total amount of money collected by both boys = 15 +14x + 50 + 7x
= 21x + 65
= 21 (5) + 65
= 105 + 65
= $170
Hence, from the above,
We can conclude that the total amount earned by both the boys is: $170

Reasoning
How can you use an equation to show that expressions are equal?
Answer:
Combine any like terms on each side of the equation i.e., x-terms with x-terms and constants with constants. Arrange the terms in the same order, usually x-term before constants.
If all of the terms in the two expressions are identical, then the two expressions are equivalent.

Focus on math practices
Model with Math What expressions can you write to represent the amount of money collected by each boy? How can you use these expressions to write an equation?
Answer:
From the given figure,
We can observe that the two boys earned cash and checks
So,
The total amount earned by any boy = The amount earned due to cash + The amount earned due to checks
Now,
The amount earned by Jaxson = The amount earned by cash + The amount earned by checks
= 15 + 14x
The amount earned by Bryon = The amount earned by cash + The amount earned by checks
= 50 + 7x
Now,
It is given that the amount earned by both boys are equal
So,
The amount earned by Jaxson = The amount earned by Bryon
15 + 14x = 50 + 7x
Rearrange the like terms
14x – 7x = 50 – 15
7x = 35
Hence, from the above,
We can conclude that the representation of the amount collected by each boy in the form of the equation is:
7x = 35

Essential Question
How do you use inverse operations to solve equations with variables on both sides?
Answer:
The “Inverse operations” allow us to undo what has been done to the variable
Example:
Solve:
x+3=8
From the above equation,
We can observe that 
3 has been added to the variable, x.
We know that,
The inverse of addition is subtraction
So,
By subtracting 3, We can undo the addition.
Now,
After 3 was added, the result was equal to 8.
We undo the addition, by subtracting 3 and see that, the starting amount was 5

Try It!

Class A was given a sunflower with a height of 8 centimeters that grows at a rate of 3\(\frac{1}{2}\) centimeters per week. Class B was given a sunflower with a height of 10 centimeters that grows at a rate of 3\(\frac{1}{4}\) centimeters per week. After how many weeks are the sunflowers the same height?
Let w= the number of weeks.
____ w + 8 = _____ w + 10
_____ w + 8 = 10
_____ w = _____
w = _____
The sunflowers are the same height after ________ weeks.
Answer:
It is given that
Class A was given a sunflower with a height of 8 centimeters that grows at a rate of 3\(\frac{1}{2}\) centimeters per week and class B was given a sunflower with a height of 10 centimeters that grows at a rate of 3\(\frac{1}{4}\) centimeters per week.
Now,
Let the number of weeks be w
So,
The height of a sunflower of class A after w weeks = 3\(\frac{1}{2}\)w + 8
We know that,
3\(\frac{1}{2}\) = \(\frac{7}{2}\)
So,
The height of a sunflower of class A after w weeks = \(\frac{7}{2}\)w + 8
Now,
The height of a sunflower of class B after w weeks = 3\(\frac{1}{4}\)w + 10
We know that,
3\(\frac{1}{4}\) = \(\frac{13}{4}\)
So,
The height of a sunflower of class A after w weeks = \(\frac{13}{4}\)w + 10
Now,
To make the height of a sunflower from both classes equal,
The height of sunflower of class A after w weeks = The height of sunflower of class B after w weeks
\(\frac{7}{2}\)w + 8 = \(\frac{13}{4}\)w + 10
Rearrange the like terms
\(\frac{7}{2}\)w – \(\frac{13}{4}\)w = 10 – 8
\(\frac{14 – 13}{4}\)w = 2
\(\frac{1}{4}\)w = 2
Multiply with 4 on both sides
\(\frac{4}{4}\)w = 2 (4)
w = 8 weeks
Hence, from the above,
We can conclude that after 8 weeks, the sunflowers of class A and class B are of the same height

Convince Me!
How can you check your work to make sure the value of the variable makes the equation true? Explain.
Answer:
To make a true equation, check your math to make sure that the values on each side of the equals sign are the same. Ensure that the numerical values on both sides of the “=” sign are the same to make a true equation.
Examples:
a) 9 = 9 is a true equation
b) 5 + 4 = 9 is a true equation

Try It!

Solve the equation 96 – 4.5y – 3.2y = 5.6y + 42.80.
Answer:
The given equation is:
96 – 4.5y – 3.2y = 5.6y + 42.80
Now,
Rearrange the like terms at one side i.e., y-terms to one side and the constant terms to other side
So,
-5.6y – 4.5y – 3.2y = 42.80 – 96
-13.3y = -53.2
13.3y = 53.2
Divide by 13.3 into both sides
So,
\(\frac{13.3}{13.3}\)y = \(\frac{53.2}{13.3}\)
y = 4
Hence, from the above,
We can conclude that the value of y is: 4

KEY CONCEPT

When two expressions represent equal quantities, they can be set equal to each other. Then you can use inverse operations and properties of equality to combine like terms and solve for the unknown.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 19
3x + 15 = 4x + 12
3x – 3x + 15 = 4x – 3x + 12
15 = x + 12
15 – 12 = x + 12 – 12
3 = x

Do You Understand?
Question 1.
Essential Question How do you use inverse operations to solve equations with variables on both sides?
Answer:
The “Inverse operations” allow us to undo what has been done to the variable
Example:
Solve:
x+3=8
From the above equation,
We can observe that 
3 has been added to the variable, x.
We know that,
The inverse of addition is subtraction
So,
By subtracting 3, We can undo the addition.
Now,
After 3 was added, the result was equal to 8.
We undo the addition, by subtracting 3 and see that, the starting amount was 5

Question 2.
Reasoning Why are inverse operations and properties of equality important when solving equations? Explain.
Answer:
An “Inverse operation” is two operations that undo each other
Ex: Addition and Subtraction or Multiplication and Division.
You can perform the same inverse operation on each side of an equivalent equation without changing the equality.
This gives us a couple of properties that hold true for all equations.

Question 3.
Model with Math Cynthia earns $680 in commissions and is paid $10.25 per hour. Javier earns $410 in commissions and is paid $12.50 per hour. What will you find if you solve for x in the equation 10.25x + 680 = 12.5x + 410?
Answer:
It is given that
Cynthia earns $680 in commissions and is paid $10.25 per hour. Javier earns $410 in commissions and is paid $12.50 per hour.
It is also given that
The representation of the given situation in the form of the equation is:
10.25x + 680 = 12.5x + 410
From the above equation,
We can observe that
10.25x is the amount paid to Cynthia per hour and x is the number of hours
Hence, from the above,
We can conclude that the variable x represents the “Number of hours”

Do You Know How?
Question 4.
Maria and Liam work in a banquet hall. Maria earns a 20% commission on her food sales. Liam earns a weekly salary of $625 plus a 10% commission on his food sales. What amount of food sales will result in Maria and Liam earning the same amount for the week?
Answer:
It is given that
Maria earns a 20% commission on her food sales. Liam earns a weekly salary of $625 plus a 10% commission on his food sales.
So,
To find the number of food sales that will result in Maria and Liam earning the same amount for the week,
20%x = $625 + 10%x
Where,
x is the number of food sales
So,
\(\frac{20}{100}\)x = $625 + \(\frac{10}{100}\)x
Rearrange the like terms
\(\frac{20 – 10}{100}\)x = $625
\(\frac{10}{100}\)x = $625
\(\frac{1}{10}\)x = $625
Multiply with 10 on both sides
So,
\(\frac{10}{10}\)x = $625 (10)
x = $6,250
Hence, from the above,
We can conclude that the number of food sales that will make the same amount in the week for Maria and Liam is: $6,250

Question 5.
Selma’s class is making care packages to give to victims of a natural disaster. Selma packs one box in 5 minutes and has already packed 12 boxes. Her friend Trudy packs one box in 7 minutes and has already packed 18 boxes. How many more minutes does each need to work in order to have packed the same number of boxes?
Answer:
It is given that
Selma’s class is making care packages to give to victims of a natural disaster. Selma packs one box in 5 minutes and has already packed 12 boxes. Her friend Trudy packs one box in 7 minutes and has already packed 18 boxes.
Now,
Let x be the number of  more minutes that each has to work so that they have the same number of boxes
So,
To find the more minutes each need to work in order to have packed the same number of boxes,
\(\frac{x}{5}\) + 12 = \(\frac{x}{7}\) + 18
Rearrange the like terms
So,
\(\frac{x}{5}\) – \(\frac{x}{7}\) = 18 – 12
\(\frac{7x – 5x}{35}\) = 6
\(\frac{2x}{35}\) = 6
Divide by 35 into both sides
So,
2x = 6 (35)
Divide by 2 into both sides
So,
x = \(\frac{6 (35)}{2}\)
x = 3 (35)
x = 105 minutes
Hence, from the above,
We can conclude that the number of more minutes that each need to work so that the number of boxes becomes equal is: 105 minutes

Question 6.
Solve the equation –\(\frac{2}{5}\)x + 3 = \(\frac{2}{3}\)x + \(\frac{1}{3}\).
Answer:
The given equation is:
–\(\frac{2}{5}\)x + 3 = \(\frac{2}{3}\)x + \(\frac{1}{3}\)
Rearrange the like terms
So,
\(\frac{2}{3}\)x + \(\frac{2}{5}\)x = 3 – \(\frac{1}{3}\)
\(\frac{10 + 6}{15}\)x = \(\frac{9 – 1}{3}\)
\(\frac{16}{15}\)x = \(\frac{8}{3}\)
Multiply with \(\frac{15}{16}\) on both sides
x = \(\frac{8}{3}\) × \(\frac{15}{16}\)
x = \(\frac{8 × 15}{3 × 16}\)
x = \(\frac{5}{2}\)
Hence, from the above,
We can conclude that the value of x is: \(\frac{5}{2}\)

Question 7.
Solve the equation -2.6b + 4 = 0.9b – 17.
Answer:
The given equation is:
-2.6b + 4 = 0.9b – 17
Rearrange the like terms
So,
0.9b + 2.6b = 17 + 4
3.5b = 21
Divide by 3.5 into both sides
So,
\(\frac{3.5}{3.5}\)b = \(\frac{21}{3.5}\)
b = 6
Hence, from the above,
We can conclude that the value of b is: 6

Practice & Problem Solving

Leveled Practice In 8 and 9, solve each equation.
Question 8.
6 – 4x = 6x – 8x + 2
6 – 4x = ____ + 2
6 = _____ + 2
____ = _____
_______ = x
Answer:
The given equation is:
6 – 4x = 6x – 8x + 2
So,
6 – 4x = 2 – 2x
Rearrange the like terms
So,
4x – 2x = 6 – 2
2x = 4
Divide by 2 into both sides
So,
\(\frac{2}{2}\)x = \(\frac{4}{2}\)
x = 2
Hence, from the above,
We can conclude that the value of x is: 2

Question 9.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 20
Answer:
The given equation is:
\(\frac{5}{3}\)x + \(\frac{1}{3}\)x = 13\(\frac{1}{3}\) + \(\frac{8}{3}\)x
Rearrange the like terms
So,
\(\frac{5 + 1}{3}\)x – \(\frac{8}{3}\)x = 13\(\frac{1}{3}\)
\(\frac{6 – 8}{3}\)x = 13\(\frac{1}{3}\)
–\(\frac{2}{3}\)x = \(\frac{40}{3}\)
Multiply with 3 on both sides
So,
-2x = 40
divide by -2 into both sides
So,
x = \(\frac{-40}{2}\)
x = -20
Hence, from the above,
We can conclude that the value of x is: -20

Question 10.
Two towns have accumulated different amounts of snow. In Town 1, the snow depth is increasing by 3\(\frac{1}{2}\) inches every hour. In Town 2, the snow depth is increasing by 2\(\frac{1}{4}\) inches every hour. In how many hours will the snowfalls of the towns be equal?
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 21
Answer:
It is given that
Two towns have accumulated different amounts of snow. In Town 1, the snow depth is increasing by 3\(\frac{1}{2}\) inches every hour. In Town 2, the snow depth is increasing by 2\(\frac{1}{4}\) inches every hour.
Now,
Let x be the number of hours
So,
To make the snowfalls of the two towns equal,
5 + 3\(\frac{1}{2}\)x = 6 + 2\(\frac{1}{4}\)x
We know that,
3\(\frac{1}{2}\) = \(\frac{7}{2}\)
2\(\frac{1}{4}\) = \(\frac{9}{4}\)
So,
\(\frac{7}{2}\)x – \(\frac{9}{4}\)x = 6 – 5
\(\frac{14 – 9}{4}\)x = 1
\(\frac{5}{4}\)x = 1
Multiply with \(\frac{4}{5}\) on both sides
So,
x = \(\frac{4}{5}\)
x = 0.8 hours
Hence, from the above,
We can conclude that after 0.8 hours, the snowfalls of the two towns will be equal

Question 11.
Solve the equation 5.3g + 9 = 2.3g + 15.
a. Find the value of g.
Answer:
The given equation is:
5.3g + 9 = 2.3g + 15
Rearrange the like terms
So,
5.3g – 2.3g = 15 – 9
3.0g = 6
Divide by 3 into both sides
\(\frac{3}{3}\)g = \(\frac{6}{3}\)
g = 2
Hence, from the above,
We can conclude that the value of g is: 2

b. Explain how you can check that the value · you found for g is correct. If your check does not work, does that mean that your result is incorrect? Explain.
Answer:
From part (a),
We get the value of g : 2
So,
Whether the value of g is correct or not, put it in the given equation
If LHS = RHS,
Then, your check is correct. Otherwise, your check is not correct
Now,
5.3g + 9 = 2.3g + 15
Put, g = 2
So,
5.3 (2) + 9 = 2.3 (2) + 15
10.6 + 9 = 4.6 + 15
19.6 = 19.6
Hence, from the above,
We can conclude that the check is correct

Question 12.
Solve the equation 6 – 6x = 5x – 9x – 2.
Answer:
The given equation is:
6 – 6x = 5x – 9x – 2
So,
6 – 6x = -4x – 2
Rearrange the like terms
So,
-4x + 6x = 6 + 2
2x = 8
Divide by 2 into both sides
So,
\(\frac{2}{2}\)x = \(\frac{8}{2}\)
x = 4
Hence, from the above,
We can conclude that the value of x is: 4

Question 13.
Model with Math The population of one town in Florida is 43,425. About 125 people move out of the town each month. Each month, 200 people on average move into town. A nearby town has a population of 45,000. It has no one moving in and an average of 150 people moving away every month. In about how many months will the population of the towns be equal? Write an equation that represents this situation and solve it.
Answer:
It is given that
The population of one town in Florida is 43,425. About 125 people move out of the town each month. Each month, 200 people on average move into town. A nearby town has a population of 45,000. It has no one moving in and an average of 150 people moving away every month.
Now,
Let the population that are moving in and moving out be x
We know that,
Moving in will be positive and Moving out will be negative
So,
The population of one town in Florida = 43,425 + 200x – 125x
The population of a nearby town = 45,000 – 150x
So,
To find out after how many months, they will be equal,
43,425 + 200x – 125x = 45,000 – 150x
43,425 + 75x = 45,000 – 150x
Rearrange the like terms
So,
150x + 75x = 45,000 – 43,425
225x = 1,575
Divide by 225 into both sides
\(\frac{225}{225}\)x = \(\frac{1,575}{225}\)
x = 7
Hence, from the above,
We can conclude that after 7 months, the population of the towns will be equal

Question 14.
Veronica is choosing between two health clubs After how many months will the total cost for each health club be the same?
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 22
Answer:
It is given that Veronica is choosing between two health clubs
Now,
Let x be the number of months so that the cost for the two health clubs will be the same
Now,
The total health cost of Yoga studio A = 22 + 24.50x
The total health cost of Yoga studio B = 47 + 18.25x
So,
To find out after how many months, the total cost for the two health clubs will be the same,
22 + 24.50x = 47 + 18.25x
Rearrange the like terms
So,
47 – 22 = 24.50x – 18.25x
25 = 6.25x
Divide by 25 into both sides
So,
\(\frac{25}{25}\) = \(\frac{6.25}{25}\)x
1 = 0.25x
\(\frac{x}{4}\) = 1
x = 4
Hence, from the above,
We can conclude that after 4 months, the total cost for the two health clubs will be the same

Question 15.
Higher-Order Thinking The price of Stock A at 9 A.M. was $12.73. Since then, the price has been increasing at the rate of $0.06 per hour. At noon, the price of Stock B was $13.48. It begins to decrease at the rate of $0.14 per hour. If the stocks continue to increase and decrease at the same rates, in how many hours will the prices of the stocks be the same?
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 23
Answer:
It is given that
The price of Stock A at 9 A.M. was $12.73. Since then, the price has been increasing at the rate of $0.06 per hour. At noon, the price of Stock B was $13.48. It begins to decrease at the rate of $0.14 per hour.
Now,
Let x be the number of hours
So,
The price of stock A = $12.73 + $0.06x (Since it is increasing)
The price of stock B = $13.48 – $0.14x (Since it is decreasing)
Now,
To find out after how many hours, the prices will be the same,
$12.73 + $0.06x = $13.48 – $0.14x
Rearrange the like terms
So,
$13.48 – $12.73 = $0.14x + $0.06x
$0.75 = $0.2x
Divide by 0.2 into both sides
So,
x = \(\frac{0.75}{0.2}\)
x = 3.75
x = 3.60 + 0.15
x = 4 hours 15 minutes
Hence, from the above,
We can conclude that after 4 hours 15 minutes, the prices of the stocks will be equal

Assessment Practice
Question 16.
In an academic contest, correct answers earn 12 points and incorrect answers lose 5 points. In the final round, School A starts with 165 points and gives the same number of correct and incorrect answers. School B starts with 65 points and gives no incorrect answers and the same number of correct answers as School A. The game ends with the two schools tied.
PART A
Which equation models the scoring in the final round and the outcome of the contest?
A. 12x + 5x – 165 = -12x + 65
B. 12x – 5x + 165 = 12x + 65
C. 5x – 12x + 165 = 12x + 65
D. 12x – 5x – 165 = 12x + 65
Answer:
It is given that
In an academic contest, correct answers earn 12 points and incorrect answers lose 5 points. In the final round, School A starts with 165 points and gives the same number of correct and incorrect answers. School B starts with 65 points and gives no incorrect answers and the same number of correct answers as School A. The game ends with the two schools tied.
Now,
Let the number of answers be x
We know that,
The points earned for the correct answers will be positive whereas, for the negative answers, they will be negative
So,
For school A,
The number of answers is:
12x – 5x = -165
12x – 5x + 165 = 0
For school B,
The number of answers is:
12x + 0 = -65
12x + 65 = 0
Now,
It is given that the two schools are tied
So,
12x – 5x + 165 = 12x + 65
Hence, from the above,
We can conclude that option B matches the above-given situation

PART B
How many answers did each school get correct in the final round?
Answer:
From part (a),
The equation that models the scoring and outcome of the contest is:
12x – 5x + 165 = 12x + 65
Now,
Rearrange the terms
So,
12x – 12x – 5x = 65 – 165
-5x = -100
5x = 100
Divide by 5 into both sides
So,
\(\frac{5}{5}\)x = \(\frac{100}{5}\)
x = 20
Hence, from the above,
We can conclude that each school gets 20 correct answers in the final round

Lesson 2.3 Solve Multistep Equations

Solve & Discuss It!
A water tank fills through two pipes. Water flows through one pipe at a rate of 25,000 gallons an hour and through the other pipe at 45,000 gallons an hour. Water leaves the system at a rate of 60,000 gallons an hour.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 24
I can… solve multistep equations and pairs of equations using more than one approach.
There are 3 of these tanks, and each tank holds 1 million gallons. Each tank is half full. Water is entering and leaving a tank at the maximum amounts. Determine the number of hours, x, it will take to fill all 3 tanks.
Answer:
It is given that
A water tank fills through two pipes. Water flows through one pipe at a rate of 25,000 gallons an hour and through the other pipe at 45,000 gallons an hour. Water leaves the system at a rate of 60,000 gallons an hour and there are 3 of these tanks, and each tank holds 1 million gallons. Each tank is half full. Water is entering and leaving a tank at the maximum amounts.
Now,
The capacity of each tank = \(\frac{1 million}{2}\) (Since the tank is half-full)
We know that,
1 million = 10 lakhs
So,
The capacity of each tank is: 5 Lakh gallons
So,
The capacity of 3 tanks = 5 Lakh gallons (3)
= 15 Lakh gallons
Now,
The rate of flow of each tank = (The rate of flow of inlet pipes) + (The rate of flow of outlet pipes)
We know that,
The rate of flow for the inlet pipe will be: Positive
The rate of flow for the outlet pipe will be: Negative
So,
The rate of flow of each tank = (45,000 + 25,000) – 60,000
= 70,000 – 60,000
= 10,000 gallons per hour
Since the three pipes are the same, the rate of flow will also be the same
So,
The rate of flow of three tanks = 10,000 (3)
= 30,000 gallons per hour
Now,
It is given that the number of hours is: x
So,
The number of hours took to fill all the three tanks = \(\frac{ The capacity of three tanks } { The rate of flow of the three tanks }\)
x = \(\frac{15,00,000}{30,000}\)
x = 50 hours
Hence, from the above,
We can conclude that the number of hours took to fill the three tanks is: 50 hours

Reasoning
Can you solve the problem in more than one way?
Answer:
Yes, we can solve the problem in more than one way
The first way:
First, calculate the capacity and the rate of flow of each tank and multiply both the quantities with 3 since it is for 3 tanks
So,
We will get the time took to fill the three tanks
The second way:
Calculate the capacity and the rate of the flow of each tank and also find the time taken to fill that tank and multiply the time taken by 3 to get the time taken to fill the three tanks

Focus on math practices
Use Structure What are two different ways to simplify the expression 4(3x + 7x + 5) so that it equals 40x – 20? Explain.
Answer:
The given expression is:
4 (3x + 7x + 5)
A)
The first way:
We know that,
The distributive property is:
a (b + c) = ab + ac
So,
4 (3x + 7x + 5)
= 4 (3x) + 4 (7x) + 4 (5)
= 12x + 28x + 20
= 40x + 20
B)
The second way:
4 (3x + 7x + 5)
First, simplify the expression in the brackets
So,
4 (3x + 7x + 5)
= 4 (10x + 5)
= 4 (10x) + 4(5)
= 40x + 20

Essential Question
How can you use the Distributive Property to solve multistep equations?
Answer:
Let A, B, and C be the three variables
Now,
We know that,
The Distributive Property of multiplication is:
A (B + C) = AB + AC
(A + B) C = AC + BC
(A + C) B = AB + BC

Try It!

Solve the equation 3(x – 5) – 5x = -25 + 6x.
3_____ + 3 ∙ ______ – 5x = – 25 + 6x
_____ – 5x = – 25 + 6x
______ x – 15 = – 25 + 6x
______ – 15 = -25 + _____ x
______ = _____ x ______
x = _____ or ______
Answer:
The given equation is:
3 (x – 5) – 5x = -25 + 6x
3 (x) – 3 (5) – 5x = -25 + 6x
3x – 15 – 5x = -25 + 6x
-15 – 2x = -25 + 6x
Rearrange the like terms
So,
-15 + 25 = 6x + 2x
8x = 10
Divide by 8 into both sides
x = \(\frac{10}{8}\)
x = \(\frac{5}{4}\)
Hence, from the above,
We can conclude that the value of x is: \(\frac{5}{4}\)

Convince Me!
Can you add x to -5x on the left side of the equation as the first step? Explain.
Answer:
No, we can’t add x to the -5x because from the given equation,
We are getting
3x – 5x
So,
We have to add 3x and not x to -5x

Try It!

Solve the equation -3(-7 – x) = \(\frac{1}{2}\)(x + 2).
Answer:
The given equation is:
-3 (-7 – x) = \(\frac{1}{2}\) (x + 2)
So,
-3 [-(x + 7)] = \(\frac{1}{2}\) (x + 2)
We know that,
– * – = +
So,
3 (x + 7) = \(\frac{1}{2}\) (x + 2)
Multiply with 2 on both sides
So,
6 (x + 7) = x + 2
6 (x) + 6 (7) = x + 2
6x + 42 = x + 2
Rearrange the like terms
6x – x = 2 – 42
5x = -40
Divide by 5 into both sides
So,
x = \(\frac{-40}{5}\)
x = -8
Hence, from the above,
We can conclude that the value of x is: -8

KEY CONCEPT

When solving multistep equations, sometimes you distribute first and then combine like terms.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 25
Sometimes you combine like terms first and then distribute.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 26

Do You Understand?
Question 1.
Essential Question How can you use the Distributive Property to solve multistep equations?
Answer:
Let A, B, and C be the three variables
Now,
We know that,
The Distributive Property of multiplication is:
A (B + C) = AB + AC
(A + B) C = AC + BC
(A + C) B = AB + BC

Question 2.
Reasoning What is the first step when solving the equation 3(3x – 5x) + 2 = -8?
Answer:
The given equation is:
3 (3x – 5x) + 2 = -8
Use the distributive property of multiplication
So,
3 (3x) – 3(5x) + 2 = -8 ——–> First step when solving the above equation

Question 3.
Use Structure How can you use the order of operations to explain why you cannot combine the variable terms before using the Distributive Property when solving the equation 7(x + 5) – x = 42?
Answer:
The given equation is:
7 (x + 5) – x = 42
To find the order of operations, We have to use the BODMAS rule
So,
From the above equation,
We will first solve the expression present in the brackets, then add, and then subtract
We know that,
We can do any operation only on the like terms
We know that,
The “Like terms” are the terms that have the same exponent
So,
For the above equation,
We can not combine the terms before using the distributive property
Now,
7 (x) + 7 (5) – x = 42
7x + 35 – x = 42
6x + 35 = 42
Subtract with 35 on both sides
6x = 42 – 35
6x = 7
Divide by 6 into both sides
So,
x = \(\frac{7}{6}\)
Hence, from the above,
We can conclude that the value of x is: \(\frac{7}{6}\)

Do You Know How?
Question 4.
Solve the equation 3x + 2 = x + 4(x + 2).
Answer:
The given equation is:
3x + 2 = x + 4 (x + 2)
3x + 2 = x + 4 (x) + 4 (2)
3x + 2 = x + 4x + 8
3x + 2 = 5x + 8
Rearrange the like terms
So,
5x – 3x = -8 + 2
2x = -6
Divide by 2 into both sides
So,
x = \(\frac{-6}{2}\)
x = -3
Hence, from the above,
We can conclude that the value of x is: -3

Question 5.
Solve the equation -3(x – 1) + 7x = 27.
Answer:
The given equation is:
-3 (x – 1) + 7x = 27
So,
-3 (x) + 3 (1) + 7x = 27
-3x + 3 + 7x = 27
4x + 3 = 27
Subtract with 3 on both sides
So,
4x = 27 – 3
4x = 24
Divide by 4 into both sides
So,
x = \(\frac{24}{4}\)
x = 6
Hence, from the above,
We can conclude that the value of x is: 6

Question 6.
Solve the equation \(\frac{1}{3}\)(x + 6) = \(\frac{1}{2}\)(x – 3).
Answer:
The given equation is:
\(\frac{1}{3}\)(x + 6) = \(\frac{1}{2}\)(x – 3)
Multiply with 6 on both sides so that we can make the fractions as integers (It is not compulsory to multiply with only 6. You can also multiply with any number that is multiple of both 2 and 3)
So,
\(\frac{6}{3}\) (x + 6) = \(\frac{6}{2}\) (x – 3)
2 (x + 6) = 3 (x – 3)
2 (x) + 2 (6) = 3 (x) – 3 (3)
2x + 12 = 3x – 9
Rearrange the like terms
So,
3x – 2x = 12 + 9
x = 21
Hence,f rom the above,
We can conclude that the value of x is: 21

Question 7.
Solve the equation 0.25(x + 4) – 3 = 28.
Answer:
The given equation is:
0.25 (x + 4) – 3 = 28
Add with 3 on both sides
So,
0.25 (x + 4) = 28 + 3
0.25 (x + 4) = 31
We know that,
0.25 = \(\frac{1}{4}\)
So,
\(\frac{x + 4}{4}\) = 31
Multiply with 4 on both sides
So,
x + 4 = 31 (4)
x + 4 = 124
Subtract with 4 on both sides
So,
x = 124 – 4
x = 120
Hence, from the above,
We can conclude that the value of x is: 120

Practice & Problem Solving

Leveled Practice In 8-10, find the value of x.
Question 8.
Lori bought sunglasses and flip-flops at a half-off sale. If she spent a total of $21 on the two items, what was the original price of the sunglasses?
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 27
The original price of the sunglasses was _________.
Answer:
It is given that
Lori bought sunglasses and flip-flops at a half-off sale. If she spent a total of $21 on the two items
Now,
Let x be the price of sunglasses
It is also given that
the price of flipflops is: $24
So,
\(\frac{1}{2}\) (x + $24) = $21
Multiply with 2 on both sides
So,
x + $24 = $21 (2)
x + $24 = $42
Subtract with $24 on both sides
So,
x = $42 – $24
x = $18
Hence, from the above,
We can conclude that the price of sunglasses is: $18

Question 9.
Use the Distributive Property to solve the equation 28 – (3x + 4) = 2(x + 6) + x.
28 – ______ x – _____ = 2x + _____ + x
24 – _____x = ______x + ______
24 – _____x = ______
_____ x = ______
x = ______
Answer:
The given equation is:
28 – (3x + 4) = 2 (x + 6) + x
By using the distributive property,
28 – 3x – 4  = 2 (x) + 2 (6) + x
24 – 3x = 2x + 12 + x
24 – 3x = 3x + 12
Rearrange the like terms
So,
3x + 3x = 24 – 12
6x = 12
Divide by 6 into both sides
So,
x = \(\frac{12}{6}\)
x = 2
Hence, from the above,
We can conclude that the value of x is: 2

Question 10.
Use the Distributive Property to solve the equation 3(x – 6) + 6 = 5x – 6.
x – _____ + 6 = 5x – ______
_____ x – _____ = 5x – _______
_____ x – _____ = _______
______ x = _______
x = ________
Answer:
The given equation is:
3 (x – 6) + 6 = 5x – 6
By using the Distributive property,
3 (x) – 3 (6) + 6 = 5x – 6
3x – 18 + 6 = 5x – 6
3x – 12 = 5x – 6
Rearrange the like terms
So,
5x – 3x = 6 – 12
2x = -6
x = \(\frac{-6}{2}\)
x = -3
Hence, from the above,
We can conclude that the value of x is: -3

Question 11.
What is the solution to -2.5(4x – 4) = -6?
Answer:
The given equation is:
-2.5 (4x – 4) = -6
So,
-2.5 (4x) + 2.5 (4) = -6
-10x + 10 = -6
Subtract with 10 on both sides
So,
-10x = -6 – 10
-10x = -16
10x = 16
Divide by 10 into both sides
So,
x = \(\frac{16}{10}\)
x = 1.6
Hence, from the above,
We can conclude that the solution of the given equation is: 1.6

Question 12.
What is the solution to the equation 3(x + 2) = 2(x + 5)?
Answer:
The given equation is:
3 (x + 2) = 2 (x + 5)
So,
3 (x) + 3 (2) = 2 (x) + 2 (5)
3x + 6 = 2x + 10
Rearrange the like terms
So,
3x – 2x = 10 – 6
x = 4
Hence, from the above,
We can conclude that the solution of the given equation is: 4

Question 13.
Solve the equation \(\frac{1}{6}\)(x – 5) = \(\frac{1}{2}\)(x + 6).
Answer:
The given equation is:
\(\frac{1}{6}\)(x – 5) = \(\frac{1}{2}\)(x + 6)
Multiply with 6 on both sides
So,
x – 5 = 3 (x + 6)
x – 5 = 3 (x) + 3 (6)
x – 5 = 3x + 18
Rearrange the like terms
So,
x – 3x = 18 + 5
-2x = 23
Divide by -2 into both sides
So,
x = –\(\frac{23}{2}\)
Hence, from the above,
We can conclude that the value of x for the given equation is: –\(\frac{23}{2}\)

Question 14.
Solve the equation 0.6(x + 2) = 0.55(2x + 3).
Answer:
The given equation is:
0.6 (x + 2) = 0.55 (2x + 3)
So,
0.6 (x) + 0.6 (2) = 0.55 (2x) + 0.55 (3)
0.6x + 1.2 = 1.10x + 1.65
Rearrange the like terms
So,
1.10x – 0.6x = 1.2 – 1.65
0.5x = -0.45
Divide by 0.5 into both sides
So,
x = \(\frac{-0.45}{0.5}\)
x = -0.9
Hence, from the above,
We can conclud ethat the value of x is: -0.9

Question 15.
Solve the equation 4x – 2(x – 2) = -9 + 5x – 8.
Answer:
The given equation is:
4x – 2 (x – 2) = -9 + 5x – 8
So,
4x – 2 (x) + 2 (2) = -9 + 5x – 8
4x – 2x + 4 = 5x – 17
2x + 4 = 5x – 17
Rearrange the like terms
So,
5x – 2x = 17 + 4
3x = 21
Divide by 3 into both sides
So,
x = \(\frac{21}{3}\)
x = 7
Hence, from the above,
We can conclude that the value of x is: 7

Question 16.
Use the Distributive Property to solve the equation 2(m + 2) = 22. Describe what it means to distribute the 2 to each term inside the parentheses.
Answer:
The given equation is:
2 (m + 2) = 22
We know that,
By using the distributive property of multiplication,
A (B + c) = AB + AC
So,
2 (m) + 2 (2) = 22
2m + 4 = 22
2m = 22 – 4
2m = 18
m = \(\frac{18}{2}\)
m = 9
Hence, from the above,
We can conclude that the value of m is: 9

Question 17.
What is Peter’s number?
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 28
Answer:
Let peter’s number be x
So,
According to the given statement,
-3 (x – 12) = -54
3 (x – 12) = 54
3x – 3 (12) = 54
3x – 36 = 54
3x = 36 + 54
3x = 90
x = \(\frac{90}{3}\)
x = 30
Hence, from the above,
We acn conclude that peter’s number is: 30

Question 18.
Higher Order Thinking Use the Distributive Property to solve the equation \(\frac{4x}{5}\) – x = \(\frac{x}{10}\) – \(\frac{9}{2}\)
Answer:
The given equation is:
\(\frac{4x}{5}\) – x = \(\frac{x}{10}\) – \(\frac{9}{2}\)
Rearrange the like terms
So,
\(\frac{4x}{5}\) – x – \(\frac{x}{10}\) = –\(\frac{9}{2}\)
\(\frac{7x}{10}\) – x = –\(\frac{9}{2}\)
–\(\frac{3x}{10}\) = –\(\frac{9}{2}\)
Multiplywith \(\frac{10}{3}\) on both sides
x = \(\frac{9 × 10}{2 × 3}\)
x = 15
Hence, from the above,
We can conclude that the value of x is: 15

Assessment Practice
Question 19.
How many solutions does the equation -2(x + 4) = -2(x + 4) – 6 have?
Answer:
The given equation is:
-2 (x + 4) = -2 (x + 4) – 6
So,
-2 (x) – 2 (4) = -2 (x) – 2 (4) – 6
-2x – 8 = -2x – 8 – 6
Rearrange the like terms
So,
-2x + 2x – 8 + 8 = -6
0 = -6
Hence, from the above,
We can conclude that there are no solutions for the given equation

Question 20.
Solve the equation 3(x + 4) = 2x + 4x – 6 for x.
Answer:
The given equation is:
3 (x + 4) = 2x + 4x – 6
So,
3 (x) + 3 (4) = 6x – 6
3x + 12 = 6x – 6
Rearrange the like terms
So,
6x – 3x = 12 + 6
3x = 18
x = \(\frac{18}{3}\)
x = 6
Hence, from the above,
We can conclude that the solution of the given equation is: 6

Lesson 2.4 Equations with No Solutions or Infinitely Many Solutions

Explore It!
The Great Karlo called twins Jasmine and James onto the stage. Jasmine, multiply your age by 3 and add 6. Then multiply this sum by 2. James, multiply your age by 2 and add 4. Then multiply this sum by 3. I predict you will both get the same number!
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 29
I can… determine the number of solutions an equation has.

A. Write expressions to represent Great Karlo’s instructions to each twin.
Answer:
It is given that
The Great Karlo called the twins Jasmine and James onto the stage. Jasmine, multiply your age by 3 and add 6. Then multiply this sum by 2. James, multiply your age by 2 and add 4. Then multiply this sum by 3.
Now,
Great Karlo’s instructions to Jasmine:
Let the age of Jasmine be x
Step 1:
Multiply your age by 3 and add 6
3x + 6
Step 2:
Multiply step 1 with 2
2 (3x +6)
So,
The expression representing the age of Jasmine is: 2 (3x + 6)
Great Karlo’s instructions to James
Let the age of James be x
Step 1:
Multiply your age by 2 and add 4
2x + 4
Step 2:
Multiply step 1 with 3
3 (2x +4)
So,
The expression representing the age of James is: 3 (2x + 4)
Hence, from the above,
We can conclude that the expressions that represent the Great Karlo’s instruction to each twin are:
For Jasmine —–> 2 (3x + 6)
For James ——-> 3 (2x + 4)

B. Choose 4 whole numbers for the twins’ age and test each expression. Make a table to show the numbers you tried and the results.
Answer:
It is given that the great Karlo predicted that the twins will get the same number
So,
2 (3x + 6) = 3 (2x + 4)
2 (3x) + 2 (6) = 3 (2x) + 3 (4)
6x + 12 = 6x + 12
Hence,
The table to show the numbers tried for Jasmine’s and James ages and the results are:

C. What do you notice about your results?
Answer:
From the table that is present in part (b),
We can observe that the ages of Jasmine and James are the same

Focus on math practices
Make Sense and Persevere Choose three more values and use them to evaluate each expression. What do you notice? Do you think this is true for all values? Explain.
Answer:
The table that represents three more values of Jasmine’s and James’ ages and its results are:

Hence, from the above table,
We can observe that the ages of Jasmine and James are the same
Hence, from the above,
We can conclude that for any type of the whole number, the ages of Jasmine and James are the same

Essential Question
Will a one-variable equation always have only one solution?
Answer:
Every linear equation that is a conditional equation has one solution. However, not every linear equation in one variable has a single solution. There are two other cases: no solution and the solution set of all real numbers

Try It!

How many solutions does the equation
3x + 15 = 2x + 10 + x + 5 have?
The equation has ______ solutions.
3x + 15 = 2x + 10 + x + 5
3x + 15 = _____ x + ______
3x – _____ + 15 = 3x – _____ + 15
______ = _______
Answer:
The given equation is:
3x + 15 = 2x + 10 + x + 5
So,
3x + 15 = 3x + 15
Subtract with 3x on both sides
So,
15 = 15
Hence, from the above,
We can conclude that the given equation has infinitely many solutions

Convince Me!
If the value of x is negative, would the equation still be true? Explain.
Answer:
For the given equation,
3x + 15 = 2x + 10 + x + 5,
The solutions are infinite i..e, for any value of x, the given equation will be true i.e., for both positive and negative values of x, the equation will be true
Hence, from the above,
We can conclude that the given equation would still be true even if the value of x is negative

Try It!

How many solutions does the equation 4x + 8 = 0.1x + 3 + 3.9x have? Explain.
Answer:
The given equation is:
4x + 8 = 0.1x + 3 + 3.9x
So,
4x + 8 = 4x + 3
Subtract with 4x on both sides
So,
8 = 3
Hence, from the above,
We can conclude that the given equation has no solutions

Try It!

Determine the number of solutions each equation has without solving. Explain your reasoning.
a. 3x + 1.5 = 2.5x + 4.7
Answer:
The give equation is:
3x + 1.5 = 2.5x + 4.7
Rearrange the like terms
So,
3x – 2.5x = 4.7 – 1.5
0.5x = 3.2
Divide by 0.5 into both sides
So,
x = \(\frac{3.2}{0.5}\)
x = 6.4
Hence, from the above,
We can conclude that the given equation ahs only 1 solution

b. 3(x + 2) = 3x – 6
Answer:
The given equation is:
3 (x + 2) = 3x – 6
So,
3 (x) + 3 (2) = 3x – 6
3x + 6 = 3x – 6
Subtract with 3x on both sides
So,
6 = -6
Hence, from the above,
We can conclude that the given equation has no solutions

c. 9x – 4 = 5x – 4 + 4x
Answer:
The given equation is:
9x – 4 = 5x – 4 + 4x
So,
9x – 4 = 9x – 4
Subtract with 9x on both sides
So,
-4 = -4
4 = 4
Hence, from the above,
We can conclude that the given equation has infinitely many solutions

KEY CONCEPT

A one-variable equation has infinitely many solutions when solving results in a true statement, such as 2 = 2.
A one-variable equation has one solution when solving results in one value for the variable, such as x = 2.
A one-variable equation has no solution when solving results in an untrue statement, such as 2 = 3.

Do You Understand?
Question 1.
Essential Question Will a one-variable equation always have only one solution?
Answer:
Every linear equation that is a conditional equation has one solution. However, not every linear equation in one variable has a single solution. There are two other cases: no solution and the solution set of all real numbers

Question 2.
Use Structure Kaylee writes the equation 6x + 12 = 2(3x + 6). Can you find the number of solutions this equation has without solving for x? Explain.
Answer:
The given equation is:
6x + 12 = 2 (3x + 6)
So,
6x + 2 = 2 (3x) + 2 (6)
6x + 12 = 6x + 12
Subtract with 12 on both sides
So,
12 = 12
Hence, from the above,
We can conclude that the given equation has infinitely many solutions

Question 3.
Construct Arguments The height of an experimental plant after x days can be represented by the formula 3(4x + 2). The height of a second plant can be represented by the formula 6(2x + 2). Is it possible that the two plants will ever be the same height? Explain.
Answer:
It is given that
The height of an experimental plant after x days can be represented by the formula 3(4x + 2). The height of a second plant can be represented by the formula 6(2x + 2)
So,
Now,
To find out whether the two plants will ever be the same height or not,
3 (4x + 2) = 6 (2x + 2)
So,
3 (4x) + 3 (2) = 6 (2x) + 6 (2)
12x + 6 = 12x + 12
Subtract with 12x on both sides
So,
6 = 12
So,
The given equation has no solution
Hence, from the above,
We can conclude that it is not possible the two plants will ever be the same height

Do You Know How?
Question 4.
How many solutions does the equation 3(2.4x + 4) = 4.1x + 7 + 3.1x have? Explain.
Answer:
The given equation is:
3 (2.4x + 4) = 4.1x + 7 + 3.1x
So,
3 (2.4x) + 3 (4) = 7.2x + 7
7.2x + 12 = 7.2x + 7
Subtract with 7.2x on both sides
So,
12 = 7
Hence, from the above,
We can conclude that the given equation has no solutions

Question 5.
How many solutions does the equation 7x + 3x – 8 = 2(5x – 4) have? Explain.
Answer:
The given equation is:
7x + 3x – 8 = 2 (5x – 4)
So,
10x – 8 = 2 (5x) – 2 (4)
10x – 8 = 10x – 8
Subtract with 10x on both sides
So,
-8 = -8
8 = 8
Hence, from the above,
We can conclude that the given equation has infinitely many solutions

Question 6.
Todd and Agnes are making desserts. Todd buys peaches and a carton of vanilla yogurt. Agnes buys apples and a jar of honey. They bought the same number of pieces of fruit. Is there a situation in which they pay the same amount for their purchases? Explain.
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 30
Answer:
It is given that
Todd and Agnes are making desserts. Todd buys peaches and a carton of vanilla yogurt. Agnes buys apples and a jar of honey. They bought the same number of pieces of fruit.
Now,
Let the number of pieces of fruit be x
So,
The amount purchased by Todd = $1.25x + $4
The amount purchased by Agnes = $1x + $6
Now,
To find whether they pay the same amount for purchase or not,
$1.25x + $4 = $1x + $6
Rearrange the like terms
So,
$1.25x – $1x = $6 – $4
$0.25x = $2
Divide by 0.25 into both sides
So,
x = \(\frac{2}{0.25}\)
x = 8
Hence, from the above,
We can conclude that if there are 8 fruits, then Todd and Agnes will pay the same amount for purchase

Practice & Problem Solving

Leveled Practice In 7 and 8, complete the equations to find the number of solutions.
Question 7.
Classify the equation 33x + 99 = 33x – 99 as having one solution, no solution, or infinitely many solutions.
33x + 99 = 33x – 99
33x – ______ + 99 = 33x – _____ – 99
99 ______ – 99
Since 99 is _______ equal to -99, the equation has _______ solution(s).
Answer:
The given equation is:
33x + 99 = 33x – 99
Subtract with 33x on both sides
So,
33x – 33x + 99 = 33x – 33x – 99
99 = -99
We know that,
99 ≠ -99
Hence, from the above,
We can conclude that there are no solutions for the given equation

Question 8.
Solve 4(4x + 3) = 19x + 9 – 3x + 3. Does the equation have one solution, no solution, or infinitely many solutions?
4(4x + 3) = 19x + 9 – 3x + 3
4 • ______ + 4 • ______ = 19x + 9 – 3x + 3
16x + 12 = _______ + _______
16x – ______ + 12 = 16x ______ + 12
12 _______ 12
Since 12 is ________ equal to 12, the equation has ________ solution(s).
Answer:
The given equation is:
4 (4x + 3) = 19x + 9 – 3x + 3
So,
4 (4x) + 4 (3) = 16x + 12
16x + 12 = 16x + 12
Subtract with 16x on both sides
So,
16x – 16x + 12 = 16x – 16x + 12
12 = 12
Hence, from the above,
We can conclude that the given equation has infinitely many solutions

Question 9.
Generalize What does it mean if an equation is equivalent to 0 = 0? Explain.
Answer:
If an equation is equivalent to 0 = 0, then
The equation is true for all the values of x
Hence,
That equation has infinitely many solutions

Question 10.
Solve 4x + x + 4 = 8x – 3x + 4. Does the equation have one solution, no solution, or infinitely many solutions? If one solution, write the solution. Explain.
Answer:
The given equation is:
4x + x + 4 = 8x – 3x + 4
So,
5x + 4 = 5x + 4
Subtract with x on both sides
So,
5x – 5x + 4 = 5x – 5x + 4
4 = 4
Hence, from the above,
We can conclude that the given equation has infinitely many solutions

Question 11.
Reasoning Two rival dry cleaners both advertise their prices. Let x equal the number of items dry cleaned. Store A’s prices are represented by the expression 15x – 2. Store B’s prices are represented by the expression 3(5x + 7). When do the two stores charge the same rate? Explain.
Answer:
It  is given that
Two rival dry cleaners both advertise their prices. Let x equal the number of items dry cleaned. Store A’s prices are represented by the expression 15x – 2. Store B’s prices are represented by the expression 3(5x + 7)
So,
To find when the two stores charge the same rate,
15x – 2 = 3 (5x + 7)
So,
15x – 2 = 3 (5x) + 3 (7)
15x – 2 = 15x – 21
Subtract with 15x on both sides
So,
15x – 15x – 2 = 15x – 15x – 21
-2 = -21
2 = 21
So,
The equation has no solution
Hence, from the above,
We can conclude that the two stores will never charge the same rate

Question 12.
Reasoning How is solving an equation with no solution similar to solving an equation that has an infinite number of solutions?
Answer:
No solution would mean that there is no answer to the equation. It is impossible for the equation to be true no matter what value we assign to the variable. Infinite solutions would mean that any value for the variable would make the equation true.

Question 13.
Solve 0.9x + 5.1x – 7 = 2(2.5x – 3). How many solutions does the equation have?
Answer:
The given equation is:
0.9x + 5.1x – 7 = 2 (2.5x – 3)
So,
6.0x – 7 = 2 (2.5x) – 2 (3)
6x – 7 = 5x – 6
Rearrange the like terms
So,
6x – 5x = 7 – 6
x = 1
Hence, from the above,
We can conclude that the given equation has only 1 solution

Question 14.
Critique Reasoning Your friend solved the equation 4x + 12x – 6 = 4(4x + 7) and got x = 34.
What error did your friend make? What is the correct solution?
4x + 12x – 6 = 4 (4x + 7).
16x – 6 = 16x + 28
16x – 16x – 6 = 16x – 16x + 28
x – 6= 28
x – 6 + 6 = 28 + 6
x = 34
Answer:
The given equation is:
4x + 12x – 6 = 4 (4x + 7)
So,
16x – 6 = 4 (4x) + 4 (7)
16x – 6 = 16x + 28
Subtract with 16x on both sides
So,
16x – 16x – 6 = 16x – 16x + 28
-6 = 28
So,
From the above,
We can observe that after subtracting the given equation with 16x, there are no x terms.
So,
We can’t get the value of x but your friend takes variable x after subtracting 16x from the given equation even though there is no possibility for the x-term
Hence, from the above,
We can conclude that the correct solution for the given equation is: No solutions for the given equation

Question 15.
Solve 49x + 9 = 49x + 83.
a. Does the equation have one solution, no solution, or infinitely many solutions?
Answer:
The given equation is:
49x + 9 = 49x + 83
Subtract with 49x on both sides
So,
49x – 49x + 9 = 49x – 49x + 83
9 = 83
Hence, from the above,
We can conclude that the given equation has no solutions

b. Write two equations in one variable that have the same number of solutions as this equation.
Answer:
The two equations in one variable that have the same number of solutions as the equation that is present in part (a) are:
A) 10x + 8 = 10x – 25
B) 5 (3x + 10) = 15x + 40

Question 16.
Classify the equation 6(x + 2) = 5(x + 7) as having one solution, no solution, or infinitely many solutions.
Answer:
The given equation is:
6 (x + 2) = 5 (x + 7)
So,
6 (x) + 6 (2) = 5 (x) + 5 (7)
6x + 12 = 5x + 35
Rearrange the like terms
So,
6x – 5x = 35 – 12
x = 23
Hence, from the above,
We can conclude that the given equation has only 1 solution

Question 17.
Solve 6x + 14x + 5 = 5(4x + 1). Write a word problem that this equation, or any of its equivalent forms, represents.
Answer:
The given equation is:
6x + 14x + 5 = 5 (4x + 1)
So,
20x + 5 = 5 (4x) + 5 (1)
20x + 5 = 20x + 5
Subtract with 20x on both sides
So,
20x – 20x + 5 = 20x – 20x + 5
5 = 5
Hence, from the above,
We can conclude that the given equation is true for any value of x i..e, the given equation has infinitely many solutions

Question 18.
Classify the equation 170x – 1,000 = 30(5x – 30) as having one solution, no solution, or infinitely many solutions.
Answer:
The given equation is:
170x – 1,000 = 30 (5x – 30)
So,
170x – 1,000 = 30 (5x) – 30 (30)
170x – 1,000 = 150x – 900
Rearrange the like terms
So,
170x – 150x = 1,000 – 900
20x = 100
x= \(\frac{100}{20}\)
x = 5
Hence, from the above,
We can conclude that the given equation has only 1 solution

Question 19.
Higher Order Thinking Write one equation that has one solution, one equation that has no solution, and one equation that has infinitely many solutions.
Answer:
The example representation of the equation that has one solution is:
20x + 5 = 15x – 4
The example representation of the equation that has no solutions is:
3 (6x – 2) = 9 (2x – 4)
The example representation of the equation that has infinitely many solutions is:
4 (2x – 6) = 8 (x – 3)

Question 20.
Solve 4(4x – 2) + 1 = 16x – 7.
Answer:
The given equation is:
4 (4x – 2) + 1 = 16x – 7
So,
4 (4x) – 4 (2) + 1 = 16x – 7
16x – 8 + 1 = 16x – 7
16x – 7 = 16x – 7
Subtract with 16x on both sides
So,
16x – 16x – 7 = 16x – 16x – 7
-7 = -7
7 = 7
Hence, from the above,
We can conclude that the given equation is true for all the values of x i..e, the given equation has infinitely many solutions

Question 21.
Solve 6x + 26x – 10 = 8(4x + 10).
Answer:
The given equation is:
6x + 26x – 10 = 8 (4x + 10)
So,
32x – 10 = 8 (4x) + 8 (10)
32x – 10 = 32x + 80
Subtract with 32x on both sides
So,
32x – 32x – 10 = 32x – 32x + 80
-10 = 80
Hence, from the above,
We can conclude that the given equation has no solution

Question 22.
Classify the equation 64x – 16 = 16(4x – 1) as having one solution, no solution, or infinitely many solutions.
Answer:
The given equation is:
64x – 16 = 16 (4x – 1)
So,
64x – 16 = 16 (4x) – 16 (1)
64x – 16 = 64x – 16
Subtract with 64x on both sides
So,
64x – 64x – 16 = 64x – 64x – 16
– 16 = -16
16 = 16
Hence, from the above,
We can conclude that the given equation has infinitely many solutions

Question 23.
Classify the equation 5(2x + 3) = 3(3x + 12) as having one solution, no solution, or infinitely many solutions.
Answer:
The given equation is:
5 (2x + 3) = 3 (3x + 12)
So,
5 (2x) + 5 (3) = 3 (3x) + 3 (12)
10x + 15 = 9x + 36
Rearrange the like terms
So,
10x – 9x = 36 – 15
x = 21
Hence, from the above,
We can conclude that the given equation has only 1 solution

Assessment Practice
Question 24.
Which of the following best describes the solution to the equation 4(2x + 3) = 16x + 12 – 8x?
A. The equation has one solution.
B. The equation has infinitely many solutions.
C. The equation has no solution.
D. The equation has two solutions.
Answer:
The given equation is:
4 (2x + 3) = 16x + 12 – 8x
So,
4 (2x) + 4 (3) = 4x + 12
8x + 12 = 4x + 12
Rearrange the like terms
So,
8x – 4x = 12 – 12
4x = 0
x = 0
So,
The given equation has only 1 solution
Hence, from the above,
We can conclude that option A matches with the solution of the given equation

Question 25.
Which of the following statements are true about the equation 10x + 45x – 13 = 11(5x + 6)? Select all that apply.
☐ The operations that can be used to solve the equation are addition and multiplication.
☐ The operations that can be used to solve the equation are multiplication and division.
☐ The equation has infinitely many solutions.
☐ The equation has a solution of x = 53.
☐ The equation has no solution.
Answer:
Let the given options be named: A, B, C, D, and E
Now,
The given equation is:
10x + 45x – 13 = 11 (5x + 6)
So,
55x – 13 = 11 (5x) + 11 (6)
55x – 13 = 55x + 66
Subtract with 55x on both sides
So,
55x – 55x – 13 = 55x – 55x + 66
-13 = 26
Hence, from the above,
We can conclude that option A and option E matches with the situation for the given equation

Topic 2 MID-TOPIC CHECKPOINT

Question 1.
Vocabulary How can you determine the number of solutions for an equation? Lesson 2-4
Answer:
A one-variable equation has infinitely many solutions when solving results in a true statement, such as 2 = 2.
A one-variable equation has one solution when solving results in one value for the variable, such as x = 2.
A one-variable equation has no solution when solving results in an untrue statement, such as 2 = 3.

Question 2.
Solve the equation –\(\frac{2}{3}\)d – \(\frac{1}{4}\)d = -22 for d. Lesson 2-1
Answer:
The given equation is:
–\(\frac{2}{3}\)d – \(\frac{1}{4}\)d = -22
So,
\(\frac{8 + 3}{12}\)d = 22
\(\frac{11}{12}\)d = 22
Multiply with \(\frac{12}{11}\) on both sides
So,
d = 22 × \(\frac{12}{11}\)
d = \(\frac{22 × 12}{11}\)
d = 24
Hence, from the above,
We can conclude that the value of d is: 24

Question 3.
Edy has $450 in her savings account. She deposits $40 each month. Juan has $975 in his checking account. He writes a check for $45.45 each month for his cell phone bill. He also writes a check for $19.55 each month for his water bill. After how many months will Edy and Juan have the same amount of money in their accounts? Lesson 2-2
Answer:
It is given that
Edy has $450 in her savings account. She deposits $40 each month. Juan has $975 in his checking account. He writes a check for $45.45 each month for his cell phone bill. He also writes a check for $19.55 each month for his water bill.
Now,
Let the number of months be x
So,
The amount of money in the account of Edy = $450 + $40x
The amount of money in the account of Juan = $975 – $45.45x – $19.55x
Now,
To find after how many months they will have the same amount of money in their accounts,
$450 + $40x = $975 – $45.45x – $19.55x
$450 + $40x = $975 – $65x
Rearrange the like terms
So,
$65x + $40x = $975 – $450
$105x = $525
Divide by 105 into both sides
So,
x = \(\frac{$25}{105}\)
x = 5 months
Hence, from the above,
We can conclude that after 5 months, Edy and Jian will have the same amount of money in their accounts

Question 4.
Which equation has infinitely many solutions? Lesson 2-4
A. \(\frac{3}{4}\)x + x – 5 = 10 + 2x
Answer:
The given equation is:
\(\frac{3}{4}\)x + x – 5 = 10 + 2x
\(\frac{3 + 4}{4}\)x – 5 = 10 + 2x
\(\frac{7}{4}\)x – 5 = 10 + 2x
Rearrange the like terms
So,
\(\frac{7}{4}\)x – 2x = 10 + 5
–\(\frac{1}{4}\)x = 15
Multiply with -4 on both sides
So,
x = -60
Hence, from the above,
We acn conclude that the given equation has only 1 solution

B. 3x – 2.7 = 2x + 2.7 + x
Answer:
The given equation is:
3x – 2.7 = 2x + x + 2.7
3x – 2.7 = 3x + 2.7
Subtract with 3x on both sides
So,
-2.7 = 2.7
Hence, from the above,
We can conclude that the given equation has no solutions

C. 9x + 4.5 – 2x = 2.3 +7x + 2.2
Answer:
The given equation is:
9x + 4.5 – 2x = 2.3 + 7x + 2.2
7x + 4.5 = 7x + 4.5
Subtract with 7x on both sides
So,
4.5 = 4.5
Hence, from the above,
We can conclude that the given equation has infinitely many solutions

D. \(\frac{1}{5}\) x – 7 = \(\frac{3}{4}\) + 2x – 25\(\frac{3}{4}\)
Answer:
The given equation is:
\(\frac{1}{5}\) x – 7 = \(\frac{3}{4}\) + 2x – 25\(\frac{3}{4}\)
We know that,
25\(\frac{3}{4}\) = \(\frac{103}{4}\)
So,
\(\frac{1}{5}\) x – 7 = \(\frac{3}{4}\) + 2x – \(\frac{103}{4}\)
Rearrange the like terms
So,
\(\frac{1}{5}\)x – 2x = 7 – \(\frac{103}{4}\)
–\(\frac{9}{5}\)x = –\(\frac{75}{4}\)
Multiply with –\(\frac{5}{9}\) on both sides
So,
x = \(\frac{75 × 5}{4 × 9}\)
x = \(\frac{125}{4}\)
Hence, from the above,
We can conclude that the given equation has only 1 solution

Question 5.
Solve the equation -4(x – 1) + 6x = 2(17 – x) for x. Lesson 2.3
Answer:
The given equation is:
-4 (x – 1) + 6x = 2 (17 – x)
So,
-4 (x) + 4 (1) + 6x = 2 (17) – 2 (x)
-4x + 4 + 6x = 34 – 2x
2x + 4 = 34 – 2x
Rearrange the like terms
So,
2x + 2x = 34 – 4
4x = 30
Divide by 4 on both sides
So,
x = \(\frac{30}{4}\)
x = \(\frac{15}{2}\)
Hence, from the above,
We can conclude that the value of x is: \(\frac{15}{2}\)

Question 6.
Hakeem subtracted 8 from a number, then multiplied the difference by \(\frac{4}{5}\). The result was 20. Write and solve an equation to find the number, x. Lesson 2-3
Answer:
It is given that
Hakeem subtracted 8 from a number, then multiplied the difference by \(\frac{4}{5}\). The result was 20.
Now,
Let the number be x
So,
According to Hakeem,
The expression that represents the given situation is:
\(\frac{4}{5}\) (x – 8) = 20
Multiply with \(\frac{5}{4}\) on both sides
So,
x – 8 = \(\frac{5 × 20}{4}\)
x – 8 = 25
Add with 8 on both sides
So,
x = 25 + 8
x = 34
Hence, from the baove,
We can conclude that Hakeem’s number is: 34

Topic 2 MID-TOPIC PERFORMANCE TASK

Hector is competing in a 42-mile bicycle race. He has already completed 18 miles of the race and is traveling at a constant speed of 12 miles per hour when Wanda starts the race. Wanda is traveling at a constant speed of 16 miles per hour.

PART A
Write and solve an equation to find when Wanda will catch up to Hector.
Answer:
It is given that
Hector is competing in a 42-mile bicycle race. He has already completed 18 miles of the race and is traveling at a constant speed of 12 miles per hour when Wanda starts the race. Wanda is traveling at a constant speed of 16 miles per hour.
Now,
Let the time be x
We know that,
Speed = \(\frac{Distance} {Time}\)
So,
Time = \(\frac{Distance}{Speed}\)
Now,
Time taken by Hector to complete a bicycle race = \(\frac{42 – 18}{12}\)
x = \(\frac{24}{12}\)
x = 2 hours
Now,
Time taken by Wanda to complete the bicycle race = \(\frac{The total distance of race}{The speed traveled by Wanda}\)
x = \(\frac{42}{16}\)
x = \(\frac{21}{8}\) hours
x = 2.625 hours
Now,
The time that took Wanda to catch up to Hector = The time taken by Wanda to complete the race – The tie taken by Hector to complete the race
= 2.625 – 2
= 0.625
= 0.625 (60 minutes)
= 37.5 minutes
Hence, from the above,
We can conclude that Wanda will catch up to Hector after 37.5 minutes of Hector completing the race

PART B
Will Wanda catch up to Hector before the race is complete? Explain.
Answer:
From part (a),
The time taken by Hector to complete the race is: 2 hours
The time taken by Wanda to complete the race is: 2.625 hours
So,
From the above times,
We can observe that the race is completed at 2 hours
Hence, from the above,
We can conclude that Wanda can’t catch up to Hector before the race is complete

PART C
At what constant speed could Wanda travel to catch up with Hector at the finish line? Explain.
Answer:
We know that,
Speed = \(\frac{Distance}{Time}\)
So,
The speed at which Wanda travel to catch up to Hector = \(\frac{The distance of the race}{The time taken by Hector to complete the race}\)
= \(\frac{42}{2}\)
= 21 miles per hour
Hence, from the above,
We can conclude that at 21 miles per hour speed, Wanda could catch up to Hector

3-Act Mathematical Modeling: Powering Down

3-ACT MATH
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 150.1

АСТ 1

Question 1.
After watching the video, what is the first question that comes to mind?
Answer:
After watching the video,
The first question that comes to mind is:
what will be the battery percentage you should have to complete your work?

Question 2.
Write the Main Question you will answer.
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 150.2
Answer:
The main question you will answer is:
what will be the battery percentage you should have to complete your work?

Question 3.
Construct Arguments Predict an answer to this Main Question. Explain your prediction.
Answer:
The answer to the main question is: 100%
Reason for the prediction:
We don’t know how much work has left. So, it is better to have a battery percentage of 100%

Question 4.
On the number line below, write a time that is too early to be the answer. Write a time that is too late.
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 150.3
Answer:
The time that is too early to be the answer for the above problem is: 5 minutes
The time that is too late to be the answer for the above problem is: Greater than the time that battery percentage is 100%

Question 5.
Plot your prediction on the same number line.

ACT 2

Question 6.
What information in this situation would be helpful to know? How would you use that information?
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 150.4
Answer:
The information in this situation that would be helpful to know is:
A) The time is taken for battery percentage to be full

Question 7.
Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
Answer:

Question 8.
Model with Math
Represent the situation using mathematics. Use your representation to answer the Main Question.
Answer:

Question 9.
What is your answer to the Main Question? Is it earlier or later than your prediction? Explain why.
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 150.5
Answer:

ACT 3

Question 10.
Write the answer you saw in the video.
Answer:

Question 11.
Reasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 150.6
Answer:

Question 12.
Make Sense and Persevere Would you change your model now that you know the answer? Explain.
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 150.7
Answer:

Act 3

Reflect

Question 13.
Model with Math
Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 150.8
Answer:

Question 14.
Look for Relationships What pattern did you notice in the situation? How did you use that pattern?
Answer:

SEQUEL

Question 15.
Be Precise After 35 minutes, he started charging his phone. 21 minutes later, the battery is at 23%. Explain how you would determine when the phone will be charged to 100%.
Answer:

Lesson 2.5 Compare Proportional Relationships

Solve & Discuss It!

Mei Li is going apple picking. She is choosing between two places. The cost of a crate of apples at each place is shown.
Where should Mei Li go to pick her apples? Explain.
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 150.9
Answer:
It is given that Mei Li is going apple picking. She is choosing between two places.
So,
In Annie’s Apple Orchard,
The cost of 20lb of apples is: $7.25
In Franklin’s fruit Orchard,
The cost of 12lb of apples is: $5
We know that,
Where the cost of 1lb of apples is low, Mei Li will go there to buy the apples
Now,
In Annie’s Apple Orchard,
The cost of 1lb of apples = \(\frac{$7.25}{20}\)
= $0.3625
In Franklin’s fruit Orchard,
The cost of 1lb of apples = \(\frac{$5}{12}\)
= $0.4166
So,
The cost of 1lb of apples in Annie’s Orchard < The cost f 1lb of apples in Franklin’s fruit Orchard
Hence, from the above,
We can conclude that Mei Li should go to pick apples from Annie’s Apple Orchard

Construct Arguments
What information provided can be used to support your answer?
Answer:
From the given figure,
The information provided that can be used to support your answer is:
The weight of the apples is inversely proportional to the price of the apples
So,
In Annie’s Apple Orchard, the weight of the apples is high when compared to the weight of the apples in franklin’s fruit Orchard
Hence,
The price of the apples is low in Annie’s Apple Orchard when compared to Franklin’s fruit Orchard

Focus on math practices
Model with Math Which representation did you use to compare prices? Explain why.
Answer:
The relation that is used to compare the prices of apples is:
Weight of the apples ∝ \(\frac{1}{Price of the apples}\)

? Essential Question
How can you compare proportional relationships represented in different ways?
Answer:
To compare proportional relationships represented in different ways, find the unit rate, or the constant of proportionality, for each representation.

Try It!
The graph represents the rate at which Marlo makes origami birds for a craft fair. The equation y = 2.5x represents the number of birds, y, Josh makes in x minutes. Who makes birds at a faster rate?
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.1
Answer:
It is given that
The graph represents the rate at which Marlo makes origami birds for a craft fair. The equation y = 2.5x represents the number of birds, y, Josh makes in x minutes.
So,
The rate that birds made by Josh = \(\frac{y}{x}\)
= 2.5
Now,
From the graph,
The rate that birds made by Marlo = \(\frac{Time taken to make birds by Marlo}{The number of birds}\)
= \(\frac{40}{8}\) (Here, we can take any value that is present in the graph. For example,\(\frac{20}{4}\), \(\frac{10}{2}\) etc., )
= 5
So,
The rate that birds made by Marlo > The rate that birds made by Josh
Hence, from the above,
We can conclude that Marlo makes birds at a faster rate

Convince Me!
If you were to graph the data for Josh and Marlo on the same coordinate plane, how would the two lines compare?
Answer:
When we graph the data for Josh and Marlo on the same coordinate plane,
We can observe that the two graphs will be the lines that are parallel to each other and the rate of change  of Marlo will be greater than the rate of change of Josh

Try It!
The distance covered by the fastest high-speed train in Japan traveling at maximum speed is represented on the graph. The fastest high-speed train in the United States traveling at maximum speed covers 600 kilometers in 2\(\frac{1}{2}\) hours. Which train has a greater maximum speed? Explain.
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.2
Answer:
It is given that
The distance covered by the fastest high-speed train in Japan traveling at maximum speed is represented on the graph. The fastest high-speed train in the United States traveling at maximum speed covers 600 kilometers in 2\(\frac{1}{2}\) hours.
Now,
We know that,
Speed = \(\frac{Distance}{Time}\)
So,
The speed of the fastest high-speed train in the United states = 600 / \(\frac{5}{2}\)
We know that,
2\(\frac{1}{2}\) = \(\frac{5}{2}\)
So,
The speed of the fastest high-speed train in the United states = \(\frac{600 × 2}{5}\)
= 240 kilometers per hour
Now,
From the given graph,
The speed of the fastest high-speed train in Japan = \(\frac{The difference between any two distances from the graph}{The  difference between the values of the time that corresponds to the taken value of distances}\)
= \(\frac{1000 – 650}{3 – 2}\)
= 350 kilometers per hour
So,
The speed of the fastest high-speed train in Japan > The speed of the fastest high-speed train in the United States
Hence, from the above,
We can conclude that the fastest high-speed train in Japan has a maximum speed

KEY CONCEPT
To compare proportional relationships represented in different ways, find the unit rate, or the constant of proportionality, for each representation.
The representations below show the rental cost per hour for canoes at three different shops.
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.3

Do You Understand?

Question 1.
?Essential Question How can you compare proportional relationships represented in different ways?
Answer:
To compare proportional relationships represented in different ways, find the unit rate, or the constant of proportionality, for each representation.

Question 2.
How can you find the unit rate or constant of proportionality for a relationship represented in a graph?
Answer:
In a graph,
The unit rate or constant of proportionality for a relationship is represented by:
\(\frac{The value of y}{The value of x}\) or \(\frac{The difference between any 2 values of y}{The difference between the values of x that is corresponded to the values of x}\)

Question 3.
Generalize Why can you use the constant of proportionality with any representation?
Answer:
We can use the constant of proportionality to find the rate of change between the physical quantities that have a proportional relationship
Ex:
Speed Vs Distance, Speed Vs Time, etc

Do You Know How?

Question 4.
Amanda babysits and Petra does yard work on weekends. The graph relating Amanda’s earnings to the number of hours she babysits passes through the points (0, 0) and (4, 24). The table below relates Petra’s earnings to the number of hours she does yard work.
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.4
Who earns more per hour?
Answer:
It is given that
Amanda babysits and Petra does yard work on weekends. The graph relating Amanda’s earnings to the number of hours she babysits passes through the points (0, 0) and (4, 24). The table below relates Petra’s earnings to the number of hours she does yard work.
Now,
The Earnings per hour of Amanda = \(\frac{24 – 0}{4 – 0}\)
= \(\frac{24}{4}\)
= 6
The Earnings per hour of Petra = \(\frac{15}{3}\)
= 5
So,
The Earnings per hour of Amanda > The Earnings per hour of Petra
Hence, from the above,
We can conclude that Amanda earns more

Question 5.
Milo pays $3 per pound for dog food at Pat’s Pet Palace. The graph below represents the cost per pound of food at Mark’s Mutt Market. At which store will Milo pay a lower price per pound for dog food?
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.5
Answer:
It is given that
Milo pays $3 per pound for dog food at Pat’s Pet Palace. The graph below represents the cost per pound of food at Mark’s Mutt Market.
So,
Now,
The cost per pound of food at Mark’s Mutt Market = \(\frac{Any value of cost from the given graph}{The value of weight that corresponds to the selected cost}\)
= \(\frac{5}{1}\)
= $5
So,
The cost per pound of food at Pat’s Pet Palace < The cost per pound of food at Mark’s Mutt Market
Hence, from the above,
We can conclude that at Pat’s Pet Palace, Milo will pay a lower price per pound for dog food

Practice & Problem Solving

Leveled Practice For 6 and 7, complete the information to compare the rates.

Question 6.
Sam and Bobby want to know who cycled faster. The table shows the total miles Sam traveled over time. The graph shows the same relationship for Bobby. Who cycled faster.
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.6
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.67
Find the unit rate (constant of proportionality) for Bobby.
Use (Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.7) and (Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.7) to find the constant of proportionality.
The unit rate (constant of proportionality) is Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.8
So Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.10 cycled faster.
Answer:
It is given that
Sam and Bobby want to know who cycled faster. The table shows the total miles Sam traveled over time. The graph shows the same relationship for Bobby.
We know that,
Speed = \(\frac{Distance}{Time}\)
So,
For Sam, from the table,
Speed = \(\frac{20}{2}\) miles per hour
= 10 miles per hour
Now,
For Bobby, from the graph,
Speed = \(\frac{Any value of the distance from the graph}{The value of time that corresponds to the distance that we have taken}\)
= \(\frac{72}{8}\)
= 9 miles per hour
So,
The speed of Sam > The speed of Bobby
Hence, from the above,
We can conclude that Sam cycled faster

Question 7.
Model with Math The equation y = 15x can be used to determine the amount of money, y, Pauli’s Pizzeria makes by selling x pizzas. The graph shows the money Leo’s Pizzeria takes in for different numbers of pizzas sold. Which pizzeria makes more money per pizza?
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.100
Pauli’s Pizzeria takes in Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.11 per pizza.
Leo’s Pizzeria takes in Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.11 per pizza.
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.11‘s Pizzeria takes in more money per pizza.
Answer:
It is given that
The equation y = 15x can be used to determine the amount of money, y, Pauli’s Pizzeria makes by selling x pizzas. The graph shows the money Leo’s Pizzeria takes in for different numbers of pizzas sold
So,
The money earned by Pauli’s Pizzeria = \(\frac{y}{x}\)
= 15 (From the given equation y = 15x)
Now,
From the given graph,
The money earned by Leo’s Pizzeria = \(\frac{Any value of the amount made from the graph}{The value of pizzas sold that corresponds to the value of the amount that we have considered}\)
= \(\frac{96}{8}\)
= 12
So,
The money earned by Pauli’s Pizzeria > The money earned by Leo’s Pizzeria
Hence, from the above,
We can conclude that Pauli’s Pizzeria takes in more money per pizza

Question 8.
The graph shows the amount of savings over time in Eliana’s account. Lana, meanwhile, puts $50 each week into her savings account. If they both begin with $0, who is saving at the greater rate?
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.12
Answer:
It is given that
The graph shows the amount of savings over time in Eliana’s account. Lana, meanwhile, puts $50 each week into her savings account
So,
The amount of savings over time in Lana’s account = \(\frac{Any value of total savings in the graph}{The corresponding value of time to that savings amount}\)
= \(\frac{94}{2}\)
= $47
So,
The amount of savings over time in Elina’s account > The amount of savings over time in Lana’s account
Hence, from the above,
We can conclude that Elina is saving money at a greater rate

Question 9.
Make Sense and Persevere Beth, Manuel, and Petra are collecting sponsors for a walk-a-thon. The equation y = 20x represents the amount of money Beth raises for walking x miles. The table shows the relationship between the number of miles Manuel walks and the amount of money he will raise. Petra will earn $15 for each mile that she walks.
a. In order to compare the proportional relationships, what quantities should you use to find the unit rate?
Answer:
In order to compare the proportional relationships,
The quantities you should use to find the unit rate is:
A) The number of miles walked
B) The amount of money raised for the corresponding number of miles

b. Compare the amount of money raised per mile by the three people.
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.13
Answer:
It is given that
Beth, Manuel, and Petra are collecting sponsors for a walk-a-thon. The equation y = 20x represents the amount of money Beth raises for walking x miles. The table shows the relationship between the number of miles Manuel walks and the amount of money he will raise. Petra will earn $15 for each mile that she walks.
So,
The amount of money raised by Beth = \(\frac{y}{x}\)
= $20 (From the equation y = 20x)
Now,
The amount of money raised by Manuel = \(\frac{Any value of the money raised in the table}{The number of miles walked that corresponds to the value of money raised}\)
= \(\frac{$45}{3}\)
= $15
So,
The amount of money raised by Beth > The amount of money raised by Manuel = The amount of money raised by Petra
Hence, from the above,
We can conclude that Beth raised more amount of money when compared to Manuel and Petra

Question 10.
Higher-Order Thinking Winston compares the heights of two plants to see which plant grows more per day. The table shows the height of Plant 1, in centimeters, over 5 days. The graph shows the height of Plant 2, in centimeters, over 10 days. Winston says that since Plant 1 grows 6 cm per day and Plant 2 grows 4 cm per day, Plant 1 grows more per day.
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.14
a. Do you agree with Winston? Explain your response.
Answer:
It is given that
Winston compares the heights of two plants to see which plant grows more per day. The table shows the height of Plant 1, in centimeters, over 5 days. The graph shows the height of Plant 2, in centimeters, over 10 days. Winston says that since Plant 1 grows 6 cm per day and Plant 2 grows 4 cm per day, Plant 1 grows more per day.
So,
From the given information,
The height growth of plant 1 > The height growth of plant 2
Hence, from the above,
You can agree with Winston

b. What errors might Winston have made?
Answer:
For plant 1,
The height growth per day = \(\frac{Any value of height}{The value of days correspond to the value of height}\)
= \(\frac{6}{2}\)
= 3 cm
For plant 2,
The height growth per day = \(\frac{Any value of height}{The value of days correspond to the value of height}\)
= \(\frac{4}{2}\)
= 2 cm
But,
It is given that
Winston says that Plant 1 grows 6 cm per day and Plant 2 grows 4 cm per day
But according to the calculation,
Plant 1 grows 3 cm per day and plant 2 grows 2 cm per day
So,
The calculation of the height growth of the plants are the errors made by Winston

Assessment Practice

Question 11.
Ashton, Alexa, and Clara want to know who types the fastest. The equation y = 39x models the rate at which Ashton can type, where y is the number of words typed and x is the time in minutes. The table shows the relationship between words typed and minutes for Alexa. The graph shows the same relationship for Clara. Who types the fastest?
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.15
Answer:
It is given that
Ashton, Alexa, and Clara want to know who types the fastest. The equation y = 39x models the rate at which Ashton can type, where y is the number of words typed and x is the time in minutes. The table shows the relationship between words typed and minutes for Alexa. The graph shows the same relationship for Clara.
So,
The rate at which Ashton can type = \(\frac{y}{x}\)
= 39 words per minute (From the equation y = 39x)
The rate at which Alexa can type = \(\frac{Any value of the words typed from the table}{The value of minds corresponds to the words typed}\)
= \(\frac{78}{2}\)
= 39 words per minute
The rate at which Clara can type = \(\frac{Any value of the words typed from the graph}{The value of minds corresponds to the words typed}\)
= \(\frac{78}{2}\)
= 39 words per minute
So,
The rate at which Ashton can type = The rate at which Alexa can type = the rate at which Clara can type
Hence, from the above,
We can conclude that no one is the fastest

Lesson 2.6 Connect Proportional Relationships and Slope

ACTIVITY

Solve & Discuss It!

In the fall, Rashida earns money as a soccer referee for her town’s under-10 soccer league. So far, she has worked 5 games and has been paid $98.50. She will work a total of 14 games this fall. How can Rashida determine how much she will earn refereeing soccer games this fall?
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.20
Answer:
It is given that
In the fall, Rashida earns money as a soccer referee for her town’s under-10 soccer league. So far, she has worked 5 games and has been paid $98.50. She will work a total of 14 games this fall.
So,
The amount of money paid for 1 game = \(\frac{The amount of money paid for 5 games}{5}\)
= \(\frac{$98.50}{5}\)
= $19.70
So,
The amount of money paid for 14 games to Rashida = (The total number of games) × (The amount of money paid for 1 game)
= 14 × $19.70
= $275.80
Hence, from the above,
We can conclude that by finding out the money paid to a game for Rashida, Rashida can find total money earned by refereeing soccer games in the fall

Look for Relationships
How is the number of games Rashida works related to her earnings?
Answer:
From the above,
We can observe that Rashida earns more money by refereeing more soccer games
Hence, from the above,
We can conclude that
The number of games Rashida works ∝ The earnings of Rashida

Focus on math practices
Reasoning: How would Rashida’s earnings change if she were paid by the hour instead of by the game?
Answer:
Rashida’s earnings would increase if she were paid by the hour instead of by the game
Example:
From the above,
We can observe that
The money earned by Rashida per game = $17.90
But, if a game will continue for 2 hours and the amount of money that is per game will also be applicable to this situation, then
The amount of money earned by Rashida for this game = $17.90 × 2 = $35.80
Hence, from the above,
We can conclude that Rashida can earn more if she were paid by the hour instead of by the game

? Essential Question
What is the slope?
Answer:
The slope of a line is a measure of its steepness. Mathematically, the slope is calculated as “rise over run” (change in y divided by change in x).
The representation of the slope mathematically is:
Slope = \(\frac{Rise}{Run}\)

Try It!

Jack graphs how far he plans to bike over a 3-day charity ride. Find the slope of the line.
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.21
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.22
Answer:
It is given that Jack graphs how far he plans to bike over a 3-day charity ride
Now,
From the given graph,
The given points are: (3, 90), and (2, 60)
Compare the given points with (x1, y1), and (x2, y2)
We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{y2  – y1}{x2 – x1}\)
So,
The slope of the line = \(\frac{60 – 90}{2 – 3}\)
= 30
Hence, from the above,
We can conclude that the slope of the line is: 30

Convince Me!
How do the unit rate and constant of proportionality relate to the slope of a line?
Answer:
The relative steepness of the line is called slope. The slope of a graph is the same as the constant of proportionality of the equation. A line with a steeper slope has a larger value for k.

Try It!
The graph shows the proportions of red and blue food coloring that Taylor mixes to make the purple frosting. What is the slope of the line? Tell what it means in the problem situation.
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.23
Answer:
It is given that
The graph shows the proportions of red and blue food coloring that Taylor mixes to make the purple frosting.
Now,
From the given graph,
The given points are: (50, 70), and (25, 35)
Compare the given points with (x1, y1), and (x2, y2)
We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{y2  – y1}{x2 – x1}\)
So,
The slope of the line = \(\frac{70 – 35}{50 – 25}\)
= \(\frac{35}{25}\)
= \(\frac{7}{5}\)
Hence, from the above,
We can conclude that
For every 7 parts of red food coloring, we have to mix 5 parts of blue food coloring to make the purple frosting

KEY CONCEPT

Slope is the measure of the steepness of a line. It represents the ratio of the rise (that is, the vertical distance) to the run (the horizontal distance) between two points on the line. In proportional relationships, slope is the same as the unit rate and constant of proportionality.
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.24
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 160.25

Do You Understand?

Question 1.
? Essential Question
What is the slope?
Answer:
The slope of a line is a measure of its steepness. Mathematically, the slope is calculated as “rise over run” (change in y divided by change in x).
The representation of the slope mathematically is:
Slope = \(\frac{y2  – y1}{x2 – x1}\) (or) Sloe = \(\frac{Rise}{Run}\)

Question 2.
Reasoning How is the slope related to a unit rate?
Answer:
The slope is the unit rate, which is the coefficient of x. For a table, the change in y divided by the change in x is the unit rate or slope.

Question 3.
Look for Relationships Why is the slope between any two points on a straight line always the same?
Answer:
The ratio of the rise over run describes the slope of all straight lines. This ratio is constant between any two points along a straight line, which means that the slope of a straight line is constant, too, no matter where it is measured along the line.

Do You Know How?

Question 4.
What is the slope of the line?
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 180.1
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 180.1
We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{ Any value of y}{The value of x that corresponds to the taken value of y}\)
= \(\frac{Price ($)}{Grapes (lb)}\)
So,
The slope of the line = \(\frac{6}{2}\)
= 3
Hence, from the above,
We can conclude that the slope of the line is: 3

Question 5.
The scale of a model airplane is shown in the graph.
a. Find the slope of the line using \(\frac{y2  – y1}{x2 – x1}\)
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 180.2
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 180.2
Now,
From the given graph,
The given points are: (6, 10), and (3, 5)
Compare the given points with (x1, y1), and (x2, y2)
We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{y2  – y1}{x2 – x1}\)
So,
The slope of the line = \(\frac{5 – 10}{3 – 6}\)
= \(\frac{5}{3}\)
Hence, from the above,
We can conclude that the slope of the line is: \(\frac{5}{3}\)

b. What does the slope mean in the problem situation?
Answer:
From part (a),
The slope is: \(\frac{5}{3}\)
So,
From the above slope,
We can conclude that for every 3 cm, the model airplane can fly 5 feet

Practice & Problem Solving

Leveled Practice in 6 and 7, find the slope of each line.

Question 6.
The graph shows the number of soda bottles a machine can make over time. Use the two points shown to find the number of soda bottles the machine can make per minute.
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 180.3
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 180.4
The machine Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 180.5 can make soda bottles each minute.
Answer:
It is given that
The graph shows the number of soda bottles a machine can make over time
Now,
The given graph is:
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 180.4
Now,
From the given graph,
The given points are: (6, 150), and (2, 50)
Compare the given points with (x1, y1), and (x2, y2)
We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{y2  – y1}{x2 – x1}\)
So,
The slope of the line = \(\frac{50 – 150}{2 – 6}\)
= \(\frac{100}{4}\)
= 25
Hence, from the above,
We can conclude that the machine can make 25 soda bottles each minute

Question 7.
Find the slope of the line.
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 180.55
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 180.55
We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{ Any value of y}{The value of x that corresponds to the taken value of y}\)
= \(\frac{Items}{Time in min}\)
So,
The slope of the line = \(\frac{50}{10}\)
= 5
Hence, from the above,
We can conclude that the slope of the line is: 5

Question 8.
Reasoning How can you find the slope of the line that passes through the points (0,0) and (2, 4)? Explain.
Answer:
The given points are: (0, 0), and (2, 4)
Compare the given points with (x1, y1), and (x2, y2)
We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{y – y1}{x2 – x1}\)
So,
The slope of the line = \(\frac{4 – 0}{2 – 0}\)
= \(\frac{4}{2}\)
= 2
Hence, from the above,
We can conclude that the slope of the line is: 2

Question 9.
The points (2.1, -4.2) and (2.5, -5) form a proportional relationship. What is the slope of the line that passes through these two points?
Answer:
It is given that the points (2.1, -4.2) and (2.5, -5) form a proportional relationship
Now,
The given points are: (2.1, -4.2), and (2.5, -5)
Compare the given points with (x1, y1), and (x2, y2)
We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{y – y1}{x2 – x1}\)
So,
The slope of the line = \(\frac{-5 + 4.2}{2.5 – 2.1}\)
= \(\frac{-0.8}{0.4}\)
= -2
Hence, from the above,
We can conclude that the slope of the line that passes through the given points is: -2

Question 10.
Find the slope of the line.
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 180.6
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 180.6
Now,
From the graph,
We can observe that
The given points are: (-3, 7), and (-1, 2)
Compare the given points with (x1, y1), and (x2, y2)
We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{y – y1}{x2 – x1}\)
So,
The slope of the line = \(\frac{2 – 7}{-1 + 3}\)
= \(\frac{-5}{2}\)
= –\(\frac{5}{2}\)
Hence, from the above,
We can conclude that the slope of the line that passes through the given points is: –\(\frac{5}{2}\)

Question 11.
The graph shows the number of Calories Natalia burned while running.
a. What is the slope of the line?
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 180.7
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations 180.7
We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{ Any value of y}{The value of x that corresponds to the taken value of y}\)
= \(\frac{Calories}{Time in min}\)
So,
The slope of the line = \(\frac{70}{7}\)
= 10
Hence, from the above,
We can conclude that the slope of the line is: 10

b. What does the slope tell you?
Answer:
From part (a),
We can observe that
The slope of the line is: 10
So,
From the given slope,
We can conclude that Natalia burns 10 calories per minute while running

Question 12.
Critique Reasoning A question on a test provides this graph and asks students to find the speed at which the car travels. Anna incorrectly says that the speed of the car is \(\frac{1}{64}\) mile per hour.
a. What is the speed of the car?
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 180.8
Answer:
The given graph is:
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 180.8
We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{ Any value of y}{The value of x that corresponds to the taken value of y}\)
= \(\frac{Distance in miles}{Time in hours}\)
So,
The slope of the line = \(\frac{256}{4}\)
= 64
Hence, from the above,
We can conclude that the speed of the car is: 64 miles per hour

b. What error might Anna have made?
Answer:
From part (a),
We can observe that the speed of the car is: 64 miles per hour
Bt,
According to Anna,
The speed of the car is: \(\frac{1}{64}\) miles per hour
So,
The error made by Anna is that she takes the slope in the form of \(\frac{x}{y}\) but the actual form of the slope is \(\frac{y}{x}\)

Question 13.
Higher-Order Thinking You use a garden hose to fill a wading pool. If the water level rises 11 centimeters every 5 minutes and you record the data point of (10, y), what is the value of y? Use slope to justify your answer.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 180.9
Answer:
It is given that
You use a garden hose to fill a wading pool. If the water level rises 11 centimeters every 5 minutes and you record the data point of (10, y)
We know that,
Slope = \(\frac{Rise}{Run}\)
So,
From the given information,
We can write the slope as:
Slope = \(\frac{11}{5}\)
Now,
Compare the given point with (x, y)
So,
The slope of the line = \(\frac{y}{x}\)
= \(\frac{y}{10}\)
So,
\(\frac{y}{10}\) = \(\frac{11}{5}\)
Multiply with 10 on both sides
So,
y = \(\frac{11 × 10}{5}\)
y = 22
Hence, from the above,
We can conclude that the value of y is: 22

Assessment Practice

Question 14.
The points (15, 21) and (25, 35) form a proportional relationship.
a. Find the slope of the line that passes through these points.
Answer:
It is given that the points (15, 21) and (25, 35) form a proportional relationship.
Now,
The given points are: (15, 21), and (25, 35)
Compare the given points with (x1, y1), and (x2, y2)
We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{y – y1}{x2 – x1}\)
So,
The slope of the line = \(\frac{35 – 21}{25 – 15}\)
= \(\frac{14}{10}\)
= \(\frac{7}{5}\)
Hence, from the above,
We can conclude that the slope of the line that passes through the given points is: \(\frac{7}{5}\)

b. Which graph represents this relationship?
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 180.10
Answer:
We know that,
The representation of the equation when two points form  a proportionate relationship is:
y = kx
and the line have to pass through the origin i.e., (0, 0)
So,
From the given graphs,
The graphs B and C have the possibility to become the graph of the given points
Now,
We know that,
Slope = \(\frac{Rise}{Run}\)
From graph B,
Slope = \(\frac{42}{30}\)
= \(\frac{7}{5}\)
From graph C,
Slope = \(\frac{30}{42}\)
= \(\frac{5}{7}\)
Hence, from the above,
We can conclude that the graph B represents the given relationship

Lesson 2.7 Analyze Linear Equations: y = mx

ACTIVITY

Explore It!

A group of college students developed a solar-powered car and entered it in a race. The car travels at a constant speed of 100 meters per 4 seconds.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 180.11
A. What representation can show the distance the car will travel over time?
Answer:
It is given that
A group of college students developed a solar-powered car and entered it in a race. The car travels at a constant speed of 100 meters per 4 seconds.
We know that,
Speed = \(\frac{Distance}{Time}\)
It is given that speed is constant
So,
Distance ∝ Time
So,
The greater the distance, the greater the time
Hence, from the above,
We can conclude that
The representation that can show the distance the car will travel over time is:
Distance ∝ Time

B. What expression can show the distance the car will travel over time?
Answer:
From part (a),
We can observe that
Distance ∝ Time (Since the speed is constant)
Hence,
The expression that can show the distance the car will travel over time is:
Distance = k (Time)
Where,
k is a constant

C. Compare the representation and the expression. Which shows the distance traveled over time more clearly? Explain.
Answer:
From part (a),
The representation that can show the distance traveled over time is:
Distance ∝ Time
The expression that can show the distance traveled over time is:
Distance = k (Time)
Now,
From the representation and the expression,
We can observe that the expression shows the distance traveled over time more clearly because for any value of distance and time, the value of the expression is constant
Hence, from the above,
We can conclude that the expression shows the distance traveled over time more clearly

Focus on math practices
Be Precise How would the representation or expression change if the speed was converted to miles per minute?
Answer:
From part (a),
The representation is:
Distance ∝ Time
The expression is:
Distance = k (Time)
Now,
Even if the speed was converted to miles per minute, there will be no change in the representation and the expression because miles per minute is a unit of speed and it won’t affect the overall situation of the representation and the expression

? Essential Question
How does slope relate to the equation for a proportional relationship?
Answer:
The steepness of the slope for directly proportional relationships increases as the value of the constant m (y = mx) increases.

Try It!
Write an equation to describe the relationship shown in the graph.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 180.12. The equation of the line is y = Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 180.13x.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 180.14
Answer:
The given graph is:
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 180.14
Now,
From the given graph,
The points are: (3, 60), and (4, 80) [We can take any 2 ordered pairs from the graph like (0, 0), and (1, 20); (2, 40), and (3, 60), etc]
Now,
Compare the given points with (x1, y1), and (x2, y2)
We know that,
Slope can be represented as “m”
So,
m = \(\frac{y – y1}{x2 – x1}\)
So,
m = \(\frac{80 – 60}{4 – 3}\)
= \(\frac{20}{1}\)
= 20
We know that,
The equation of the line is:
y = mx
Hence, from the above,
We can conclude that
The equation of the line is: y = 20x

Convince Me!
How do the equations y = mx and y = kx compare?
Answer:
We can compare y = kx to the slope-intercept form of a line, y = mx + b. We can see that y = kx is a linear equation with slope k and y-intercept 0. This tells us that the graph of a direct variation is a line that passes through the origin, point (0,0).

Try It!
a. Write the equation of the line.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 180.15
Answer:
The given graph is:
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 180.15
Now,
From the given graph,
The points are: (10, 4), and (-10, -4)
Now,
Compare the given points with (x1, y1), and (x2, y2)
We know that,
Slope can be represented as “m”
So,
m = \(\frac{y – y1}{x2 – x1}\)
So,
m = \(\frac{-4 – 4}{-10 – 10}\)
= \(\frac{-8}{-20}\)
= \(\frac{2}{5}\)
We know that,
The equation of the line is:
y = mx
So,
y = \(\frac{2}{5}\)x
Multiply with 5 on both sides
So,
5y = 2x
Hence, from the above,
We can conclude that
The equation of the line is: 5y = 2x

b. Graph the line y = -3x.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 180.16
Answer:
The given equation is:
y = -3x
Hence,
The representation of the given equation in the coordinate plane is:

KEY CONCEPT

The equation for a proportional relationship is y = mx where m represents the slope of the line.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 180.17

Do You Understand?

Question 1.
Essential Question How does slope relate to the equation for a proportional relationship?
Answer:
The steepness of the slope for directly proportional relationships increases as the value of the constant m (y = mx) increases.

Question 2.
Look for Relationships What do the graphs of lines in the form y = mx have in common? How might they differ?
Answer:
The graphs of lines in the form y = mx are all straight lines that pass through the origin

Question 3.
Use Structure The table below shows the distance a train traveled over time. How can you determine the equation that represents this relationship?
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 181.2
Answer:
It is given that
The table below shows the distance a train traveled over time.
Now,
Verify whether \(\frac{Distance}{Time}\) is constant or not
Now,
From the given table,
For 25 m and 2s,
\(\frac{Distance}{Time}\) = \(\frac{25}{2}\)
For 50m and 4s,
\(\frac{Distance}{Time}\) = \(\frac{50}{4}\) = \(\frac{25}{2}\)
Since,
\(\frac{Distance}{Time}\) is constant
Speed is also constant
So,
The representation of the equation that describes the given relationship is:
Distance = k (Time)
So,
y = mx [ Compare the above equation with y = mx ]
Where
m is a constant slope
So,
y = \(\frac{25}{2}\)x
2y = 25x
Hence, from the above,
We can conclude that the representation of the equation that represents the given situation is: 2y = 25x

Do You Know How?

Question 4.
The relationship between a hiker’s elevation and time is shown in the graph.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 181.3
a. Find the constant of proportionality of the line. Then find the slope of the line.
Answer:
It is given that
The relationship between a hiker’s elevation and time is shown in the graph.
Now,
We know that,
The constant of proportionality and the slope are the same
So,
Slope of the line (m) = \(\frac{y}{x}\)
So,
From the given graph,
\(\frac{y}{x}\) = \(\frac{120}{4}\)
= 30
So,
m = 30
Hence, from the above,
We can conclude that the slope of the line is: 30

b. Write the equation of the line.
Answer:
We know that,
The equation of the line is:
y = mx
From part (a),
m = 30
Hence, from the above,
We can conclude that the equation of the line is: y = 30x

Question 5.
Graph the equation y = –\(\frac{1}{2}\)x.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 181.4
Answer:
The given equation is:
y = –\(\frac{1}{2}\)x
Hence,
The representation of the given equation in the coordinate plane is:

Practice & Problem Solving

Question 6.
Leveled Practice Resting heart rate is a measure of how fast the heart beats when a person is not performing physical activity. The graph shows the number of heartbeats over time for a given person.
a. Use two sets of coordinates to write an equation to describe the relationship.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 181.5
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 181.6
Answer:
It is given that
Resting heart rate is a measure of how fast the heart beats when a person is not performing physical activity. The graph shows the number of heartbeats over time for a given person.
Now,
From the given graph,
The points are: (3, 210), and (4, 280)
Now,
Compare the given points with (x1, y1), and (x2, y2)
We know that,
Slope can be represented as “m”
So,
m = \(\frac{y – y1}{x2 – x1}\)
So,
m = \(\frac{280 – 210}{4 – 3}\)
= \(\frac{70}{1}\)
= 70
We know that,
The equation of the line is:
y = mx
Hence, from the above,
We can conclude that
The equation of the line that describes the given situation is: y = 70x

b. Interpret the equation in words.
The heart’s resting heart rate is Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 182.1 beats each minute.
Answer:
From part (a),
The equation of the line that describes the given situation is: y = 70x
Hence, from the above,
We can conclude that the heart’s resting heart rate is 70 beats each minute

Question 7.
Model with Math The graph relates the number of gallons of white paint to the number of gallons of red paint Jess used to make the perfect pink. Write an equation that describes the relationship.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 182.10
Answer:
It is given that
The graph relates the number of gallons of white paint to the number of gallons of red paint Jess used to make the perfect pink.
Now,
The given graph is:
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 182.10
Now,
Slope of the given line (m) = \(\frac{y}{x}\)
m = \(\frac{4}{1}\)
m = 4
We know that,
The equation of the line is:
y = mx
Hence, from the above,
We can conclude that the equation of the line that represents the given situation is: y = 4x

Question 8.
Critique Reasoning Franco made this graph to show the equation y = -x. Is the graph correct? Explain.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 182.100
Answer:
The given graph is:
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 182.100
Now,
We know that,
Slope of the line (m) = \(\frac{y}{x}\)
m = \(\frac{4}{4}\)
m = 1
We know that,
The equation of the line is:
y = mx
So,
The equation of the line is:
y = x
But,
Franco made this graph to show the equation y = -x
Hence, from the above,
We can conclude that the graph of Franco is not correct

Question 9.
The graph shows a proportional relationship between the variables x and y.
a. Write an equation to model the relationship.
b. Reasoning Explain how you know if an equation or a graph represents a proportional relationship.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 183.1
Answer:
a.
The given graph is:
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 183.1
We know that,
The slope of the line (m) = \(\frac{y}{x}\)
= \(\frac{96}{8}\)
= 12
We know that,
The equation of the line is:
y = mx
So,
The equation of the line to the given relationship is:
y = 12x
Hence, from the above,
We can conclude that the equation of the line that represents the given situation is: y = 12x

b.
If the relationship between two quantities is a proportional relationship, this relationship can be represented by the graph of a straight line through the origin with a slope equal to the unit rate. For each point (x, y) on the graph, ž is equal to k, where k is the unit rate.

Question 10.
Model with Math Graph the equation y = -5x on the coordinate plane.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 183.50
Answer:
The given equation is:
y = -5x
Hence,
The representation of the given equation in the coordinate plane is:

Question 11.
Graph the equation y = \(\frac{3}{5}\)x on the coordinate plane.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 183.4
Answer:
The given equation is:
y = \(\frac{3}{5}\)x
Hence,
The representation of the given equation in the coordinate plane is:

Question 12.
Higher-Order Thinking A movie theater sends out a coupon for 70% off the price of a ticket.
a. Write an equation for the situation, where y is the price of the ticket with the coupon and x is the original price.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 183.5
Answer:
It is given that
A movie theater sends out a coupon for 70% off the price of a ticket.
So,
The proportionality constant of the given situation = \(\frac{70}{100}\)
= \(\frac{7}{10}\)
We know that,
Proportionality constant = Slope
So,
Slope (m) = \(\frac{7}{10}\)
We know that,
The equation of the line is:
y = mx
So,
y = \(\frac{7}{10}\)
10y = 7x
Hence, from the above,
We can conclude that the equation of the line for the given situation is: 10y = 7x

b. Graph the equation and explain why the line should only be in the first quadrant.
Answer:
From part (a),
The equation of the line is:
10y = 7x
So,
The representation of the given equation in the coordinate plane is:

From the graph,
We can observe that
The graph should only be in 1st quadrant because the values of x and y are both positive

Assessment Practice

Question 13.
An equation and a graph of proportional relationships are shown. Which has the greater unit rate? y = \(\frac{47}{2}\)x
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 183.6
Answer:
The given graph is:
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 183.6
From the given graph,
Slope (m) = \(\frac{y}{x}\)
= \(\frac{282}{6}\)
= 47
Now,
The given equation is:
y = \(\frac{47}{2}\)x
So,
Slope (m) = \(\frac{y}{x}\)
= \(\frac{47}{2}\)
Now,
When we compare the rates or slopes,
47 > \(\frac{47}{2}\)
Hence, from the above,
We ca conclude that the unit rate of the graph is greater than the unit rate of the equation

Question 14.
Car X travels 186 miles in 3 hours.
PART A Write the equation of the line that describes the relationship between distance and time.
Answer:
It is given that car X travels 186 miles in 3 hours.
Now,
We know that,
Speed = \(\frac{Distance}{Time}\)
We know that,
The equation of the line is:
y = mx
Where,
m = \(\frac{Distance}{Time}\)
So,
The equation of the line is:
y = \(\frac{186}{3}\)x
y = 62x
Hence, from the above,
We can conclude that the equation of the line that descries the relationship between distance and time is:
y = 62x

PART B Which graph represents the relationship between distance and time for Car X?
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 183.7
Answer:
From part (a),
The equation of the line that describes the relationship between distance and time is:
y = 62x
Where,
62 —-> The value of \(\frac{y}{x}\) (or) m
So,
From the above graphs,
We can observe that,
m = 62 is possible from graphs C and D
But,
We know that,
The equation y = mx passes through the origin
Hence, from the above,
We can conclude that the graph C represents the relationship between distance and time for car X

Lesson 2.8 Understand the y-Intercept of a Line

Solve and Discuss It!

Eight-year-old Alex is learning to ride a horse. The trainer says that a horse ages 5 years for every 2 human years. The horse is now 50 years old in human years. How can you determine the age of the horse, in human years, when Alex was born?
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 184.1
Answer:
It is given that
Eight-year-old Alex is learning to ride a horse. The trainer says that a horse ages 5 years for every 2 human years. The horse is now 50 years old in human years.
So,
When Alex is 8 years old,
The age of the horse in human years is: 50 years
Now,
For every 2 human years, the horse ages 5 years
So,
So,
For Alex,
The number of times his age increases = \(\frac{8}{2}\)
= 4 times
So,
The increase in the age of the horse when Alex is 8 years old = 5 × 4 = 20 years
So,
The age of the horse when Alex born = The present age of the horse – The increased age of the horse
= 50 – 20
= 30 years
Hence, from the above,
We can conclude that the age of the horse when Alex is born is: 30 years

Focus on math practices
Use Structure A veterinarian says that cat ages 8 years for every 2 human years. If a cat is now 64 years old in cat years, how old is the cat in human years?
Answer:
It is given that
A veterinarian says that a cat ages 8 years for every 2 human years.
Now,
Let the age of the cat in human years be x
So,
\(\frac{The age of the cat in cat years}{The age of the cat in human years}\) = \(\frac{The increase of the age of the cat for the increase of human years}{The increase of the age of human for the increase of human years}\)
\(\frac{64}{x}\) = \(\frac{8}{2}\)
Divide by 64 into both sides
So,
\(\frac{64}{x × 64}\) = \(\frac{8}{2 × 64}\)
\(\frac{1}{x}\) = \(\frac{1}{16}\)
x = 16 years
Hence, from the above,
We can conclude that the age of cat in human years is: 16 years

? Essential Question
What is the y-intercept and what does it indicate?
Answer:
The slope and y-intercept values indicate characteristics of the relationship between the two variables x and y. The slope indicates the rate of change in y per unit change in x. The y-intercept indicates the y-value when the x-value is 0.

Try It!
Prices for a different bowling alley are shown in the graph. How much does this bowling alley charge for shoe rental? The line crosses the y-axis as Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 190.1
The y-intercept is Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 190.2
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 190.3
Answer:
It is given that
The prices for a different bowling alley are shown in the graph
So,
y-coordinate of the point where the line crosses the y-axis is the y-intercept
So,
From the given graph,
The given line crosses the y-axis at (0, 3)
We know that,
The y-intercept is the value of y when the value of x is 0
Hence, from the above,
We can conclude that
The given passes through (0, 3)
The y-intercept is: 3

Convince Me!
In these examples, why does the y-intercept represent the cost to rent bowling shoes?
Answer:
In this example,
From the slope,
We can determine the cost of each game
We know that,
y-coordinate of the point where the line crosses the y-axis is the y-intercept and that y-intercept is the cost to rent bowling shoes because the cost won’t ever be zero

Try It!
What is the y-intercept of each graph? Explain.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 190.4
Answer:
Let the given graphs be named as graph A and graph B respectively
We know that,
y-coordinate of the point where the line crosses the y-axis is the y-intercept
So,
From graph A,
The y-intercept is: (0, 2)
From graph B,
The y-intercept is: (0, -0.5)

KEY CONCEPT
The y-intercept is the y-coordinate of the point on a graph where the line crosses the y-axis.
When the line crosses through the origin, the y-intercept is 0.
When the line crosses above the origin, the y-intercept is positive.
When the line crosses below the origin, the y-intercept is negative.
Envision Math Common Core 8th Grade Answers Topic 2 Analyze And Solve Linear Equations 191.2

Do You Understand?

Question 1.
? Essential Question What is the y-intercept and what does it indicate?
Answer:
The slope and y-intercept values indicate characteristics of the relationship between the two variables x and y. The slope indicates the rate of change in y per unit change in x. The y-intercept indicates the y-value when the x-value is 0.

Question 2.
Look for Relationships Chelsea graphs a proportional relationship. Bradyn graphs a line that passes through the origin. What do you know about the y-intercept of each student’s graph? Explain your answer.
Answer:
It is given that
Chelsea graphs a proportional relationship. Bradyn graphs a line that passes through the origin
So,
From the given situation,
We can observe that
The graph of Chelsea may pass through the origin or may not pass through the origin i.e., the y-intercept may be zero, positive, or negative
The graph of Braydon passes through the origin i.e., the y-intercept is zero

Question 3.
Generalize When the y-intercept is positive, where does the line cross the y-axis on the graph? When it is negative?
Answer:
When the y-intercept is positive, the line crosses above the origin,
When the y-intercept is negative, the line crosses below the origin

Do You Know How?

Question 4.
What is the y-intercept shown in the graph?
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 192.5
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 192.5
From the given graph,
We can observe that the line passes through the origin
Hence, from the above,
We can conclude that the value of the y-intercept is: 0

Question 5.
The graph shows the relationship between the remaining time of a movie and the amount of time since Kelly hit “play.” What is the y-intercept of the graph and what does it represent?
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 192.8
Answer:
It is given that
The graph shows the relationship between the remaining time of a movie and the amount of time since Kelly hit “play.”
Now,
We know that,
y-coordinate of the point where the line crosses the y-axis is the y-intercept
So,
From the graph,
We can observe that the line crosses the y-axis at (0, 1.8)
Hence, from the above,
We can conclude that the y-intercept of the graph is: 1.8
The y-intercept represents the remaining time of a movie in the given situation

Practice & Problem Solving

Question 6.
Leveled Practice Find the y-intercept of the line. The y-intercept is the point where the graph crosses the Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 192.9-axis.
The line crosses the y-axis at the point Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 192.10
The y-intercept is Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 192.9
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 192.100
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 192.100
Now,
We know that,
y-coordinate of the point where the line crosses the y-axis is the y-intercept
So,
From the graph,
We can observe that
The line crosses the y-axis at the point (0, 7)
Hence, from the above,
We can conclude that
The y-intercept is the point where the graph crosses the y-axis
The y-intercept for the given graph is: 8

Question 7.
Find the y-intercept of the graph.
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 192.99
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 192.99
Now,
We know that,
y-coordinate of the point where the line crosses the y-axis is the y-intercept
So,
From the graph,
We can observe that
The line crosses the y-axis at the point (0, -4)
Hence, from the above,
We can conclude that
The y-intercept for the given graph is: -4

Question 8.
Find the y-intercept of the graph.
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 192.101
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 192.101
Now,
We know that,
y-coordinate of the point where the line crosses the y-axis is the y-intercept
Now,
From the graph,
We can observe that
The equation of the line is:
y = kx
From the above equation,
We can say that the line passes through the origin
So,
The line crosses the y-axis at the point (0, 0)
Hence, from the above,
We can conclude that
The y-intercept of the given graph is: 0

Question 9.
The graph represents the height y, in meters, of a hot air balloon x minutes after beginning to descend. How high was the balloon when it began its descent?
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 192.11
Answer:
It is given that
The graph represents the height y, in meters, of a hot air balloon x minutes after beginning to descend
Now,
We know that,
y-coordinate of the point where the line crosses the y-axis is the y-intercept
So,
From the graph,
We can observe that the line crosses the y-axis at (0, 80)
The y-intercept of the graph gives us information about the height of the balloon when it began its descent
Hence, from the above,
We can conclude that the height of the balloon when it began its descent is: 80 m

Question 10.
Model with Math The graph represents the amount of gasoline in a canister after Joshua begins to fill it at a gas station pump. What is the y-intercept of the graph and what does it represent?
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 192.12
Answer:
It is given that
The graph represents the amount of gasoline in a canister after Joshua begins to fill it at a gas station pump.
Now,
We know that,
y-coordinate of the point where the line crosses the y-axis is the y-intercept
Now,
From the graph,
We can observe that the line passes through the origin
So,
The line crosses the y-axis at the point (0, 0)
Hence, from the above,
We can conclude that
The y-intercept of the given graph is: 0
The y-intercept of the given graph represents the amount of gas in gallons at the starting time

Question 11.
The line models the temperature on a certain winter day since sunrise.
a. What is the y-intercept of the line?
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 192.13
Answer:
It is given that
The line models the temperature on a certain winter day since sunrise.
Now,
We know that,
y-coordinate of the point where the line crosses the y-axis is the y-intercept
So,
From the graph,
We can observe that the line crosses the y-axis at (0, 4)
Hence, from the above,
We can conclude that the y-intercept of the given line is: 4

b. What does the y-intercept represent?
Answer:
The y-intercept of the graph gives us information about the starting temperature on a certain winter day at sunrise

Question 12.
Higher-Order Thinking Your friend incorrectly makes this graph as an example of a line with a y-intercept of 3.
a. Explain your friend’s possible error.
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 192.14
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 192.14
Now,
From the given graph,
We can observe that the line crosses the y-axis at: (0, 4)
So,
The y-intercept of the graph is: 4
But,
Your friend incorrectly makes this graph as an example of a line with a y-intercept of 3.
Hence, from the above,
We can conclude that the y-intercept of the given graph is 4 but not 3

b. Draw a line on the graph that does represent a y-intercept of 3.
Answer:
Let the equation with the y-intercept of 3 is:
y = x + 3
Hence,
The representation of the graph that does represent a y-intercept of 3 in the coordinate plane is:

Assessment Practice

Question 13.
For each graph, draw a line through the point such that the values of the x-intercept and y-intercept are additive inverses.
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 192.15
Answer:
Let the graphs be named as graph A and graph B respectively
Now,
The given graphs are:
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 192.15
So,
From graph A,
We can observe that the x-intercept is 3 and the y-intercept is 3
We know that,
The “Additive inverse” of a number ‘a’ is the number that, when added to ‘a’, yields zero. This number is also known as the opposite (number), sign change, and negation.
So,
The additive inverses of the x-intercept and y-intercept are: (-3, -3)
From graph B,
We can observe that the x-intercept is -3 and the y-intercept is -3
We know that,
The “Additive inverse” of a number ‘a’ is the number that, when added to ‘a’, yields zero. This number is also known as the opposite (number), sign change, and negation.
So,
The additive inverses of the x-intercept and y-intercept are: (3, 3)
Hence,
The representation of the additive inverses of the x and y-intercepts in the coordinate plane is:

Question 14.
Which statements describe the graph of a proportional relationship? Select all that apply.
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 193.1 The y-intercept is always at the point (0, 1).
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 193.1 The line always crosses the y-axis at (0, 0).
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 193.1 The y-intercept is 0.
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 193.1 The y-intercept is 1.
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 193.1 The line does NOT cross the y-axis.
Answer:
Let the options be named as A, B, C, D, and E respectively
Now,
We know that,
The representation of the proportional relationship is:
y = kx
So,
From the equation,
We can say that the equation passes through the origin and the y-intercept is 0
Hence, from the above,
We can conclude that options B and C describes the proportional relationship

Lesson 2.9 Analyze Linear Equations: y = mx + b

ACTIVITY

Explain It!

Xiu and Jon take the tram from the base camp to the mountain summit. After about six and a half minutes in the tram, Jon says, “Cool! We are a mile above sea level.” Xiu says, “We passed the one-mile mark a couple of minutes ago.”
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 194.1

A. Construct an argument to defend Xiu’s statement.

B. What mistake could Jon have made? Explain.
Answer:

Focus on math practices
Reasoning Can you use the equation y = mx to represent the path of the tram? Is there a proportional relationship between x and y? Explain.

? Essential Question
What is the equation of a line for a nonproportional relationship?
Answer:
Linear equations can be written in the form y = mx + b. When b ≠ 0, the relationship between x and y is nonproportional.

Try It!
Write a linear equation in slope-intercept form for the graph shown.
The y-intercept of the line is Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 195.1
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 195.2
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 195.3
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 195.3
From the given graph,
We can observe that,
The points are: (8, 8), and (4, 5)
Compare the given points with (x1, y1), (x2, y2)
We know that,
The y-intercept of the line is the point that crosses the y-axis
So,
From the given graph,
The y-intercept is: 2
We know that,
The linear equation in the slope-intercept form is:
y = mx + c
Where,
m is the slope
c is the y-intercept
Now,
m = \(\frac{y2 – y1}{x2 – x1}\)
= \(\frac{5 – 8}{4 – 8}\)
= \(\frac{3}{4}\)
So,
The linear equation in the slope-intercept form is:
y = \(\frac{3}{4}\)x + 2
y = \(\frac{3x + 8}{4}\)
4y = 3x + 8
Hence, from the above,
We can conclude that the linear equation in the slope-intercept form is:
4y = 3x + 8

Convince Me!

What two values do you need to know to write an equation of a line, and how are they used to represent a line?
Answer:
To write an equation of a line in the slope-intercept form,
The two values you need to know are:
A) Slope of a line and it is represented as “m”
B) The y-intercept of a line and is represented as “c”

KEY CONCEPT

The equation of a line that represents a nonproportional relationship can be written in slope-intercept form, y = mx + b, where m is the slope of the line and b is the y-intercept.

Do You Understand?

Question 1.
? Essential Question What is the equation of a line for a nonproportional relationship?
Answer:
Linear equations can be written in the form y = mx + b. When b ≠ 0, the relationship between x and y is nonproportional.

Question 2.
Use Structure The donations by a restaurant to a certain charity, y, will be two-fifths of its profits, x, plus $50. How can you determine the equation in slope-intercept form that shows the relationship between x and y without graphing the line?
Answer:
It is given that
The donations by a restaurant to a certain charity, y, will be two-fifths of its profits, x, plus $50.
So,
Donations to a certain charity by a restaurant = The part of the profits of a restaurant + $50
y = \(\frac{2}{5}\)x + $50
Compare the above equation with
y = mx + c
Where,
m is the slope of a line
c is the y-intercept of a line
So,
When we compare the equation,
The slope of a line is (m): \(\frac{2}{5}\)
The y-intercept of a line is (c) : $50

Question 3.
Be Precise Priya will graph a line with the equation y = \(\frac{3}{4}\)x – 4. She wants to know what the line will look like before she graphs the line. Describe the line Priya will draw, including the quadrants the line will pass through.
Answer:
It is given that
Priya will graph a line with the equation y = \(\frac{3}{4}\)x – 4. She wants to know what the line will look like before she graphs the line.
Now,
Compare the given equation with
y = mx + c
Where,
m is the slope of the line
c is the y-intercept
So,
By comparing,
We get,
m = \(\frac{3}{4}\)
c = -4
Now,
From the y-intercept,
We can say that the y-intercept lies below the origin i.e., in the 3rd quadrant
From the slope of the line,
We can say that the value of m lies in the 1st quadrant
Hence, from the above,
We can conclude that the line drawn by Priya will be in the 4th quadrant for the above values of c and m

Do You Know How?

Question 4.
Chrissie says the equation of the line shown on the graph is y = \(\frac{1}{2}\)x – 5. George says that the equation of the line is y = \(\frac{1}{2}\)x + 5. Which student is correct? Explain.
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 199.1
Answer:
It is given that
Chrissie says the equation of the line shown on the graph is y = \(\frac{1}{2}\)x – 5. George says that the equation of the line is y = \(\frac{1}{2}\)x + 5.
Now,
The given graph is
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 199.1
From the given graph,
The y-intercept is: 5
Now,
When we observe the given two equations,
The slope is the same and the y-intercepts are different and the correct y-intercept must be 5
Hence, from the above,
We can conclude that George is correct

Question 5.
Fara wants to rent a tent for an outdoor celebration. The cost of the tent is $500 per hour, plus an additional $100 set-up fee.
a. Draw a line to show the relationship between the number of hours the tent is rented, x, and the total cost of the tent, y.
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 199.2
Answer:
It is given that
Fara wants to rent a tent for an outdoor celebration. The cost of the tent is $500 per hour, plus an additional $100 set-up fee.
Now,
The total cost of the rent = The cost of the rent per hour + Additional set-up fee
So,
y = 500x + 100
Hence,
The representation of the above equation in the coordinate plane is:

b. What is the equation of the line in slope-intercept form?
Answer:
We know that,
The total cost of the rent = The cost of the rent per hour + Additional set-up fee
So,
y = 500x + 100
Where,
x is the number of hours
The above equation is in the form of
y = mx + c
Which is the slope-intercept form of the equation
Hence, from the above,
We can conclude that the equation of the line in the slope-intercept form is:
y = 500x + 100

Practice & Problem Solving

Question 6.
Leveled Practice What is the graph of the equation y = 2x + 4?
The y-intercept is Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 199.3, which means the line crosses the y-axis at the point (Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 194). Plot this point.
The slope of the line is positive, so it goes Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 199.3 from left to right.
Start at the y-intercept. Move up Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 199.3, and then move right Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 199.3
You are now at the point (Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 194). Plot this point. Draw a line to connect the two points.
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 199.5
Answer:
The given equation is:
y = 2x + 4
So,
The representation of the given equation in the coordinate plane is:

Compare the given equation with
y = mx + c
Wher,
m is the slope of a line
c is the y-intercept of a line
So,
The y-intercept of the given graph is 4 which means the line crosses the y-axis at the point (0, 4)
The slope of the line is positive, so it goes up from left to right.
Start at the y-intercept. Move up 2 units, and then move right 2 units
So,
You are now at the point (3, 10).

Question 7.
Write an equation for the line in slope-intercept form.
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 199.6
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 199.6
We know that,
The equation of the line in the slope-intercept form is:
y = mx + c
Now,
From the given graph,
We can observe that the y-intercept is: -3
Now,
The given points from the graph to find the slope are: (-2, -2), and (4, -5)
Now,
SLope (m) = \(\frac{-5  – (-2)}{4 – (-2)}\)
m = \(\frac{-3}{6}\)
m = –\(\frac{1}{2}\)
So,
The equation of the line in the slope-intercept form is:
y = –\(\frac{1}{2}\)x – 3
Hence, from the above,
We can conclude that the equation of the line in the slope-intercept form is:
y = –\(\frac{1}{2}\)x – 3

Question 8.
Write an equation for the line in slope-intercept form.
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 199.7
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 2 Analyze And Solve Linear Equations 199.7
We know that,
The equation of the line in the slope-intercept form is:
y = mx + c
Now,
From the given graph,
We can observe that the y-intercept is: 4
Now,
The given points from the graph to find the slope are: (1, 1), and (0, 4)
Now,
SLope (m) = \(\frac{4 – 1}{0 – 1}\)
m = \(\frac{3}{-1}\)
m = -3
So,
The equation of the line in the slope-intercept form is:
y = -3x + 4
Hence, from the above,
We can conclude that the equation of the line in the slope-intercept form is:
y = -3x + 4

Question 9.
The line models the cost of renting a kayak. Write an equation in slope-intercept form for the line, where x is the number of hours the kayak is rented and y is the total cost of renting the kayak.
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 199.8
Answer:
It is given that
The line models the cost of renting a kayak
where,
x is the number of hours the kayak is rented and y is the total cost of renting the kayak.
Nw,
The given graph is:
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 199.8
From the given graph,
We can observe that
The y-intercept of the graph is: 5
We know that,
The equation of the line in the slope-intercept form is:
y = mx + c
Where,
m is the slope
c is the y-intercept
Now,
From the given graph,
The points to find the graph are: (3, 40), and (2, 30)
So,
Slope (m) = \(\frac{30 – 40}{2 – 3}\)
= 10
So,
The equation of the line in the slope-intercept form is:
y = mx + c
So,
y = 10x + 5
Hence, from the above,
We can conclude that
The equation of the line in the slope-intercept form is:
y = 10x + 5

Question 10.
Graph the equation y = 3x – 5.
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 199.9
Answer:
The given equation is:
y = 3x – 5
Hence,
The representation of the given equation in the coordinate plane is:

Question 11.
Amy began with $25 in her bank account and spent $5 each day. The line shows the amount of money in her bank account. She incorrectly wrote an equation for the line in slope-intercept form as y = -5x + 5.
a. What is the correct equation for the line in slope-intercept form?
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 199.10
Answer:
It is given that
Amy began with $25 in her bank account and spent $5 each day. The line shows the amount of money in her bank account
Now,
The given graph is:
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 199.10
From the given graph,
The y-intercept is: 25
We know that,
The equation of the line in the slope-intercept form is:
y = mx + c
Now,
The given points to find the slope are: (5, 0), and (1, 20)
So,
Slope (m) = \(\frac{20 – 0}{5 – 1}\)
= \(\frac{20}{5}\)
= 4
So,
The equation of the line in the slope-intercept form is:
y = 4x + 25
Hence, from the above,
We can conclude that the equation of the line in the slope-intercept form is:
y = 4x + 25

b. Critique Reasoning What mistake might Amy have made?
Answer:
Answer:
The mistakes might made by Amy are:
A) The value of y-intercept is 25 and the value of x-intercept is: 5
B) The slope is not negative as it moves down from top to bottom

Question 12.
Higher-Order Thinking The line represents the cost of ordering concert tickets online.
a. Write an equation for the line in slope-intercept form, where x is the number of tickets and y is the total cost.
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 199.11
Answer:
It is given that
The line represents the cost of ordering concert tickets online.
Now,
The given graph is:
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 199.11
From the given graph,
The y-intercept is: 10
We know that,
The equation of the line in the slope-intercept form is:
y = mx + c
Now,
The given points to find the slope are: (1, 33.25), and (0, 12.25)
So,
Slope (m) = \(\frac{12.25 – 33.25}{0 – 1}\)
= \(\frac{21}{1}\)
= 21
So,
The equation of the line in the slope-intercept form is:
y = 21x + 10
Hence, from the above,
We can conclude that the equation of the line in the slope-intercept form is:
y = 21x + 10

b. Explain how you can write an equation for this situation without using a graph.
Answer:
We know that,
The total cost of ordering concert tickets online = (The cost of 1 Ticket) × (The number of Tickets) + Processing fee
Let the number of tickets be x
Let the total cost of ordering concert tickets online be y
So,
y = 21x + 10
Hence, from the above,
We can conclude that the equation for this situation without using a graph is:
y = 21x + 10

c. Is this graph a good representation of the situation? Explain.
Answer:
Yes,
The given graph is good for the given situation because the equation of the line is the same for this situation with using the graph and without using the graph

Assessment Practice

Question 13.
What should you do first to graph the equation y = \(\frac{2}{5}\)x – 1?
A. Plot the point (0, 0).
B. Plot the point (2, 5).
C. Plot a point at the x-intercept.
D. Plot a point at the y-intercept.
Answer:
The given equation is:
y = \(\frac{2}{5}\)x – 1
Compare the above equation with
y = mx + c
Hence, from the above,
We can conclude that the first step to draw the graph for the given equation is:
Plot a point at the y-intercept

Question 14.
Write an equation for the line in slope-intercept form.
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 200
Answer:
The given graph is:
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 200
From the given graph,
We can observe that
The y-intercept of the graph is: 8
We know that,
The equation of the line in the slope-intercept form is:
y = mx + c
Where,
m is the slope
c is the y-intercept
Now,
From the given graph,
The points to find the graph are: (4, 0), and (0, 8)
So,
Slope (m) = \(\frac{8 – 0}{0 – 4}\)
= -2
So,
The equation of the line in the slope-intercept form is:
y = mx + c
So,
y = -2x + 8
Hence, from the above,
We can conclude that
The equation of the line in the slope-intercept form is:
y = -2x + 8

TOPIC 2 REVIEW

? Topic Essential Question

How can you analyze connections between linear equations and use them to solve problems?
Answer:
Assuming that your two equations are distinct (neither is merely a multiple of the other), we can use the “elimination by addition and subtraction” method or substitution method to eliminate one variable, leaving us with an equation in one variable, solve this 1-variable (Ex: in x) equation, and then use the resulting value in the other

Vocabulary Review

Complete each definition and provide an example of each vocabulary word.

Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 200.1

Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 200.12

Question 1.
The change in y divided by the change in x is the ____
Answer:
The change in y divided by the change in x is defined as the “Slope of a line”
Example:
Slope = \(\frac{y}{x}\)
= \(\frac{2}{5}\)

Question 2.
The point on the graph where the line crosses the y-axis is the ____ of a line.
Answer:
The point on the graph where the line crosses the y-axis is the “y-intercept” of a line. In the y-intercept, the value of x is 0
Example:
The point on the graph where the line crosses the y-axis is at (0, 2)
So,
The y-intercept is: 2

Question 3.
The ____ of a line is y = mx + b. The variable m in the equation stands for the __. The variable b in the equation stands for the ___
Answer:
The “Slope-intercept form” of a line is
y = mx + b
The variable m in the equation stands for the x-intercept.
The variable b in the equation stands for the y-intercept

Use Vocabulary in Writing
Paddle boats rent for a fee of $25, plus an additional $12 per hour. What equation, in y = mx + b form, represents the cost to rent a paddle boat for x hours? Explain how you write the equation. Use vocabulary words in your explanation.
Answer:
It is given that
Paddleboats rent for a fee of $25, plus an additional $12 per hour.
Where,
x represents the cost to rent a paddleboat for x hours
Now,
The total cost to rent a paddleboat = The cost of a paddleboat per hour + $12
y = $25x + $12
Hence, from the above,
We can conclude that the equation of the line for this situation is:
y = $25x + $12

Concepts and Skills Review

LESSON 2.1 Combine Like Terms to Solve Equations

Quick Review
You can use variables to represent unknown quantities. To solve an equation, collect like terms to get one variable on one side of the equation. Then use inverse operations and properties of equality to solve the equation.

Practice
Solve each equation for x.

Question 1.
2x + 6x = 1,000
Answer:
The given equation is:
2x + 6x = 1,000
So,
8x = 1,000
Divide by 8 into both sides
x = \(\frac{1,000}{8}\)
x = 125
Hence, from the above,
We can conclude that the value of x is: 125

Question 2.
2\(\frac{1}{4}\)x + 2\(\frac{1}{2}\)x = 44
Answer:
The given equation is:
2\(\frac{1}{4}\)x + 2\(\frac{1}{2}\)x = 44
We know that,
2\(\frac{1}{4}\) = \(\frac{9}{4}\)
2\(\frac{1}{2}\) = \(\frac{5}{2}\)
So,
\(\frac{9}{4}\)x + \(\frac{5}{2}\)x = 44
\(\frac{19}{4}\)x = 44
Multiply with \(\frac{4}{19}\) on both sides
So,
x = 44 × \(\frac{4}{19}\)
x = \(\frac{88}{19}\)
Hence, from the above,
We can conclude that the value of x is: \(\frac{88}{19}\)

Question 3.
-2.3x – 4.2x = -66.3
Answer:
The given equation is:
-2.3x – 4.2x = -66.3
So,
-6.5x = -66.3
6.5x = 66.3
Divide by 6.5 into both sides
So,
x = \(\frac{66.3}{6.5}\)
x = \(\frac{51}{5}\)
x = 10.2
Hence, from the above,
We can conclude that the value of x is: 10.2

Question 4.
Javier bought a microwave for $105. The cost was 30% off the original price. What was the price of the microwave before the sale?
Answer:
It is given that
Javier bought a microwave for $105. The cost was 30% off the original price
So,
The price of the microwave before the sale = The price of the microwave + 30% of the price of the microwave
= $105 + \(\frac{30}{100}\) ($105)
= $105 (\(\frac{130}{100}\))
= \(\frac{13650}{100}\)
= $136.5
Hence, from the above,
We can conclude that the price of the microwave before the sale is: $136.5

LESSON 2.2 Solve Equations with Variables on Both Sides

Quick Review
If two quantities represent equal amounts and have the same variables, you can set the expressions equal to each other. Collect all the variables on one side of the equation and all the constants on the other side. Then use inverse operations and properties of equality to solve the equation.

Practice
Solve each equation for x.

Question 1.
3x + 9x = 6x + 42
Answer:
The given equation is:
3x + 9x = 6x + 42
12x = 6x + 42
Rearrange the like terms
So,
12x – 6x = 42
6x = 42
So,
x = \(\frac{42}{6}\)
x = 7
Hence, from the above,
We can conclude that the value of x is: 7

Question 2.
\(\frac{4}{3}\)x + \(\frac{2}{3}\)x = \(\frac{1}{3}\)x + 5
Answer:
The given equation is:
\(\frac{4}{3}\)x + \(\frac{2}{3}\)x = \(\frac{1}{3}\)x + 5
So,
\(\frac{6}{3}\)x = \(\frac{1}{3}\)x + 5
\(\frac{6}{3}\)x – \(\frac{1}{3}[latex]x = 5
[latex]\frac{5}{3}\)x = 5
Multiply with \(\frac{3}{5}\) on both sides
So,
x = 5 × \(\frac{3}{5}\)
x = 3
Hence, from the above,
We can conclude that the value of x is: 3

Question 3.
9x – 5x + 18 = 2x + 34
Answer:
The given equation is:
9x – 5x + 18 = 2x + 34
So,
4x + 18 = 2x + 34
Rearrange the like terms
So,
4x – 2x = 34 – 18
2x = 16
Divide by 2 into both sides
So,
x = \(\frac{16}{2}\)
x = 8
Hence, from the above,
We can conclude that the value of x is: 8

Question 4.
Megan has $50 and saves $5.50 each week. Connor has $18.50 and saves $7.75 each week. After how many weeks will Megan and Connor have saved the same amount?
Answer:
It is given that
Megan has $50 and saves $5.50 each week. Connor has $18.50 and saves $7.75 each week.
Now,
Let x be the number of weeks
So,
The money saved by Megan = $50 + $5.50x
The money saved by Connor = $18.50 + $7.75x
So,
To find out after how many weeks Megan and Connor have saved the same amount,
$50 + $5.50x = $18.50 + $7.75x
Rearrange the like terms
So,
$50 – $18.50 = $7.75x – $5.50x
$31.05 = $2.25x
Divide by 2.25 into both sides
So,
x = \(\frac{31.05}{2.25}\)
x = 13.8
x = 14 weeks 1 day
x ≅ 14 weeks
Hence, from the above,
We can conclude that after approximately 14 weeks, Megan and Connor have saved the same amount

LESSON 2.3 Solve Multistep Equations

Quick Review
When solving multistep equations, sometimes the Distributive Property is used before you collect like terms. Sometimes like terms are collected, and then you use the Distributive Property.

Practice Solve each equation for x.

Question 1.
4(x + 4) + 2x = 52
Answer:
The given equation is:
4 (x + 4) + 2x = 52
So,
4 (x) + 4 (4) + 2x = 52
4x + 16 + 2x = 52
6x + 16 = 52
Rearrange the like terms
So,
6x = 52 – 16
6x = 36
x = \(\frac{36}{6}\)
x = 6
Hence, from the above,
We can conclude that the value of x is: 6

Question 2.
8(2x + 3x + 2) = -4x + 148
Answer:
The given equation is:
8 (2x + 3x + 2) = -4x + 148
So,
8 (5x + 2) = -4x + 148
8 (5x) + 8 (2) = -4x + 148
40x + 16 = -4x + 148
Rearrange the like terms
So,
40x + 4x = 148 – 16
44x = 132
x = \(\frac{132}{4}\)
x = 3
Hence, from the above,
We can conclude that the value of x is: 3

Question 3.
Justin bought a calculator and a binder that were both 15% off the original price. The original price of the binder was $6.20. Justin spent a total of $107.27. What was the original price of the calculator?
Answer:
It is given that
Justin bought a calculator and a binder that were both 15% off the original price. The original price of the binder was $6.20. Justin spent a total of $107.27.
So,
Total spent money of Justin = The original price of binder + The original price of a calculator
Let the original price of the calculator be x
So,
$6.20 + 30% of $6.20 + x + 30% of x = $107.27
$6.20 + \(\frac{3}{10}\) ($6.20) + x + \(\frac{3}{10}\) of x = $107.27
$6.20 + 1.86 + 1.3x = $107.27
$8.06 + 1.3x = $107.27
1.3x = $107.27 – $8.06
1.3x = 99.21
x = \(\frac{99.21}{1.3}\)
x = 76.31
Hence, from the above,
We can conclude that the original price of the calculator is: $76.31

LESSON 2.4 Equations with No Solutions or Infinitely Many Solutions

Quick Review
When solving an equation results in a statement that is always true, there are infinitely many solutions. When solving an equation produces a false statement, there are no solutions. When solving an equation gives one value for a variable, there is one solution.

Practice
How many solutions does each equation have?

Question 1.
x + 5.5 + 8 = 5x – 13.5 – 4x
Answer:
The given equation is:
x + 5.5 + 8 = 5x – 13.5 – 4x
So,
x + 13.5 = x – 13.5
Subtract with x on both sides
So,
13.5 = -13.5
Hence, from the above,
we can conclude that there are no solutions for the given equation

Question 2.
4(\(\frac{1}{2}\)x + 3) = 3x + 12 – x
Answer:
The given equation is:
4(\(\frac{1}{2}\)x + 3) = 3x + 12 – x
So,
4 × \(\frac{1}{2}\)x + 4 (3) = 3x + 12 – x
2x + 12 = 2x + 12
Subtract with 2x on both sides
So,
12 = 12
Hence, from the above,
We can conclude that there are infinitely many solutions for the given equation

Question 3.
2(6x + 9 – 3x) = 5x + 21
Answer:
The given equation is:
2 (6x + 9 – 3x) = 5x + 21
So,
2 (3x + 9) = 5x + 21
2 (3x) + 2 (9) = 5x + 21
6x + 18 = 5x + 21
Rearrange the like terms
So,
6x – 5x = 21 – 18
x = 3
Hence, from the above,
We can conclude that there is only 1 solution for the given equation

Question 4.
The weight of Abe’s dog can be found using the expression 2(x + 3), where x is the number of weeks. The weight of Karen’s dog can be found using the expression 3(x + 1), where x is the number of weeks. Will the dogs ever be the same weight? Explain.
Answer:
It is given that
The weight of Abe’s dog can be found using the expression 2(x + 3), where x is the number of weeks. The weight of Karen’s dog can be found using the expression 3(x + 1), where x is the number of weeks.
Now,
To find out whether the weight of the dogs will be the same or not,
2 (2x + 3) = 3 (3x + 1)
So,
2 (2x) + 2 (3) = 3 (3x) + 3 (1)
4x + 6 = 9x + 3
Rearrange the like terms
So,
9x – 4x = 6 – 3
5x = 3
x = \(\frac{3}{5}\)
So,
There is only 1 solution for the given equation
Hence, from the above,
We can conclude that the weights of the dogs will be the same

LESSON 2.5 Compare Proportional Relationships

Quick Review
To compare proportional relationships, compare the rate of change or find the unit rate.

Practice

Question 1.
Two trains are traveling at a constant rate. Find the rate of each train. Which train is traveling at the faster rate?
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 200.2
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 200.13
Answer:
We know that,
Unit rate = \(\frac{y}{x}\)
We know that,
Speed = \(\frac{Distance}{Time}\)
Now,
For Train A,
Unit rate = \(\frac{A value of Distance}{The value of time that corresponds to the Distance}\)
= \(\frac{50}{2}\)
= 25 miles per hour
For Train B,
Unit rate = \(\frac{y}{x}\)
= \(\frac{20}{1}\)
= 20 miles per hour
So,
Unit rate of Train A > Unit rate of Train B
Hence, from the above,
We can conclude that Train A is the fastest

Question 2.
A 16-ounce bottle of water from Store A. costs $1.28. The cost in dollars, y, of a bottle of water from Store B is represented by the equation y = 0.07x, where x is the number of ounces. What is the cost per ounce of water at each store? Which store’s bottle of water costs less per ounce?
Answer:
It is given that
A 16-ounce bottle of water from Store A. costs $1.28. The cost in dollars, y, of a bottle of water from Store B is represented by the equation y = 0.07x, where x is the number of ounces.
So,
The cost per ounce of water of store A = \(\frac{The cost of a 16-ounce bottle of water}{16}\)
= \(\frac{$1.28}{16}\)
= $0.08
The cost per ounce of water of store B = \(\frac{y}{x}\)
= $0.07
So,
The cost per ounce of water of store A > The cost per ounce of water of store B
Hence, from the above,
We can conclude that the cost per ounce of water of store B costs less per ounce

LESSON 2.6 Connect Proportional Relationships and Slope

Quick Review
The slope of a line in a proportional relationship is the same as the unit rate and the constant of proportionality.

Practice

Question 1.
The graph shows the proportions of blue paint and yellow paint that Briana mixes to make green paint. What is the slope of the line? Tell what it means in the problem situation.
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 202.4
Answer:
It is given that
The graph shows the proportions of blue paint and yellow paint that Briana mixes to make green paint.
Now,
The given graph is:
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 202.4
So,
From the graph,
The slope of the given line = \(\frac{y}{x}\)
= \(\frac{5}{6}\)
Hence, from the above slope of the line,
We can conclude that for 5 parts of yellow paint, we have to mix 6 parts of blue paint to make green paint

LESSON 2.7 Analyze Linear Equations: y = mx

Quick Review
A proportional relationship can be represented by an equation in the form y = mx, where m is the slope.

Practice
A mixture of nuts contains 1 cup of walnuts for every 3 cups of peanuts.

Question 1.
Write a linear equation that represents the relationship between peanuts, x, and walnuts, y.
Answer:
It is given that
A mixture of nuts contains 1 cup of walnuts for every 3 cups of peanuts.
We know that,
Slope (m) = \(\frac{y}{x}\)
m = \(\frac{1}{3}\)
We know that,
The linear equation that represents the relationship between peanuts and walnuts is:
y = mx
So,
y = \(\frac{1}{3}\)x
x = 3y
Hence, from the above,
We can conclude that the linear equation that represents the relationship between peanuts and walnuts is:
x = 3y

Question 2.
Graph the line.
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 202.5
Answer:
The linear equation that represents the relationship between peanuts and walnuts is:
x = 3y
Hence,
The representation of the linear equation in the coordinate plane is:

LESSON 2.8 Understand the y-Intercept of a Line

Quick Review
The y-intercept is the y-coordinate of the point where a line crosses the y-axis. The y-intercept of a proportional relationship is 0.

Practice
The equation y = 5 +0.5x represents the cost of getting a car wash and using the vacuum for x minutes.
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 203.1

Question 1.
What is the y-intercept?
Answer:
We know that,
The equation of the line in the y-intercept form is:
y = mx + c
Where,
m is the slope
c is the y-intercept
Now,
The given equation is:
y = 5 + 0.5x
Hence, from the above,
We can conclude that the y-intercept is: 5

Question 2.
What does the y-intercept represent?
Answer:
The y-intercept in the given situation represents that the initial cost of getting a car wash using the Vaccum

LESSON 2.9 Analyze Linear Equations: y = mx + b

Quick Review
An equation in the form y = mx + b, where b=0, has a slope of m and a y-intercept of b. This form is called the slope-intercept form. There is not a proportional relationship between x and y in these cases.

Practice

Question 1.
Graph the line with the equation y = \(\frac{1}{2}\)x – 1.
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 205.1
Answer:
The given equation is:
y = \(\frac{1}{2}\)x – 1
Hence,
The representation of the given equation in the coordinate plane is:

Question 2.
What is the equation of the line?
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 205.2
Answer:
The given graph is:
Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 205.2
From the given graph,
We can observe that
The y-intercept is: 3
Now,
We know that,
The equation of the line in the slope-intercept form is:
y = mx + c
Now,
To find the slope,
The points are: (0, 3), and (3, 0)
So,
Slope (m) = \(\frac{0 – 3}{3 – 0}\)
= \(\frac{-3}{3}\)
= -1
Hence, from the above,
We can conclude that the equation of the line in the slope-intercept form is:
y = -x + 3

Topic 2 Fluency Practice

Pathfinder

Each block below shows an equation and a possible solution. Shade a path from START to FINISH. Follow the equations that are solved correctly. You can only move up, down, right, or left.

Envision Math Common Core 8th Grade Answer Key Topic 2 Analyze And Solve Linear Equations 206.1

Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers

Go through the enVision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers regularly and improve your accuracy in solving questions.

Envision Math Common Core 6th Grade Answers Key Topic 2 Integers and Rational Numbers

?Topic Essential Question What are integers and rational numbers? How are points graphed on a coordinate plane?
Answer:
An “Integer” can be written as a fraction by giving it a denominator of one. So, any integer is a rational number
“Rational numbers” are those numbers that are integers and can be expressed in the form of \(\frac{x}{y}\) where both numerator and denominator are integers
To graph or plot points, we use two perpendicular lines called the x-axis and the y-axis. The horizontal number line is the x-axis and the vertical line is the y-axis.  Every point in the coordinate plane is represented by an ordered pair of x and y coordinates.

3-ACT MATH

The ULTIMATE THROW
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 1
The Ultimate Throw
Have you ever played ultimate? It’s a team sport played with a flying disc. The goal is to score the most points by passing the disc to your opponent’s end zone. Ultimate is played by millions of people across the globe, from casual games to professional leagues.
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 1.1
There are many ways to throw a flying disc. It takes a lot of practice to learn each type of throw. If you want the disc to travel a specific path and distance, you need to try different throws with different amounts of spin and power. Think about this during the 3-Act Mathematical Modeling lesson.

enVision STEM Project

Did You Know?
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 3.1

Your Task: Improve Your School

Now that you have defined the problem, or improvement needed, you and your classmates will apply the engineering design process to propose solutions.
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 3.2

Topic 2 GET READY!

Review What You Know!

Vocabulary

Choose the best term from the box to complete each definition.

decimal
denominator
fraction
numerator

Question 1.
A ____ names part of a whole, part of a set, or a location on a number line.
Answer:
We know that,
A “Fraction” names part of a whole, part of a set, or a location on a number line.
Hence, from the above,
We can conclude that
The best term that is suitable for the given definition is: Fraction

Question 2.
The number above the fraction bar that represents the part of the whole is the ____
Answer:
We know that,
The number above the fraction bar that represents the part of the whole is the “Numerator”
Hence, from the above,
We can conclude that
The best term that is suitable for the given definition is: Numerator

Question 3.
The number below the fraction bar that represents the total number of equal parts in one whole is the ____
Answer:
We know that,
The number below the fraction bar that represents the total number of equal parts in one whole is the “Denominator”
Hence, from the above,
We can conclude that
The best term that is suitable for the given definition is: Denominator

 Integers and Rational Numbers 1

Fractions and Decimals
Write each fraction as a decimal.

Question 4.
\(\frac{2}{5}\)
Answer:
The given fraction is: \(\frac{2}{5}\)
Hence, from the above,
We can conclude that
The representation of the given fraction in the form of a decimal is:

Question 5.
\(\frac{3}{4}\)
Answer:
The given fraction is: \(\frac{3}{4}\)
Hence, from the above,
We can conclude that
The representation of the given fraction in the form of a decimal is:

Question 6.
\(\frac{10}{4}\)
Answer:
The given fraction is: \(\frac{10}{4}\)
Hence, from the above,
We can conclude that
The representation of the given fraction in the form of a decimal is:

Question 7.
\(\frac{12}{5}\)
Answer:
The given fraction is: \(\frac{12}{5}\)
Hence, from the above,
We can conclude that
The representation of the given fraction in the form of a decimal is:

Question 8.
\(\frac{3}{5}\)
Answer:
The given fraction is: \(\frac{3}{5}\)
Hence, from the above,
We can conclude that
The representation of the given fraction in the form of a decimal is:

Question 9.
\(\frac{15}{3}\)
Answer:
The given fraction is: \(\frac{15}{3}\)
Hence, from the above,
We can conclude that
The representation of the given fraction in the form of a decimal is:

Division with Decimals

Divide.

Question 10.
1.25 ÷ 0.5
Answer:
The given Division Expression is: 1.25 ÷ 0.5
Now,
By using the Long Division,

Hence, from the above,
We can conclude that
1.25 ÷ 0.5 = 2.5

Question 11.
13 ÷ 0.65
Answer:
The given Division Expression is: 13 ÷ 0.65
Now,
By using the Long Division,

Hence, from the above,
We can conclude that
13 ÷ 0.65 = 20

Question 12.
12.2 ÷ 0.4
Answer:
The given Division Expression is: 12.2 ÷ 0.4
Now,
By using the Long Division,

Hence, from the above,
We can conclude that
12.2 ÷ 0.4 = 30

Ordered Pairs
Write the ordered pair for each point shown on the graph.

Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 5.1

Question 13.
J
Answer:
The given coordinate plane is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 5.1
Now,
We know that,
“Ordered pairs” are often used to represent two variables. The number which corresponds to the value of x is called the x-coordinate and the number which corresponds to the value of y is called the y-coordinate.
Hence, from the above,
We can conclude that
The ordered pair J is: (4, 3)

Question 14.
K
Answer:
The given coordinate plane is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 5.1
Now,
We know that,
“Ordered pairs” are often used to represent two variables. The number which corresponds to the value of x is called the x-coordinate and the number which corresponds to the value of y is called the y-coordinate.
Hence, from the above,
We can conclude that
The ordered pair K is: (0, 6)

Question 15.
L
Answer:
The given coordinate plane is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 5.1
Now,
We know that,
“Ordered pairs” are often used to represent two variables. The number which corresponds to the value of x is called the x-coordinate and the number which corresponds to the value of y is called the y-coordinate.
Hence, from the above,
We can conclude that
The ordered pair L is: (6, 8)

Question 16.
M
Answer:
The given coordinate plane is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 5.1
Now,
We know that,
“Ordered pairs” are often used to represent two variables. The number which corresponds to the value of x is called the x-coordinate and the number which corresponds to the value of y is called the y-coordinate.
Hence, from the above,
We can conclude that
The ordered pair M is: (7, 1)

 Integers and Rational Numbers 2

Plot each point on the coordinate plane.

Question 17.
A(6, 2)
Answer:
The given point is: A (6, 2)
Hence,
The representation of the given point in a coordinate plane is:

Question 18.
B(1, 3)
Answer:
The given point is: B (1, 3)
Hence,
The representation of the given point in a coordinate plane is:

Question 19.
C(5, 7)
Answer:
The given point is: C (5, 7)
Hence,
The representation of the given point in a coordinate plane is:

Question 20.
D(3, 4)
Answer:
The given point is: D (3, 4)
Hence,
The representation of the given point in a coordinate plane is:

Explain

Question 21.
Les said that the quotient of 3.9 ÷ 0.75 is 0.52. Explain how you know Les is incorrect without completing the division.
Answer:
It is given that
Les said that the quotient of 3.9 ÷ 0.75 is 0.52
Now,
The given Division Expression is: 3.9 ÷ 0.75
Now,
By using the Long Division,

Now,
When we observe the numbers after the decimal point in the given division expression,
We can say that there are 2 numbers after the decimal point n the numerator and 1 number after the decimal point in the denominator
So,
The numbers after the decimal point in the quotient will be only 1 number
But,
There are 2 numbers after the decimal point
Hence, from the above,
We can conclude that Les is incorrect

Language Development

Use the graphic organizer to help you understand new vocabulary terms.

Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 6.1

Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 6.2

Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 6.3
Answer:


Topic 2 PICK A PROJECT

PROJECT 2A
What places would you like to visit in the United States?
PROJECT: MAKE A TRAVEL BROCHURE
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 6.33

PROJECT 2B
If you were to solve a puzzle, what type would you choose?
PROJECT: DESIGN A CONNECT-THE-DOTS PUZZLE
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 6.4

PROJECT 2C
What are some exercises for staying fit and having fun?
PROJECT: RECORD AN EXERCISE VIDEO
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 6.5

PROJECT 2D
If you were going to make a commercial, what type of product would you feature?
PROJECT: WRITE YOUR OWN COMMERCIAL
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 6.6

Lesson 2.1 Understand Integers

ACTIVITY

Explain It!
Sal recorded the outdoor temperature as -4°F at 7:30 A.M. At noon, it was 22°F. Sal said the temperature changed by 18°F because 22 – 4 = 18.

Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 6.9
A. Critique Reasoning is Sal right or wrong? Explain.
Answer:
It is given that
Sal recorded the outdoor temperature as -4°F at 7:30 A.M. At noon, it was 22°F. Sal said the temperature changed by 18°F because 22 – 4 = 18.
So,
The total temperature from 7:30 A.M to noon = (The temperature at 7:30 A.M) + (The temperature at noon
= -4 + 22
= 18° F
Hence, from the above,
We can conclude that Sal is right

B. Construct Arguments What was the total temperature change from 7:30 A.M. until noon? Use the thermometer to help justify your solution.
Answer:
It is given that
It is given that
Sal recorded the outdoor temperature as -4°F at 7:30 A.M. At noon, it was 22°F. Sal said the temperature changed by 18°F because 22 – 4 = 18.
Now,
The given figure is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 6.9
Now,
Using the thermometer,
The temperature at 7:30 A.M is: -4° F
Now,
The temperature at noon is: 18° F
So,
The change in temperature from 7: 30 A.M to until noon by using the thermometer is: 18° F
Hence, from the above,
We can conclude that
The total temperature change from 7:30 A.M. until noon is: 18° F

 Integers and Rational Numbers 3

Focus on math practices
Reasoning 0°C is the temperature at which water freezes. Which is colder, 10°C or -10°C? Explain.
Answer:
It is given that
0°C is the temperature at which water freezes
Now,
We know that,
A temperature below 0°C is a negative temperature
Hence, from the above,
We can conclude that
-10°C is colder than 0°C

Visual Learning

? Essential Question What are integers and how are they used to represent real-world quantities?
Answer:
“Integers” are a set of numbers that include the positive whole numbers (1, 2, 3, 4, 5, …), their opposites (-1, -2, -3, -4, -5, …) and zero.
You can use integers to help represent many real-world situations, such as Increases and decreases in temperature

Try It!
Label the integers on the number line.
The opposite of 4 isEnvision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 7. The opposite of -4 isEnvision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 7.

Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 8.2
Answer:
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
Now,
The completed number line is:

Hence, from the above,
We can conclude that
By using the given definition,
The opposite of 4 is: -4
The opposite of -4 is: 4

Convince Me! How do you know that two numbers are opposites?
Answer:
Two numbers are opposites if they have the same absolute value but different signs. Opposites are the same distance from 0 on a number line, and they are on opposite sides of 0

Try It!
Which number is greater, -4 or -2? Explain.
Answer:
The given numbers are: -4, and -2
Now,
We know that,
In the positive numbers i.e., the numbers that are to the right side of 0,
4 > 2
So,
In the negative numbers, i.e., the numbers that are to the left side of 0,
-4 < -2
Hence, from the above,
We can conclude that,
The greater number is: -2

Try It!
Which integer represents each situation?
a. A $10 debt
Answer:
We know that,
A “Debt” is a negative value
Hence, from the above,
We can conclude that
The representation of the given situation as an integer is: -$10

b. Six degrees below zero
Answer:
We know that,
In a vertical number line,
The numbers above zero are positive
The numbers below zero are negative
Hence, from the above,
We can conclude that
The representation of the given situation as an integer is: -6° C

c. Deposit of $25
Answer:
We know that,
A “Deposit” is a positive value
Hence, from the above,
We can conclude that
The representation of the given situation as an integer is: $25

KEY CONCEPT
Integers are all of the counting numbers, their opposites, and 0. Opposites are integers that are the same distance from 0 and on opposite sides of 0 on a number line.
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 8.4

Do You Understand?

Question 1.
? Essential Question What are integers and how are they used to represent real-world quantities?
Answer:
“Integers” are a set of numbers that include the positive whole numbers (1, 2, 3, 4, 5, …), their opposites (-1, -2, -3, -4, -5, …) and zero.
You can use integers to help represent many real-world situations, such as Increases and decreases in temperature

Question 2.
Reasoning What do you know about two different integers that are opposites?
Answer:
Let us consider two integers a and b such that a > b (or) a < b
Now,
The opposites of the two integers are: -a and -b
Now,
In the number line,
The numbers after 0 increase from left to right
The numbers before 0 decrease from right to left
So,
We can conclude that
-a < -b (or) -a > -b
Hence, from the above,
We can conclude that
The two different integers that are opposites can be represented as:
-a < -b or -a > -b

Question 3.
How do you read -17?
Answer:
The given number is: -17
Hence,
We can conclude that
-17 can read as:
a. The opposite of 17
b. The number that is on the left side of the number line

Question 4.
Construct Arguments which amount represents a debt of two hundred fifty dollars, $250 or -$250? Explain.
Answer:
We know that,
A “Debt” is a negative number
Hence, from the above,
We can conclude that
The representation of a debt of two hundred fifty dollars in the form of an integer is: -$250

Question 5.
Generalize when comparing two negative integers, how can you determine which integer is the greater number?
Answer:
We know that,
In the number line,
The numbers that are on the left side of 0 are called “Negative Numbers”
The “Negative Numbers” decrease when moves from right to left
Hence, from the above,
We can conclude that
When comparing two negative integers,
The  negative number that is nearest to zero is the greater number

Do You Know How?

In 6-17, write the opposite of each integer.

Question 6.
1
Answer:
The given integer is: 1
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
Hence, from the above,
We can conclude that
The opposite of 1 is: -1

Question 7.
-1
Answer:
The given integer is: -1
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
Hence, from the above,
We can conclude that
The opposite of -1 is: 1

Question 8.
-11
Answer:
The given integer is: -11
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
Hence, from the above,
We can conclude that
The opposite of -11 is: 11

Question 9.
30
Answer:
The given integer is: 30
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
Hence, from the above,
We can conclude that
The opposite of 30 is: -30

Question 10.
0
Answer:
The given integer is: 0
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
There is not any opposite value for 0
Hence, from the above,
We can conclude that
The opposite of 0 is: 0

Question 11.
-16
Answer:
The given integer is: -16
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
Hence, from the above,
We can conclude that
The opposite of -16 is: 16

Question 12.
-(-8)
Answer:
The given integer is: -(-8)
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
Hence, from the above,
We can conclude that
The opposite of -(-8) is: -8

Question 13.
28
Answer:
The given integer is: 28
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
Hence, from the above,
We can conclude that
The opposite of 28 is: -28

Question 14.
-(-65)
Answer:
The given integer is: -(-65)
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
Hence, from the above,
We can conclude that
The opposite of -(-65) is: -65

Question 15.
98
Answer:
The given integer is: 98
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
Hence, from the above,
We can conclude that
The opposite of 98 is: -98

Question 16.
100
Answer:
The given integer is: 100
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
Hence, from the above,
We can conclude that
The opposite of 100 is: -100

Question 17.
-33
Answer:
The given integer is: -33
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
Hence, from the above,
We can conclude that
The opposite of -33 is: 33

In 18-20, write the integers in order from least to greatest.

Question 18.
2, -3, 0, -4
Answer:
The given integers are: 2, -3, 0, -4
Now,
We know that,
In a number line,
The numbers after zero increase from left to right i.e., the farthest number after zero will be a greater number
The numbers before zero decrease from right to left i.e., the nearest number to zero will be a greater number
Hence, from the above,
We can conclude that
The given integers from the least to the greatest is: -4, -3, 0, 2

Question 19.
4, 12, -12, -11
Answer:
The given integers are: 4, 12, -12, -11
Now,
We know that,
In a number line,
The numbers after zero increase from left to right i.e., the farthest number after zero will be a greater number
The numbers before zero decrease from right to left i.e., the nearest number to zero will be a greater number
Hence, from the above,
We can conclude that
The given integers from the least to the greatest is: -12, -11, 4, 12

Question 20.
-5, 6, -7, -8
Answer:
The given integers are: -5, 6, -7, -8
Now,
We know that,
In a number line,
The numbers after zero increase from left to right i.e., the farthest number after zero will be a greater number
The numbers before zero decrease from right to left i.e., the nearest number to zero will be a greater number
Hence, from the above,
We can conclude that
The given integers from the least to the greatest is: -8, -7, -5, 6

Practice & Problem Solving

Scan for Multimedia

In 21-24, use the pictures at the right.
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 8.10

Question 21.
Generalize Which integer represents sea level? Explain.
Answer:
We know that,
We will represent the real-world situations always like above the sea level and below the sea level
Now,
When we observe a number line,
We will say the numbers above zero and the numbers below zero
Now,
When we compare the 2 definitions,
We can say that
The integer that represents the sea level is: zero
Hence, from the above,
We can conclude that
The integer that represents the sea level is: Zero

Question 22.
Use a negative integer to represent the depth to which a dolphin may swim.
Answer:
From the given figure,
We can observe that
A dolphin can swim to 150 feet below the sea level
Now,
We know that,
The values below the sea level are all “Negative numbers”
Hence, from the above,
We can conclude that
The representation of the depth to which a dolphin may swim in the form of an integer is: -150 feet

Question 23.
Which of these animals can travel at the greatest distance from sea level?
Answer:
We know that,
The values that are above the sea level are “Positive values” and the value that is farthest from the sea level is the greatest value
The values that are below the sea level are “Negative values” and the value that is nearest to the sea level is the greatest value
Hence, from the above,
We can conclude that
The animal that can travel at the greatest distance above sea level is: Griffon
The animal that can travel at the greatest distance below sea level is: Sperm whale

Question 24.
Order the elevations of the animals as integers from least to greatest.
Answer:
The given figure is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 8.10
Now,
We know that,
We know that,
The values that are above the sea level are “Positive values” and the value that is farthest from the sea level is the greatest value
The values that are below the sea level are “Negative values” and the value that is nearest to the sea level is the greatest value
Hence, from the above,
We can conclude that
The elevations of the animals as integers from the least to the greatest is:
Sperm whale, Dolphin, Migrating bird, Griffon

In 25-30, plot each point on the number line below.

Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 10.1

Question 25.
G(-10)
Answer:
The given point is: G (-10)
Hence,
The representation of the given point on the number line is:

Question 26.
H(8)
Answer:
The given point is: H (8)
Hence,
The representation of the given point on the number line is:

Question 27.
I(-1)
Answer:
The given point is: I (-1)
Hence,
The representation of the given point on the number line is:

Question 28.
J(9)
Answer:
The given point is: J (9)
Hence,
The representation of the given point on the number line is:

Question 29.
K(6)
Answer:
The given point is: K (6)
Hence,
The representation of the given point on the number line is:

Question 30.
L(-3)
Answer:
The given point is: L (-3)
Hence,
The representation of the given point on the number line is:

In 31-36, write the integer value that each point represents. Then use the number line to help write its opposite.
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 10.3

Question 31.
A
Answer:
The given number line is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 10.3
Now,
From the given figure,
We can observe that
The value of A is: -7
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
So,
The opposite value of A is: 7
Hence, from the above,
We can conclude that
The integer value that A represents is: -7
The opposite value of -7 is: 7

Question 32.
B
Answer:
The given number line is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 10.3
Now,
From the given figure,
We can observe that
The value of B is: 4
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
So,
The opposite value of B is: -4
Hence, from the above,
We can conclude that
The integer value that B represents is: 4
The opposite value of 4 is: -4

Question 33.
C
Answer:
The given number line is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 10.3
Now,
From the given figure,
We can observe that
The value of C is: 0
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
The opposite of 0 is 0 i.e., there is not any opposite value for 0
So,
The opposite value of C is: 0
Hence, from the above,
We can conclude that
The integer value that C represents is: 0
The opposite value of 0 is: 0

Question 34.
D
Answer:
The given number line is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 10.3
Now,
From the given figure,
We can observe that
The value of D is: -2
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
So,
The opposite value of D is: 2
Hence, from the above,
We can conclude that
The integer value that D represents is: -2
The opposite value of -2 is: 2

Question 35.
E
Answer:
The given number line is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 10.3
Now,
From the given figure,
We can observe that
The value of E is: 2
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
So,
The opposite value of E is: -2
Hence, from the above,
We can conclude that
The integer value that E represents is: 2
The opposite value of 2 is: -2

Question 36.
F
Answer:
The given number line is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 10.3
Now,
From the given figure,
We can observe that
The value of F is: -5
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
So,
The opposite value of F is: 5
Hence, from the above,
We can conclude that
The integer value that F represents is: -5
The opposite value of -5 is: 5

Question 37.
Write the opposite of each integer.
A. 5
Answer:
The given integer is: 5
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
Hence, from the above,
We can conclude that
The opposite of 5 is: -5

B. -13
Answer:
The given integer is: -13
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
Hence, from the above,
We can conclude that
The opposite of -13 is: 13

C. -(-22)
Answer:
The given integer is: -(-22)
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
Hence, from the above,
We can conclude that
The opposite of -(-22) is: -22

D. -31
Answer:
The given integer is: -31
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
Hence, from the above,
We can conclude that
The opposite of -31 is: 31

E. -50
Answer:
The given integer is: -50
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
Hence, from the above,
We can conclude that
The opposite of -50 is: 50

F. -(-66)
Answer:
The given integer is: -(-66)
Now,
We know that,
The opposite of any integer ‘a’ is −a Similarly, the opposite of any integer ‘−a’ is ‘−(−a)’ = ‘a’
Hence, from the above,
We can conclude that
The opposite of -(-66) is: -66

Question 38.
Compare the integers and write the integer with the greater value.
A. -5, 1
Answer:
The given integers are: -5, 1
Now,
We know that,
In a number line,
The numbers that are after zero are “Positive numbers” and the numbers that are farthest to zero are the greater numbers
The numbers that are before zero are “Negative numbers” and the numbers that are nearest to zero are the least numbers
The “Positive numbers” are always greater than “Negative numbers”
Hence, from the above,
We can conclude that
1 > -5
The greater value is: 1

B. -6, -7
Answer:
The given integers are: -6, -7
Now,
We know that,
In a number line,
The numbers that are after zero are “Positive numbers” and the numbers that are farthest to zero are the greater numbers
The numbers that are before zero are “Negative numbers” and the numbers that are nearest to zero are the least numbers
The “Positive numbers” are always greater than “Negative numbers”
Hence, from the above,
We can conclude that
-6 > -7
The greater value is: -6

C. -9, 8
Answer:
The given integers are: -9, 8
Now,
We know that,
In a number line,
The numbers that are after zero are “Positive numbers” and the numbers that are farthest to zero are the greater numbers
The numbers that are before zero are “Negative numbers” and the numbers that are nearest to zero are the least numbers
The “Positive numbers” are always greater than “Negative numbers”
Hence, from the above,
We can conclude that
8 > -9
The greater value is: 8

D. -12, -(-10)
Answer:
The given integers are: -12, -(-10)
So,
The given integers are: -12, 10
Now,
We know that,
In a number line,
The numbers that are after zero are “Positive numbers” and the numbers that are farthest to zero are the greater numbers
The numbers that are before zero are “Negative numbers” and the numbers that are nearest to zero are the least numbers
The “Positive numbers” are always greater than “Negative numbers”
Hence, from the above,
We can conclude that
10 > -12
The greater value is: 10

E. -(-9), 11
Answer:
The given integers are: -(-9), 11
So,
The given integers are: 9, 11
Now,
We know that,
In a number line,
The numbers that are after zero are “Positive numbers” and the numbers that are farthest to zero are the greater numbers
The numbers that are before zero are “Negative numbers” and the numbers that are nearest to zero are the least numbers
The “Positive numbers” are always greater than “Negative numbers”
Hence, from the above,
We can conclude that
11 > 9
The greater value is: 11

F. -(-4), 3
Answer:
The given integers are: -(-4), 3
So,
The given integers are: 4, 3
Now,
We know that,
In a number line,
The numbers that are after zero are “Positive numbers” and the numbers that are farthest to zero are the greater numbers
The numbers that are before zero are “Negative numbers” and the numbers that are nearest to zero are the least numbers
The “Positive numbers” are always greater than “Negative numbers”
Hence, from the above,
We can conclude that
4 > 3
The greater value is: 4

Question 39.
The display at the right shows the daily low temperatures for several consecutive days in a New England city. Write the temperatures in order from least to greatest. On which day was it the coldest?
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 16.1
Answer:
It is given that
The display at the right shows the daily low temperatures for several consecutive days in a New England city.
Now,
The given temperatures are:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 16.1
Now,
We know that,
In a number line,
The numbers that are after zero are “Positive numbers” and the numbers that are farthest to zero are the greater numbers
The numbers that are before zero are “Negative numbers” and the numbers that are nearest to zero are the least numbers
The “Positive numbers” are always greater than “Negative numbers”
So,
The temperatures in order from the last to the greatest is:
-7° < -5° < 4° < 7°
The coldest day from the given days is: Wednesday
Hence, from the above,
We can conclude that
The temperatures in order from the last to the greatest is:
-7° < -5° < 4° < 7°
The coldest day from the given days is: Wednesday

Question 40.
In a bank account, a paid-out expense is called a debit, and a deposit is called a credit. Would you use positive or negative integers to represent credits? Debits? Explain.
Answer:
It is given that
In a bank account, a paid-out expense is called a debit, and a deposit is called a credit.
Now,
We know that,
A “Paid-out expense” or “Debit” is a negative value
A “Deposit” or “Credit” is a positive value
Hence, from the above,
We can conclude that
We will use “Positive integers” to represent “Credits”
We will use “Negative integers” to represent “Debits”

Question 41.
Higher-Order Thinking Atoms have negatively charged particles called electrons and positively charged particles called protons. If an atom loses an electron, it has a positive electric charge. If it gains an electron, it has a negative electric charge. Which integer would represent the electric charge of an atom that has an equal number of electrons and protons?
Answer:
It is given that
Atoms have negatively charged particles called electrons and positively charged particles called protons. If an atom loses an electron, it has a positive electric charge. If it gains an electron, it has a negative electric charge.
Now,
We know that,
“Protons” and “Electrons” have the same values but opposite signs
So,
The value of the integer that has an equal number of electrons and protons =  (The number of Protons) – (The number of Electrons)
= 0 (Since both have the same values but with opposite signs)
Hence, from the above,
We can conclude that
The value of the integer that has an equal number of electrons and protons is: 0

Assessment Practice

Question 42.
Marco goes on a recreational scuba diving expedition. What is a possible diving depth for his expedition?
A. 0 meters
B. 40 meters
C. 400 meters
D. -40 meters
Answer:
It is given that
Marco goes on a recreational scuba diving expedition.
Now,
In terms of sea level,
The “Height” represents the positive values
The “Depth” represents the negative values
Hence, from the above,
We can conclude that
The possible diving depth for Marco’s expedition is:

Question 43.
Fill in the bubbles to match each integer with its opposite.
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 16.3
Answer:
The bubbles that matched with each integer and its opposite are:

Lesson 2.2 Represent Rational Numbers on the Number Line

ACTIVITY

Explore It!
The locations of four animals relative to sea level are shown.
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 16.4

A. What can you say about the animals and their positions relative to sea level?
Answer: It is given that
The locations of four animals relative to sea level are shown.
Now,
The given figure is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 16.4
Now,
We know that,
The sea level is considered as zero
The values above the sea level are positive
The values below the sea level are negative
Now,
From the given figure,
We can observe that
The animals that are below the sea level are: Dolphin, Sea Turtle, and Shark
The animals that are above sea level are: Seagull
Hence, from the above,
We can conclude that
The animals that are below the sea level are: Dolphin, Sea Turtle, and Shark
The animals that are above sea level are: Seagull

B. How can you use a number line to represent the locations of the animals?

Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 16.5
Answer:
The given figure is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 16.4
Now,
In terms of a number line,
The sea level is considered zero
The values above zero (Sea level) are positive
The values below zero (Sea level) are negative
So,
The representations of the locations of the given animals in a number line are:

Hence, from the above,
We can conclude that
The representation of the locations of the animals in a number line is:

Focus on math practices
Generalize How is representing the locations of negative fractions and decimals like representing the locations of positive fractions and decimals? How is it different?
Answer:
We know that,
In a number line,
The values above zero are “Positive numbers”
The values below zero are “Negative numbers”
Hence,
If the fractions and decimals are positive, then they are above zero in a number line
If the fractions and decimals are negative, then they are below zero in a number line

Visual Learning

? Essential Question How can you plot, compare, and order rational numbers using a number line?
Answer:
Just as positive and negative integers can be represented on a number line, so can positive and negative rational numbers. You can use a number line to help you compare and order negative rational numbers.
When comparing the values of two numbers, you can use a number line to determine which number is greater. The number on the right is always greater than the number on the left

Try It!
How can you find and position –\(\frac{5}{4}\) and -1.75 on the number lines? Write –\(\frac{5}{4}\) and -1.75 as mixed numbers, then plot the points on the number lines.
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 18.1

Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 18.30
Answer:
The given numbers are: –\(\frac{5}{4}\) and -1.75
Now,
The representation of –\(\frac{5}{4}\) into a mixed number is:

The representation of -1.75 into a mixed number is:

So,
The representation of –\(\frac{5}{4}\) on a number line is:

So,
The representation of -1.75 on a number line is:

Hence, from the above,
We can conclude that
The representation of the given numbers as mixed numbers are:

The representation of the given numbers on a number line is:

Convince Me! Why is it helpful to rename –\(\frac{5}{4}\) and -1.75 as mixed numbers when plotting these points on number lines?
Answer:
The given numbers are: –\(\frac{5}{4}\) and -1.75
Now,
The representations of the given numbers into mixed numbers are:

Now,
When we observe the fraction part in the mixed numbers,
We can draw a conclusion as to how much nearer to the whole numbers the fraction part is
Now,
We know that,
Between 2 numbers,
\(\frac{1}{4}\) is near to the previous number
\(\frac{1}{2}\) is between 2 numbers
\(\frac{3}{4}\) is near to the next number
Hence,
We will rewrite the numbers into mixed numbers to facilitate the marking of the numbers on the number line

Try It!
If \(\frac{1}{4}\) is ordered within the list of numbers in the example above, between which two numbers would it be placed?
Answer:
The given numbers in Example 2 are:
\(\frac{2}{3}\), 1.75 and -0.75
Now,
The given numbers from the given information will be:
\(\frac{2}{3}\), \(\frac{1}{4}\), 1.75 and -0.75
Now,
Convert all the given numbers into decimal numbers to compare the numbers from the least to the greatest
So,
The representation of the given fractions into decimal numbers are:
\(\frac{2}{3}\) = 0.66
\(\frac{1}{4}\) = 0.25
So,
The order of the given numbers from the least t the greatest is:
-0.75 < \(\frac{1}{4}\) < \(\frac{2}{3}\) < 1.75
Hence, from the above,
We can conclude that
\(\frac{1}{4}\) would be placed between -0.75 and \(\frac{2}{3}\)

Try It!
At 10:00 P.M. one winter night, the temperature was -3°C. At midnight, the temperature was -7°C. Use <, >, or = to compare the two temperatures and explain their relationship.
Answer:
It is given that
At 10:00 P.M. one winter night, the temperature was -3°C. At midnight, the temperature was -7°C.
Now,
We know that,
The values before zero are called “Negative Numbers”
The values before zero decrease as we move from right to left
The farthest value at the leftmost side of a number line is the “least value”
So,
-3° C > -7° C
Hence, from the above,
We can conclude that
The temperature at 10 P.M is greater than the midnight

KEY CONCEPT
A rational number can be expressed as a fraction in the form \(\frac{a}{b}\) and –\(\frac{a}{b}\) are integers and b is not 0.

Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 18.5
The numbers, in order from least to greatest, are: -1.75, \(\frac{3}{5}\), 1.25

Do You Understand?

Question 1.
? Essential Question How can you plot, compare, and order rational numbers using a number line?
Answer:
Just as positive and negative integers can be represented on a number line, so can positive and negative rational numbers. You can use a number line to help you compare and order negative rational numbers.
When comparing the values of two numbers, you can use a number line to determine which number is greater. The number on the right is always greater than the number on the left

Question 2.
Generalize Why are whole numbers rational numbers? Use 15 as an example.
Answer:
Rational numbers are the numbers that can be written in the form of \(\frac{p}{q}\) where q ≠ 0.  “Whole numbers” are the numbers that start from 0 to ∞. Whole numbers can be written in the form of \(\frac{0}{1}\), \(\frac{1}{1}\), \(\frac{2}{1}\), etc. Thus, every whole number is a rational number but every rational number is not a whole number.
Example:
15 = \(\frac{15}{1}\)
We know that,
15 is a “Whole number”
So,
We can write 15 in the form of \(\frac{p}{q}\) i.e., a “Rational number”

Question 3.
Vocabulary Why are integers rational numbers? Give an example.
Answer:
An “Integer” can be written as a fraction by giving it a denominator of one, So, any integer is a rational number
Examples: 17 and -34 are integers as well as rational numbers

Question 4.
Reasoning Explain how the inequality -4°C > -9°C describes how the temperatures are related.
Answer:
The given inequality is: -4°C > -9°C
Now,
We know that,
In a number line,
The values before zero are “Negative values”
The negative values on a number line decrease from right to left
So,
In the negative values,
The large number will be a small number and vice-versa
Hence, from the above,
We can conclude that
-4° C is greater than -9° C

Do You Know How?

In 5-7, write the number positioned at each point.
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 19.1

Question 5.
A
Answer:
The given number line is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 19.1
Hence,
The representation of the value of A on the given number line is:

Question 6.
B
Answer:
The given number line is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 19.1
Hence,
The representation of the value of B on the given number line is:

Question 7.
C
Answer:
The given number line is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 19.1
Hence,
The representation of the value of C on the given number line is:

In 8-11, plot the points on the number line below.

Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 19.2

Question 8.
P at -1\(\frac{1}{4}\)
Answer:
The given point is: P (-1\(\frac{1}{4}\))
Now,
The representation of -1\(\frac{1}{4}\) into a decimal number is:

-1\(\frac{1}{4}\) = -1.25
Hence,
The representation of P on the given number line is:

Question 9.
Q at 0.25
Answer:
The given point is: Q (0.25)
Hence,
The representation of Q on the given number line is:

Question 10.
R at -0.75
Answer:
The given point is: R (-0.75)
Hence,
The representation of R on the given number line is:

Question 11.
S at –\(\frac{1}{4}\)
Answer:
The given point is: S (-\(\frac{1}{4}\))
Now,
The representation of –\(\frac{1}{4}\) into a decimal number is:

In 12-14, use the number line to help order the numbers from least to greatest.

Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 20.3

Question 12.
1.25, –\(\frac{3}{2}\), -1.25, 1\(\frac{1}{2}\)
Answer:
The given numbers are:
1.25, –\(\frac{3}{2}\), -1.25, 1\(\frac{1}{2}\)
Now,
The representation of the given numbers into decimal numbers are:
1.25, -1.5, -1.25, 1.5
Hence, from the above,
We can conclude that
The numbers that are from the least to the greatest are: -1.5, -1.25, 1.25, 1.5

Question 13.
– 0.5, \(\frac{1}{2}\), -0.75, \(\frac{3}{4}\)
Answer:
The given numbers are:
– 0.5, \(\frac{1}{2}\), -0.75, \(\frac{3}{4}\)
Now,
The representation of the given numbers into decimal numbers are:
-0.5, 0.5, -0.75, 0.75
Hence, from the above,
We can conclude that
The numbers that are from the least to the greatest are: -0.75, -0.5, 0.5, 0.75

Question 14.
-1.5, -0.75, -1,2
Answer:
The given numbers are:
-1.5, -0.75, -1,2
Hence, from the above,
We can conclude that
The numbers that are from the least to the greatest are: -1.5, -1.2, -0.75

Practice & Problem Solving

Scan for Multimedia

In 15-20, write the number positioned at each point.

Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 20.6

Question 15.
A
Answer:
The given number line is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 20.6
Now,
We know that,
Each line in the given number line represents 0.25 units
Hence, from the above,
We can conclude that
The value of A in the given number line is: -3.25

Question 16.
B
Answer:
The given number line is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 20.6
Now,
We know that,
Each line in the given number line represents 0.25 units
Hence, from the above,
We can conclude that
The value of B in the given number line is: -4.5

Question 17.
C
Answer:
The given number line is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 20.6
Now,
We know that,
Each line in the given number line represents 0.25 units
Hence, from the above,
We can conclude that
The value of C in the given number line is: 1.25

Question 18.
D
Answer:
The given number line is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 20.6
Now,
We know that,
Each line in the given number line represents 0.25 units
Hence, from the above,
We can conclude that
The value of D in the given number line is: -5.75

Question 19.
E
Answer:
The given number line is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 20.6
Now,
We know that,
Each line in the given number line represents 0.25 units
Hence, from the above,
We can conclude that
The value of E in the given number line is: 0.5

Question 20.
F
Answer:
The given number line is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 20.6
Now,
We know that,
Each line in the given number line represents 0.25 units
Hence, from the above,
We can conclude that
The value of F in the given number line is: -2.5

Question 21.
Plot the numbers on the number line below.
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 21.1

A. -5\(\frac{1}{2}\)
Answer:
The given fraction is: -5\(\frac{1}{2}\)
Now,
The given number line is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 21.1
Now,
The representation of the given fraction into a decimal number is: -5.5
From the given number line,
We can observe that
Each line in the number line represents 0.1 units
Hence,
The representation of the given fraction in the given number line is:

B. -6.3
Answer:
The given number is: -6.3
Now,
The given number line is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 21.1
Now,
From the given number line,
We can observe that
Each line in the number line represents 0.1 units
Hence,
The representation of the given decimal number in the given number line is:

C. -5.8
Answer:
The given number is: -5.8
Now,
The given number line is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 21.1
Now,
From the given number line,
We can observe that
Each line in the number line represents 0.1 units
Hence,
The representation of the given decimal number in the given number line is:

D. -6\(\frac{7}{10}\)
Answer:
The given fraction is: -6\(\frac{7}{10}\)
Now,
The given number line is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 21.1
Now,
The representation of the given fraction into a decimal number is: -6.7
From the given number line,
We can observe that
Each line in the number line represents 0.1 units
Hence,
The representation of the given fraction in the given number line is:

E. -4.9
Answer:
The given number is: -4.9
Now,
The given number line is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 21.1
Now,
From the given number line,
We can observe that
Each line in the number line represents 0.1 units
Hence,
The representation of the given decimal number in the given number line is:

F. -6\(\frac{9}{10}\)
Answer:
The given fraction is: -6\(\frac{9}{10}\)
Now,
The given number line is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 21.1
Now,
The representation of the given fraction into a decimal number is: -6.9
From the given number line,
We can observe that
Each line in the number line represents 0.1 units
Hence,
The representation of the given fraction in the given number line is:

Question 22.
Use <, >, or = to compare.
A. \(\frac{1}{10}\) Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 22 0.09
Answer:
The given numbers are: \(\frac{1}{10}\), 0.09
Now,
Convert –\(\frac{1}{10}\) into a decimal number
So,
\(\frac{1}{10}\) = 0.10
Hence, from the above,
We can conclude that

B. –1.44 Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 22 -1\(\frac{1}{4}\)
Answer:
The given numbers are: -1.44, -1\(\frac{1}{4}\)
Now,
Convert -1\(\frac{1}{4}\) into a decimal number
So,
1\(\frac{1}{4}\) = 1.25
Hence, from the above,
We can conclude that

C. –\(\frac{2}{3}\) Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 22 -0.8
Answer:
The given numbers are: -0.8, –\(\frac{2}{3}\)
Now,
Convert –\(\frac{2}{3}\) into a decimal number
So,
\(\frac{2}{3}\) = 0.66
Hence, from the above,
We can conclude that

D. 0.5 Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 22 \(\frac{2}{4}\)
Answer:
The given numbers are: 0.5, \(\frac{2}{4}\)
Now,
Convert \(\frac{2}{4}\) into a decimal number
So,
\(\frac{2}{4}\) = 0.5
Hence, from the above,
We can conclude that

E. -2\(\frac{3}{4}\) Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 22 -2.25
Answer:
The given numbers are: -2\(\frac{3}{4}\), -2.25
Now,
Convert 2\(\frac{3}{4}\) into a decimal number
So,
2\(\frac{3}{4}\) = 2.75
Hence, from the above,
We can conclude that

F. –\(\frac{3}{5}\) Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 22 -0.35
Answer:
The given numbers are: –\(\frac{3}{5}\), -0.35
Now,
Convert \(\frac{3}{5}\) into a decimal number
So,
\(\frac{3}{5}\) = 0.60
Hence, from the above,
We can conclude that

Question 23.
Order the numbers from least to greatest.
A. -6, 8, -9, 13
Answer:
The given numbers are: -6, 8, -9, 13
Now,
We know that,
In a number line,
The positive values are greater than negative values
The positive values increase from left to right
The negative values decrease from right to left
Hence, from the above,
We can conclude that
The order of the given numbers from the least to the greatest is: -9, -6, 8, 13

B. –\(\frac{4}{5}\), –\(\frac{1}{2}\), 0.25, -0.2
Answer:
The given numbers are: –\(\frac{4}{5}\), –\(\frac{1}{2}\), 0.25, -0.2
Now,
We know that,
In a number line,
The positive values are greater than negative values
The positive values increase from left to right
The negative values decrease from right to left
Now,
The representation of the given numbers into a decimal number is: -0.8, -0.5, 0.25, -0.2
Hence, from the above,
We can conclude that
The order of the given numbers from the least to the greatest is: -0.8, -0.5, -0.2, 0.25

C. 4.75, -2\(\frac{1}{2}\), –\(\frac{8}{3}\), –\(\frac{9}{2}\)
Answer:
The given numbers are: 4.75, -2\(\frac{1}{2}\), –\(\frac{8}{3}\), –\(\frac{9}{2}\)
Now,
We know that,
In a number line,
The positive values are greater than negative values
The positive values increase from left to right
The negative values decrease from right to left
Now,
The representation of the given numbers into a decimal number is: 4.75, -2.5, -2.66, -4.5
Hence, from the above,
We can conclude that
The order of the given numbers from the least to the greatest is: -4.5, -2.66, -2.5, 4.75

D. 4, -3, -8, -1
Answer:
The given numbers are 4, -3, -8, -1
Now,
We know that,
In a number line,
The positive values are greater than negative values
The positive values increase from left to right
The negative values decrease from right to left
Hence, from the above,
We can conclude that
The order of the given numbers from the least to the greatest is: -8, -3, -1, 4

E. –\(\frac{1}{4}\), 0.5, \(\frac{3}{4}\), –\(\frac{1}{2}\)
Answer:
The given numbers are: –\(\frac{1}{4}\), 0.5, \(\frac{3}{4}\), –\(\frac{1}{2}\)
Now,
We know that,
In a number line,
The positive values are greater than negative values
The positive values increase from left to right
The negative values decrease from right to left
Now,
The representation of the given numbers into a decimal number is: -0.25, 0.5, 0.75, -0.5
Hence, from the above,
We can conclude that
The order of the given numbers from the least to the greatest is: -0.5, -0.25, 0.5, 0.75

F. –\(\frac{4}{5}\), –\(\frac{5}{4}\), –\(\frac{3}{2}\), 1.5
Answer:
The given numbers are: –\(\frac{4}{5}\), –\(\frac{5}{4}\), –\(\frac{3}{2}\), 1.5
Now,
We know that,
In a number line,
The positive values are greater than negative values
The positive values increase from left to right
The negative values decrease from right to left
Now,
The representation of the given numbers into a decimal number is -0.8, -1.25, -1.5, 1.5
Hence, from the above,
We can conclude that
The order of the given numbers from the least to the greatest is: -1.5, -1.25, -0.8, 1.5

Question 24.
Make Sense and Persevere What is the least number of points you must plot to have examples of all four sets of numbers, including at least one positive integer and one negative integer? Explain.
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 23.1
Answer:
Let’s define the sets:
Integers: The set of all whole numbers.
Rational numbers: Numbers that can be written as the quotient of two integer numbers.
Natural numbers: The set of positive integers.
Whole numbers: All the numbers that can be made by adding (or subtracting) 1 a given number of times.
Hence, from the above,
We can conclude that
We can use only one example for all four sets of numbers, including at least one positive integer and one negative integer

Question 25.
Reasoning Suppose you plot the locations of the animals on a number line. Which animal would be represented by the point farthest from 0 on the number line? Explain.
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 23.2
Answer:
It is given that
You plot the locations of the animals on a number line
Now,
The given table is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 23.2
Now,
The representation of the given data present in the table into a decimal number is:
Bloodbelly comb jelly         –        -0.8 km
Deep-sea anglerfish           –        -0.66 km
Fanfin anglerfish                 –        -2.25 km
Gulper eel                          –         -1.1 km
Pacific blackdragon            –         -0.3 km
Slender snipe eel                –         -0.6 km
Now,
We know that,
In a number line,
The negative decrease from right to left
The negative number that is at the end of the number line on the left side is the “Least number”
Hence, from the above,
We can conclude that
The animal which is farthest from 0 at the left side of the number line is: Fanfin anglerfish
The animal which is farthest from 0 at the right side of the number line is: Pacific blackdragon

Question 26.
Which animal is closest to a depth of -0.7 km?
Answer:
From Question 25,
The given information is:
Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers 23.2
Now,
The representation of the given data present in the table into a decimal number is:
Bloodbelly comb jelly         –        -0.8 km
Deep-sea anglerfish           –        -0.66 km
Fanfin anglerfish                 –        -2.25 km
Gulper eel                          –         -1.1 km
Pacific blackdragon            –         -0.3 km
Slender snipe eel                –         -0.6 km
Hence, from the above,
We can conclude that
The animal that is closest to a depth of -0.7 km is: Deep-sea anglerfish

Question 27.
The change in the value of a stock is represented by the rational number -5.90. Describe, in words, what this means.
Answer:
It is given that
The change in the value of a stock is represented by the rational number -5.90.
Now,
From the given information,
We can observe that
The given number is a negative number.
Now,
We know that,
When a stock is given by a negative number, it represents the decrease in stock and when a stock is given by a positive number, it represents the increase in stock.
As we are given number is a negative number,
We can just say that
The stock value is being decreased by 5.90.
Hence, from the above,
We can conclude that
The representation of the given information in words is:
The stock value is being decreased by 5.90.

Question 28.
Construct Arguments A classmate ordered these numbers from greatest to least. Is he correct? Construct an argument to justify your answer.
4.4, 4.2, -4.42, -4.24
Answer:
It is given that
A classmate ordered these numbers from greatest to least and the given order of these numbers is:
4.4, 4.2, -4.42, -4.24
Now,
We know that,
The positive values are greater than the negative values
In a number line,
The negative values decrease from right to left
The positive values increase from left to right
So,
The order of the numbers ordered by a classmate from the greatest to the least is:
4.4, 4.2, -4.24, -4.42
Hence, from the above,
We can conclude that
The classmate is not correct

Question 29.
Make Sense and Persevere Order -3.25, -3\(\frac{1}{8}\), -3\(\frac{3}{4}\), and -3.1 from least to greatest. Explain.
Answer:
The given numbers are: -3.25, -3\(\frac{1}{8}\), -3\(\frac{3}{4}\), and -3.1
Now,
The representation of the given numbers into a decimal number is:
-3.25, -3.12, -3.75, and -3.1
Hence, from the above,
We can conclude that
The order of the given numbers from the least to the greatest is:
-3.75, -3.25, -3.12, -3.1

Question 30.
Higher Order Thinking Suppose \(\frac{a}{b}\), \(\frac{c}{d}\), and \(\frac{e}{f}\) represent three rational numbers. If \(\frac{a}{b}\) is less than \(\frac{c}{d}\), and \(\frac{c}{d}\) is less than \(\frac{e}{f}\), compare \(\frac{a}{b}\) and \(\frac{e}{f}\). Explain.
Answer:
It is given that
Suppose \(\frac{a}{b}\), \(\frac{c}{d}\), and \(\frac{e}{f}\) represent three rational numbers. If \(\frac{a}{b}\) is less than \(\frac{c}{d}\), and \(\frac{c}{d}\) is less than \(\frac{e}{f}\)
Now,
According to the given information,
\(\frac{a}{b}\) < \(\frac{c}{d}\) and \(\frac{c}{d}\) < \(\frac{e}{f}\)
So,
\(\frac{a}{b}\) < \(\frac{c}{d}\) < \(\frac{e}{f}\)
Hence, from the above,
We can conclude that
\(\frac{a}{b}\) < \(\frac{e}{f}\)

Assessment Practice

Question 31.
Which could be a value for n?
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 2.80
A. –\(\frac{1}{2}\)
B. –\(\frac{1}{3}\)
C. –\(\frac{1}{4}\)
D. –\(\frac{1}{6}\)
Answer:
The given number line is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 2.80
Now,
From the given figure,
We can observe that
Each line in the given number line is: 0.1 units
So,
The representation of the value of n on the given number line is:

Hence, from the above,
We can conclude that
The value of n on the given number line is:

Question 32.
Which inequality does NOT represent the correct position of two numbers on a number line?
A. 4\(\frac{1}{2}\) > \(\frac{25}{4}\)
B. -4\(\frac{1}{2}\) > –\(\frac{25}{5}\)
C. -6 < -5
D. –\(\frac{1}{2}\) < \(\frac{1}{2}\)
Answer:
The given inequalities are:
A. 4\(\frac{1}{2}\) > \(\frac{25}{4}\)
B. -4\(\frac{1}{2}\) > –\(\frac{25}{5}\)
C. -6 < -5
D. –\(\frac{1}{2}\) < \(\frac{1}{2}\)
So,
The representation of the given inequalities in the form of a decimal number is:
A. 4.5 > 6.25
B. -9.5 > -5
C. -6 < -5
D. -0.5 < 0.5
Hence, from the above,
We can conclude that
The inequality that does not represent the correct position of two numbers on a number line is:

Lesson 2.3 Absolute Values of Rational Numbers

Solve & Discuss It!
A portion of a bank account statement is shown below How would you interpret the value of the ending balance? Explain.
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 29.2
Answer:
It is given that
A portion of a bank account statement is shown
Now,
The given figure is:

Now,
From the given figure,
We can observe that
The Ending Balance is: -$30
Now,
We know that,
A negative balance in your Debit Account means you owe money to the bank. It probably means you have used more than what you had in your account
Hence, from the above,
We can conclude that
The Ending Balance of -$30 in your account statement represents a negative balance in your Debit Account which means you owe money to the bank. It probably means you have used more than what you had in your account

Focus on math practices
Reasoning What is an example of a bank account balance that represents an amount owed greater than $40?
Answer:
We know that,
The money owed should be represented as “Negative”
Now,
Let the bank account balance that represents an amount be x
So,
According to the given information,
x > -$40
Hence, from the above,
We can conclude that
The representation of the bank account balance that represents an amount greater than 40$ is:
x > -$40

Visual learning

? Essential Question
How are absolute values used to describe quantities?
Answer:
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive

Try It!
The students in a science class recorded the change in the water level of a local river. During which week did the water level change by the greatest amount?
Use absolute values to represent the change in the water level.
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 29.4
The water level changed by the greatest amount in WeekEnvision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 30.
Answer:
It is given that
The students in a science class recorded the change in the water level of a local river
Now,
The given table is:

Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
So,
The representation of the changes in the water level is:

Hence, from the above,
We can conclude that

Convince Me! Can a lesser number represent a greater change in water level than a greater number? Explain.
Answer:
The given table is:

Now,
From the given table,
We can observe that
The greatest number in the given table represents the greatest change in the water level
Hence, from the above,
We can conclude that
A lesser number can not represent a greater change in water level than a greater number

Try It!
A bank has two customers with overdrawn accounts. Which balance is the greater number? Which balance is the lesser amount owed?
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 29.5
Answer:
It is given that
A bank has two customers with overdrawn accounts
Now,
The given information is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 29.5
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
So,
The absolute value of V.Wong’s Account’s balance is: $19.45
The absolute value of J.Olson’s Account’s balance is: $23.76
Now,
We know that,
In a number line,
The value of a negative number decrease from right to left and the negative number that is farthest to zero is the least number
We represent the amount of money owed as “Negative”
So,
The greater number is: $23.76
The balance that is the lesser amount owed is: $-19.45
Hence, from the above,
We can conclude that
The greater number is: $23.76
The balance that is the lesser amount owed is: $-19.45

KEY CONCEPT
The absolute value of a number is its distance from 0 on a number line. Distance is always positive. The absolute value of any number, n, is written |n|.
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 30.1
-4 and 4 are opposites as they are the same distance from 0.

Do You Understand?

Question 1.
? Essential Question How are absolute values used to describe quantities?
Answer:
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive

Question 2.
Construct Arguments Explain why – 7 has a greater absolute value than the absolute value of 6.
Answer:
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
So,
The absolute value of -7 is:
|-7| = 7
So,
In the number line,
We know that,
The positive values increase from left to right
Hence, from the above,
We can conclude that
According to the Property of representation of Integers in a number line,
7 > 6

Question 3.
Reasoning Give an example of a balance that has a greater integer value than a balance of -$12 but represents a debt of less than $5.
Answer:
It is given that
Give an example of a balance that has a greater integer value than a balance of -$12 but represents a debt of less than $5.
Now,
Let x be the balance that has a greater integer value than a balance of -$12, but represents a debt of less than $5.
Now,
We are given that a balance that has a greater integer value than a balance of -$12
So,
x >-12
Now,
We are given that balance represents a debt of less than $5.
So,
Debt < -5
So,
x < -5
So,
-5 >x >-12
So,
x ∈ [-6,-11]
Hence, from the above,
We can conclude that
The balance that has a greater integer value than a balance of -$12, but represents a debt of less than $5 is:
-5 > x > -12

Question 4.
Of the three elevations, -2 feet, -12 feet, and 30 feet, which represents the least number? Which represents the farthest distance from sea level?
Answer:
It is given that
There are three elevations namely: -2 feet, -12 feet, and 30 feet
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
So,
The absolute values of the three elevations are: 2 feet, 12 feet, and 30 feet
Now,
We know that,
In a number line,
The positive values increase from left to right
So,
The least number is: 2 feet
The farthest distance from the sea level is: 30 feet
Hence, from the above,
We can conclude that
The least number is: 2 feet
The farthest distance from the sea level is: 30 feet

Do You Know How?

In 5-14, find each absolute value.

Question 5.
|-9|
Answer:
The given number is |-9|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |-9| is: 9

Question 6.
|5\(\frac{3}{4}\)|
Answer:
The given number is |5\(\frac{3}{4}\)|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |5\(\frac{3}{4}\)| is: 5\(\frac{3}{4}\)

Question 7.
|-5.5|
Answer:
The given number is |-5.5|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |-5.5| is: 5.5

Question 8.
|82.5|
Answer:
The given number is |82.5|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |82.5| is: 82.5

Question 9.
|-14\(\frac{1}{3}\)|
Answer:
The given number is |-14\(\frac{1}{3}\)|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |-14\(\frac{1}{3}\)| is: 14\(\frac{1}{3}\)

Question 10.
|-7.75|
Answer:
The given number is |-7.75|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |-7.75| is: 7.75

Question 11.
|–19|
Answer:
The given number is |-19|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |-19| is: 19

Question 12.
|-2\(\frac{1}{2}\)|
Answer:
The given number is |-2\(\frac{1}{2}\)|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |-2\(\frac{1}{2}\)| is: 2\(\frac{1}{2}\)

Question 13.
|24|
Answer:
The given number is |24|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |24| is: 24

Question 14.
|35.4|
Answer:
The given number is |35.4|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |35.4| is: 35.4

In 15-17, use the absolute value of each account balance to determine which account has the greater overdrawn amount.

Question 15.
Account A: -$5.42
Account B: -$35.76
Answer:
The given information is:
Account A: -$5.42
Account B: -$35.76
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
So,
The absolute value of each account balance is:
Account A: $5.42
Account B: $35.76
Now,
We know that,
In a number line,
The positive values increase from left to right
So,
The account that has the greater overdrawn amount is: Account B
Hence, from the above,
We can conclude that
The account that has the greater overdrawn amount is: Account B

Question 16.
Account A: – $6.47
Account B: -$2.56
Answer:
The given information is:
Account A: -$6.47
Account B: -$2.56
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
So,
The absolute value of each account balance is:
Account A: $6.47
Account B: $2.56
Now,
We know that,
In a number line,
The positive values increase from left to right
So,
The account that has the greater overdrawn amount is: Account A
Hence, from the above,
We can conclude that
The account that has the greater overdrawn amount is: Account A

Question 17.
Account A: -$32.56
Account B: -$29.12
Answer:
The given information is:
Account A: -$32.56
Account B: -$29.12
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
So,
The absolute value of each account balance is:
Account A: $32.56
Account B: $29.12
Now,
We know that,
In a number line,
The positive values increase from left to right
So,
The account that has the greater overdrawn amount is: Account A
Hence, from the above,
We can conclude that
The account that has the greater overdrawn amount is: Account A

Practice & Problem Solving

In 18-33, find each absolute value.

Question 18.
|-46|
Answer:
The given number is |-46|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |-46| is: 46

Question 19.
|0.7|
Answer:
The given number is |0.7|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |0.7| is: 0.7

Question 20.
|-\(\frac{2}{3}\)|
Answer:
The given number is |-\(\frac{2}{3}\)|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |-\(\frac{2}{3}\)| is: \(\frac{2}{3}\)

Question 21.
|-7.35|
Answer:
The given number is |-7.35|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |-7.35| is: 7.35

Question 22.
|-4\(\frac{3}{4}\)|
Answer:
The given number is |-4\(\frac{3}{4}\)|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |-4\(\frac{3}{4}\)| is: 4\(\frac{3}{4}\)

Question 23.
|-54.5|
Answer:
The given number is |-54.5|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |-54.5| is: 54.5

Question 24.
|27\(\frac{1}{4}\)|
Answer:
The given number is |27\(\frac{1}{4}\)|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |27\(\frac{1}{4}\)| is: 27\(\frac{1}{4}\)

Question 25.
|–13.35|
Answer:
The given number is |-13.35|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |-13.35| is: 13.35

Question 26.
|14|
Answer:
The given number is |14|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |14| is: 14

Question 27.
|-11.5|
Answer:
The given number is |-11.5|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |-11.5| is: 11.5

Question 28.
|-6.3|
Answer:
The given number is |-6.3|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |-6.3| is: 6.3

Question 29.
|3.75|
Answer:
The given number is |3.75|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |3.75| is: 3.75

Question 30.
|-8.5|
Answer:
The given number is |-8.5|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |-8.5| is: 8.5

Question 31.
|15|
Answer:
The given number is |15|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |15| is: 15

Question 32.
|-6\(\frac{3}{4}\)|
Answer:
The given number is |-6\(\frac{3}{4}\)|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |-6\(\frac{3}{4}\)| is: 6\(\frac{3}{4}\)

Question 33.
|-5.3|
Answer:
The given number is |-5.3|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
Hence, from the above,
We can conclude that
The absolute value of |-5.3| is: 5.3

In 34-37, order the numbers from least to greatest.

Question 34.
|-12|, |11\(\frac{3}{4}\)|, |-20.5|, |2|
Answer:
The given numbers are:
|-12|, |11\(\frac{3}{4}\)|, |-20.5|, |2|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
So,
The absolute values of the given numbers are:
12, 11\(\frac{3}{4}\), 20.5, 2
So,
The absolute values of the given numbers in the decimal form are:
12, 11.75, 20.5, 2
Now,
We know that,
In a number line,
The positive values increase from left to right
Hence, from the above,
We can conclude that
The order of the numbers from the least to the greatest is:
2, 11.75, 12, 20.5

Question 35.
|10|, |-3|, |0|, |-5.25|
Answer:
The given numbers are:
|10|, |-3|, |0|, |-5.25|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
So,
The absolute values of the given numbers are:
10, 3, 0, 5.25
Now,
We know that,
In a number line,
The positive values increase from left to right
Hence, from the above,
We can conclude that
The order of the numbers from the least to the greatest is:
0, 3, 5.25, 10

Question 36.
|-6|, |-4|, |11|, |0|
Answer:
The given numbers are:
|-6|, |-4|, |11|, |0|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
So,
The absolute values of the given numbers are:
6, 4, 11, 0
Now,
We know that,
In a number line,
The positive values increase from left to right
Hence, from the above,
We can conclude that
The order of the numbers from the least to the greatest is:
0, 4, 6, 11

Question 37.
|4|, |-3|, |–18|, |-3.18|
Answer:
The given numbers are:
|4|, |-3|, |–18|, |-3.18|
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
So,
The absolute values of the given numbers are:
4, 3, 18, 3.18
Now,
We know that,
In a number line,
The positive values increase from left to right
Hence, from the above,
We can conclude that
The order of the numbers from the least to the greatest is:
3, 3.18, 4, 18

Alberto and Rebecca toss horseshoes at a stake. Whoever’s horseshoe is closer to the stake wins a point.

Question 38.
Reasoning What integer best describes the location of Alberto’s horseshoe in relation to the stake? What integer best describes the location of Rebecca’s horseshoe?
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 35.1
Answer:
It is given that
Alberto and Rebecca toss horseshoes at a stake. Whoever’s horseshoe is closer to the stake wins a point.
Now,
The given figure is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 35.1
Now,
From the given figure,
We can observe that
The stake is at zero position
The location of Alberto’s horseshoe is at: -3
The location of Rebecca’s horseshoe is at: 2
So,
The location of Alberto’s horseshoe is: 3 feet
The location of Reecca’s horseshoe is: 2 feet
Hence, from the above,
We can conclude that
The integer that best describes the location of Alberto’s horseshoe in relation to the stake is: 3 feet
The integer that best describes the location of Rebecca’s horseshoe is: 2 feet

Question 39.
Critique Reasoning Alberto says that -3 is less than 2, so he wins a point. Is Alberto correct? Explain.
Answer:
It is given that
Alberto says that -3 is less than 2, so he wins a point
Now,
The given figure is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 35.1
Now,
The number of points between Alberto’s horseshoe and Rebecca’s horseshoe = |(The location of Alberto’s horseshoe) + (The location of Rebecca’s horseshoe)|
= |-3 + 2|
= |-1|
= 1
Hence, from the above,
We can conclude that Alberto is correct

Question 40.
A model with Math
Find the distance from Alberto’s horseshoe to Rebecca’s horseshoe. Explain.
Answer:
The given figure is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 35.1
So,
The distance between Alberto’s horseshoe and Rebecca’s horseshoe = |(The location of Alberto’s horseshoe) + (The location of Rebecca’s horseshoe)|
= |-3 + 2|
= |-1|
= 1 feet
Hence, from the above,
We can conclude that
The distance from Alberto’s horseshoe to Rebecca’s horseshoe is: 1 feet

Question 41.
Higher-Order Thinking Let a = any rational number. Is the absolute value of a difference if a is a positive number or a negative number? Explain.
Answer:
It is given that
“a” is any rational number
Now,
Let
a = 6 (or) a = -5
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
So,
|a| = |6| (or) |a| = |-5|
|a| = 6 (or) |a| = 5
So,
|6 – 5| = |1| =1
Hence, from the above,
We can conclude that
The absolute value of a difference is positive even if “a” is a positive number or a negative number

Question 42.
Construct Arguments Samuel and Leticia are playing a game. After the first round of the game, Samuel’s score was -19, and Leticia’s score was 21. The score with the greater absolute value wins each round. Who won the first round? Explain.
Answer:
It is given that
Samuel and Leticia are playing a game. After the first round of the game, Samuel’s score was -19, and Leticia’s score was 21. The score with the greater absolute value wins each round
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
So,
The abso;ute value of Samuel’s score was:
|-19| = 19
The absolute value of Leticia’s score was:
|21| = 21
Now,
We know that,
The positive values increase from left to right
So,
Leticia won the first round
Hence, from the above,
We can conclude that
Leticia won the first round

Question 43.
Use Structure Ana and Chuyen are exploring underwater sea life while on a helmet diving adventure. Ana’s location is -30 feet below sea level, and Chuyen’s location is -12 feet below sea level. Which girl is located farther from sea level?
Answer:
It is given that
Ana and Chuyen are exploring underwater sea life while on a helmet diving adventure. Ana’s location is -30 feet below sea level, and Chuyen’s location is -12 feet below sea level
Now,
We know that,
In a number line,
The negative values decrease from right to left
The negative value that is farthest to the number line is the least value
Hence, from the above,
We can conclude that
Ana is located farther from sea level

Question 44.
Marie’s account balance is – $45.62. Tom’s account balance is $42.55. Which balance represents the greater number? Which balance represents the lesser amount owed?
Answer:
It is given that
Marie’s account balance is – $45.62. Tom’s account balance is $42.55
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
So,
The absolute value of Marie’s account balance is: $45.62
The absolute value of Tom’s account balance is: $42.55
So,
The balance that represents the greater number is: Marie’s account balance
The balance that represents the lesser amount owed is: Tom’s account balance
Hence, from the above,
We can conclude that
The balance that represents the greater number is: Marie’s account balance
The balance that represents the lesser amount owed is: Tom’s account balance

Question 45.
In New York, the Federal Reserve gold vault is located at a depth of |-80| feet below ground. The treasure at Oak Island is believed to be at a depth of |-134| feet. Which is farther below ground, the gold vault or the Oak Island treasure?
Answer:
It is given that
In New York, the Federal Reserve gold vault is located at a depth of |-80| feet below ground. The treasure at Oak Island is believed to be at a depth of |-134| feet
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
So,
The absolute value of the depth of the Federal reserve gold vault is: 80 feet
The absolute value of the depth of the treasure at Oak Island is: 134 feet
So,
The treasure at oak Island is farther below ground
Hence, from the above,
We can conclude that
The treasure at Oak Island is farther below ground

Question 46.
Two scuba divers are swimming below sea level. The locations of the divers can be represented by -30 feet and -42 feet. Which measure represents the location that is closest to sea level?
Answer:
It is given that
Two scuba divers are swimming below sea level. The locations of the divers can be represented by -30 feet and -42 feet
Now,
We know that,
In a number line,
The negative values decrease from left to right
Hence, from the above,
We can conclude that
The measure that represents the location that is closest to sea level is: -30 feet

Assessment Practice

Question 47.
The table at the right shows the scores at the end of the first round of a golf tournament. The scores are relative to par.
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 38.1
PART A
Par is represented as 0. Using absolute value, show the distance each score is from par.
Answer:
It is given that
The table at the right shows the scores at the end of the first round of a golf tournament. The scores are relative to par.
Now,
The given table is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 38.1
Hence,
The representation of the distance each score is from Par in a number line is:

PART B
The golfer with the least score wins the round. Who won the first round of the tournament? Explain.
Answer:
It is given that
The golfer with the least score wins the round
The given table is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 38.1
Now,
From part (a),
The representation of the scores on a number line is:

Now,
From the given table,
We can observe that
The golfer who has the least score is: Kate
Hence, from the above,
We can conclude that
Kate won the first round of the tournament

Topic 2 MID-TOPIC CHECKPOINT

Question 1.
Vocabulary Describe the relative locations of the rational numbers -(-\(\frac{a}{b}\)) and \(\frac{a}{b}\) on a number line. Lessons 2.1 and 2.2
Answer:
The given rational numbers are: \(\frac{a}{b}\) and -(-\(\frac{a}{b}\))
Now,
We know that,
– × – = +
So,
The given rational numbers are: \(\frac{a}{b}\) and \(\frac{a}{b}\)
Now,
From the given fractions,
We can observe that both are positive rational numbers
Now,
We know that,
In a number line,
The values after zero are the “positive Values”
Hence, from the above,
We can conclude that
The relative locations of the given rational numbers on a number line are: At the right side of the number line

Question 2.
Marc deposited $175 in a new bank account. After buying some furniture, he was overdrawn by $55. Select all the true statements about Marc’s account. Lesson 2.1
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 41 To start, Marc had a negative balance.
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 41 In this situation, 0 represents an empty bank account.
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 41 When Marc was overdrawn, he had a negative balance.
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 41 After buying furniture, Marc had a positive balance.
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 41 The lowest balance in the account was -$55.
Answer:
It is given that
Marc deposited $175 in a new bank account. After buying some furniture, he was overdrawn by $55.
Hence, from the above,
We can conclude that
All the true statements about Marc’s account are:

Question 3.
What number is represented on the number line? Give your answer as a decimal and as a fraction. Lesson 2.2
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 41.1
Answer:
The given number line is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 41.1
Now,
From the given number line,
We can observe that
Each line in the given number line represents 0.25 units
So,
The number that is represented on the number line is: -2.25
Hence, from the above,
We can conclude that
The number that is represented on the number line as a decimal number is: -2.25
The number that is represented on the number line as a fraction is: -2\(\frac{1}{4}\)

Question 4.
The absolute value of a number is 52. Select all the integers that this number could be. Lesson 2.3
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 42 -52
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 42 -25
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 42 25
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 42 –\(\frac{1}{52}\)
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 42 52
Answer:
It is given that
The absolute value of a number is 52
Hence, from the above,
We can conclude that
All the integers that the number could be:

Question 5.
The table shows the location of four treasure chests relative to sea level. How can you use the number line to find the treasure chest that is farthest from sea level? Lesson 2.2
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 42.1
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 42.3
Answer:
It is given that
The table shows the location of four treasure chests relative to sea level
Now,
The given table is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 42.1
Now,
The representation of \(\frac{5}{4}\) feet in a decimal form is: 1.25 feet
Now,
The given number line is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 42.3
Now,
From the given number line,
We can observe that
Each unit represents 0.25 on the given number line
Hence,
The representation of the locations of the treasure chests on the given number line is:

Question 6.
Three customers have accounts owing money. The table shows the account balances that represent what the customers owe. Which customer owes the least amount of money? Lesson 2.3
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 42.4
Answer:
It is given that
Three customers have accounts owing money. The table shows the account balances that represent what the customers owe.
Now,
The given table is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 42.4
Now,
We know that,
In a number line,
The negative value that is farthest from zero is the least value
The negative values decrease from right to left
Hence, from the above,
We can conclude that
The customet that owes the least amount of money is: B Barker

How well did you do on the mid-topic checkpoint? Fill in the stars. Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 42.5

Topic 2 MID-TOPIC PERFORMANCE TASK

Warren and Natasha started a dog-walking business. During their first week, they paid $10 to make their business cards and $6 for a 4.5-pound box of doggie treats. Warren walked a dog for 15 minutes, and Natasha walked a dog for 30 minutes.

PART A
Which integers represent the dollar amounts either spent or earned during the first week Warren and Natasha were in business? Select all that apply.
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 43 $5
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 43 -$5
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 43 $10
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 43 -$10
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 43 -$6
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 43.1
Answer:
It is given that
Warren and Natasha started a dog-walking business. During their first week, they paid $10 to make their business cards and $6 for a 4.5-pound box of doggie treats. Warren walked a dog for 15 minutes, and Natasha walked a dog for 30 minutes.
Now,
The given table is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 43.1
So,
The amount of money Warren and Natasha spent during the first week = (The amount spent on business cards) + 9the amount spent on a 4.5-pound box of doggie treats)
= $10 + $6
= $16
So,
The amount earned by Warren = $5
The amount earned by Natasha = $10
Hence, from the above,
We can conclude that
The integers that represent the dollar amounts either spent or earned during the first week Warren and Natasha were in business are:

PART B
At the end of each week, Warren records the weight in pounds of doggie treats eaten as a negative rational number. Plot the numbers of pounds eaten each week on the number line. Order the numbers from most pounds eaten to fewest pounds eaten.
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 43.2
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 43.3
Answer:
It is given that
At the end of each week, Warren records the weight in pounds of doggie treats eaten as a negative rational number
Now,
The given table is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 43.2
Now,
The representation of the number of pounds eaten into a decimal number are:
–\(\frac{3}{2}\) = -1.5
–\(\frac{2}{3}\) = -0.6
–\(\frac{5}{4}\) = -1.25
Now,
We know that,
In a number line,
The negative numbers decrease from right to left
So,
The representation of the number of pounds eaten on a number line is:

So,
The order of the numbers from most pounds eaten to fewest pounds eaten is:
-1.5 < -1.25 < -0.66 < -0.5
–\(\frac{3}{2}\) < –\(\frac{5}{4}\) < –\(\frac{2}{3}\) < -0.5
Hence, from the above,
We can conclude that
The representation of the number of pounds eaten on a number line is:

The order of the numbers from most pounds eaten to fewest pounds eaten is:
–\(\frac{3}{2}\) < –\(\frac{5}{4}\) < –\(\frac{2}{3}\) < -0.5

PART C
Find the absolute value for the number of pounds of doggie treats eaten each week. Which two weeks had the greatest number of pounds eaten?
Answer:
The given table from part B is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 43.2
Now,
We know that,
“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
So,
The absolute values for the number of pounds eaten are:
|-\(\frac{3}{2}\)| = \(\frac{3}{2}\)
|-\(\frac{2}{3}\)| = \(\frac{2}{3}\)
|-0.5| = 0.5
|-\(\frac{5}{4}\)| = \(\frac{5}{4}\)
So,
The representation of the absolute values for the number of pounds eaten into a decimal number is:
1.5, 0.66, 0.5, 1.25
Now,
From the given table,
We can observe that
Week 1 and Week 4 had the greatest number of pounds eaten
Hence, from the above,
We can conclude that
a. The absolute values for the number of pounds eaten are:
|-\(\frac{3}{2}\)| = \(\frac{3}{2}\)
|-\(\frac{2}{3}\)| = \(\frac{2}{3}\)
|-0.5| = 0.5
|-\(\frac{5}{4}\)| = \(\frac{5}{4}\)
b. Week 1 and Week 4 had the greatest number of pounds eaten

Lesson 2.4 Represent Rational Numbers on the Coordinate Plane

Solve & Discuss It!

Point B has the same x-coordinate as point A, but its y-coordinate is the opposite of the y-coordinate of point A. Plot point B and write its coordinates.
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 43.4
Answer:
It is given that
Point B has the same x-coordinate as point A, but its y-coordinate is the opposite of the y-coordinate of point A.
Now,
The given coordinate plane is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 43.4
Now,
From the given coordinate plane,
We can observe that A (3, 5)
Now,
We know that,
The opposite of y-coordinate in the coordinate plane is: -y
So,
The coordinates for point B is: B (3, -5)
Hence, from the above,
We can conclude that
The coordinates for point B is: B (3, -5)
The representation of B (3, -5) in the coordinate plane is:

Make Sense and Persevere How can you use what you know about integers and graphing points on a coordinate plane to plot point B?
Answer:
From the above problem,
We know that,
The coordinates of point B are: B (3, -5)
Now,
We know that,
A coordinate plane consists of 2 number lines i.e., 1 horizontal number line and 1 vertical number line, and both intersect at zero
Now,
We know that,
In a number line,
The right side of zero and the upside of zero will be positive numbers
The left side of zero and the downside of zero will be negative numbers
Hence, from the above,
We can conclude that
B (3, -5) will be located on the right side of the horizontal number line and the downside of the vertical number line

Focus on math practices
Generalize Two points that have the same x-coordinate but opposite y-coordinates. Across which axis do they form mirror images of each other?
Answer:
It is given that
Two points that have the same x-coordinate but opposite y-coordinates.
So,
The representation of the point and its mirror image respectively are:
(x, y) ——–> (x, -y)
Hence, from the above,
We can conclude that
The mirror images of the given points will form across the x-axis

Visual Learning

? Essential Question How can you graph a point with rational coordinates on a coordinate plane?
Answer:
Graphing Points with Rational Number Coordinates:
a. Locate “a” on the x-axis.
b. From “a” on the x-axis, if “b” is positive, move up “b” units, and if “b” is negative, move down “b” units.
c. The location should now line up to “a” on the x-axis and to “b” on the y-axis, so we draw a point at the location. This is the point (a, b).

Try It!
Graph point P(-2, -3) on the coordinate plane shown.
Start at the originEnvision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 50.
The x-coordinate is negative, so moveEnvision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 51 units to the left.
Then use y-coordinate to moveEnvision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 51 units down.
Draw and label the point.
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 53.5
Answer:
The given point is: P (-2, -3)
Now,

Hence,
The representation of the given point in the coordinate plane is:

Convince Me! How do the signs of the coordinates relate to the quadrant in which a point is located? Explain for each of the four quadrants.
Answer:
a. If both x and y are positive, then the point lies in the “First quadrant”
b. If x is negative and y is positive, then the point lies in the “Second quadrant”
c. If both x and y are negative, then the point lies in the “Third quadrant”
d. If x is positive and y is negative, then the point lies in the “Fourth quadrant”

Try It!
What landmark is located on the map at (2, \(\frac{1}{4}\))?

Answer:
The given point is: (2, \(\frac{1}{4}\)
Now,
The given figure is:

Now,
The representation of the given point in the decimal form is: (2, 0.25)
Now,
From the given map,
Locate the given point
So,
The location of the given point in the map is:

Hence, from the above,
We can conclude that
The landmark that is located on the map at (2, \(\frac{1}{4}\)) is: FBI Building

Try It!
The coordinates of point A are (-3, 5). What are the coordinates of point B, which is a reflection of point A across the x-axis?
Answer:
It is given that
The coordinates of point A are (-3, 5)
Now,
It is given that
Point B is the reflection of point A across the x-axis
Now,
We know that,
When (x, y) reflects across the x-axis, the point will become (x, -y)
Hence, from the above,
We can conclude that
The coordinates of point B, which is a reflection of point A across the x-axis is: (-3, -5)

KEY CONCEPT
A coordinate plane is a grid that contains number lines that intersect at right angles and divide the plane into four quadrants. The horizontal number line is called the x-axis, and the vertical number line is called the y-axis.
The location of a point on a coordinate plane is written as an ordered pair (x, y).
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 52.1

Do You Understand?

Question 1.
? Essential Question How can you graph a point with rational coordinates on a coordinate plane?
Answer:
Graphing Points with Rational Number Coordinates:
a. Locate “a” on the x-axis.
b. From “a” on the x-axis, if “b” is positive, move up “b” units, and if “b” is negative, move down “b” units.
c. The location should now line up to “a” on the x-axis and to “b” on the y-axis, so we draw a point at the location. This is the point (a, b).

Question 2.
What is the y-coordinate of any point that lies on the x-axis?
Answer:
Every point on the x-axis has no distance (zero distance) from the x-axis, therefore, the y – coordinate of every point lying on the x-axis is always “Zero”

Question 3.
Look for Relationships How are the points (4, 5) and (-4, 5) related?
Answer:
The given points are: (4, 5), and (-4, 5)
Now,
We know that,
The reflection of A (x, y) across the x-axis is: B(x, -y)
The reflection of A (x, y) across the y-axis is: B(-x, y)
Now,
From the given points,
We can observe that
(-4, 5) is the reflection of (4, 5) across the y-axis
Hence, from the above,
We can conclude that
(-4, 5) is the reflection of (4, 5) across the y-axis

Question 4.
Construct Arguments On a larger map, the coordinates for the location of another Washington, D.C. landmark are (8, -10). In which quadrant of the map is this landmark located? Explain.
Answer:
It is given that
On a larger map, the coordinates for the location of another Washington, D.C. landmark are (8, -10)
Now,
We know that,
a. If both x and y are positive, then the point lies in the “First quadrant”
b. If x is negative and y is positive, then the point lies in the “Second quadrant”
c. If both x and y are negative, then the point lies in the “Third quadrant”
d. If x is positive and y is negative, then the point lies in the “Fourth quadrant”
Hence, from the above,
We can conclude that
The location of another Washington D. C of the map is located in the fourth quadrant

In 5-7, graph and label each point on the coordinate plane.
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 53.4

Question 5.
A(-4, 1)
Answer:
The given point is: A (-4, 1)
Hence,
The representation of the given point in the coordinate plane is:

Question 6.
B(4, 3)
Answer:
The given point is: B (4, 3)
Hence,
The representation of the given point in the coordinate plane is:

Question 7.
C(0, -2)
Answer:
The given point is: C (0, -2)
Hence,
The representation of the given point in the coordinate plane is:

Question 8.
What ordered pair gives the coordinates of point P above?
Answer:
The given coordinate plane is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 53.4
Now,
From the given coordinate plane,
We can observe that
Point P is in the 4th quadrant
The coordinates of point P is: P (3, -2)
Hence, from the above,
We can conclude that
The ordered pair that gives the coordinates of Point P is: P (3, -2)

In 9 and 10, use the map in Example 2 and write the ordered pair of each location.

Question 9.
White House
Answer:
The given map is:

Now,
From the given map,
We can observe that
The coordinates of the White House are: (1, 1)
Hence, from the above,
We can conclude that
The coordinates of the White House are: (1, 1)

Question 10.
Lincoln Memorial
Answer:
The given map is:

Now,
From the given map,
We can observe that
The coordinates of the Lincoln Memorial are: (-1, -0.5)
Hence, from the above,
We can conclude that
The coordinates of the White House are: (-1, -0.5)

In 11 and 12, use the map in Example 2 and write the landmark located at each ordered pair.

Question 11.
(0.5, 0)
Answer:
The given point is: (0.5, 0)
Now,
The given map is:

Now,
From the given map,
We can observe that
The landmark that is located at (0.5, 0) is: Ellipse
Hence, from the above,
We can conclude that
The landmark that is located at (0.5, 0) is: Ellipse

Question 12.
(\(\frac{3}{4}\), –\(\frac{1}{2}\))
Answer:
The given point is: (\(\frac{3}{4}\), –\(\frac{1}{2}\))
Now,
The given map is:

Now,
From the given map,
We can observe that
The landmark that is located at (\(\frac{3}{4}\), –\(\frac{1}{2}\)) is: Washington Monument
Hence, from the above,
We can conclude that
The landmark that is located at (\(\frac{3}{4}\), –\(\frac{1}{2}\)) is: Washington Monument

Practice & Problem Solving

In 13-20, graph and label each point.

Question 13.
A(1, -1)
Answer:
The given point is: A (1, -1)
Hence,
The representation of the given point in the coordinate plane is:

Question 14.
B(4, 3)
Answer:
The given point is: B (4, 3)
Hence,
The representation of the given point in the coordinate plane is:

Question 15.
C(-4, 3)
Answer:
The given point is: C (-4, 3)
Hence,
The representation of the given point in the coordinate plane is:

Question 16.
D(5, -2)
Answer:
The given point is: D (5, -2)
Hence,
The representation of the given point in the coordinate plane is:

Question 17.
E(-2.5, 1.5)
Answer:
The given point is: E (-2.5, 1.5)
Hence,
The representation of the given point in the coordinate plane is:

Question 18.
F(2, 1.5)
Answer:
The given point is: F (2, 1.5)
Hence,
The representation of the given point in the coordinate plane is:

Question 19.
G(-2, -1\(\frac{1}{2}\))
Answer:
The given point is: G (-2, -1\(\frac{1}{2}\))
Hence,
The representation of the given point in the coordinate plane is:

Question 20.
H(1\(\frac{1}{2}\), -1)
Answer:
The given point is: H (1\(\frac{1}{2}\), -1)
Hence,
The representation of the given point in the coordinate plane is:

In 21-26, write the ordered pair for each point.
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 55.1

Question 21.
P
Answer:
The given coordinate plane is:
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 55.1
Now,
From the given coordinate plane,
We can observe that
The coordinates of point P from the given coordinate plane is: P (0, -8)
Hence, from the above,
We can conclude that
The coordinates of point P from the given coordinate plane is: P (0, -8)

Question 22.
Q
Answer:
The given coordinate plane is:
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 55.1
Now,
From the given coordinate plane,
We can observe that
The coordinates of point Q from the given coordinate plane is: Q (5, -3)
Hence, from the above,
We can conclude that
The coordinates of point Q from the given coordinate plane is: Q (5, -3)

Question 23.
R
Answer:
The given coordinate plane is:
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 55.1
Now,
From the given coordinate plane,
We can observe that
The coordinates of point R from the given coordinate plane is: R (-8, 0)
Hence, from the above,
We can conclude that
The coordinates of point R from the given coordinate plane is: R (-8, 0)

Question 24.
S
Answer:
The given coordinate plane is:
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 55.1
Now,
From the given coordinate plane,
We can observe that
The coordinates of point S from the given coordinate plane is: S (-2.5, -0.5)
Hence, from the above,
We can conclude that
The coordinates of point S from the given coordinate plane is: S (-2.5, -0.5)

Question 25.
T
Answer:
The given coordinate plane is:
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 55.1
Now,
From the given coordinate plane,
We can observe that
The coordinates of point T from the given coordinate plane is: T (1.5, 2.5)
Hence, from the above,
We can conclude that
The coordinates of point T from the given coordinate plane is: T (1.5, 2.5)

Question 26.
U
Answer:
The given coordinate plane is:
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 55.1
Now,
From the given coordinate plane,
We can observe that
The coordinates of point U from the given coordinate plane is: U (-1, -0.5)
Hence, from the above,
We can conclude that
The coordinates of point U from the given coordinate plane is: U (-1, -0.5)

In 27-30, use the map at the right.
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 56.1

Question 27.
Which building is located in Quadrant III?
Answer:
The given map is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 56.1
Now,
We know that,
a. If both x and y are positive, then the point lies in the “First quadrant”
b. If x is negative and y is positive, then the point lies in the “Second quadrant”
c. If both x and y are negative, then the point lies in the “Third quadrant”
d. If x is positive and y is negative, then the point lies in the “Fourth quadrant”
So,
From the given map,
We can observe that
The building that is located in Quadrant III is: Fire House
Hence, from the above,
We can conclude that
The building that is located in Quadrant III is: Fire House

Question 28.
Which two places have the same x-coordinate?
Answer:
The given map is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 56.1
Now,
We know that,
a. If both x and y are positive, then the point lies in the “First quadrant”
b. If x is negative and y is positive, then the point lies in the “Second quadrant”
c. If both x and y are negative, then the point lies in the “Third quadrant”
d. If x is positive and y is negative, then the point lies in the “Fourth quadrant”
So,
From the given map,
We can observe that
The two buildings that have the same x-coordinate are: Swimming pool and clubhouse
Hence, from the above,
We can conclude that
The two buildings that have the same x-coordinate are: Swimming pool and clubhouse

Question 29.
Use Structure The city council wants the location of the entrance to a new city park to be determined by the reflection of the school entrance across the y-axis. What are the coordinates of the entrance to the new city park on this map?
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 56.1
Answer:
It is given that
The city council wants the location of the entrance to a new city park to be determined by the reflection of the school entrance across the y-axis
Now,
The given map is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 56.1
Now,
From the given map,
We can observe that
The coordinates of the school entrance are: (4, 6)
Now,
We know that,
The reflection of (x, y) across the y-axis is: (-x, y)
So,
The coordinates of the entrance to a new city park are: (-4, 6)
Hence, from the above,
We can conclude that
The coordinates of the entrance to a new city park are: (-4, 6)

Question 30.
Higher-Order Thinking You are at the market square (0, 0) and want to get to the doctor’s office. Following the gridlines, what is the shortest route?
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 56.1
Answer:
It is given that
You are at the market square (0, 0) and want to get to the doctor’s office.
Now,
The given map is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 56.1
Now,
The representation of the distance of the Market and doctor’s office in the map are:

Now,
From the given map,
We can observe that
The shortest route can be determined by using the Pythagoras Theorem since a right triangle is formed
So,
The shortest route between the market and doctor’s office = \(\sqrt{4² + 2²}\)
= \(\sqrt{20}\)
= 4.47 units
Hence, from the above,
We can conclude that
The shortest route between the market and the doctor’s office is: 4.47 units

In 31-36, use the coordinate plane at the right.
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 56.5

Question 31.
What is located at (-0.7,-0.2)?
Answer:
The given point is: (-0.7, -0.2)
Now,
The given coordinate plane is:

Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 56.5
Now,
From the given coordinate plane,
We can observe that the landmark that is located at (-0.7, -0.2) is: Pond
Hence, from the above,
We can conclude that
The pond is located at (-0.7, -0.2)

Question 32.
What is located at (\(\frac{3}{10}\), –\(\frac{1}{5}\))?
Answer:
The given point is: (\(\frac{3}{10}\), –\(\frac{1}{5}\))
Now,
The given coordinate plane is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 56.5
Now,
The representation of the given point in the form of a decimal number is: (0.3, -0.2)
Now,
From the given coordinate plane,
We can observe that the landmark that is located at (\(\frac{3}{10}\), –\(\frac{1}{5}\)) is: Start of Hiking Trail
Hence, from the above,
We can conclude that
The start of Hiking Trail is located at (\(\frac{3}{10}\), –\(\frac{1}{5}\))

Question 33.
Be Precise Write the ordered pair to locate the end of the hiking trail in two different ways.
Answer:
The given coordinate plane is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 56.5
Now,
From the given coordinate plane,
We can observe that
The coordinates to locate the end of the hiking trail is: (0.2, -0.8)
Hence, from the above,
We can conclude that
The coordinates to locate the end of the hiking trail is: (0.2, -0.8)

Question 34.
What are the coordinates of the information center? Explain.
Answer:
The given coordinate plane is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 56.5
Now,
From the given coordinate plane,
We can observe that
The coordinates of the Information center are: (-0.2, 0.7)
Hence, from the above,
We can conclude that
The coordinates of the Information center are: (-0.2, 0.7)

Question 35.
What are the coordinates of the point that is a reflection across the x-axis of the pond?
Answer:
From Question 31,
We can observe that,
The location of the pond in the coordinate plane is: (-0.7, -0.2)
Now,
The given coordinate plane is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 56.5
Now,
We know that,
The reflection of (x, y) across the x-axis is: (x, -y)
So,
The coordinates of the point that is a reflection across the x-axis of the pond are: (-0.7, 0.2)
Hence, from the above,
We can conclude that
The coordinates of the point that is a reflection across the x-axis of the pond are: (-0.7, 0.2)

Question 36.
Use Structure Which picnic areas are located at points that are reflections of each other across one of the axes of the coordinate plane?
Answer:
The given coordinate plane is:
Envision Math Common Core 6th Grade Answers Topic 2 Integers and Rational Numbers 56.5
Now,
From the given coordinate plane,
We can observe that
The coordinates of Picnic area 1 are: (-0.8, 0.6)
The coordinates of Picnic area 2 are: (0.6, 0.9)
The coordinates of Picnic area 3 are: (0.6, 0.3)
The coordinates of Picnic area 4 are: (-0.8, -0.6)
Now,
We know that,
The reflection of (x, y) across the x-axis is: (x, -y)
The reflection of (x, y) across the y-axis is: (-x, y)
Hence, from the above,
We can conclude that
Picnic area 4 is the reflection of Picnic area 1 across the x-axis

Assessment Practice

Question 37.
Graph and label each point on the coordinate plane at the right.
A. (\(\frac{3}{4}\), -1\(\frac{1}{2}\))
B. (-2.75, -2.25)
C. (0, 2\(\frac{1}{4}\))
D. (-1.75, 2)
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 58.1
Answer:
The given points are:
A. (\(\frac{3}{4}\), -1\(\frac{1}{2}\))
B. (-2.75, -2.25)
C. (0, 2\(\frac{1}{4}\))
D. (-1.75, 2)
Hence,
The representation of the given points in the coordinate plane are:

3-Act Mathematical Modeling: The Ultimate Throw

3-ACT MATH

The ULTIMATE THROW

ACT 1

Question 1.
After watching the video, what is the first question that comes to mind?
Answer:

Question 2.
Write the Main Question you will answer.
Answer:

Question 3.
Make a prediction to answer this Main Question.
The person who threw the flying disc farther is Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 65
Answer:

Question 4.
Construct Arguments Explain how you arrived at your prediction.
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 65.1
Answer:

ACT 2

Question 5.
What information in this situation would be helpful to know? How would you use that information?
Answer:

Question 6.
Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
Answer:

Question 7.
A Model with Math
Represent the situation using mathematics. Use your representation to answer the Main Question.
Answer:

Question 8.
What is your answer to the Main Question? Does it differ from your prediction? Explain.
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 65.6
Answer:

АСТ 3

Question 9.
Write the answer you saw in the video.
Answer:

Question 10.
Reasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 65.8
Answer:

Question 11.
Make Sense and Persevere Would you change your model now that you know the answer? Explain.
Answer:

Reflect

Question 12.
Model with Math
Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
Answer:

Question 13.
Make Sense and Persevere When did you struggle most while solving the problem? How did you overcome that obstacle?
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 65.9
Answer:

SEQUEL

Question 14.
Reasoning Suppose each person walks to the other person’s disc. They throw each other’s discs toward the starting point. Where do you think each disc will land?
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 65.10
Answer:

Lesson 2.5 Find Distances on the Coordinate Plane

Solve & Discuss It!

Graph the points on the coordinate plane below. What picture do you make when you connect the points in order?
(3, 3), (0, 0), (-4,-4), (-9,0), (-4, 4), (0, 0), (3, -3), (3, 3)
Name a pair of points that are the same distance from the x-axis. Explain your choice.
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 66.1
Answer:
The given points are:
(3, 3), (0, 0), (-4,-4), (-9,0), (-4, 4), (0, 0), (3, -3), (3, 3)
Now,
Represent the given points in the coordinate plane and connect the points in the given order
So,
The representation of the given points in the coordinate plane is:

Now,
From the given coordinate plane,
We can observe that
When we connect the given points in order,
The picture you made looks like a composite figure of a rhombus and a triangle
Now,
The pair of points that are the same distance from the x-axis is:
(3, 3), (3, -3); and (-4, 4), (-4, -4)
Hence, from the above,
We can conclude that
a. When we connect the given points in order,
The picture you made looks like a composite figure of a rhombus and a triangle
b. The pair of points that are the same distance from the x-axis is:
(3, 3), (3, -3); and (-4, 4), (-4, -4)

Use Structure How can you use the structure of the grid to find a pair of points that are the same distance from the x-axis?
Answer:
Derived from the Pythagorean Theorem, the distance formula is used to find the distance between two points in the plane. The Pythagorean Theorem, a²+b²=c² is based on a right triangle where a and b are the lengths of the legs adjacent to the right angle, and c is the length of the hypotenuse.

Focus on math practices
Use Structure How can you use the coordinate plane to find the total length of the picture you graphed?
Answer:
We know that,
The “Linear distance” between the two points is the square root of the sum of the squared values of the x-axis distance and the y-axis distance
Ex:
The distance between (3,2) and (7,8) is \(\sqrt{52}\), or approximately 7.21 units.

? Essential Question How can you find the distance between two points on a coordinate plane?
Answer:
We know that,
The distance between two points by using the distance formula, which is an application of the Pythagorean theorem. We can rewrite the Pythagorean theorem as
d=√(x2-x1)²+(y2-y1
to find the distance between any two points.

Try It!
What is the distance from the school to the playground? Explain how you used absolute values to find the distance?

Answer:
The given map is:

Now,
From the given coordinate plane,
We can observe that
The coordinates at which the school is located are: (-4, 2)
The coordinates at which the playground is located are: (-4, 5)
Now,
Compare the given points with (x, a), and (x, b) respectively
We know that,
Let (x, a), and (y, b) are the two points
Now,
The absolute distance of (x, a) = | x + a|
The absolute distance of (y, b) = | y + b|
So,
The absolute distance between (x, a), and (y, b) = |x + a| + |y + b|
So,
The absolute distance between (-4, 2), and (-4, 5) = | -4 + 2| + |-4 +5|
= |-2| + |1|
= 2 + 1
= 3 units
Hence, from the above,
We can conclude that
The distance from the school to the playground is: 3 units

Convince Me! To find the distance from the school to the playground, do you add or subtract the absolute values of the y-coordinates? Explain.
Answer:
Let (x, a), and (y, b) are the two points
Now,
The absolute distance of (x, a) = | x + a|
The absolute distance of (y, b) = | y + b|
So,
The absolute distance between (x, a), and (y, b) = |x + a| + |y + b|
Hence, from the above,
We can conclude that
To find the distance from the school to the playground, we added the absolute values of the y-coordinates

Try It!
What is the total distance of the Coulters’ return trip after their day at the water park?

Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 67
Answer:
The given figure is:

Now,
From the given figure,
we can observe that
The total distance of the Coulters’ return trip after their day at the water park = (The absolute distance of Water park) + (The absolute distance of Coulters’ home)
Now,
We know that,
Let (x, a), and (y, b) are the two points
Now,
The absolute distance of (x, a) = | x + a|
The absolute distance of (y, b) = | y + b|
So,
The absolute distance between (x, a), and (y, b) = |x + a| + |y + b|
Now,
From the given points,
We can observe that the x-coordinates are the same for the Water park and the Coulters’ home
So,

Hence, from the above,
We can conclude that
The total distance of the Coulters’ return trip after their day at the Water Park is: 151 miles

Try It!
Point D is in Quadrant IV and is the same distance from point B as point A. What are the coordinates of point D?

Answer:

KEY CONCEPT
You can use absolute values to find distances between points on a coordinate plane.
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 67.2

Do You Understand?

Question 1.
?Essential Question How can you find the distance between two points on a coordinate plane?
Answer:
We know that,
The distance between two points by using the distance formula, which is an application of the Pythagorean theorem. We can rewrite the Pythagorean theorem as
d=√(x2-x1)²+(y2-y1
to find the distance between any two points.

Question 2.
Look for Relationships To find the distance between two points using their coordinates, when do you add their absolute values and when do you subtract them?
Answer:
Case 1:
Let the coordinates of points be (x, y), and (a, b)
So,
The absolute value of (x, y) is: |x + y|
The absolute value of (a, b) is: |a + b|
Hence, from the above,
We can conclude that
We will add the absolute values when both the coordinates of the given points are positive
Case 2:
Let the coordinates of points be (x, -y), and (-a, b)
So,
The absolute value of (x, -y) is: |x – y|
The absolute value of (-a, b) is: |b – a|
Hence, from the above,
We can conclude that
We will subtract the absolute values when any one of the coordinates of the given points are negative

Question 3.
Reasoning Can you use absolute value to find the distance between Li’s house and Tammy’s house in
Example 1? Explain.

Answer:
The given figure is:

Now,
From the given figure,
We can observe that
The coordinates at which Li’s House is located = (-4, -3)
The coordinates at which Tammy’s house is located = (2, 0)
Now,
We know that,
The absolute distance between two points is: |x + a| + |y + b|
Hence, from the above,
We can conclude that
You can use absolute value to find the distance between Li’s house and Tammy’s house that is present in Example 1

Do You Know How?

In 4-9, find the distance between each pair of points.

Question 4.
(-5, 2) and (-5, 6)
Answer:
The given points are: (-5, 2) , and (-5, 6)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The absolute distance between two points = |x + a| + |y + b|
So,
The distance between (-5, 2), and (-5, 6) = |-5 + 2| + |-5 + 6|
= 3 + 1
= 4 units
Hence, from the above,
We can conclude that
The distance between the given pair of points is: 4 units

Question 5.
(4.5, -3.3) and (4.5, 5.5)
Answer:
The given points are: (4.5, -3.3), and (4.5, 5.5)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The absolute distance between two points = |x + a| – |y + b|
So,
The distance between (4.5, -3.3), and (4.5, 5.5) = |4.5 – 3.3| – |4.5 + 5.5|
= |1.2 – 10|
= 8.8 units
Hence, from the above,
We can conclude that
The distance between the given pair of points is: 11.2 units

Question 6.
(5\(\frac{1}{2}\), -7\(\frac{1}{2}\)) and (5\(\frac{1}{2}\), -1\(\frac{1}{2}\))
Answer:
The given points are: (5\(\frac{1}{2}\), -7\(\frac{1}{2}\)) and (5\(\frac{1}{2}\), -1\(\frac{1}{2}\))
Now,
Convert the coordinates of the given points into decimal numbers
So,
The given points are: (5.5, -3.5), and (5.5, -1.5)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The absolute distance between two points = |x + a| – |y + b|
So,
The distance between (5.5, -3.5), and (5.5, -1.5) = |5.5 – 3.5| – |5.5 – 1.5|
= |2 – 4|
= 2 units
Hence, from the above,
We can conclude that
The distance between the given pair of points is: 2 units

Question 7.
(-2\(\frac{1}{4}\), -8 ) and (7\(\frac{3}{4}\), -8)
Answer:
The given points are: (-2\(\frac{1}{4}\), -8 ) and (7\(\frac{3}{4}\), -8)
Now,
Convert the coordinates of the given points into a decimal number
So,
The given points are: (-2.25, -8), and (7.75, -8)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The absolute distance between two points = |x + a| – |y + b|
So,
The distance between (-2.25, -8), and (7.75, -8) = |-2.25 – 8| – |7.75 – 8|
= |10.25 – 0.25|
= 10 units
Hence, from the above,
We can conclude that
The distance between the given pair of points is: 10 units

Question 8.
(5\(\frac{1}{4}\), -3\(\frac{1}{4}\)) and (5\(\frac{1}{4}\), -6\(\frac{1}{4}\))
Answer:
The given points are: (5\(\frac{1}{4}\), -3\(\frac{1}{4}\)) and (5\(\frac{1}{4}\), -6\(\frac{1}{4}\))
Now,
Convert the coordinates of the given points into a decimal number
So,
The given points are: (5.25, -3.25), and (5.25, -6.25)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The absolute distance between two points = |x + a| + |y + b|
So,
The distance between (5.25, -3.25), and (5.25, -6.25) = |5.25 – 3.25| + |5.25 – 6.25|
= 2 + 1
= 3 units
Hence, from the above,
We can conclude that
The distance between the given pair of points is: 3 units

Question 9.
(-1\(\frac{1}{2}\), –6\(\frac{1}{2}\)) and (-2\(\frac{1}{2}\), -6\(\frac{1}{2}\))
Answer:
The given points are: (-1\(\frac{1}{2}\), –6\(\frac{1}{2}\)) and (-2\(\frac{1}{2}\), -6\(\frac{1}{2}\))
Now,
Convert the coordinates of the given points into a decimal number
So,
The given points are: (-1.5, -6.5), and (-2.5, -6.5)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The absolute distance between two points = |x + a| + |y + b|
So,
The distance between (-1.5, -6.5), and (-2.5, -6.5) = |-1.5 – 6.5| – |-2.5 – 6.5|
= |8 – 9|
= 1 units
Hence, from the above,
We can conclude that
The distance between the given pair of points is: 1 units

Practice & Problem Solving

Scan for Multimedia

Leveled Practice In 10-15, find the distance between each pair of points.

Question 10.
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 70.1
Answer:
The given points are: (-2, 8), and (7, 8)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between two points = |x + a| + |y + b|
So,

Hence, from the above,
We can conclude that
The distance between the given points is: 21 units

Question 11.
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 70.2
Answer:
The given points are: (-6.1, -8.4), and (-6.1, -4.2)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between two points = |x + a| – |y + b|
Now,
Since the x-coordinates are the same, subtract the y-coordinates to find the distance between the given points
So,

Hence, from the above,
We can conclude that
The distance between the given points is: 4.2 units

Question 12.
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 70.3
Answer:
The given points are: (12\(\frac{1}{2}\), 3\(\frac{3}{4}\)), and (-4\(\frac{1}{2}\), 3\(\frac{3}{4}\))
Now,
Convert the coordinates of the given points into a decimal number
So,
The given points are: (12.5, 3.75), and (-4.5, 3.75)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between two points = |x + a| + |y + b|
Now,
Since y-coordinates are the same, add the x-coordinates to find the distance between the given points
So,

Hence, from the above,
We can conclude that
The distance between the given points is: 17 units

Question 13.
(-5, -3) and (-5, -6)
Answer:
The given points are: (-5, -3) , and (-5, -6)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The absolute distance between two points = |x + a| – |y + b|
So,
The distance between (-5, -3), and (-5, -6) = |-5 – 3| – |-5 – 6|
= |8 – 11|
= 3 units
Hence, from the above,
We can conclude that
The distance between the given pair of points is: 3 units

Question 14.
(-5.4, 4.7) and (0.6, 4.7)
Answer:
The given points are: (-5.4, 4.7), and (0.6, 4.7)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The absolute distance between two points = |x + a| – |y + b|
So,
The distance between (-5.4, 4.7), and (0.6, 4.7) = |-5.4 + 4.7| – |0.6 + 4.7|
= |-0.7 – 5.3|
= 6 units
Hence, from the above,
We can conclude that
The distance between the given pair of points is: 6 units

Question 15.
(7\(\frac{1}{2}\), -5\(\frac{3}{4}\)) and (7\(\frac{1}{2}\), -1\(\frac{1}{4}\))
Answer:
The given points are: (7\(\frac{1}{2}\), -5\(\frac{3}{4}\)) and (7\(\frac{1}{2}\), -1\(\frac{1}{4}\))
Now,
Convert the coordinates of the given points into a decimal number
So,
The given points are: (7.5, -5.75), and (7.5, -1.25)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The absolute distance between two points = |x + a| – |y + b|
So,
The distance between (7.5, -5.75), and (7.5, -1.25) = |7.5 – 5.75| – |7.5 – 1.25|
= |1.75 – 6.25|
= 4.5 units
Hence, from the above,
We can conclude that
The distance between the given pair of points is: 4.5 units

In 16-19, use the map at the right.
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 80.1

Question 16.
Find the distance from roller coaster 1 to the swings.
Answer:
The given map is:

Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 80.1
Now,
From the given map,
We can observe that
The coordinates of roller coaster 1 are: (-6, 7)
The coordinates of the swings are: (1, 7)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between the two points = |x + a| – |y – b|
So,
The distance from roller coaster 1 to the swings = |-6 + 7| – |1 + 7|
= |1 – 8|
= 7 units
Hence, from the above,
We can conclude that
The distance from roller coaster 1 to the swings is: 7 units

Question 17.
Find the distance from the Ferris wheel to the roller coaster 3.
Answer:
The given map is:
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 80.1
Now,
From the given map,
We can observe that
The coordinates of the Ferris Wheel are: (-6, 2)
The coordinates of roller coaster 3 are: (-6, -8)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between the two points = |x + a| + |y – b|
So,
The distance from Ferris Wheel to the roller coaster 3 = |-6 + 2| – |-6 – 8|
= |-4 + 14|
= 10 units
Hence, from the above,
We can conclude that
The distance from Ferris Wheel to the roller coaster 3 is: 10 units

Question 18.
Find the total distance from roller coaster 2 to roller coaster 3 and then to the water slide.
Answer:
The given map is:
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 80.1
Now,
From the given map,
We can observe that
The coordinates of roller coaster 2 are: (-6, -3)
The coordinates of roller coaster 3 are: (-6, -8)
The coordinates of the Water slide are: (8, -8)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between the two points = |x + a| + |y – b|
So,
The distance from Ferris Wheel to the roller coaster 3 = |-6 + 2| – |-6 – 8|
= |-4 + 14|
= 10 units
Hence, from the above,
We can conclude that
The distance from Ferris Wheel to the roller coaster 3 is: 10 units

Question 19.
Higher Order Thinking Is the distance from the merry-go-round to the water slide the same as the distance from the water slide to the merry-go-round? Explain.
Answer:
The given map is:
Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 80.1
Now,
From the given map,
We can observe that
The coordinates of merry-go-round are: (8, 2)
The coordinates of the Water slide are: (8, -8)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between the two points = |x + a| + |y – b|
So,
The distance from merry-go-round to the Water slide  = |8 + 2| – |8 – 8|
= |0 + 10|
= 10 units
So,
The distance from the Water slide to merry-go-round = |8 – 8| – |8 + 2|
= |0 – 10|
= 10 units
Hence, from the above,
We can conclude that
The distance from merry-go-round to the Water slide and the distance from the Water slide to merry-go-round are the same

In 20 and 21, use the coordinate plane at the right.

The graph shows the locations of point G and point H. Point J is graphed at (n, -3). The distance from point H to point J is equal to the distance from point H to point G.

Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 81.1

Question 20.
What is the distance from point H to point J?
Answer:
It is given that
The graph shows the locations of point G and point H. Point J is graphed at (n, -3). The distance from point H to point J is equal to the distance from point H to point G.
Now,
The given coordinate plane is:

Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 81.1
Now,
From the given coordinate plane,
We can observe that
The coordinates of G are: (0, -y)
It is given that
The coordinates of point J are: (n, -3)
Now,
Compare points H and J with (x, a), and (y, b)
Now,
We know that,
The distance between 2 points = | x + a| – |y + b|
So,
The distance between points H and J = |n – 6| – |n – 3|
= |n – 6 – n + 3|
= 3 units
Hence, from the above,
We can conclude that
The distance from point G to point H is: 3 units

Question 21.
What is the value of n?
Answer:
It is given that
The graph shows the locations of point G and point H. Point J is graphed at (n, -3). The distance from point H to point J is equal to the distance from point H to point G.
Now,
The given coordinate plane is:

Envision Math Common Core Grade 6 Answers Topic 2 Integers and Rational Numbers 81.1
Now,
From Exercise 20,
We know that,
The distance from point G to point H is: 3 units
Now,
|n – 6 + n – 3| = 3
|2n – 9| = 3
2n = 3 + 9
2n = 12
n = \(\frac{12}{2}\)
n = 6 units
Hence, from the above,
We can conclude that
The value of n is: 6 units

Question 22.
Use Structure Suppose a, b, and c are all negative numbers. How do you find the distance between points (a, b) and (a, c)?
Answer:
It is given that
a, b, and c are all negative numbers.
Now,
The given points are: (a, b), and (a, c)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between 2 points = |x + a| – |y + b|
So,
The distance between the given points = |-a + (-b)| – |-a + (-c)|
= |-a – b| – |-a – c|
= |-a – b + a + c|
= |c – b| units
Hence, from the above,
We can conclude that
The distance between points (a, b), and (a, c) is: |c – b| units

Question 23.
A scientist graphed the locations of the epicenter of an earthquake and all of the places where people reported feeling the earthquake. She positioned the epicenter at (-1, 8) and the farthest location reported to have felt the quake was positioned at (85, 8). If each unit on the graph represents 1 mile, how far from its epicenter was the earthquake felt?
Answer:
It is given that
A scientist graphed the locations of the epicenter of an earthquake and all of the places where people reported feeling the earthquake. She positioned the epicenter at (-1, 8) and the farthest location reported to have felt the quake was positioned at (85, 8) and each unit on the graph represents 1 mile
Now,
The given points are: (-1, 8), (85, 8)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between 2 points = |x + a| – |y + b|
So,
The distance between the given points = |-1 + 8| – |85 + 8|
= |7 – 93|
= 86 miles
Hence, from the above,
We can conclude that
The earthquake felt 86 miles away from its epiccenter

Question 24.
The rectangle ABCD shown on the coordinate plane represents an overhead view of a piece of land. Each unit represents 1,000 feet. What are the dimensions of the rectangular piece of land, in feet?
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 82.1
Answer:
It is given that
The rectangle ABCD shown on the coordinate plane represents an overhead view of a piece of land. Each unit represents 1,000 feet
Now,
The given coordinate plane is:
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 82.1
Now,
From the given coordinate plane,
We can observe that
The dimensions of the rectangle present in the given coordinate plane are:
A (0, 0), B (0, 4), C (5, 5), and D (5, 0)
So,
The dimensions of the rectangular piece of land, in feet are:
A (0, 0), B (0, 4,000), C (5,000, 5,000), and D (5,000, 0)
Hence, from the above,
We can conclude that
The dimensions of the rectangular piece of land, in feet are:
A (0, 0), B (0, 4,000), C (5,000, 5,000), and D (5,000, 0)

Assessment Practice

Question 25.
You are given the following ordered pairs.
(3.5, -1) (-1.5, 3) (-3, 3) (3.5, 2.5) (-1.5, -1.5)
PART A
Graph the ordered pairs on the coordinate plane.
Answer:
The given ordered pairs are:
(3.5, -1), (-1.5, 3), (-3, 3), (3.5, 2.5), (-1.5, -1.5)
Hence,
The representation of the given ordered pairs in the coordinate plane is:

PART B
Find the two ordered pairs on the coordinate plane that are 4.5 units apart.
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 82.3
Answer:
The given ordered pairs are:
A (3.5, -1), B(-1.5, 3), C(-3, 3), D(3.5, 2.5), E(-1.5, -1.5)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between 2 points = |x – a| + |y + b|
So,
The distance of AB = 6.41 units
The distance of AC = 7.63 units
The dsitance of AD = 3.5 units
The distance of AE = 5.02 units
The distance of BC = 4.5 units
The distance of BD = 5.02 units
The distance of BE = 4.5 units
Hence, from the above,
We can conclude that
The two ordered pairs on the coordinate plane that are 4.5 units apart is:
B(-1.5, 3), and C (-3, 3); B(-1.5, 3), and E (-1.5, -1.5)

Lesson 2.6 Represent Polygons on the Coordinate Plane

Solve & Discuss It!

ACTIVITY

Draw a polygon with vertices at A(-1, 6), B(-7, 6), C(-7, -3), and D(-1, -3). Then find the perimeter of the polygon.
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 82.5
Answer:
The vertices of a given polygon are:
A(-1, 6), B(-7, 6), C(-7, -3), and D(-1, -3)
So,
The representtaion of the given vertices of a polygon in the coordinate plane are:

Now,
We know that,
The “Perimeter” of a figure is the sum of all of the side lengths of a given figure
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance betwee 2points = |x + a| + |y + b|
So,
The distace of AB = |-1 + 6| + |-7 + 6|
= |5 + 1|
= 6 units
The distance of BC = |-7 + 6| + |-7 – 3|
= |1 – 10|
= 9 units
The distace of CD = |-7 – 3| + |-1 – 3|
= |10 – 4|
= 6 units
The distace of DA = |-1 – 3| + |-1 + 6|
= |5 + 4|
= 9 units
So,
The perimeter of the given polygon = 6 + 6 + 9 + 9
= 30 units
Hence, from the above,
We can conclude that
The perimeter of the given polygon is: 30 units

Use Structure How can you use the coordinate plane to draw the polygon and find its perimeter?
Answer:
You can use ordered pairs to represent vertices of polygons. To draw a polygon in a coordinate plane, plot and connect the ordered pairs
To find the perimeter of a regular polygon, we take the length of each side, , and multiply it by the number of sides

Focus on math practices
Construct Arguments What type of polygon did you draw? Use a definition to justify your answer.
Answer:
The representation of the vertices of the given polyon is:

Now,
From the given coordinate plane,
We can observe that
The opposite side lengths are equal and all the angle measures are 90°
Now,
We know that,
The polygon that has the same opposite side lengths is called as “Rectangle”
Hence, from the above,
We can conclude that
The type of polyon did you draw is: Rectangle

Visual Learning

? Essential Question How is distance used to solve problems about polygons in a coordinate plane?
Answer:
To find the distance between two points with the same x-coordinates, subtract their y-coordinates. To find the distance between two points with the same y-coordinates, subtract their x-coordinates
To find the distance between the different x and y-coordinates, subtract the x-coordinates of the 2 points and y-coordinates of the 2 points and add both the values

Try It!
The archaeologist later decides to extend the roped-off area so that the new perimeter goes from A to B to the food tent to the working tent and then back to A. How much rope does she need now?
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 85.5
Answer:
It is given that
The archaeologist later decides to extend the roped-off area so that the new perimeter goes from A to B to the food tent to the working tent and then back to A
Now,
The given figure is:

Now,
From the given figure,
We can observe that
The coordinates of A are: (-4, 6)
The coordinates of B are: (2, 6)
The coordinates of Food Tent are: (2, -2)
The coordinates of Working Tent are: (-4, -2)
Now,
Compare the given points with (x, a), and (y, b)
So,

Hence, from the above,
We can concldue that
The length of rope the archaeologist needs now is: 28 meters

Convince Me! How could you use the formula for the perimeter of a rectangle to find the perimeter of the larger rectangle using two of the distances?
Answer:
We know that,
The perimeter of a rectangle = 2(l + w)
Now,
If the perimeter and the length of a rectangle are known, then
Width = P/2 – l,
where,
l = length, w = width, and P = perimeter of the rectangle.
Now,
If the perimeter and the width are known, then
Length(L) = P/2 – w.

Try It!
The rancher needs to replace the fence for the holding pen for the horses. How much fencing does he need?
The rancher needs Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 85.6 yards of fencing.
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 85.7
Answer:
It is given that
The rancher needs to replace the fence for the holding pen for the horses
Now,
The given figure is:
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 85.7
Now,
From the given figure,
The coordinates of the fence that is used for the holding pen for the houses is:
A (4.5, 14), B (15, 14), C (15, 8), D (10, 8), E (10, 3), F (15, 3), G (15, -2.25), and H (4.5, -2.25)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between 2 points = |x + a| + |y + b|
Now,
The distance of AB = |4.5 – 15| = 11.5 units
The distance of BC = |14 – 8| = 6 units
The distance of CD = |15 – 10| = 5 units
The distance of DE = |8 – 3| = 5 units
The distance of EF = |10 – 15| = 5 units
The distance of FG = | 3 + 2.25| = 5.25 units
The distance of GH = |15 – 4.5| = 11.5 units
The distance of HA = |14 + 2.25| = 16.25 units
Now,
We know that,
To find the length of fencing, we have to find the perimeter of the given polyon
So,
The length of the fencing that Rancher needed = 11.5 + 6 + 5 + 5 + 5 + 5.25 + 11.5 + 16.25
= 65.5 yards
Hence, from the above,
We can conclude that
The rancher needs 65.5 yards of fencing

Try It!
Joaquin says that quadrilateral ADEF is a square. Is he correct? Explain.
Answer:
It is given that
Joaquin says that quadrilateral ADEF is a square
Now,
The given figure is:

Now,
From the given figure,
We can observe that
The coordinates of A are: (-5, 2)
The coordinates of D are: (7, 2)
The coordinates of E are: (7, -8)
The coordinates of F are: (-5, -8)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between 2 points = |x + a| + |y + b|
Now,
The distance of AD = |-5 + 2| + |7 + 2|
= 12 units
The distance of DE = |7 + 2| + |7 – 8|
= 10 units
The distance of EF = |7 – 8| + |-5 – 8|
= 12 units
The disatnce of FD = |-5 – 8| – |-5 + 2|
= 10 units
Now,
From the given side lengths,
We can observe that
The opposite side lengths are equal
Now,
we know that,
The rectangle has the opposite side lengths
Hence, from the above,
We can conclude that
Joaquin is not correct

KEY CONCEPT
You can represent polygons on a coordinate plane and solve problems by using absolute values to find side lengths.
Add or subtract absolute values to find the length of each side.
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 88.1
AB: |-3| + |2| = 3 + 2 = 5 units
BC: |4| – |2| = 4 – 2 = 2 units
CD: |-3| + |2| = 3 + 2 = 5 units
DA: |4| – |2| = 4 – 2 = 2 units

Do You Understand?

Question 1.
?Essential Question How is distance used to solve problems about polygons in a coordinate plane?
Answer:
To find the distance between two points with the same x-coordinates, subtract their y-coordinates. To find the distance between two points with the same y-coordinates, subtract their x-coordinates
To find the distance between the different x and y-coordinates, subtract the x-coordinates of the 2 points and y-coordinates of the 2 points and add both the values

Question 2.
Reasoning In Example 1, why do you add absolute values to find the distance from A to B but subtract absolute values to find the distance from B to C?
Answer:
From Example 1,
We know that,
The coordinates of A, B, and C are:
A (-4, 6), B (2, 6), and C (2, 1)
Now,
We know that,
The absolute distance between given points = |x + a| + |y + b|
Now,
When the y-coordinates are the same,
The absolute distance between 2 points = |x + y|
So,
The absolute distance from A and B is positive because the x-coordinates move from left to right (or) vice-versa
When the x-coordinates are the same,
The absolute distance between 2 points = |a – b|
So,
The absolute distance from B and C is negative because the y-coordinates move from top to bottom (or) vice-versa

Question 3.
Construct Arguments Could you add or subtract the absolute values of coordinates to find the length of the diagonal AC of rectangle ABCD in Example 1? Explain.
Answer:
From Example 1,
We know that,
A (-4, 6), B (2, 6), C (2, 1), and D (-4, 1)
Now,
We know that,
The distance between 2 points = |x + a| + |y + b|
So,
The distance of the diagonal AC of rectangle ABCD is solved by the addition of the absolute values of the coordinates of A and C
Hence, from the above,
We can conclude that
We add the absolute values of coordinates to find the length of the diagonal AC of rectangle ABCD in Example 1

Do You Know How?

Question 4.
Find the perimeter of rectangle MNOP with vertices M(-2,5), N(-2, -4), 0(3,-4), and P(3, 5).
Answer:
The given vertices of rectangle MNOP are:
M(-2,5), N(-2, -4), 0(3,-4), and P(3, 5)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between 2 points = |x + a| + |y + b|
Now,
The disatnce of MN = |-2 + 5| + |-2 – 4|
= |3 + 6|
= 9 units
The distance of NO = |-2 – 4| + |3 – 4|
= |6 – 1|
= 5 units
The distance of OP = |3 – 4| + |3 + 5|
= |1 + 8|
= 9 units
The distance of PM = |3 + 5| + |-2 + 5|
= |8 – 3|
= 5 units
Now,
The perimeter of the rectangle MNOP = MN + NO + OP + PM
= 9 + 5 + 9 + 5
= 28 units
Hence, from the above,
We can conclude that
The perimeter of the given rectangle MNOP is: 28 units

Question 5.
Jen draws a polygon with vertices E(-2, 3.5), F(3, 3.5), G(3, -1.5), and H(-2, -1.5). Is EFGH a square? Justify your answer.
Answer:
It is given that
Jen draws a polygon with vertices E(-2, 3.5), F(3, 3.5), G(3, -1.5), and H(-2, -1.5)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between 2 points = |x + a| + |y + b|
Now,
The distance of EF = |3 + 2|
= 5 units
The distance of FG = |3.5 + 1.5|
= 5 units
The distance of GH = |3 + 2|
= 5 units
The distance of HE = |3.5 + 1.5|
= 5 units
Now,
We know that,
The “Square” has all the same side lengths
Hence, from the above,
We can conclude that
EFGH is a square

Question 6.
Square ABCD has vertices A(-4.5, 4), B(3.5, 4), C(3.5, -4), and D(-4.5, -4). What is the area of square ABCD?
Answer:
It is given that
Square ABCD has vertices A(-4.5, 4), B(3.5, 4), C(3.5, -4), and D(-4.5, -4)
Now,
We know that,
The square has the same side lengths
So,
The distance between any 2 points is sufficient to find the area of square ABCD
Now,
We know that,
The area of a square = Side²
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between 2 points = |x + a| + |y + b|
Now,
The distance of AB =|3.5 + 4.5|
= 8 units
So,
The area of square ABCD = Side²
= 8²
= 64 units²
Hence, from the above,
We can conclude that
The area of square ABCD is: 64 units²

Practice & Problem Solving

Scan for Multimedia

Leveled Practice
In 7 and 8, find the perimeter of each rectangle.

Question 7.
Rectangle JKLM: J(-3, 8), K(-3,-1), L(4, -1), M(4,8)
JK = |8| + |-1| = Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 90
KL = |–3| + |4| = Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 90
Perimeter = Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 90 units
Answer:
The given vertices of Rectangle JKLM are: J(-3, 8), K(-3,-1), L(4, -1), M(4,8)
Now,
We know that,
The side lengths of the parallel sides are the same
So,
In Rectangle JKLM,
JK = LM and KL = MJ
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between 2 points = |x + a| + |y + b|
Now,

Hence, from the above,
We can conclude that
The perimeter of Rectangle JKLM is: 16 units

Question 8.
Rectangle WXYZ: W(-3, -2), X(4, -2), Y(4, -5), Z(-3, -5)
WX = |-3| + |4| = Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 90
XY = |–5| – |-2| = Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 90
Perimeter = Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 90 units
Answer:
The given vertices of Rectangle WXYZ are: W(-3, -2), X(4, -2), Y(4, -5), Z(-3, -5)
Now,
We know that,
The side lengths of the parallel sides are the same
So,
In Rectangle WXYZ,
WX = YZ and XY = ZW
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between 2 points = |x + a| + |y + b|
Now,

Hence, from the above,
We can conclude that
The perimeter of Rectangle JKLM is: 10 units

Question 9.
Triangle JKL has vertices J(0, 0), K(5, 0), and L(0, -3). Is triangle JKL equilateral? Justify your answer.
Answer:
It is given that
Triangle JKL has vertices J(0, 0), K(5, 0), and L(0, -3)
Now,
We know that,
The “Equilateral Triangle” is a triangle that has the same side lengths
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between 2 points = |x + a| + |y + b|
So,
The distance of JK = |0 + 0| + |5 + 0|
= 5 units
The distance of KL = |5 + 0| + |0 – 3|
= 8 units
The distance of LJ = |0 – 3| + |0 + 0|
= 3 units
So,
From the given side lengths,
JK ≠ L ≠ LJ
Hence, from the above,
Hence, from the above,
We can conclude that
Triangle JKL is not an equilateral triangle

Question 10.
Polygon WXYZ has vertices W(-1.5, 1.5), X(6, 1.5), Y(6, -4.5), and Z(-1.5, -4.5). Is WXYZ a rectangle? Justify your answer.
Answer:
It is given that
Polygon WXYZ has vertices W(-1.5, 1.5), X(6, 1.5), Y(6, -4.5), and Z(-1.5, -4.5)
Now,
The representation of the given polygon WXYZ with its side lengths is:

Now,
From the above figure,
We can observe that
The side lengths of the opposite sides are equal
Now,
We know that,
A “Rectangle” has the same opposite side lengths
Hence, from the above,
We can conclude that
Polygon WXYZ is a rectangle

Question 11.
What are the perimeter and area of rectangle ABCD?
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 91
Answer:
The given figure is:
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 91
Now,
From the given figure,
We can observe that
The coordinates of the rectangle ABCD are:
A (-3, -7), B (4, -7), C (4, 6), and D (-3, 6)
Now,
We know that,
In a Rectangle,
The side lengths of the opposite sides are the same
So,
AB = CD and BC = DA
Now,
We know that,
The perimeter of a rectangle = 2 (l + w)
The area of a rectangle = l × w
Now,
The distance of AB = |-3 – 7| + |4 – 7|
= |-10 + 3|
= 7 units
The disatnce of BC = |4 – 3| + |6 + 6|
= |1 + 12|
=13 units
So,
The perimeter of Rectangle ABCD = 2 (AB + BC)
= 2 (7 + 13)
= 2 (20)
= 40 units
The area of Rectangle ABCD = AB × CD
= 7 × 13
= 91 units²
Hence, from the above,
We can conclude that
The perimeter of Rectangle BACD is: 40 units
The area of Rectangle BACD is: 91 units²

Question 12.
Mike used a coordinate plane to design the patio shown at the right. Each unit on the grid represents 1 yard. To buy materials to build the patio, Mike needs to know its perimeter. What is the perimeter of the patio?
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 92
Answer:
It is given that
Mike used a coordinate plane to design the patio shown at the right. Each unit on the grid represents 1 yard.
Now,
The given figure is:
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 92
Now,
From the given figure,
We can observe that
The coordinates of the given patio are:
A (3, 2), B (3, 4), C (7, 4), D (7, 7), E (9, 7), and F (9, 2)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between 2 points = |x + a| + |y + b|
Now,
The distance of AB = |4 – 2| = 2 units
The disatnce of BC = |7 – 3| = 4 units
The distance of CD = |7 – 4| = 3 units
The distance of DE = |9 – 7| = 2 units
The distance of EF = |7 – 2| = 5 units
The distance of FA = |9 – 3| = 6 units
Now,
The perimeter of the patio = 2 + 4 + 3 + 2 + 5 + 6
= 22 units
Hence, from the above,
We can conclude that
The perimeter of the patio is: 22 units

Question 13.
Jordan started at her home at point H. She ran to the bank (B), the library (1), the post office (P), the café (C), her school (S), and then back to her home, as shown. The coordinates represent the position, in miles, of each of these locations with respect to the center of town, which is located at the origin. What is the total distance that Jordan ran?
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 92.1
Answer:
It is given that
Jordan started at her home at point H. She ran to the bank (B), the library (1), the post office (P), the café (C), her school (S), and then back to her home, as shown. The coordinates represent the position, in miles, of each of these locations with respect to the center of town, which is located at the origin
Now,
The given figure is:
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 92.1
Now,
From the given figure,
We can observe that
The coordinates are:
H (\(\frac{-3}{4}\), \(\frac{1}{2}\)), B (1, \(\frac{1}{2}\)), L (1, –\(\frac{1}{4}\)), P (\(\frac{1}{2}\), –\(\frac{1}{4}\)), C (\(\frac{1}{2}\), –\(\frac{3}{4}\)), S (-\(\frac{3}{4}\), –\(\frac{3}{4}\))
Now,
The representation of the given coordinates with its side lengths is:

So,
The total distance that Jordan ran = 1.75 + 0.75 + 0.70 + 1.25 + 1.25
= 5.7 miles
Hence, from the above,
We can conclude that
The total distance that Jordan ran is: 5.7 miles

Question 14.
Use Structure Ana drew a plan for a rectangular piece of material that she will use for a quilt. The vertices are (-1.2, -3.5), (-1.2, 4.4), and (5.5, 4.4). What are the coordinates of the fourth vertex?
Answer:
It is given that
Ana drew a plan for a rectangular piece of material that she will use for a quilt. The vertices are (-1.2, -3.5), (-1.2, 4.4), and (5.5, 4.4)
Now,
Let the vertices be A (-1.2, -3.5), B (-1.2, 4.4), C (5.5, 4.4), and D (x, y)
Now,
We know that,
In a Rectangle,
The opposite side lengths are the same
So,
For the given quilt,
AB = CD and BC = DA
Now,
If you observe carefully the vertices given, you will see that there are two vertex with same coordinates in x axis: (-1.2, -3.5), and (-1.2, 4.4),
And one vertex has other coordinate in x axis, it is: (5.5, 4.4)
So,
To form a rectangle, you need a fourth vertex that will have same coordinate in x than (5.5, 4.4) and same coordinate in y axis as (-1.2, -3.5),
So,
The fourth vertex coordinate is (5.5, -3.5)
Hence, from the above,
We can conclude that
The coordinates of the fourth vertex are: (5.5, -3.5)

Question 15.
Mr. Janas is building a pool in his backyard. He sketches the rectangular pool on a coordinate plane. The vertices of the pool are A(-5, 7), B(1,7), C(1, -1), and D(-5, -1). If each unit represents 1 yard, how much area of the backyard is needed for the pool?
Answer:
It is given that
Mr. Janas is building a pool in his backyard. He sketches the rectangular pool on a coordinate plane. The vertices of the pool are A(-5, 7), B(1,7), C(1, -1), and D(-5, -1)
Now,
The representation of the given vertices of a rectangular pool and its side lengths in the coordinate plane is:

Now,
We know that,
In a Rectangle,
The opposite side lengths have the same value
Now,
We know that,
The area of a Rectangle = Length × Width
So,
The area of the rectangular swimming pool = AB × BC
= 6 × 8
= 48 yard²
Hence, from the above,
We can conclude that
The area of the given rectangular swimming pool is: 48 yard²

Question 16.
Vocabulary Why is absolute value used to find distances on a coordinate plane?
Answer:
We know that the absolute value of a point on the number line (or the absolute value of the coordinate of a tree in your backyard) tells you the distance between that point (or tree) and the number zero at origin of your coordinate system

Question 17.
Higher Order Thinking A square on a coordinate plane has one vertex at (-0.5, -2) and a perimeter of 10 units. If all of the vertices are located in Quadrant III, what are the coordinates of the other three vertices?
Answer:
It is given that
A square on a coordinate plane has one vertex at (-0.5, -2) and a perimeter of 10 units and all of the vertices are located in Quadrant III
Now,
We know that,
The perimeter of square is:

where,
b is the length side of the square
Now,
According to the given information,

Now,
Divide the above equation by 4 both sides
So,

Now,
A (-0.5, -2) —> The given coordinates of one vertex
Now,
we know that
All of the vertices are located in Quadrant III
So,
The other three vertices are located at the left and down of vertex A
Now,
The coordinate of vertex B located at 2.5 units at left of vertex A
The coordinate of vertex C located at 2.5 units at left and 2.5 units down of vertex A
The coordinate of vertex D located at 2.5 units down of vertex A
So,
B(-0.5-2.5,-2) —–> B(-3, -2)
C(-0.5-2.5,-2-2.5) —–> C(-3, -4.5)
D(-0.5,-2-2.5) —–> D(-0.5, -4.5)
Hence, from the above,
We can conclude that
The coordinates of the other three vertices are:
B (-3, -2), C (-3, -4.5), and D (-0.5, -4.5)

Assessment Practice

Question 18.
You are given the following points on a coordinate plane: A(-1\(\frac{1}{2}\), –\(\frac{1}{2}\)), B(-1\(\frac{1}{2}\), -3), and C (4, -3).
PART A
Using absolute value, find the distance (number of units) between points A and B.
Answer:
The given points are:
A(-1\(\frac{1}{2}\), –\(\frac{1}{2}\)), B(-1\(\frac{1}{2}\), -3), and C (4, -3)
Now,
The representation of the given coordinates along with its side lengths in the coordinate plane is:

So,
From the above figure,
We can observe that
The distance between points A and B is: 2.5 units
Hence, from the above,
We can conclude that
The distance between points A and B is: 2.5 units
PART B
Select all the coordinates that are 8 units from point C.
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 94 (12, -3)
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 94 (12, -11)
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 94 (4, -3)
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 94 (-4, -3)
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 94 (4,-11)
Answer:
It is given that
The coordinates of Point C are: C (4, -3)
Now,
When the coordiantes of point C moves 8 units towards right,
C (4 + 8, -3) = C (12, -3)
When the coordiantes of point C moves 8 units towards left,
C (4 – 8, -3) = C (-4, -3)
Hence, from the above,
We can conclude that
All the coordinates that are 8 units from point C are:

Topic 2 REVIEW

? Topic Essential Question
What are integers and rational numbers? How are points graphed on a coordinate plane?
Answer:
An “Integer” can be written as a fraction by giving it a denominator of one. So, any integer is a rational number
“Rational numbers” are those numbers that are integers and can be expressed in the form of \(\frac{x}{y}\) where both numerator and denominator are integers
To graph or plot points, we use two perpendicular lines called the x-axis and the y-axis. The horizontal number line is the x-axis and the vertical line is the y-axis.  Every point in the coordinate plane is represented by an ordered pair of x and y coordinates.

Vocabulary Review

Complete each definition and then provide an example of each vocabulary word.

Vocabulary

absolute value
opposite
ordered pair
rational number

Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 95.1
Answer:

Use Vocabulary in Writing

Explain how the points A(9, –\(\frac{2}{5}\)) and B(9, \(\frac{2}{5}\)) are related. Use vocabulary words in your explanation.
Answer:
The given points are:
A(9, –\(\frac{2}{5}\)) and B(9, \(\frac{2}{5}\))
Now,
Compare the given points with (x, a), and (y, b)
Now,
From the given points,
We can observe that
a. The x-coordinates are the same
b. The y-coordinates are opposite to each other
c. The y-coordinates and x-coordinates are rational numbers since they can be written in the form of \(\frac{x}{y}\)

Concepts and Skills Review

LESSON 2.1 Understand Integers

Quick Review
Integers are all of the counting numbers, their opposites, and 0. Opposites are integers located on opposite sides of 0 and the same distance from 0 on a number line.

Example
For each point on the number line, write the integer and its opposite.
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 150
A: 4,-4 B: 0,0 C: -6,6
The opposite of the opposite of a number is the number itself.

Practice
For each point on the number line, write the integer and its opposite.
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 151
Question 1.
A
Answer:
The given number line is:
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 151
Now,
From the given number line,
We can observe that
Each line in the given number line represents 1 unit
Now,
We know that,
The opposite value of a is: -a
The opposite value of -a is: a
Hence, from the above,
We can conclude that
The value of A from the given number line is: 3
The opposite value of A is: -3

Question 2.
B
Answer:
The given number line is:
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 151
Now,
From the given number line,
We can observe that
Each line in the given number line represents 1 unit
Now,
We know that,
The opposite value of a is: -a
The opposite value of -a is: a
Hence, from the above,
We can conclude that
The value of B from the given number line is: -1
The opposite value of B is: 1

Question 3.
C
Answer:
The given number line is:
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 151
Now,
From the given number line,
We can observe that
Each line in the given number line represents 1 unit
Now,
We know that,
The opposite value of a is: -a
The opposite value of -a is: a
Hence, from the above,
We can conclude that
The value of C from the given number line is: 6
The opposite value of C is: -6

Question 4.
D
Answer:
The given number line is:
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 151
Now,
From the given number line,
We can observe that
Each line in the given number line represents 1 unit
Now,
We know that,
The opposite value of a is: -a
The opposite value of -a is: a
Hence, from the above,
We can conclude that
The value of D from the given number line is: -7
The opposite value of D is: 7

Question 5.
E
Answer:
The given number line is:
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 151
Now,
From the given number line,
We can observe that
Each line in the given number line represents 1 unit
Now,
We know that,
The opposite value of a is: -a
The opposite value of -a is: a
Hence, from the above,
We can conclude that
The value of E from the given number line is: -5
The opposite value of E is: 5

Question 6.
F
Answer:
The given number line is:
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 151
Now,
From the given number line,
We can observe that
Each line in the given number line represents 1 unit
Now,
We know that,
The opposite value of a is: -a
The opposite value of -a is: a
Hence, from the above,
We can conclude that
The value of F from the given number line is: 1
The opposite value of F is: -1

LESSON 2.2 Represent Rational Numbers on the Number Line

Quick Review
Rational numbers are numbers that can be written as a quotient \(\frac{a}{b}\), where a and b are integers and b does not equal 0. You can use number lines to represent, compare, and order rational numbers.

Example
Compare and order -0.1, 0.75, and –\(\frac{1}{4}\) from least to greatest.
Plot the numbers on a number line.
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Number 159
so –\(\frac{1}{4}\) < -0.1 < 0.75, and their order from least to greatest is –\(\frac{1}{4}\), -0.1, 0.75.

Practice
In 1-3, plot each rational number on the number line.

Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 160

Question 1.
\(\frac{3}{4}\)
Answer:
The given point is: \(\frac{3}{4}\)
Now,
The given number line is:
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 160
Now,
From the given number line,
We can observe that
Each line in the given number line represents 0.1 unit
Now,
The representation of \(\frac{3}{4}\) in the decimal form is: 0.75
Hence,
The representation of the given point on the given number line is:

Question 2.
–\(\frac{2}{5}\)
Answer:
The given point is: –\(\frac{2}{5}\)
Now,
The given number line is:
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 160
Now,
From the given number line,
We can observe that
Each line in the given number line represents 0.1 unit
Now,
The representation of –\(\frac{2}{5}\) in the decimal form is: -0.4
Hence,
The representation of the given point on the given number line is:

Question 3.
0.5
Answer:
The given point is: 0.5
Now,
The given number line is:
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 160
Now,
From the given number line,
We can observe that
Each line in the given number line represents 0.1 unit
Hence,
The representation of the given point on the given number line is:

In 4-7, use <, >, or = to compare.

Question 4.
0.25 Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 170 \(\frac{1}{4}\)
Answer:
The given numbers are: 0.25 and \(\frac{1}{4}\)
Now,
The conversion of \(\frac{1}{4}\) into a decimal number is: 0.25
Hence, from the above,
We can conclude that

Question 5.
1\(\frac{5}{8}\) Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 170 1.6
Answer:
The given numbers are: 1\(\frac{5}{8}\) and 1.6
Now,
The conversion of 1\(\frac{5}{8}\) into a decimal number is: 1.625
Hence, from the above,
We can conclude that

Question 6.
3.65 Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 170 3\(\frac{3}{4}\)
Answer:
The given numbers are: 3.65 and 3\(\frac{3}{4}\)
Now,
The conversion of 3\(\frac{3}{4}\) into a decimal number is: 3.75
Hence, from the above,
We can conclude that

Question 7.
–\(\frac{2}{3}\) Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 170 \(\frac{3}{4}\)
Answer:
The given numbers are: –\(\frac{2}{3}\) and \(\frac{3}{4}\)
Now,
The conversion of \(\frac{3}{4}\) into a decimal number is: 0.75
The conversion of –\(\frac{2}{3}\) into a decimal number is: -0.66
Hence, from the above,
We can conclude that

LESSON 2.3 Absolute Values of Rational Numbers

Quick Review
The absolute value of a number is its distance from 0 on the number line. Distance is always positive. Absolute values are never negative.

Example
Find the absolute values and order |3|, |4|, |-2|, |–5| from least to greatest.
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 95.2

Practice
In 1-4, find each value.

Question 1.
|-9|
Answer:
The given absolute number is: |-9|
Now,
We know that,
The absolute value of any number is positive
Hence, from the above,
We can conclude that
The value of |-9| is: 9

Question 2.
|-2|
Answer:
The given absolute number is: |-2|
Now,
We know that,
The absolute value of any number is positive
Hence, from the above,
We can conclude that
The value of |-2| is: 2

Question 3.
|4|
Answer:
The given absolute number is: |4|
Now,
We know that,
The absolute value of any number is positive
Hence, from the above,
We can conclude that
The value of |4| is: 4

Question 4.
-|-10|
Answer:
The given absolute number is: -|-10|
Now,
We know that,
The absolute value of any number is positive
Hence, from the above,
We can conclude that
The value of -|-10| is: -10

In 5-8, order the values from least to greatest.

Question 5.
|-3|, |-2|, |10|
Answer:
The given absolute numbers are: |-3|, |-2|, and |10|
Now,
We know that,
The absolute value of any number is positive
So,
The values of the given absolute numbers are: 3, 2, and 10
So,
The order of the given absolute numbers from the least to the greatest is: 2, 3, and 10
Hence, from the above,
We can conclude that
The order of the given absolute numbers from the least to the greatest is: 2, 3, and 10

Question 6.
|-7|, |0|, |-5|
Answer:
The given absolute numbers are: |-7|, |0|, and |-5|
Now,
We know that,
The absolute value of any number is positive
So,
The values of the given absolute numbers are: 7, 0, and 5
So,
The order of the given absolute numbers from the least to the greatest is: 0, 5, and 7
Hence, from the above,
We can conclude that
The order of the given absolute numbers from the least to the greatest is: 0, 5, and 7

Question 7.
|-18.5|, |18|, |-12.5|
Answer:
The given absolute numbers are: |-18.5|, |18|, and |-12.5|
Now,
We know that,
The absolute value of any number is positive
So,
The values of the given absolute numbers are: 18.5, 18, and 12.5
So,
The order of the given absolute numbers from the least to the greatest is: 12.5, 18, and 18.5
Hence, from the above,
We can conclude that
The order of the given absolute numbers from the least to the greatest is: 12.5, 18, and 18.5

Question 8.
|26|, |-20|, |-24.5|
Answer:
The given absolute numbers are: |26|, |-20|, and |-24.5|
Now,
We know that,
The absolute value of any number is positive
So,
The values of the given absolute numbers are: 26, 20, and 24.5
So,
The order of the given absolute numbers from the least to the greatest is: 20, 24.5, and 26
Hence, from the above,
We can conclude that
The order of the given absolute numbers from the least to the greatest is: 20, 24.5, and 26

LESSON 2.4 Represent Rational Numbers on the Coordinate Plane

Quick Review
An ordered pair (x, y) of numbers gives the coordinates that locate a point on a coordinate plane. Coordinates can be whole numbers, fractions, mixed numbers, or decimals.

Example
Explain how to plot any point with coordinates (x, y).
• Start at the origin, (0, 0).
• Use the x-coordinate to move right (if positive) or left (if negative) along the x-axis.
• Then use the y-coordinate of the point to move up (if positive) or down (if negative) following the y-axis.
• Draw and label the point on the coordinate plane. Explain how to name the location of a point on a coordinate plane. Follow the grid line from the point to the x-axis to name the x-coordinate, and follow the grid line from the point to the y-axis to name the y-coordinate.

Practice
In 1-6, give the ordered pair for each point.
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 98.1

Question 1.
U
Answer:
The given coordinate plane is:
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 98.1
Now,
From the given coordinate plane,
We can observe that
The coordinates of point U are: (0, 2.5)
Hence, from the above,
We can conclude that
The ordered pair for point U is: (0, 2.5)

Question 2.
V
Answer:
The given coordinate plane is:
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 98.1
Now,
From the given coordinate plane,
We can observe that
The coordinates of point V are: (-2, 1.5)
Hence, from the above,
We can conclude that
The ordered pair for point V is: (-2, 1.5)

Question 3.
W
Answer:
The given coordinate plane is:
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 98.1
Now,
From the given coordinate plane,
We can observe that
The coordinates of point W are: (-4, -1)
Hence, from the above,
We can conclude that
The ordered pair for point W is: (-4, -1)

Question 4.
X
Answer:
The given coordinate plane is:
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 98.1
Now,
From the given coordinate plane,
We can observe that
The coordinates of point X are: (2.5, 0)
Hence, from the above,
We can conclude that
The ordered pair for point X is: (2.5, 0)

Question 5.
Y
Answer:
The given coordinate plane is:
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 98.1
Now,
From the given coordinate plane,
We can observe that
The coordinates of point Y are: (2, -1.5)
Hence, from the above,
We can conclude that
The ordered pair for point Y is: (2, -1.5)

Question 6.
Z
Answer:
The given coordinate plane is:
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 98.1
Now,
From the given coordinate plane,
We can observe that
The coordinates of point Z are: (-1.5, -3)
Hence, from the above,
We can conclude that
The ordered pair for point Z is: (-1.5, -3)

LESSONS 2.5 AND 2.6 Find Distances and Represent Polygons on the Coordinate Plane

Quick Review
You can use absolute value to find the distance between two points that share the same X- or y-coordinate. When the y-coordinates are the same, use the x-coordinates to find the distance. When the x-coordinates are the same, use the y-coordinates. If the points are in different quadrants, add their absolute values. If the points are in the same quadrant, subtract their absolute values.
You can use what you know about finding the distance between two points to find the lengths of the sides of a polygon on a coordinate plane.

Example
Find the length of side AB.
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 98.2
The ordered pairs for points A and B are A(-3, 2) and B(-1, 2). The points are in the same quadrant, so subtract the absolute values of the x-coordinates.
|-3| – |-1| = 3 – 1 = 2 units
The length of side AB is 2 units.

Practice

In 1-6, find the remaining side lengths of polygon ABCDEF. Then find the polygon’s perimeter.
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 98.2
Answer:
The given coordinate plane is:
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 98.2
Now,
From the given coordinate plane,
The coordinates of the vertices are:
A (-3, 2) B (-1, 2), C (-1, 1), D (1, 1), E (1, -2), F (-3, -2)

Question 1.
Length of BC
Answer:
The given coordinates of points B and C are: B (-1, 2), and C (-1, 1)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between 2 points = |x + a| + |y + b|
So,
The length of BC = |2 – 1|
= 1 unit
Hence, from the above,
We can conclude that
The length of BC is: 1 unit

Question 2.
Length of CD
Answer:
The given coordinates of points C and D are: C (-1, 1), and D (1, 1)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between 2 points = |x + a| + |y + b|
So,
The length of CD = |1 + 1|
= 2 units
Hence, from the above,
We can conclude that
The length of CD is: 2 units

Question 3.
Length of DE
Answer:
The given coordinates of points D and E are: D (1, 1), and E (1, -2)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between 2 points = |x + a| + |y + b|
So,
The length of DE = |2 + 1|
= 3 units
Hence, from the above,
We can conclude that
The length of DE is: 3 units

Question 4.
Length of EF
Answer:
The given coordinates of points E and F are: E (1, -2), and F (-3, -2)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between 2 points = |x + a| + |y + b|
So,
The length of EF = |3 + 1|
= 4 units
Hence, from the above,
We can conclude that
The length of EF is: 4 units

Question 5.
Length of FA
Answer:
The given coordinates of points F and A are: F (-3, -2), and A (-3, 2)
Now,
Compare the given points with (x, a), and (y, b)
Now,
We know that,
The distance between 2 points = |x + a| + |y + b|
So,
The length of FA = |2 + 2|
= 4 units
Hence, from the above,
We can conclude that
The length of FA is: 4 units

Question 6.
Perimeter of ABCDEF
Answer:
We know that,
The perimeter of any polygon is the sum of its side lengths
So,
The perimeter of polygon ABCDEF = AB + BC + CD+ DE+ EF+ FA
= 2 + 1 + 2 + 3 + 4 + 4
= 16 units
Hence, from the above,
We can conclude that
The perimeter of the given polygon ABCDEF is: 16 units

In 7 and 8, polygon QRST has vertices Q(-4, -1), R(-4,5), S(2,5), and T(2, -1).

Question 7.
Draw and label polygon QRST on the coordinate plane.
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 96.1
Answer:
The given vertices of the polygon QRST are:
Q (-4, -1), R (-4,5), S (2,5), and T (2, -1)
Hence,
The representation of the given vertices along with its side lengths in the given coordinate plane is:

Question 8.
Construct an argument to justify whether or not polygon QRST is a square.
Answer:
From Exercise 7,
The representation of polygon QRST along with its side lengths in the given coordinate plane is:

Now,
From the given coordinate plane,
We can observe that
The sidelengths of all the sides are the same
Now,
We know that,
A “Square” has the same side lengths on all the four sides
Hence, from the above,
We can conclude that
Polygon QRST is a square

Topic 2 Fluency Practice

Hidden Clue
For each ordered pair, simplify the two coordinates. Then locate and label the corresponding point on the graph. Draw line segments to connect the points in alphabetical order. Use the completed picture to help answer the riddle below.
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 98.4
Envision Math Common Core 6th Grade Answer Key Topic 2 Integers and Rational Numbers 98.5

enVision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers

enVision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers

Go through the enVision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers regularly and improve your accuracy in solving questions.

enVision Math Common Core 5th Grade Answers Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers

Essential Question: What are the standard procedures for estimating and finding products of multi-digit numbers?

Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 1
enVision STEM Project: Water Usage
Do Research Use the Internet or other sources to find how much water is used for household activities like taking a shower or bath, using a dishwasher, hand washing dishes, and using a washing machine.
Journal: Write a Report Include what you found. Also in your report:

  • Choose 3 of the activities. Estimate how many times each activity is done each week in your household.
  • Estimate the weekly water usage for each activity. Organize your results in a table.
  • Make up and solve multiplication problems based on your data.

Fluently Multiply Multi-Digit Whole Numbers 1

Review What You Know

Vocabulary

Choose the best term from the box. Write it on the blank.
• multiple
• equation
• exponent
• power
• factor
• product

Question 1.
The answer to a multiplication problem is the ____.

Answer:
The answer to a multiplication problem is the product.

Explanation:
In the above-given question,
given that,
the answer to a multiplication problem is called the product.
for example:
15 x 2 = 30.
15 is called multiplicand.
2 is the multiplier.
30 is the product.
So the answer to a multiplication problem is called the product.

Question 2.
A number sentence that shows two expressions with the same value is a(n) _____

Answer:
A number sentence that shows two expressions with the same value is an equation.

Explanation:
In the above-given question,
given that,
A number sentence that shows two expressions with the same value is an equation.
for example:
4 + 8 = 12.
5 + 6 = 12.
so the number sentence that shows two expressions with the same value is an equation.

Queen 3.
A(n) ___ tells the number of times the base is used as a(n) ___.

Answer:
A(n) tells the number of times the base is used as an exponent.

Explanation:
In the above-given question,
given that,
A(n) ___ tells the number of times the base is used as a(n).
for example:
5².
where 2 is the exponent.
5 is base.

Question 4.
50 is a(n) ____ of 10 because 5 × 10 = 50.

Answer:
50 is a(n) base of 10 because 5 x 10 = 50.

Explanation:
In the above-given question,
given that,
50 is a(n) base of 10 because 5 x 10 = 50.
for example:
5 x a (n) = 10.
a(n) = 10 x 5.
a(n) = 50.

Operations

Find each sum or difference.

Question 5.
9,007 + 3,128

Answer:
9007 + 3128 = 12,135.

Explanation:
In the above-given question,
given that,
the two numbers are 9007 and 3128.
add the two numbers.
9007 + 3128 = 12,135.

Question 6.
7,904 – 3,199

Answer:
7904 – 3199 = 4705.

Explanation:
In the above-given question,
given that,
the two numbers are 7904 and 3199.
subtract the two numbers.
7904 – 3199 = 4705.

Question 7.
27,924 – 13,868

Answer:
27924 – 13,868 = 14,056.

Explanation:
In the above-given question,
given that,
the two numbers are 27924 and 13868.
subtract the two numbers.
27924 – 13868 =14,056.

Question 8.
9.27 + 3.128

Answer:
9.27 + 3.128 = 12.398.

Explanation:
In the above-given question,
given that,
the two numbers are 9.27 and 3.128.
add the two numbers.
9.27 + 3.128 = 12.398.

Question 9.
119.04 – 86.5

Answer:
119.04 – 86.5 = 32.54.

Explanation:
In the above-given question,
given that,
the two numbers are 119.04 and 86.5.
subtract the two numbers.
119.04 – 86.5 = 32.54.

Fluently Multiply Multi-Digit Whole Numbers 2

Question 10.
165.2 – 133.18

Answer:
165.2 – 133.18 = 32.02.

Explanation:
In the above-given question,
given that,
the two numbers are 165.2 and 133.18.
subtract the two numbers.
165.2 – 133.18 = 32.02.

Round Whole Numbers and Decimals

Round each number to the place of the underlined digit.

Question 11.
14.3

Answer:
14.3.

Explanation:
In the above-given question,
given that,
the number is 14.3.
the underlined digit is 4.
if the tenths place is greater than 5.
then add by 1.
if tenths place is less than 5.
then write the same number as zeros.
so the answer is 14.3.

Question 12.
385.7

Answer:
395.7.

Explanation:
In the above-given question,
given that,
the number is 385.7.
the underlined digit is 8.
if the tenths place is greater than 5.
then add by 1.
if tenths place is less than 5.
then write the same number as zeros.
so the answer is 395.7.

Question 13.
0.545

Answer:
0.500.

Explanation:
In the above-given question,
given that,
the number is 0.545.
the underlined digit is 5.
if the tenths place is greater than 5.
then add by 1.
if tenths place is less than 5.
then write the same number as zeros.
so the answer is 0.500.

Question 14.
496.533

Answer:
497.533.

Explanation:
In the above-given question,
given that,
the number is 496.533.
the underlined digit is 6.
if the tenths place is greater than 5.
then add by 1.
if tenths place is less than 5.
then write the same number as zeros.
so the answer is 496.533.

Question 15.
496.353

Answer:
496.000.

Explanation:
In the above-given question,
given that,
the number is 496.353.
the underlined digit is 6.
if the tenths place is greater than 5.
then add by 1.
if tenths place is less than 5.
then write the same number as zeros.
so the answer is 496.000.

Question 16.
1,857.205

Answer:
1857.215.

Explanation:
In the above-given question,
given that,
the number is 1857.205.
the underlined digit is 0.
if the tenths place is greater than 5.
then add by 1.
if tenths place is less than 5.
then write the same number as zeros.
so the answer is 1857.205.

Compare Decimals

Question 17.
Write the numbers in order from least to greatest. 8.062 8.26 8.026 8.6

Answer:
The numbers in order from least to greatest is 8.026, 8.062, 8.26, and 8.6.

Explanation:
In the above-given question,
given that,
the numbers are 8.062, 8.26, 8.026, and 8.6.
the numbers in order from least to greatest is
8.026, 8.062, 8.26, and 8.6.

Question 18.
Write the numbers in order from greatest to least. 0.115 0.15 0.005 0.5

Answer:
The numbers in order from greatest to least are 0.005, 0.115, 0.15, and 0.5.

Explanation:
In the above-given question,
given that,
the numbers are 0.115, 0.15, 0.005, and 0.5.
the numbers in order from greatest to least are
0.115, 0.15, 0.005, and 0.5.

pick a Project

PROJECT ЗА
What puts the bounce in a bouncy ball?
Project: Make a Business Plan
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 2

PROJECT 3B
How can you build a fort?
Project: Build a Model Fort
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 21

PROJECT 3C
How many people can a ferry carry?
Project: Design a Prototype Ferry
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 22

3-ACT MATH PREVIEW

Math Modeling

Morning Commute

Before watching the video, think:
Train conductors don’t wear this kind of hat anymore. Even paper tickets are less common now that some train lines use an app to purchase tickets. What are some other ways we have updated transportation as part of our modern society? All aboard!
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 23

Fluently Multiply Multi-Digit Whole Numbers 3

Lesson 3.1 Multiply Greater Numbers by Powers of 10

Activity

At Izzy’s Party Store, party invitations come in packages of 8. How many invitations are in 10 packages? 100 packages? 1,000 packages? Solve this problem any way you choose.

Answer:
The number of invitations is in 10 packages = 80.
the number of invitations is in 100 packages = 800.
the number of invitations is in 1000 packages = 8000.

Explanation:
In the above-given question,
given that,
At Izzy’s Party Store, party invitations come in packages of 8.
8 x 10 = 80.
8 x 100 = 800.
8 x 1000 = 8000.
so the number of invitations is in 10 packages = 80.
the number of invitations is in 100 packages = 800.
the number of invitations is in 1000 packages = 8000.

You can use appropriate tools. Place-value blocks are useful for picturing problems that involve powers of 10.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 24

Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 24.1

Look Back! What patterns do you notice in your work above?

Visual Learning Bridge

Essential Question
How Can You Use Patterns and Mental Math to Question Multiply a Whole Number by a Power of 10?

A.
The value of each place in a number is 10 times the value of the place to the right. The place-value chart shows this relationship for the number 4. Look for patterns.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 24.2
10 times greater than 4
10 times greater than 40
10 times greater than 400
10 times greater than 4,000
10 times greater than 40,000
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 24.3

B.
Find 32 × 10,000 by using place-value relationships.
Multiply 32 by 1; 10; 100; 1,000; and 10,000.
32 × 1 = 32 ones = 32
32 × 10 = 32 tens = 320
32 × 100 = 32 hundreds = 3,200
32 × 1,000 = 32 thousands = 32,000
32 × 10,000 = 32 ten thousands = 320,000
Pattern
Pattern The product ends with the same number of zeros as the power of 10.

C.
Instead of using the standard form, write each power of 10 using exponents.
32 × 1 = 32 × 100 = 32
32 × 10 = 32 × 101 = 320
32 × 100 = 32 × 102 = 3,200
32 × 1,000 = 32 × 103 = 32,000
32 × 10,000 = 32 × 104 = 320,000

Pattern
The exponent tells how many additional zeros the product will end with.

Convince Me! Critique Reasoning Nellie says that 60 × 1,000 is 6,000 because there are three zeros in 1,000. Kara says that 60 × 1,000 = 60,000. Whose thinking is correct? Explain.

Answer:
Kara says is correct.

Explanation:
In the above-given question,
given that,
Nellie says that 60 × 1,000 is 6,000.
Kara says that 60 × 1,000 = 60,000.
so kara says is correct.

Guided Practice

Do You Understand?

Question 1.
How many zeros will there be in the product of 39 × 1,000? How many zeros will there be in the product of 50 × 1,000?

Answer:
There are 3 zeros in the 39000.
there are 4 zeros in the 50000.

Explanation:
In the above-given question,
given that,
39 x 1000 = 39000.
50 x 1000 = 50000.
so there are 3 zeros in the 39000.
there are 4 zeros in the 50000.

Question 2.
Explain how to find the product of 90 × 104.

Answer:
The product of 90 x 104

Explanation:
In the above-given question,
given that,
90 x 104
90 x 10 x 10 x 10 x 10.
90 x 100 x 100.
90 x 10000.
900000.

Do You Know How?

In 3-5, use reasoning to fill in the missing numbers.

Question 3.
60 × 1 = ____
60 × 100 = ____
60 × 10,000 = ____

Answer:
60 x 1 = 60.
60 x 100 = 6000.
60 x 1000 = 60000.

Explanation:
In the above-given question,
given that,
60 x 1 = 60.
60 x 100 = 6000.
60 x 1000 = 60000.

Question 4.
13 × ___ = 13,000

Answer:
13 x 1000 = 13000.

Explanation:
In the above-given question,
given that,
13 x 10 x 10 x 10.
13 x 1000 = 13000.

Question 5.
___ × 104 = 100,000

Answer:
10 × 104 = 100,000.

Explanation:
In the above-given question,
given that,
10 x 1 =10.
104 = 10 x 10 x 10 x 10 =10000.
10 x 10000 = 100,000.

Independent Practice

Leveled Practice In 6-13, find each product.

Question 6.
89 × 1
89 × 10
89 × 100
89 × 1,000
89 × 10,000

Answer:
89 x 1 = 89.
89 x 10 = 890.
89 x 100 = 8900.
89 x 1000 = 89000.
89 x 10,000 = 890,000.

Explanation:
In the above-given question,
given that,
89 x 1 = 89.
89 x 10 = 890.
89 x 100 = 8900.
89 x 1000 = 89000.
89 x 10,000 = 890,000.

Question 7.
30 × 1
30 × 10
30 × 100
30 × 1,000
30 × 10,000

Answer:
30 × 1 = 30.
30 × 10 = 300.
30 × 100 = 3000.
30 × 1,000 = 30,000.
30 × 10,000 = 300,000.

Explanation:
In the above-given question,
given that,
30 × 1 = 30.
30 × 10 = 300.
30 × 100 = 3000.
30 × 1,000 = 30,000.
30 × 10,000 = 300,000.

Question 8.
41 × 1
41 × 101
41 × 102
41 × 103
41 × 104

Answer:
41 × 1 = 41.
41 × 101
41 × 102
41 × 103
41 × 104

Explanation:
In the above-given question,
given that,
41 × 1 = 41.
41 × 101= 4100.
41 × 102 = 42000.
41 × 103= 43000.
41 × 104= 430000.

Question 9.
90 × 1
90 × 101
90 × 102
90 × 103
90 × 104

Answer:
90 x 1 = 90.
90 × 101
90 × 102
90 × 103
90 × 104

Explanation:
In the above-given question,
given that,
90 x 1 = 90.
90 × 101
90 × 102
90 × 103
90 × 104

Question 10.
4 × 103

Answer:
4 x 103 = 4000.

Explanation:
In the above-given question,
given that,
4 x 103.
4 x 10 x 10 x 10.
4 x 1000.
4000.

Question 11.
85 × 100

Answer:
85 x 100 = 8500.

Explanation:
In the above-given question,
given that,
85 x 100.
8500.

Question 12.
16 × 102

Answer:
16 x 10 x 10 = 1600.

Explanation:
In the above-given question,
given that,
16 x 102.
16 x 10 x 10.
1600.

Question 13.
103 × 38

Answer:
10 x 10 x 10 x 38 = 38000.

Explanation:
In the above-given question,
given that,
103 × 38.
10 x 10 x 10 x 38.
100 x 10 x 38.
38 x 1000.
38000.

In 14-19, use reasoning to fill in the missing numbers.

Question 14.
52 × 10- = 520,000

Answer:
52 x 10 4 = 520,000.

Explanation:
In the above-given question,
given that,
52 x 10 x 10 x 10 x 10.
52 x 100 x 100.
52 x 10000.
520,000.

Question 15.
68,637 = 101 × ___

Answer:
10 x 68637 = 686370.

Explanation:
In the above-given question,
given that,
68637 x 10.
686370.

Question 16.
___ = 382 × 104

Answer:
382 x 10000 = 3820000.

Explanation:
In the above-given question,
given that,
382 x 10 x 10 x 10 x 10.
382 x 104.
382 x 100 x 100.
3820000.

Question 17.
___ = 103 × 80

Answer:
80 x 103 = 80000.

Explanation:
In the above-given question,
given that,
80 x 103.
80 x 10 x 10 x 10.
80 x 1000.
80000.

Question 18.
10 × 374 = 37,400

Answer:
10 x 374 x 10 = 37400.

Explanation:
In the above-given question,
given that,
10 x 374 x 10.
100 x 374.
37400.

Question 19.
500,000 = 50 × 10-

Answer:
50 x 10000 = 50,000.

Explanation:
In the above-given question,
given that,
50 x 10 x 10 x 10 x 10.
50 x 100 x 100.
50 x 10000.
50,000.

Problem Solving

Question 20.
At a football championship game, the home team gave a football to each of the first 100 fans who arrived at the stadium. Each football cost the team $28. How much did the team pay for the footballs it gave away?

Answer:
The team pay for the footballs it gave away = $2800.

Explanation:
In the above-given question,
given that,
At a football championship game,
the home team gave a football to each of the first 100 fans who arrived at the stadium.
Each football cost the team $28.
28 x 100 = 2800.
so the team pay for the football it gave away = $2800.

Question 21.
Construct Arguments Without multiplying, tell which expression is greater, 93 × 103 or 11 × 104? How do you know?

Answer:
The expression 93 x 103 is greater.

Explanation:
In the above-given question,
given that,
the two expressions are 93 × 103 or 11 × 104.
93 x 10 x 10 x 10.
93 x 1000.
93000.
11 x 10 x 10 x 10 x 10.
11 x 10000.
110000.

Question 22.
A truck is carrying 102 bushels of onions, 101 bushels of peaches, and 103 bushels of corn. What is the total weight of the crops?
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 50.1

Answer:
The total weight of the crops = 76,200.

Explanation:
In the above-given question,
given that,
A truck is carrying 102 bushels of onions.
101 bushels of peaches, and 103 bushels of corn.
57 x 100 = 5700.
50 x 10 = 500.
70 x 10 x 10 x 10 = 70000.
5700 + 500 + 70000 = 76,200.
so the total weight of the crops = 76200.

Question 23.
Norman bought a 16-pound bag of charcoal for $7.89 and a 10.4-pound bag of charcoal for $5.69. What was the total weight of the two bags of charcoal?

Answer:
The total weight of the two bags of charcoal = $185.416.

Explanation:
In the above-given question,
given that,
Norman bought a 16-pound bag of charcoal for $7.89.
10.4-pound bag of charcoal for $5.69.
16 x 7.89 = 126.24.
10.4 x 5.69 = 59.176.
126.24 + 59.176 = 185.416.
so the total weight of the two bags of charcoal = $185.416.

Question 24.
Higher Order Thinking There are 2,000 pounds in 1 ton. In the United States, the weight limit for a truck and its cargo is 40 tons. How many pounds is that? How did you find the answer?

Answer:
The number of pounds = 80,000.

Explanation:
In the above-given question,
given that,
There are 2,000 pounds in 1 ton.
In the United States, the weight limit for a truck and its cargo is 40 tons.
2000 x 40 = 80000.
so the number of pounds = 80,000.

Assessment Practice

Question 25.
Which is equivalent to multiplying a number by 104?
A. multiplying by 40
B. multiplying by 100
C. multiplying by 1,000
D. multiplying by 10,000

Answer:
The number is equivalent to multiplying a number by 10000.

Explanation:
In the above-given question,
given that,
multiplying by 10,000.
10 x 10 x 10 x 10 = 10,000.
so the number equivalent to multiplying a number by 10000.

Question 26.
Select the statements that are equivalent to multiplying 20 × 104.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 52 Add 10 to 20 four times.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 52 Multiply 20 by 10 four times.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 52 Multiply 10 by 20 four times.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 52 Multiply 20 by 10,000.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 52 Multiply 20 by 100,000.

Answer:
Option B is correct.

Explanation:
In the above-given question,
given that,
20 x 104.
20 x 10 x 10 x 10 x 10.
20 x 100 x 100.
200000.
so option B is correct.

Lesson 3.2 Estimate Products

Activity

Solve & Share

Answer:

A school club wants to buy shirts for each of its 38 members. Each shirt costs $23. About how much money will all the shirts cost? Solve this problem any way you choose.

Answer:
The much money will all the shirts cost = $874.

Explanation:
In the above-given question,
given that,
A school club wants to buy shirts for each of its 38 members.
Each shirt costs $23.
38 x 23 = 874.
so the much money will all the shirts cost = $874.

Are you asked for an exact answer or an estimate?
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 52.1

Look Back! Construct Arguments How can you use number sense to tell that the exact answer has to be greater than $600? Explain how you know.

Visual Learning Bridge

Essential Question How Can You Estimate Products?

You can use rounding to estimate.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 522

A.
A store needs at least $75,000 in sales per month to make a profit. If the store is open every day in March and sales average $525 per day, will the store make a profit in March?

B.
Use Rounding to Estimate
$525 rounds to $500.
31 rounds to 30.
Find 30 × 500.
30 × 500 = 15,000
You know that 3 × 5 = 15.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 52.3

C.
Both numbers used to estimate were less than the actual numbers, so 15,000 is an underestimate. The store will actually have more than $15,000 worth of sales.
So, the store will make a profit in March.

Convince Me! Critique Reasoning A different store needs to make at least $20,000 to make a profit in March. They average $685 a day for the month. James used rounding and estimation to say, “$685 is almost $700. $700 × 30 days is $21,000. I think it is going to be a close call!” What do you think?

Answer:
$685 x 30 = 20,550.

Explanation:
In the above-given question,
given that,
A different store needs to make at least $20,000 to make a profit in March.
They average $685 a day for the month.
James used rounding and estimation to say, “$685 is almost $700.
$700 x 30 = $21000.
$685 x 30 = 20,550.
so they can make the profit.

Another example
Estimate 24 × 398.
25 and 4 are compatible numbers because their product is easy to compute mentally.
25 × 4 = 100
25 × 40 = 1,000
25 × 400 = 10,000
So, 10,000 is a good estimate for 24 × 398.
You can also use compatible numbers to estimate.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 52.4
Both numbers used to estimate were greater than the actual numbers.
So, 10,000 is an overestimate.

Guided Practice

Do You Understand?

Question 1.
Number Sense Each egg carton holds one dozen eggs. Michael’s chicken farm fills 121 egg cartons. He thinks that there were over 1,500 eggs. Is he correct? Use an estimate to find out.

Answer:
Yes, the estimation was correct.

Explanation:
In the above-given question,
given that,
Each egg carton holds one dozen eggs.
1 dozen = 12.
Michael’s chicken farm fills 121 egg cartons.
121 is near to 125.
125 x 12 = 1500.
so the estimation was correct.

Do You Know How?

In 2-5, estimate. Then, tell if your estimate is an overestimate or underestimate.

Question 2.
29 × 688

Answer:
29 x 688 = 19,952.

Explanation:
In the above-given question,
given that,
the two numbers are 29 and 688.
multiply the numbers.
29 x 688 = 19,952.

Question 3.
210 × 733

Answer:
210 x 733 = 153930.

Explanation:
In the above-given question,
given that,
the two numbers are 210 and 733.
multiply the numbers.
210 x 733 = 153930.

Question 4.
43 × 108

Answer:
43 x 108 = 4644.

Explanation:
In the above-given question,
given that,
the two numbers are 43 and 108.
multiply the numbers.
43 x 108 = 4644.

Question 5.
380 × 690

Answer:
380 x 690 = 262200.

Explanation:
In the above-given question,
given that,
the two numbers are 380 and 690.
multiply the numbers.
380 x 690 = 262200.

Independent Practice

Leveled Practice In 6-17, estimate each product.

Question 6.
180 × 586

Answer:
180 x 586 = 1,05,480.

Explanation:
In the above-given question,
given that,
the two numbers are 180 and 586.
multiply the numbers.
180 x 586 = 1,05,480.

Question 7.
300 × 118

Answer:
300 x 118 = 35400.

Explanation:
In the above-given question,
given that,
the two numbers are 300 and 118.
multiply the numbers.
300 x 118 = 35400.

Question 8.
19 × 513

Answer:
19 x 513 = 9,747.

Explanation:
In the above-given question,
given that,
the two numbers are 19 and 513.
multiply the numbers.
19 x 513 = 9,747.

Question 9.
38 × 249

Answer:
38 x 249 = 9462.

Explanation:
In the above-given question,
given that,
the two numbers are 38 and 249.
multiply the numbers.
38 x 249 = 9462.

Question 10.
11 × 803

Answer:
11 x 803 = 8833.

Explanation:
In the above-given question,
given that,
the two numbers are 11 and 803.
multiply the numbers.
11 x 803 = 8833.

Question 11.
44 × 212

Answer:
44 x 212 = 9328.

Explanation:
In the above-given question,
given that,
the two numbers are 44 and 212.
multiply the numbers.
44 x 212 = 9328.

Question 2.
790 × 397

Answer:
790 x 397 = 313630.

Explanation:
In the above-given question,
given that,
the two numbers are 790 and 397.
multiply the numbers.
790 x 397 = 313630.

Question 13.
42 × 598

Answer:
42 x 598 = 25,116.

Explanation:
In the above-given question,
given that,
the two numbers are 42 and 598.
multiply the numbers.
42 x 598 = 25,116.

Question 14.
25 × 191

Answer:
25 x 191 = 4775.

Explanation:
In the above-given question,
given that,
the two numbers are 25 and 191.
multiply the numbers.
25 x 191 = 4775.

Question 15.
408 × 676

Answer:
408 x 676 = 275808.

Explanation:
In the above-given question,
given that,
the two numbers are 408 and 676.
multiply the numbers.
408 x 676 = 275808.

Question 16.
290 × 12

Answer:
290 x 12 = 3,480.

Explanation:
In the above-given question,
given that,
the two numbers are 290 and 12.
multiply the numbers.
290 x 12 = 3,480.

Question 17.
854 × 733

Answer:
854 x 733 = 6,25,982.

Explanation:
In the above-given question,
given that,
the two numbers are 854 and 733.
multiply the numbers.
854 x 733 = 6,25,982.

Problem Solving

Question 18.
Reasoning Estimate 530 × 375. Is the estimated product closer to 150,000 or 200,000? Explain.

Answer:
The estimated product is closer to 200,000.

Explanation:
In the above-given question,
given that,
530 x 375 = 198,750.
198750 is equal to 200,000.

Question 19.
Vocabulary Is 500 an underestimate or overestimate for the product of 12 and 53?

Answer:
500 is an underestimate for the product 12 and 53.

Explanation:
In the above-given question,
given that,
12 x 53 = 636.
10 x 50 = 500.
500 is an underestimate for the product 12 and 53.

Question 20.
Samuel needs to estimate the product of 23 × 495. Explain two different methods Samuel can use to estimate.

Answer:
23 x 495 = 11,385.
25 x 500 = 12500.

Explanation:
In the above-given question,
given that,
the product of 23 and 495.
23 x 495 = 11,385.
25 x 500 = 12500.

Question 21.
Rebekah said that 103 is 30 because 10 + 10 + 10 = 30. Do you agree? Explain.

Answer:
No, I do not agree with it.

Explanation:
In the above-given question,
given that,
Rebekah said that 103 is 30 because 10 + 10 + 10 = 30.
10 + 10 + 10 = 30.
30 is no equal to 103.
so I do not agree with it.

Question 22.
Higher Order Thinking Abby counts 12 large boxes and 18 small boxes of pencils in the supply cabinet. Each large box contains 144 pencils. Each small box contains 24 pencils. Estimate the total number of pencils. Is your estimate an overestimate or an underestimate? Explain why it might be better to have an underestimate rather than an overestimate.

Answer:
The total number of pencils = 2160.

Explanation:
In the above-given question,
given that,
Abby counts 12 large boxes and 18 small boxes of pencils in the supply cabinet.
Each large box contains 144 pencils.
Each small box contains 24 pencils.
144 x 12 =1728.
24  x 18 = 432.
1728 + 432 = 2160.
so the total number of pencils = 2160.

Question 23.
Susan used rounding to estimate 24 × 413 and found 20 × 400. Jeremy used compatible numbers and found 25 × 400. Whose method gives an estimate closer to the actual product? Explain.

Is your answer reasonable?
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 53.1

Answer:
Jeremy used compatible numbers and found 25 x 400 = 10000.

Explanation:
In the above-given question,
given that,
Susan used rounding to estimate 24 × 413 and found 20 × 400.
Jeremy used compatible numbers and found 25 × 400.
24 x 413 = 9912.
20 x 400 = 8000.
25 x 400 = 10000.
so jeremy used compatible numbers and found 25 x 400 = 10000.

Assessment Practice

Question 24.
Lance has 102 packages of sports cards. Each package has 28 cards. Use rounding to estimate. About how many cards does Lance have?
A. 2,000
B. 2,500
C. 3,000
D. 3,500

Answer:
The number of cards does Lance has = 3000.

Explanation:
In the above-given question,
given that,
Lance has 102 packages of sports cards.
Each package has 28 cards.
102 x 28 = 2856.
2856 is near to 3000.
so the number of cards does Lance has = 3000.

Question 25.
Which does NOT show a reasonable estimate of 24 338?
A. 6,000
B. 7,000
C. 7,500
D. 10,000

Answer:
The reasonable estimate is 10,000.

Explanation:
In the above-given question,
given that,
the two numbers are 24 and 338.
338 x 24 = 8,112.
8112 is near to 10,000.
so the reasonable estimate is 10,000.

Lesson 3.3 Multiply by 1-Digit Numbers

Activity

Solve & Share

Suppose a school ordered 7 boxes of books. There are 25 books in each box. How can you use paper and pencil to find how many books were ordered? How can you check if your answer is reasonable? Solve these problems using any strategy you choose.

Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 56.1

Answer:
The number of books that were ordered = 175.

Explanation:
In the above-given question,
given that,
Suppose a school ordered 7 boxes of books.
There are 25 books in each box.
7 x 25 = 175.
so the number of books that were ordered = 175.

You can make sense and persevere. Formulating a plan can help you solve problems. Show your work!

Look Back! Without finding the exact answer, how do you know that the answer to the problem above is less than 210?

Visual Learning Bridge

Glossary

Essential Question
What Is a Common Way to Essential Question Record Multiplication?

A.
Ms. Stockton ordered 6 boxes of T-shirts with the school name on them. Each 20 box contains 26 T-shirts. How many T-shirts did Ms. Stockton order?

You can multiply using partial products. You can write and add the partial products in any order.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 50.8

B.
One Way to Record Multiplication
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 50.9

C.
Another Way to Record Multiplication
You can multiply each place value in order, beginning with the ones. Regroup if needed. Add any regrouped values to each place value.
Step 1: Multiply by the ones.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 50.10
Step 2: Multiply by the tens.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 50.11
Mrs. Stockton ordered 156 T-shirts.

Convince Me! Critique Reasoning A student did the calculation at the right. What did this student do wrong? What is the correct answer?
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 50.12

Another example!
Find 4 × 156.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 52.14

Guided Practice

Do You Understand?

Question 1.
Use place value to explain each step in finding 3 × 2,746.

Answer:
The product is 8238.

Explanation:
In the above-given question,
given that,
the numbers are 3 and 2746.
3 x 2746 = 8238.
6 x 3 ones = 18; 18 = 1 ten and 8 ones.
4 x 3 tens = 12 tens; 12 tens + 1 ten = 13 tens = 1 hundred 3 tens.
7 x 3 hundreds = 21 hundreds + 1 hundred = 22 hundreds; 2 thousands 2 hundreds.
2 x 3 thousands = 6 thousands + 2 thousands; 8 thousands.
so the product is 8238.

Do You Know How?

For 2-5, find each product. Estimate to check if your answer is reasonable.

Question 2.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 53.2

Answer:
23 x 4 = 92.

Explanation:
In the above-given question,
given that,
the two numbers are 23 and 4.
multiply the numbers.
23 x 4 = 92.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-1

Question 3.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 53.22

Answer:
378 x 2 = 756.

Explanation:
In the above-given question,
given that,
the two numbers are 378 and 2.
multiply the numbers.
378 x 2 = 756.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-2

Question 4.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 53.3

Answer:
157 x 5 = 785.

Explanation:
In the above-given question,
given that,
the two numbers are 157 and 5.
multiply the numbers.
157 x 5 = 785.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-3

Question 5.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 53.4

Answer:
1746 x 3 = 5238.

Explanation:
In the above-given question,
given that,
the two numbers are 1746 and 3.
multiply the numbers.
1746 x 3 = 5238.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-4

Independent Practice

For 6-13, find each product. Estimate to check if your answer is reasonable.

Question 6.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 53.5

Answer:
519 x 4 = 2076.

Explanation:
In the above-given question,
given that,
the two numbers are 519 and 4.
multiply the numbers.
519 x 4 = 2076.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-5

Question 7.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 53.6

Answer:
28 x 3 = 84.

Explanation:
In the above-given question,
given that,
the two numbers are 28 and 3.
multiply the numbers.
28 x 3 = 84.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-6

Question 8.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 53.7

Answer:
72 x 5 = 360.

Explanation:
In the above-given question,
given that,
the two numbers are 72 and 5.
multiply the numbers.
72 x 5 = 360.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-7

Question 9.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 53.8

Answer:
138 x 5 = 690.

Explanation:
In the above-given question,
given that,
the two numbers are 138 and 5.
multiply the numbers.
138 x 5 = 690.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-8

Question 10.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 53.9

Answer:
27 x 3 = 81.

Explanation:
In the above-given question,
given that,
the two numbers are 27 and 3.
multiply the numbers.
27 x 3 = 81.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-9

Question 11.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 53.10

Answer:
123 x 9 = 1107.

Explanation:
In the above-given question,
given that,
the two numbers are 123 and 9.
multiply the numbers.
123 x 9 = 1107.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-10

Question 12.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 53.11

Answer:
1445 x 5 = 7225.

Explanation:
In the above-given question,
given that,
the two numbers are 1445 and 5.
multiply the numbers.
1445 x 5 = 7225.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-11

Question 13.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 53.13

Answer:
2204 x 6 = 13224.

Explanation:
In the above-given question,
given that,
the two numbers are 2204 and 6.
multiply the numbers.
2204 x 6 = 13224.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-12

Problem Solving

For 14-16, use the information in the pictures below to find each mass.

Question 14.
Elephant Seal

Answer:
The mass of Elephant Seal = 3480 kg.

Explanation:
In the above-given question,
given that,
the weight of elephants is 8 times as of zebra.
the weight of zebra is 435 kg.
435 x 8 = 3480 kg.

Question 15.
Sports Car

Answer:
The weight of the sports car = 1740 kg.

Explanation:
In the above-given question,
given that,
the weight of the sports car is 4 times as of zebra.
the weight of zebra is 435 kg.
435 x 4 =1740.
so the weight of the sports car = 1740 kg.

Question 16.
Bison

Answer:
The weight of the Bison = 870 kg.

Explanation:
In the above-given question,
given that,
the weight of the sports car is 2 times as of zebra.
the weight of zebra is 435 kg.
435 x 2 =870.
so the weight of the sports Bison = 870 kg.

 

Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 53.14

Question 17.
Model with Math Last year, Anthony’s grandmother gave him 33 silver coins and 16 gold coins to start a coin collection. Now Anthony has six times as many coins in his collection. How many coins does Anthony have in his collection? Complete the bar diagram to show your work.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 53.15

Answer:
The number of coins does Anthony has in his collection = 294.

Explanation:
In the above-given question,
given that,
Last year, Anthony’s grandmother gave him 33 silver coins and 16 gold coins to start a coin collection.
Now Anthony has six times as many coins in his collection.
33 x 6 = 198.
16 x 6 = 96.
198 + 96 = 294.
so the number of coins does Anthony have in his collection = 294.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-20

Question 18.
Vocabulary Use Distributive or Commutative to complete the definition.
According to the ____ Property of Multiplication, factors can be multiplied in any order and the product remains the same.

Answer:
By using the commutative property, factors can be multiplied in any order.

Explanation:
In the above-given question,
given that,
By using the commutative property, factors can be multiplied in any order.
for example:
2 + 3 + 5.
5 + 5 = 10.
so the product remains the same.

Question 19.
Higher Order Thinking Do you think you could use a multiplication algorithm to multiply a 4-digit number by a 1-digit number? Explain.

Answer:
Yes, we can use a 4-digit number by a 1-digit number.

Explanation:
In the above-given question,
given that,
we can use a 4-digit number by a 1-digit number.
for example:
1234 x 1 = 1234.
so we can multiply a 4-digit number by a 1-digit number.

Assessment Practice

Question 20.
Find the product.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 54.1

Answer:
768 x 8 = 6114.

Explanation:
In the above-given question,
given that,
the two numbers are 768 and 8.
multiply the numbers.
768 x 8 = 6114.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-13

Question 21.
Find the product.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 54.2

Answer:
1945 x 3 = 5835.

Explanation:
In the above-given question,
given that,
the two numbers are 1945 and 3.
multiply the numbers.
1945 x 3 = 5835.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-14

Lesson 3.4 Multiply 2-Digit by 2-Digit Numbers

Solve & Share

Ms. Silva has 12 weeks to train for a race. Over the course of one week, she plans to run 15 miles. If she continues this training, how many miles will Ms. Silva run before the race? Solve this problem using any strategy you choose.

Answer:
The number of miles will Ms. Silva run before the race = 180 miles.

Explanation:
In the above-given question,
given that,
Ms. Silva has 12 weeks to train for a race.
Over the course of one week, she plans to run 15 miles.
12 x 15 = 180.
so the number of miles will Ms. Silva run before the race = 180 miles.

You can use partial products to help make sense of and solve the problem. Show your work in the space below!
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 54.3

Look Back! Critique Reasoning Dwayne estimated 60 miles as an answer to the above problem. Is this estimate reasonable? If not, what mistake do you think Dwayne made?

Visual Learning Bridge

Essential Question What Is a Common Way to Record Multiplication?

A.
A ferry carried 37 cars per trip on the weekend. If the ferry made 11 trips on Saturday and 13 trips on Sunday, how many cars did it carry on the weekend?
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 54.4

You can add to find 24 trips were made on Saturday and Sunday. So the ferry carried 37 × 24 cars on the weekend.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 54.5

B.
Use Partial Products
Use the area model to find the partial products for 24 × 37.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 54.6
The ferry carried 888 cars on the weekend.

C.
Use the Standard Algorithm
Step 1: Multiply by the ones.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 54.7
Step 2: Multiply by the tens.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 54.8
The ferry carried 888 cars.

Convince Me! Make Sense and Persevere What are ways you can estimate to check the reasonableness of the answer?

Guided Practice

Do You Understand?

Question 1.
Janet said that the standard algorithm is just a shortcut for partial products. Do you agree? Explain.

Answer:
Yes, I will agree.

Explanation:
In the above-given question,
given that,
Janet said that the standard algorithm is just a shortcut for partial products.
for example:
37 x 24 = 148 is the standard algorithm.
37 x 24 = 888 is the partial products.
so i will agree.

Do You Know How?

For 2, use an algorithm or partial products to find the product. Estimate to check if your answer is reasonable.

Question 2.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 54.9

Answer:
41 x 23 = 943.

Explanation:
In the above-given question,
given that,
the two numbers are 41 and 23.
40 + 1 = 43.
20 + 3 = 23.
20 x 40 = 800.
20 x 1 = 20.
800 + 20 = 820.
3 x 40 = 120.
3 x 1 = 3.
120 + 3 = 123.
820 + 123 = 943.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-15

Independent Practice

Leveled Practice For 3-14, use an algorithm or partial products to find the product. Use and draw area models as needed.

Use estimation to check if your answers are reasonable.
Envision Math Common Core Grade 5 Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 54.10

Question 3.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 54.11

Answer:
16 x 22 = 352.

Explanation:
In the above-given question,
given that,
the two numbers are 16 and 22.
10 + 6 = 16.
20 + 2 = 22.
10 x 20 = 200.
20 x 6 = 120.
200 + 120 = 320.
2 x 10 = 20.
2 x 6 = 12.
20 + 12 = 32.
320 + 32 = 352.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-16

Question 4.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 54.12

Answer:
15 x 16 = 240.

Explanation:
In the above-given question,
given that,
the two numbers are 16 and 15.
10 + 5 = 15.
10 + 6 = 16.
10 x 10 = 100.
10 x 5 = 50.
100 + 50 = 150.
6 x 10 = 60.
6 x 5 = 30.
60 + 30 = 90.
150 + 90 = 240.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-17

Question 5.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 54.13

Answer:
27 x 12 = 324.

Explanation:
In the above-given question,
given that,
the two numbers are 27 and 12.
20 + 7 = 27.
10 + 2 = 12.
10 x 20 = 200.
10 x 7 = 70.
200 + 70 = 270.
2 x 20 = 40.
2 x 7 = 14.
40 + 14 = 54.
270 + 54 = 324.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-18

Question 6.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 54.14

Answer:
18 x 15 = 270.

Explanation:
In the above-given question,
given that,
the two numbers are 18 and 15.
10 + 8 = 18.
10 + 5 = 15.
10 x 10 = 100.
10 x 8 = 80.
100 + 80 = 180.
5 x 10 = 50.
5 x 8 = 40.
50 + 40 = 90.
180 + 90 = 270.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-19

Question 7.
53 × 17

Answer:
53 x 17 = 901.

Explanation:
In the above-given question,
given that,
the two numbers are 53 and 17.
multiply the numbers.
53 x 17 = 901.

Question 8.
81 × 46

Answer:
81 x 46 = 901.

Explanation:
In the above-given question,
given that,
the two numbers are 81 and 46.
81 x 46 = 901.

Question 9.
15 × 16

Answer:
15 x 16 = 240.

Explanation:
In the above-given question,
given that,
the two numbers are 15 and 16.
15 x 16 = 240.

Question 10.
17 × 21

Answer:
17 x 21 = 357.
Explanation:
In the above-given question,
given that,
the two numbers are 17 and 21.
17 x 21 = 357.

Question 11.
12 × 22

Answer:
12 x 22 = 264.
Explanation:
In the above-given question,
given that,
the two numbers are 12 and 22.
12 x 22 = 264.

Question 12.
38 × 41

Answer:
38 x 41 = 1558.

Explanation:
In the above-given question,
given that,
the two numbers are 38 and 41.
38 x 41 = 1558.

Question 13.
42 × 52

Answer:
42 x 52 = 2184.
Explanation:
In the above-given question,
given that,
the two numbers are 42 and 52.
42 x 52 = 2184.

Question 14.
38 × 19

Answer:
38 x 19 = 722.
Explanation:
In the above-given question,
given that,
the two numbers are 38 and 19.
38 x 19 = 722.

Problem Solving

Question 15.
Number Sense The Queen Mary 2’s height above water is about the same as a 14-story building. What is the Queen Mary 2’s height above water?
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 62.1

Answer:
The Queen Mary 2’s height above water = 168 feet.

Explanation:
In the above-given question,
given that,
The Queen Mary 2’s height above water is about the same as a 14-story building.
14 x 12 = 168.
so the Queen Mary 2’s height above water = 168 feet.

Question 16.
Model with Math Write the multiplication equation illustrated by the array drawn on the grid. Find the partial products. Then calculate the final product.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 62.5

Answer:
The partial products are 18 and 15.
the final product is 270.

Explanation:
In the above-given question,
given that,
the two numbers are 18 and 15.
10 + 8 = 18.
10 + 5 = 15.
10 x 10 = 100.
10 x 8 = 80.
100 + 80 = 180.
5 x 10 = 50.
5 x 8 = 40.
50 + 40 = 90.
180 + 90 = 270.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-21

Question 17.
Higher Order Thinking An elevator can carry 15 adults or 20 children at one time. During the course of a day, the elevator carries a full passenger load 52 times. If all the passengers were children, how many more people would the elevator carry than if all the passengers were adults?

Answer:
The more people would the elevator carry than if all the passengers were adults = 1040.

Explanation:
In the above-given question,
given that,
An elevator can carry 15 adults or 20 children at one time.
During the course of a day, the elevator carries a full passenger load 52 times.
52 x 20 = 1040.
so the more people would the elevator carry than if all the passengers were adults = 1040.

Assessment Practice

Question 18.
Ten years ago, Melissa planted a tree in her backyard. She has taken a photo of the tree every week so she can see how it has grown as time passed. How many photos of the tree does Melissa now have?
A. 62 photos
B. 120 photos
C. 520 photos
D. 620 photos
There are 52 weeks in one year.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 64.1

Answer:
The number of photos of the tree does Melissa now has = 520 photos.

Explanation:
In the above-given question,
given that,
Ten years ago, Melissa planted a tree in her backyard.
She has taken a photo of the tree every week so she can see how it has grown as time passed.
there are 52 weeks in one year.
52 x 10 = 520.
so the number of photos of the tree does Melissa now have = 520 photos.

Question 19.
Mr. Morris bought sketchpads for 24 of his students. Each pad contained 50 sheets. How many sheets of paper were in all the pads?
A. 1,000 sheets
B. 1,200 sheets
C. 1,400 sheets
D. 1,600 sheets

Answer:
The number of sheets of paper was in all the pads = 1200 sheets.

Explanation:
In the above-given question,
given that,
Mr. Morris bought sketchpads for 24 of his students.
Each pad contained 50 sheets.
24 x 50 = 1200.
so the number of sheets of paper were in all the pads = 1200 sheets.

Lesson 3.5 Multiply 3-Digit by 2-Digit Numbers

Activity

Solve & Share

A local charity collected 163 cans of food each day for 14 days. How many cans did they collect in all? Explain how you found your answer.

You can use what you know about multiplying 2-digit numbers by 2-digit numbers to help solve the problem.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 64.3

Answer:
The number of cans did they collect in all = 2282 cans.

Explanation:
In the above-given question,
given that,
A local charity collected 163 cans of food each day for 14 days.
163 x 14 = 2282.
so the number of cans did they collect in all = 2282 cans.

Look Back! Make Sense and Persevere How can you check that your answer is reasonable?

Visual Learning Bridge

Essential Question How Do You Multiply 3-Digit Numbers by 2-Digit Numbers?

A.
Last month a bakery sold 389 boxes of bagels. How many bagels did the store sell last month? Find 12 × 389.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 64.4
You can show all partial products or you can use the standard algorithm.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 64.5

B.
Step 1
To use the Standard Algorithm, first multiply by the ones. Regroup as needed.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 70.20
2 × 9 ones = 18 ones or 1 ten and 8 ones
2 × 8 tens =16 tens
16 tens + 1 ten = 17 tens
17 tens = 1 hundred 7 tens
2 × 3 hundreds = 6 hundreds
6 hundreds + 1 hundred = 7 hundreds

C.
Step 2
Multiply by the tens.
Regroup as needed.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 70.1
10 × 9 ones = 90 ones
10 × 8 tens = 80 tens,
or 8 hundred 10 × 3 hundred = 30 hundred, or 3 thousand

D.
Step 3
Add to get the final product.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 70.2
The store sold 4,668 bagels last month.

Convince Me! Construct Arguments Is 300 10 a good estimate for the number of bagels sold at the bakery? Explain.

Guided Practice

Do You Understand?

Question 1.
A theater can seat 540 people at one time. How many tickets are sold if the theater sells out every seat for one 30-day month?

Answer:
The number of tickets is sold if the theater sells out every seat for one 30-day month = 16200.

Explanation:
In the above-given question,
given that,
A theater can seat 540 people at one time.
540 x 30 = 16200.
so the number of tickets are sold if the theater sells out every seat for one 30-day month = 16200.

Question 2.
Number Sense Is 500 30 a good estimate for the number of tickets sold at the theater in one month? Explain.

Answer:
Yes, it is a good estimate for the number of tickets sold at the theater in one month.

Explanation:
In the above-given question,
given that,
A theater can seat 540 people at one time.
540 is equal to 500.
500 x 30 = 15000.
so it is a good estimate for the number of tickets sold at the theater in one month.

Do You Know How?

In 3-6, find each product. Estimate to check that your answer is reasonable.

Question 3.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 71.2

Answer:
236 x 46 = 10856.

Explanation:
In the above-given question,
given that,
the two numbers are 236 and 46.
multiply the numbers.
236 x 46 = 10856.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-22

Question 4.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 71.3

Answer:
61 x 25 = 5185.

Explanation:
In the above-given question,
given that,
the two numbers are 61 and 25.
multiply the numbers.
61 x 25 = 5185.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-23

Question 5.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 71.4

Answer:
951 x 62 = 58962.

Explanation:
In the above-given question,
given that,
the two numbers are 951 and 62.
multiply the numbers.
951 x 62 = 58962.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-24

Question 6.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 71.5

Answer:
185 x 5 = 925.

Explanation:
In the above-given question,
given that,
the two numbers are 185 and 5.
multiply the numbers.
185 x 5 = 925.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-25

Independent Practice

Leveled Practice In 7-18, find each product. Estimate to check that your answer is reasonable.

Question 7.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 71.6

Answer:
51 x 10 = 510.

Explanation:
In the above-given question,
given that,
the two numbers are 51 and 10.
multiply the numbers.
51 x 10 = 510.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-26

Question 8.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 71.7

Answer:
892 x 18 = 16056.

Explanation:
In the above-given question,
given that,
the two numbers are 892 and 18.
multiply the numbers.
892 x 18 = 16056.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-27

Question 9.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 71.8

Answer:
946 x 33 = 31218.

Explanation:
In the above-given question,
given that,
the two numbers are 946 and 33.
multiply the numbers.
946 x 33 = 31218.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-28

Question 10.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 71.10

Answer:
735 x 41 = 30135.

Explanation:
In the above-given question,
given that,
the two numbers are 735 and 41.
multiply the numbers.
735 x 41 = 30135.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-29

Question 11.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 71.13

Answer:
100 x 25 = 2500.

Explanation:
In the above-given question,
given that,
the two numbers are 100 and 25.
multiply the numbers.
100 x 25 = 2500.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-30

Question 12.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 71.14

Answer:
81 x 11 = 891.

Explanation:
In the above-given question,
given that,
the two numbers are 81 and 11.
multiply the numbers.
81 x 11 = 891.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-31

Question 13.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 71.15

Answer:
106 x 7 = 742.

Explanation:
In the above-given question,
given that,
the two numbers are 106 and 7.
multiply the numbers.
106 x 7 = 742.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-32

Question 14.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 71.16

Answer:
90 x 59 = 5310.

Explanation:
In the above-given question,
given that,
the two numbers are 90 and 59.
multiply the numbers.
90 x 59 = 5310.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-33

Question 15.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 71.17

Answer:
360 x 18 = 6480.

Explanation:
In the above-given question,
given that,
the two numbers are 360 and 18.
multiply the numbers.
360 x 18 = 6480.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-34

Question 16.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 71.18

Answer:
222 x 75 = 16650.

Explanation:
In the above-given question,
given that,
the two numbers are 222 and 75.
multiply the numbers.
222 x 75 = 16650.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-35

Question 17.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 71.19

Answer:
481 x 35 = 16835.

Explanation:
In the above-given question,
given that,
the two numbers are 481 and 35.
multiply the numbers.
481 x 35 = 16835.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-36

Question 18.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 71.20

Answer:
659 x 17 = 11203.

Explanation:
In the above-given question,
given that,
the two numbers are 659 and 17.
multiply the numbers.
659 x 17 = 11203.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-37

Problem Solving

Question 19.
enVision® STEM How many times does a rabbit’s heart beat in 1 hour?

Remember, there are 60 minutes in 1 hour.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 72.1

Answer:
The number of times does a rabbit’s heartbeat in 1 hour = 212 beats.

Explanation:
In the above-given question,
given that,
1 hour = 60 minutes.
212 beats per minute.
so the number of times does a rabbit’s heartbeat in 1 hour = 212 beats.

Question 20.
In 1 hour, how many more times does a rabbit’s heart beat than a dog’s heart? Write an equation to show your work.

Answer:
The number of more times does a rabbit’s heartbeat than a dog’s heart = 112.

Explanation:
In the above-given question,
given that,
heart rate of the dog for a minute = 100.
heart rate of the rabbit for a minute = 212.
212 – 100 = 112.
so the number of more times does a rabbit’s heartbeat than a dog’s heart = 112.

Question 21.
Construct Arguments Is 3,198 a reasonable product for 727 × 44? Why or why not?

Answer:
Yes, it is a reasonable product.

Explanation:
In the above-given question,
given that,
the two numbers are 727 and 44.
multiply the two numbers.
727 x 44 = 31,988.
yes, it is a reasonable product.

Question 22.
Higher Order Thinking A garden store sells plants in flats. There are 6 plants in each tray. Each flat has 6 trays. The garden store sold 18 flats on Saturday and 21 flats on Sunday. How many plants did the garden store sell in all?

Answer:
The number of plants did the garden store sell in all = 234.

Explanation:
In the above-given question,
given that,
A garden store sells plants in flats.
There are 6 plants in each tray.
Each flat has 6 trays.
The garden store sold 18 flats on Saturday and 21 flats on Sunday.
18 x 6 = 108.
21 x 6 = 126.
108 + 126 = 234.
so the number of plants did the garden store sell in all = 234.

Assessment Practice

Question 23.
Tricia is building a rectangular patio. The patio will be 108 bricks wide and 19 bricks long. How many bricks does she need to build the patio?

Answer:
The number of bricks does she need to build the patio = 2052.

Explanation:
In the above-given question,
given that,
Tricia is building a rectangular patio.
The patio will be 108 bricks wide and 19 bricks long.
area of the rectangle = l x b.
where l = length and b = breadth.
108 x 19 = 2052.
so the number of bricks does she need to build the patio = 2052.

Question 24.
What is the product?
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 72.3
A. 1,560
B. 1,568
C. 4,268
D. 4,368

Answer:
312 x 14 = 4368.

Explanation:
In the above-given question,
given that,
the two numbers are 312 and 14.
multiply the two numbers.
312 x 14 = 4368.

Lesson 3.6 Multiply Whole Numbers with Zeros

Activity

Solve & Share

A school district is replacing all of the desks in its classrooms. There are 103 classrooms and each classroom needs 24 new desks. How many desks will the school district need to buy? Solve this problem any way you choose!

Use what you know about multiplying 3-digit and 2-digit numbers. Show your work!
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 72.4

Answer:
The number of desks will the school district need to buy = 2472.

Explanation:
In the above-given question,
given that,
A school district is replacing all of the desks in its classrooms.
There are 103 classrooms and each classroom needs 24 new desks.
103 x 24 = 2472.
so the number of desks will the school district need to buy = 2472.

Look Back! Make Sense and Persevere What is a good estimate for the problem above? Explain.

Visual Learning Bridge

Essential Question
How Can You Multiply with Zeros?

A.
An antique steam train makes one sight-seeing tour each day. If every seat is filled for each trip, how many passengers can it carry for 31 tours?
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 72.5

The standard algorithm does not change when there is a zero in a factor.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 72.6

B.
Step 1
Find 31 × 208.
Estimate:
30 × 200 = 6,000
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 80.1

C.
Step 2
Multiply by the ones.
Regroup if necessary.
Remember that multiplying with a zero gives a product of zero.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 80.2

D.
Step 3
Multiply by the tens.
Regroup if necessary.
Add to get the final product.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 80.3
The train can carry 6,448 passengers.

Convince Me! Model with Math Suppose the train fills an average of 102 seats for each tour. What is a reasonable estimate for the number of passengers that the train can carry in 28 tours? Write an equation to show your work.

Answer:
The number of passengers that the train can carry in 28 tours = 2856.

Explanation:
In the above-given question,
given that,
Suppose the train fills an average of 102 seats for each tour.
the two numbers are 102 and 28.
102 x 28 = 2856.
so the number of passengers that the train can carry in 28 tours = 2856.

Guided Practice

Do You Understand?

Question 1.
In an auditorium, there are 104 rows with 24 seats in each row. How many seats are available?

Answer:
The number of seats is available = 2496 seats.

Explanation:
In the above-given question,
given that,
there are 104 rows with 24 seats in each row.
104 x 24 = 2496.
so the number of seats are available = 2496 seats.

Question 2.
Why is it important to “estimate to check for reasonableness”?
Answer:

Do You Know How?

In 3-6, multiply to find the product. Estimate to check for reasonableness.

Question 3.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 80.33

Answer:
205  x 23 = 4715.

Explanation:
In the above-given question,
given that,
the two numbers are 205 and 23.
multiply the numbers.
205 x 23 = 4715.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-38

Question 4.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 80.4

Answer:
108 x 34 = 3672.

Explanation:
In the above-given question,
given that,
the two numbers are 108 and 34.
multiply the numbers.
108 x 34 = 3672.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-39

Question 5.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 80.5

Answer:
410 x 44 = 18040.

Explanation:
In the above-given question,
given that,
the two numbers are 410 and 44.
multiply the numbers.
410 x 44 = 18040.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-40

Question 6.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 80.6

Answer:
302 x 30 = 9060.

Explanation:
In the above-given question,
given that,
the two numbers are 302 and 30.
multiply the numbers.
302 x 30 = 9060.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-41

Independent Practice
Leveled Practice In 7-18, find each product. Estimate to check for reasonableness.

Question 7.
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 80.7

Answer:
302 x 17 = 5134.

Explanation:
In the above-given question,
given that,
the two numbers are 302 and 17.
multiply the numbers.
236 x 46 = 5134.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-42

Question 8.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 80.8

Answer:
608 x 23 = 13984.

Explanation:
In the above-given question,
given that,
the two numbers are 608 and 23.
multiply the numbers.
608 x 23 =13984.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-43

Question 9.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 80.9

Answer:
109 x 47 = 5123.

Explanation:
In the above-given question,
given that,
the two numbers are 109 and 47.
multiply the numbers.
109 x 47 = 5123.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-44

Question 10.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 80.10

Answer:
510 x 72 = 36864.

Explanation:
In the above-given question,
given that,
the two numbers are 510 and 72.
multiply the numbers.
510 x 72 = 36864.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-45

Question 11.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 80.11

Answer:
902 x 35 = 31570.

Explanation:
In the above-given question,
given that,
the two numbers are 902 and 35.
multiply the numbers.
902 x 35 = 31570.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-46

Question 12.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 80.12

Answer:
207 x 61 = 12627.

Explanation:
In the above-given question,
given that,
the two numbers are 207 and 61.
multiply the numbers.
207 x 61 = 12627.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-47

Question 13.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 80.13

Answer:
108 x 58 = 6264.

Explanation:
In the above-given question,
given that,
the two numbers are 108 and 58.
multiply the numbers.
108 x 58 = 6264.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-48

Question 13.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 80.14

Answer:
108 x 58 = 6264.

Explanation:
In the above-given question,
given that,
the two numbers are 108 and 58.
multiply the numbers.
108 x 58 = 6264.

Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-48

Question 14.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 80.15

Answer:
505 x 77 = 38885.

Explanation:
In the above-given question,
given that,
the two numbers are 505 and 77.
multiply the numbers.
505 x 77 = 38885.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-49

Question 15.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 80.16

Answer:
407 x 39 = 15873.

Explanation:
In the above-given question,
given that,
the two numbers are 407 and 39.
multiply the numbers.
407 x 39 = 15873.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-50

Question 16.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 80.17

Answer:
280 x 66 =18480.

Explanation:
In the above-given question,
given that,
the two numbers are 280 and 66.
multiply the numbers.
280 x 66 = 18480.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-51

Question 17.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 80.18

Answer:
105 x 24 =2520.

Explanation:
In the above-given question,
given that,
the two numbers are 105 and 24.
multiply the numbers.
105 x 24 =2520.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-52

Question 18.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 80.19

Answer:
360 x 48 = 17280.

Explanation:
In the above-given question,
given that,
the two numbers are 360 and 48.
multiply the numbers.
360 x 48 = 17280.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-53

Problem Solving

Question 19.
There are 27 students in Mr. Mello’s class. Find the total number of pages the students read by the end of November.

Answer:
The total number of pages the students read by the end of November = 783 pages.

Explanation:
In the above-given question,
given that,
there are 27 students in Mr. Mello’s class.
in November there are 29 days.
29 x 27 = 783.
so the total number of pages the students read by the end of November = 783 pages.

Question 20.
Each student read 41 pages in December. How many total pages did the students read by the end of December?
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 80.20

Answer:
The number of total pages did the students read by the end of December = 1271 pages.

Explanation:
In the above-given question,
given that,
Each student read 41 pages in December.
in December there are 31 pages.
41 x 31 = 1271.
so the number of total pages did the students read by the end of December = 1271 pages.

Question 21.
Meredith says that 15.17 is greater than 15.8 because 17 is greater than 8. Do you agree? Explain your reasoning.

Answer:
No, I do not agree with it.

Explanation:
In the above-given question,
given that,
Meredith says that 15.17 is greater than 15.8 because
17 is greater than 8.
15.17 is less than 15.8.
so I do not agree with it.

Question 22.
Use Structure Trudy wants to multiply 66 × 606. She says that all she has to do is find 6 × 606 and then double that number. Explain why Trudy’s method will not give the correct answer. Then show how to find the correct product.

Answer:
Yes, Trudy’s method will not give the correct answer.

Explanation:
In the above-given question,
given that,
Trudy wants to multiply 66 × 606.
She says that all she has to do is find 6 × 606.
66 x 606 =
6 x 606 =
the two values are not equal.
so Trudy’s method will not give the correct answer.

Question 23.
Higher Order Thinking Maria needs a trombone for only 12 months. Renting the trombone costs $34 per month. She can buy the trombone for $495. Should she buy or rent the trombone? Explain. How much does she pay?

Answer:
Yes, she can rent the trombone.

Explanation:
In the above-given question,
given that,
Maria needs a trombone for only 12 months.
Renting the trombone costs $34 per month.
She can buy the trombone for $495.
12 x $34 = $408.

Question 24.
Another music store rents trombones for $30 per month plus a yearly fee of $48. Which deal is better? Should Maria change her rental plan?

Answer:
Yes, maria change her rental plan.

Explanation:
In the above-given question,
given that,
Another music store rents trombones for $30 per month plus a yearly fee of $48.
30 x 48 = $1440.

Assessment Practice

Question 25.
What is the product?
Envision Math Common Core 5th Grade Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 71.20

Answer:
659 x 17 = 11203.

Explanation:
In the above-given question,
given that,
the two numbers are 659 and 17.
multiply the numbers.
659 x 17 = 11203.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-54

Lesson 3.7 Practice Multiplying Multi-Digit Numbers

Activity

Solve & Share

Which of the two car payment options will cost less for 1 year? How much less? Solve this problem any way you choose! Show all of your work
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 86.2

You can use reasoning to connect mathematics to everyday life. Think about the situations multiplication describes.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 86.3

Answer:
The

Look Back! How can you estimate the total for the year when paying monthly? When paying quarterly?

Visual Learning Bridge

Essential Question How Can You Use Multiplication to Solve Problems?

A.
What is the yearly total for water, gas, and electric? What is the yearly total for cell phones?
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 87.2
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 87.22
The standard algorithm for multiplying whole numbers involves breaking numbers apart using place value.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 87.3

B.
What is the yearly total for water, gas, and electric?
Find 4 × (760 + 510).
Estimate:
4 × (760 + 510) is about 4 × 1,200 = 4,800.
4 × (760+ 510) = 4 × 1,270
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 87.4
The yearly total for water, gas, and electric is $5,080.

C.
What is the yearly total for cell phones?
Find 12 × 271.
Estimate:
12 × 271 is about 10 × 270 = 2,700.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 87.5
The process for multiplying is the same regardless of the number of digits in 3,252 the factors.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 87.6
The yearly total for cell phones is $3,252.

Convince Me! Be Precise How are the processes for multiplying alike for the two calculations above? How are they different?

Guided Practice

Do You Understand?

Question 1.
Carlos saves 18 cents every day of the year. If there are 365 days this year, how many cents will he have saved by the end of the year? Write an equation that represents the problem. Then, solve the equation.

Answer:
The number of cents will he have saved by the end of the year = 6570.

Explanation:
In the above-given question,
given that,
Carlos saves 18 cents every day of the year.
If there are 365 days this year.
365 x 18 = 6570.
so the number of cents will he have saved by the end of the year = 6570.

Question 2.
Lila drives 129 kilometers round trip to work. How many kilometers does she drive in 31 days? Write an equation that represents the problem. Then solve the equation.

Answer:
The number of kilometers does she drive in 31 days = 3999 km.

Explanation:
In the above-given question,
given that,
Lila drives 129 kilometers round trip to work.
129 x 31 = 3999.
so the number of kilometers does she drive in 31 days = 3999 km.

Do You Know How?

In 3-6, estimate each product. Then complete each calculation. Check that your answer is reasonable.

Question 3.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 89.1

Answer:
134 x 11 = 1474.

Explanation:
In the above-given question,
given that,
the two numbers are 134 and 11.
multiply the numbers.
134 x 11 = 1474.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-55

Question 4.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 89.2

Answer:
208 x 26 = 5408.

Explanation:
In the above-given question,
given that,
the two numbers are 208 and 26.
multiply the numbers.
208 x 26 = 5408.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-56

Question 5.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 89.3

Answer:
428 x 35 = 14980.

Explanation:
In the above-given question,
given that,
the two numbers are 428 and 35.
multiply the numbers.
428 x 35 = 14980.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-57

Question 6.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 89.4

Answer:
275 x 56 = 15400.

Explanation:
In the above-given question,
given that,
the two numbers are 275 and 56.
multiply the numbers.
275 x 56 = 15400.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-58

Independent Practice
Leveled Practice In 7-22, estimate and then compute each product. Check that your answer is reasonable.

Question 7.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 89.5

Answer:
531 x 47 = 24,957.

Explanation:
In the above-given question,
given that,
the two numbers are 531 and 47.
multiply the numbers.
531 x 47 = 24,957.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-59

Question 8.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 89.6

Answer:
759 x 68 = 51,612.

Explanation:
In the above-given question,
given that,
the two numbers are 759 and 68.
multiply the numbers.
759 x 68 = 51,612.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-60

Question 9.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 89.7

Answer:
367 x 92 = 33,764.

Explanation:
In the above-given question,
given that,
the two numbers are 367 and 92.
multiply the numbers.
367 x 92 = 33,764.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-61

Question 10.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 89.8

Answer:
817 x 45 = 36,765.

Explanation:
In the above-given question,
given that,
the two numbers are 817 and 45.
multiply the numbers.
817 x 45 = 36,765.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-62

Question 11.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 89.9

Answer:
1206 x 77 = 92862.

Explanation:
In the above-given question,
given that,
the two numbers are 1206 and 77.
multiply the numbers.
1206 x 77 = 92862.

Question 12
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 89.10

Answer:
543 x 18 = 9774.

Explanation:
In the above-given question,
given that,
the two numbers are 543 and 18.
multiply the numbers.
543 x 18 = 9774.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-63

Question 13.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 89.11

Answer:
908 x 62 = 56,296.

Explanation:
In the above-given question,
given that,
the two numbers are 908 and 62.
multiply the numbers.
908 x 62 = 56,296.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-64

Question 14.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 89.12

Answer:
750 x 81 = 60,750.

Explanation:
In the above-given question,
given that,
the two numbers are 750 and 81.
multiply the numbers.
750 x 81 = 60,750.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-65

Question 15.
6,755 × 9

Answer:
6755 x 9 = 60,795.

Explanation:
In the above-given question,
given that,
the two numbers are 6755 and 9.
multiply the numbers.
6755 x 9 = 60,795.

Question 16.
869 × 46

Answer:
869 x 46 = 39,974.

Explanation:
In the above-given question,
given that,
the two numbers are 869 and 46.
multiply the numbers.
869 x 46 = 39,974.

Question 17.
922 × 81

Answer:
922 x 81 = 74,682.

Explanation:
In the above-given question,
given that,
the two numbers are 922 and 81.
multiply the numbers.
922 x 81 = 74,682.

Question 18.
783 × 14

Answer:
783 x 14 = 10,962.

Explanation:
In the above-given question,
given that,
the two numbers are 783 and 14.
multiply the numbers.
783 x 14 = 10,962.

Question 19.
684 × 15

Answer:
684 x 15 = 10,260.

Explanation:
In the above-given question,
given that,
the two numbers are 684 and 15.
multiply the numbers.
684 x 15 = 10,260.

Question 20.
650 × 22

Answer:
650 x 22 = 14,300.

Explanation:
In the above-given question,
given that,
the two numbers are 650 and 22.
multiply the numbers.
650 x 22 = 14,300.

Question 21.
2,525 × 37

Answer:
2,525 x 37 = 93,425.

Explanation:
In the above-given question,
given that,
the two numbers are 2525 and 37.
multiply the numbers.
2525 x 37 = 93,425.

Question 22.
615 × 41

Answer:
615 x 41 = 25,215.

Explanation:
In the above-given question,
given that,
the two numbers are 615 and 41.
multiply the numbers.
615 x 41 = 25,215.

Problem Solving

For 23 and 24, use the table.Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 91.2

Question 23.
Model with Math Jason frequently travels for work. This year he plans to make 15 trips to Chicago. What is the total cost for the airfare? Write an equation that represents the problem. Then, solve the equation.

Answer:
The total cost for the airfare = $7335.

Explanation:
In the above-given question,
given that,
Jason frequently travels for work.
the cost of the chicago is $489.
This year he plans to make 15 trips to Chicago.
$489 x 15 = 7335.
so the total cost of the airfare = $7335.

Question 24.
Which would cost more: 15 trips to Boston or 11 trips to New York? Explain.

Answer:
The trip would cost more.

Explanation:
In the above-given question,
given that,
the ticket cost of Boston is $178.
the ticket cost of new york is $225.
15 x 178 = 2670.
225 x 11 = 1958.
so the trip would cost more.

Question 25.
A cook at a restaurant is planning her food order. She expects to use 115 pounds of potatoes each day for 12 days. How many pounds of potatoes will she order?
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 91.4

Answer:
The number of pounds of potatoes will she order = 1380 pounds.

Explanation:
In the above-given question,
given that,
A cook at a restaurant is planning her food order.
She expects to use 115 pounds of potatoes each day for 12 days.
115 x 12 = 1380.
so the number of pounds of potatoes will she order = 1380.

Question 26.
Higher Order Thinking Carolyn bought a gallon of paint that covers 250 square feet. She wants to paint a wall that is 16 feet wide and 12 feet high. Explain whether or not she will need more than one gallon of paint.

Answer:
Yes, she needs only one gallon of paint.

Explanation:
In the above-given question,
given that,
Carolyn bought a gallon of paint that covers 250 square feet.
She wants to paint a wall that is 16 feet wide and 12 feet high.
16 x 12 = 192.
192 is less than 250.
so she needs less than one gallon of paint.

Assessment Practice

Question 27.
The product of the following expression is 7,453.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 91.5
What is the missing digit?
A. 1
B. 2
C. 4
D. 7

Answer:
Option B is the correct.

Explanation:
In the above-given question,
given that,
257 x 29 = 7453.
so option B is correct.

Question 28.
When you multiply a 3-digit number by a 2-digit number, what is the greatest number of digits the product can have?

Answer:
The greatest number of digits the product can have 4.

Explanation:
In the above-given question,
given that,
the three-digit number is 123.
the two-digit number is 10.
123 x 10 = 1230.

Lesson 3.8 Solve Word Problems Using Multiplication

Activity

Solve&S are

Kevin’s family took 239 photos on their summer vacation. Marco and his family took 12 times as many photos on their vacation. How many photos did Marco’s family take? Solve this problem any way you choose.

How can you use an equation to model the situation with math?
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 93.1

Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 93.2

Answer:
The number of photos did Marco’s family take = 2629 photos.

Explanation:
In the above-given question,
given that,
Kevin’s family took 239 photos on their summer vacation.
Marco and his family took 12 times as many photos on their vacation.
239 x 12 = 2868.
2868 – 239 = 2629.
so the number of photos did Marco’s family take = 2629 photos.

Look Back! How can you use estimation to tell if your answer is reasonable? Explain.

Visual Learning Bridge

Essential Question
How Can You Use a Bar Diagram to Solve a Multiplication Problem?
A.
In 1980, a painting sold for $1,575. In 2015, the same painting sold for 5 times as much. What was the price of the painting in 2015?
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 93.4
You can draw a bar diagram and use a variable to find the new price of the painting.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 93.5

B.
What am I asked to find?
The price of the painting in 2015.
Let p = the price of the painting in 2015.
Draw a bar diagram to represent the problem.
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 93.6

C.
Write and solve an equation using the variable.
$1,575 × 5 =p
$1,575 × 5 = $7,875.
So, p= $7,875.
In 2015, the painting sold for $7,875.
You can use repeated addition or division to check your answer!
Envision Math Common Core 5th Grade Answer Key Topic 3 Fluently Multiply Multi-Digit Whole Numbers 93.7

Convince Me! Construct Arguments How can you use estimation to justify that the answer $7,875 is reasonable?

Guided Practice

Do You Understand?

Question 1.
Write a real-world problem that uses multiplication. Then, draw a bar diagram and write an equation to solve your problem.
Answer:

Do You Know How?

In 2, write and solve an equation.

Question 2.
Sharon’s Stationery Store has 1,219 boxes of cards. May’s Market has 3 times as many boxes of cards. How many boxes of cards does May’s Market have?
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 94.50

Answer:
The number of cards does May’s market have = 2.

Explanation:
In the above-given question,
given that,
Sharon’s Stationery Store has 1,219 boxes of cards.
May’s Market has 3 times as many boxes of cards.
3 x 1219 = 3657.
3657 – 1219 = 2438.
1219 x 2 = 2438.
so the number of cards does May’s market have = 2.

Independent Practice

In 3-5, draw a bar diagram to model the situation. Then, write and solve an equation.

Question 3.
There are 14 theaters at the mall. Each theater has 175 seats. How many seats are there in all?
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 93.2

Answer:
The number of seats is there in all = 2450.

Explanation:
In the above-given question,
given that,
There are 14 theaters at the mall.
Each theater has 175 seats.
14 x 175 = 2450.
so the number of seats are there in all = 2450.

Question 4.
Brad lives 12 times as far away from the ocean as Jennie. If Jennie lives 48 miles from the ocean, how many miles from the ocean does Brad live?

Answer:
The number of miles from the ocean does Brad live = 576 miles.

Explanation:
In the above-given question,
given that,
Brad lives 12 times as far away from the ocean as Jennie.
If Jennie lives 48 miles from the ocean.
48 x 12 = 576.
so the number of miles from the ocean does Brad live = 576 miles.

Question 5.
A hardware store ordered 13 packs of nails from a supplier. Each pack contains 155 nails. How many nails did the store order?

Answer:
The number of nails did the store order = 2015 nails.

Explanation:
In the above-given question,
given that,
A hardware store ordered 13 packs of nails from a supplier.
Each pack contains 155 nails.
13 x 155 = 2015 nails.
so the number of nails did the store order = 2015 nails.

Problem Solving

Question 6.
Algebra Sandi’s school has 1,030 students. Karla’s school has 3 times as many students as Sandi’s school. Write an equation to find s, the number of students in Karla’s school. Then, solve your equation.
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 94.1

Answer:
The number of students in Karla’s school = 3090.

Explanation:
In the above-given question,
given that,
Sandi’s school has 1,030 students.
Karla’s school has 3 times as many students as Sandi’s school.
1030 x 3 = 3090.
so the number of students in Karla’s school = 3090.

Question 7.
enVision® STEM Jupiter is about 5 times the distance Earth is from the Sun. Earth is about 93,000,000 miles from the Sun. About how far is Jupiter from the Sun?
Look for a relationship to help you solve this problem.
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 94.8

Answer:
The far is Jupiter from the sun = 46,50,00,000.

Explanation:
In the above-given question,
given that,
Jupiter is about 5 times the distance Earth is from the Sun.
Earth is about 93,000,000 miles from the sun.
93,000,000 x 5 = 46,50,00,000.
so the far is hupiter from the sun = 46,50,00,000.

Question 8.
Higher Order Thinking William travels only on Saturdays and Sundays and has flown 1,020 miles this month. Jason travels every weekday and has flown 1,200 miles this month. If each man travels about the same number of miles each day, who travels more miles per day for this month? Explain.

Answer:
Jason travels more miles per day for this month.

Explanation:
In the above-given question,
given that,
William travels only on Saturdays and Sundays and has flown 1,020 miles this month.
Jason travels every weekday and has flown 1,200 miles this month.
1200 is greater than 1020.
so Jason travels more miles per day for this month.

Question 9.
Make Sense and Persevere Hwong can fit 12 packets of coffee in a small box and 50 packets of coffee in a large box. He has 10 small boxes of coffee and would like to reorganize the packets into large boxes. How many large boxes could he fill? Explain.

Answer:
The number of large boxes could he fill = 6.

Explanation:
In the above-given question,
given that,
How can fit 12 packets of coffee in a small box and 50 packets of coffee in a large box.
He has 10 small boxes of coffee and would like to reorganize the packets into large boxes.
12 x 50 = 600.
10 x 6 = 600.
so the number of large boxes could he fill = 6.

Assessment Practice

Question 10.
Martin ran 108 miles last year. Katrina ran 13 times as many miles as Martin last year. How many miles did Katrina run last year?
A. 1,008 miles
B. 1,404 miles
C. 1,806 miles
D. 2,000 miles

Answer:
The number of miles did Katrina run last year = 1404 miles.

Explanation:
In the above-given question,
given that,
Martin ran 108 miles last year. Katrina ran 13 times as many miles as Martin last year.
108 x 13 = 1404.
so option B is the correct.

Question 11.
The Erie shoe factory makes 245 pairs of shoes a day. The Columbus shoe factory makes 34 times as many pairs of shoes a day as the Erie shoe factory. How many pairs of shoes does the Columbus shoe factory make a day?
A. 7,545 pairs of shoes
B. 8,010 pairs of shoes
C. 8,330 pairs of shoes
D. 8,750 pairs of shoes

Answer:
Option C is the correct answer.

Explanation:
In the above-given question,
given that,
The Erie shoe factory makes 245 pairs of shoes a day.
The Columbus shoe factory makes 34 times as many pairs of shoes a day as the Erie shoe factory.
245 x 34 = 8,330.
so the option C is the correct.

Lesson 3.9 Critique Reasoning

Activity

Problem Solving

Solve & Share
A group of 44 students is planning a train trip to Washington, D.C. They held many fundraisers and raised $10,880. Nathan said, “We should have enough money to pay for the train tickets. There are about 50 students going on the trip and one round trip ticket costs about $200. That makes the total cost of the tickets less than $10,000.”
Does Nathan’s reasoning make sense?

Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 95.2

Answer:
The total cost of the tickets is less than $10,000.

Explanation:
In the above-given question,
given that,
A group of 44 students is planning a train trip to Washington, D.C.
They held many fundraisers and raised $10,880.
Nathan said, “We should have enough money to pay for the train tickets.
There are about 50 students going on the trip and one round trip ticket costs about $200.
50 x 200 = 10,000.
so the total cost of the ticket is less than $10000.

Thinking Habits
Be a good thinker! These questions can help you.
• What questions can I ask to understand people’s thinking?
• Are there mistakes in other people’s thinking?
• Can I improve other people’s thinking?
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 95.3

Look Back! Critique Reasoning What argument would you make to support Nathan’s estimate?

Visual Learning Bridge

Essential Question
How Can You Critique Reasoning of Others?

A.
Ms. Lynch needs to ship 89 boxes. 47 boxes weigh 150 pounds each. Each of the other boxes weighs 210 pounds.
Mia says that all 89 boxes can fit into one container. She reasons that 47 × 150 is less than 7,500 and 42 × 210 is a little more than 8,000, so the sum of their weights should be less than 15,400.
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 105

What is Mia’s reasoning to support her estimate?
Mia estimates the total weight of the lighter boxes and the total weight of the heavier boxes, then adds the two estimates.

Here’s my thinking…
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 95.4

B.
How can I critique the reasoning of others?
I can
• ask questions for clarification.
• decide if the strategy used makes sense.
• look for flaws in estimates or calculations.

C.
Mia’s reasoning has flaws. She estimated that 42 × 210 is a little more than 8,000, but a better estimate is 9,000.
She underestimated the products so her conclusion is not valid.
The weight of the heavier boxes is 8,820 pounds. The weight of the lighter boxes is 7,050 pounds.
The total weight is 15,870 pounds. The sum is greater than 15,400. Mia’s reasoning does not make sense.

Convince Me! Critique Reasoning Raul states that one way to get the cargo under the weight limit is to remove two of the heavier boxes and one of the lighter boxes. How can you decide if Raul’s reasoning makes sense?

Guided Practice

Critique Reasoning A stadium has 58 sections of seats. There are 288 seats in each section. Mary estimated the total number of seats by multiplying 60 × 300. She concluded that the stadium has fewer than 18,000 seats.

Question 1.
What is Mary’s argument? How does she support it?

Answer:
Mary’s argument was correct.

Explanation:
In the above-given question,
given that,
A stadium has 58 sections of seats.
There are 288 seats in each section.
Mary estimated the total number of seats by multiplying 60 × 300.
She concluded that the stadium has fewer than 18,000 seats.
58 x 288 = 16704.

Question 2.
Describe at least one thing you would do to critique Mary’s reasoning.

Answer:
Mary’s argument was correct.

Explanation:
In the above-given question,
given that,
A stadium has 58 sections of seats.
There are 288 seats in each section.
Mary estimated the total number of seats by multiplying 60 × 300.
She concluded that the stadium has fewer than 18,000 seats.
58 x 288 = 16704.

Question 3.
Does Mary’s conclusion make sense? Explain.

Answer:
Mary’s argument was correct.

Explanation:
In the above-given question,
given that,
A stadium has 58 sections of seats.
There are 288 seats in each section.
Mary estimated the total number of seats by multiplying 60 × 300.
She concluded that the stadium has fewer than 18,000 seats.
58 x 288 = 16704.

Independent Practice

Critique Reasoning
An office manager has $10,000 to spend on new equipment. He planned to purchase 300 lamps for $72 each. He completed the calculations at the right and concluded that there would be plenty of money left to buy additional equipment.
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 96.1

Question 4.
What does the office manager do to support his thinking?

Answer:
Yes, he completed the calculations at the right and concluded that there would be plenty of money.

Explanation:
In the above-given question,
given that,
An office manager has $10,000 to spend on new equipment.
He planned to purchase 300 lamps for $72 each.
300 x 72 = 21,600.
yes, he has enough money.

Question 5.
Describe how you could decide if the office manager’s calculation is reasonable.

Answer:
Yes, he completed the calculations at the right and concluded that there would be plenty of money.

Explanation:
In the above-given question,
given that,
An office manager has $10,000 to spend on new equipment.
He planned to purchase 300 lamps for $72 each.
300 x 72 = 21,600.
yes, he has enough money.

When you critique reasoning, you need to explain if the method used by another makes sense.
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 96.2

Question 6.
Does the office manager’s conclusion make sense? Explain.

Answer:
Yes, he completed the calculations at the right and concluded that there would be plenty of money.

Explanation:
In the above-given question,
given that,
An office manager has $10,000 to spend on new equipment.
He planned to purchase 300 lamps for $72 each.
300 x 72 = 21,600.
yes, he has enough money.

Problem Solving

Performance Task

Buying a Piano
Over the summer Kathleen sold 1,092 jars of jam at outdoor markets. She made a $12 profit on each one. She wants to use the profits to buy the Ivory-5K piano. She said, “Since 1,000 × 12 = 12,000, and 1,092 is greater than 1,000, I know my profits add up to more than $12,000. So, I can buy the piano.”
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 96.3

Question 7.
Make Sense and Persevere Does it make sense for Kathleen to find an overestimate or an underestimate to decide if she has earned enough money? Why?

Answer:
Yes, she can make the Ivory panio.

Explanation:
In the above-given question,
given that,
Kathleen sold 1,092 jars of jam at outdoor markets.
She made a $12 profit on each one.
1092 x 12 = 13,104.
She wants to use the profits to buy the Ivory-5K piano. She said, “Since 1,000 × 12 = 12,000, and 1,092 is greater than 1,000.
yes, she can make the Ivory patio.

Question 8.
Reasoning Should Kathleen use multiplication to estimate her total profits? Explain your reasoning.

Answer:
Yes, she can make the total profits.

Explanation:
In the above-given question,
given that,
Kathleen sold 1,092 jars of jam at outdoor markets.
She made a $12 profit on each one.
1092 x 12 = 13,104.
She wants to use the profits to buy the Ivory-5K piano. She said, “Since 1,000 × 12 = 12,000, and 1,092 is greater than 1,000.
yes, she can make the total profits.

When you critique reasoning, ask questions to help understand someone’s thinking.
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 96.4

Question 9.
Be Precise Is Kathleen’s estimate appropriate? Is her calculation correct? Explain.
Answer:

Question 10.
Critique Reasoning Explain whether Kathleen’s conclusion is logical. How did you decide? If it is not logical, what can you do to improve her reasoning?
Answer:

Topic 3 Fluency Practice

Activity

Follow the path

Solve each problem. Then follow multiples of 10 to shade a path from START to FINISH. You can only move up, down, right, or left.
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 99.7

Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 99.6

Answer:
The multiples of 10 are 1, 10, 20, 5, 2, 30, 40, 50, 60, 70, 80, 90, and 100.

Explanation:
In the above-given question,
given that,
53 x 20 = 1060.
70 x 89 = 6230.
84 x 40 = 3360.
60 x 90 = 5400.
10 x 570 = 5700.
80 x 14 = 1120.
50 x 30 = 1500.
70 x 12 = 840.
100 x 100 = 10000.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-01

Topic 3 Vocabulary Review

Glossary

Word List
• expression
• multiple
• overestimate
• partial products
• power
• underestimate
• variable

For each of these terms, give an example and a non-example.

Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 99.20

Answer:
power of 10 = 200.
multiple of 10 x 10 = 100.
an expression with a variable
an underestimate of 532 x 11 = 5852.

Explanation:
In the above-given question,
given that,
power of 10 = 200.
multiple of 10 x 10 = 100.
an expression with a variable
an underestimate of 532 x 11 = 5852.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-02

Write always, sometimes, or never.

Question 5.
The sum of partial products is equal to the final product.

Answer:
Always the sum of partial products is equal to the final product.

Explanation:
In the above-given question,
given that,
the sum of partial products is equal to the final product.
for example:
12 x 10 = 120.
10 + 2 = 12.
so Always the sum of partial products is equal to the final product.

Question 6.
A multiple of a number is the power of the number.

Answer:
Sometimes a multiple of a number is a power of the number.

Explanation:
In the above-given question,
given that,
multiple of a number is a power of the number.
for example:
2 x 2 = 4.

Question 7.
An underestimate results from rounding each factor to a greater number.

Answer:
Always an underestimate results from rounding each factor to a greater number.

Explanation:
In the above-given question,
given that,
An underestimate results from rounding each factor to a greater number.
for example:
12.5 round the number to tenth.
12.6.

Question 8.
A power of a number is a multiple of the number.

Answer:
Yes, the power of a number is a multiple of the number.

Explanation:
In the above-given question,
given that,
power of a number is a multiple of the number.
for example:
2 x 2 = 4.
the square root of 2 is 4.

Write T for true or F for false.

Question 9.
642 × 12 = 642 tens + 1,284 ones

Answer:
The expression is false.

Explanation:
In the above-given question,
given that,
642 x 12 = 7704.
642 + 1284 = 1926.
so the expression is false.

Question 10.
41 × 106 = 41,000,000

Answer:
The expression is true.

Explanation:
In the above-given question,
given that,
41 x 106.
41 x 10 x 10 x 10 x 10 x 10 x 10.
41000000.
so the expression is true.

Question 11.
80 × 103 = 8,000

Answer:
The expression is false.

Explanation:
In the above-given question,
given that,
80 × 103.
80 x 10 x 10 x 10.
80 x 1000.
80,000.
so the expression is false.

Question 12.
Suppose both factors in a multiplication problem are multiples of 10. Explain why the
number of zeros in the product may be different than the total number of zeros in the factors. Include an example.

Answer:
Topic 3 Reteaching

Set A
pages 81-84

Find 65 × 103.
Look at the exponent for the power of 10. Annex that number of zeros to the other factor to find the product.
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 93.90

Remember to look at the number of zeros or the exponent for the power of 10.

Question 1.
12 × 104

Answer:
The number of zeros is 4.

Explanation:
In the above-given question,
given that,
12 × 104.
12 x 10000.
120000.

Question 2.
100 × 815

Answer:
The number of zeros is 2.

Explanation:
In the above-given question,
given that,
100 x 815.
81500.
so the number of zeros is 2.

Question 3.
102 × 39

Answer:
The number of zeros is 3900.

Explanation:
In the above-given question,
given that,
102 × 39
100 x 39 = 3900.

Question 4.
6,471 × 101

Answer:
The number of zeros is 64710.

Explanation:
In the above-given question,
given that,
6471 x 10.
64710.

Set B
pages 85-88
Estimate 37 × 88.

Step 1
Round both factors.
37 is about 40 and 88 is about 90.
Step 2
Multiply the rounded factors.
40 × 90 = 3,600

Remember to either round the factors or use compatible numbers. Estimate each product.

Question 1.
7 × 396

Answer:
7 x 400 = 2800.

Explanation:
In the above-given question,
given that,
the two numbers are 7 and 396.
396 is equal to 400.
7 x 400 = 2800.

Question 2.
17 × 63

Answer:
17 x 63 = 1071.

Explanation:
In the above-given question,
given that,
the two numbers are 17 and 63.
17 x 63 = 1071.

Question 3.
91 × 51

Answer:
90 x 50 = 4500.

Explanation:
In the above-given question,
given that,
the two numbers are 91 and 51.
91 is equal to 90.
51 is equal to 50.
90 x 50 = 4500.

Question 4.
45 × 806

Answer:
45 x 806 = 36000.

Explanation:
In the above-given question,
given that,
the two numbers are 45 and 806.
806 is equal to 800.
45 x 800 = 36000.

Set C
pages 89-92
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 99.40
Think:
4 × 9 ones = 36; 36 is 3 tens 6 ones.
4 × 4 tens = 16 tens; 16 tens + 3 tens = 19 tens; 19 tens is 1 hundred 9 tens.
4 × 2 hundreds = 8 hundreds; 8 hundreds + 1 hundred = 9 hundreds

Remember to keep track of the place values. Find each product.

Question 1.
133 × 3

Answer:
133 x 3 = 399.

Explanation:
In the above-given question,
given that,
the two numbers are 133 and 3.
multiply two numbers.
133 x 3 = 399.

Question 2.
343 × 5

Answer:
343 x 5 = 1715.

Explanation:
In the above-given question,
given that,
the two numbers are 343 and 5.
multiply two numbers.
343 x 5 = 1715.

Question 3.
893 × 7

Answer:
893 x 7 = 6251.

Explanation:
In the above-given question,
given that,
the two numbers are 893 and 7.
multiply two numbers.
893 x 7 = 6151.

Question 4.
1,278 × 4

Answer:
1278 x 4 = 5112.

Explanation:
In the above-given question,
given that,
the two numbers are 1278 and 4.
multiply two numbers.
1278 x 4 = 5112.

Set D
pages 93-96
Find 17 × 35.
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 99.50

Remember that you can draw arrays or area models to represent multiplication.

Find each product.

Question 1.
21 × 13

Answer:
21 x 13 = 273.

Explanation:
In the above-given question,
given that,
the two numbers are 21 and 13.
multiply two numbers.
21 x 13 = 273.

Question 2.
34 × 52

Answer:
34 x 52 = 1768.

Explanation:
In the above-given question,
given that,
the two numbers are 34 and 52.
multiply two numbers.
34 x 52 = 1768.

Question 3.
89 × 27

Answer:
89 x 27 = 2403.

Explanation:
In the above-given question,
given that,
the two numbers are 89 and 27.
multiply two numbers.
89 x 27 = 2403.

Question 4.
78 × 47

Answer:
78 x 47 = 3666.

Explanation:
In the above-given question,
given that,
the two numbers are 78 and 47.
multiply two numbers.
78 x 47 = 3666.

Set E
pages 97-100, 101-104, 105-108

Find 53 × 406.
Estimate: 50 × 400 = 20,000
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 99.60

Remember to regroup if necessary. Estimate to check that your answer is reasonable.

Find each product.

Question 1.
54 × 9

Answer:
54 x 9 = 486.

Explanation:
In the above-given question,
given that,
the two numbers are 54 and 9.
multiply two numbers.
54 x 9 = 486.

Question 2.
76 × 59

Answer:
76 x 59 = 4484.

Explanation:
In the above-given question,
given that,
the two numbers are 76 and 59.
multiply two numbers.
76 x 59 = 4484.

Question 3.
47 × 302

Answer:
47 x 302 = 14194.

Explanation:
In the above-given question,
given that,
the two numbers are 47 and 302.
multiply two numbers.
47 x 302 = 14194.

Question 4.
32 × 871

Answer:
32 x 871 = 27,872.

Explanation:
In the above-given question,
given that,
the two numbers are 32 and 871.
multiply two numbers.
32 x 871 = 27872.

Question 5.
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 99.61

Answer:
604 x 55 = 33,220.

Explanation:
In the above-given question,
given that,
the two numbers are 604 and 55.
multiply two numbers.
604 x 55 = 33220.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-66

Question 6.
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 99.62

Answer:
7133 x 4 = 28532.

Explanation:
In the above-given question,
given that,
the two numbers are 7133 and 4.
multiply two numbers.
7133 x 4 = 28532.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-67

Set F
pages 109-112
Draw a picture and write an equation. Solve.
The length of James’s pool is 16 feet. The length of the pool at Wing Park is 4 times as long. How long is the pool at Wing Park?
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 99.63
16 × 4 = l
l = 64 feet
The length of Wing Park pool is 64 feet.

Remember that pictures and equations can help you model and solve problems. Draw a picture and write an equation. Solve.

Question 1.
Alexandria has a collection of 34 dolls. A toy store has 15 times as many dolls as Alexandria. How many dolls are in the store?

Answer:
The number of dolls is in the store = 510.

Explanation:
In the above-given question,
given that,
Alexandria has a collection of 34 dolls.
A toy store has 15 times as many dolls as Alexandria.
34 x 15 = 510.
so the number of dolls are in the store = 510.

Question 2.
A store received a shipment of 37 TVs valued at $625 each. What is the total value of the shipment?

Answer:
The total value of the shipment = $23,125.

Explanation:
In the above-given question,
given that,
A store received a shipment of 37 TVS valued at $625 each.
37 x $625 = 23,125.
so the total value of the shipment = $23,125.

Set G
pages 113-116
Think about these questions to help you critique the reasoning of others.

Thinking Habits
• What questions can I ask to understand other people’s thinking?
• Are there mistakes in other people’s thinking?
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 106

Remember you need to carefully consider all parts of an argument.

Sarah has 214 bags of beads. Each bag has enough beads for 22 bracelets. She estimates that since 200 × 20 = 4,000, there are enough beads for at least 4,000 bracelets.
Tell how you can critique Sarah’s reasoning.

Topic 3 Assessment Practice

Question 1.
Dr. Peterson works 178 hours each month. How many hours does she work in a year?
A. 2,000
B. 2,136
C. 3,000
D. 2,200

Answer:
The number of hours does she work in a year = 2136.

Explanation:
In the above-given question,
given that,
Dr. Peterson works 178 hours each month.
1 year = 365 days.
1 week = 7 days.
12 x 178 = 2136.
so option B is the correct.

Question 2.
A banana contains 105 calories. Last week, Brendan and Lea ate a total of 14 bananas. How many calories does this represent?

Answer:
The number of calories does this represent = 1470 calories.

Explanation:
In the above-given question,
given that,
A banana contains 105 calories.
Last week, Brendan and Lea ate a total of 14 bananas.
105 x 14 = 1470 calories.
so the number of calories does this represent = 1470.

Question 3.
At a warehouse, 127 delivery trucks were loaded with 48 packages on each truck.
A. Estimate the total number of packages on the trucks. Write an equation to model your work.
B. Did you calculate an overestimate or an underestimate? Explain how you know.

Answer:
The total number of packages on the trucks = 6096 trucks.

Explanation:
In the above-given question,
given that,
At a warehouse, 127 delivery trucks were loaded with 48 packages on each truck.
127 x 48 = 6096.
so the total number of packages on the trucks = 6096.

Question 4.
Is the equation below correct? Explain.
5.6 × 103 = 560
A. The equation is incorrect. The product should have 3 zeros.
B. The equation is correct. The product should have 1 zero.
C. The equation is incorrect. The product should have 0 zeros.
D. The equation is incorrect. The product should have 2 zeros.

Answer:
Option A is correct.

Explanation:
In the above-given question,
given that,
5.6 × 103 = 560.
5.6 = 560.
560 x 1000 = 560000.
so option A is correct.

Question 5.
The latest mystery novel costs $24. The table shows the sales of this novel by a bookstore.
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 99.64
A. What was the dollar amount of sales of the mystery novel on Saturday? Write an equation to model your work.
B. What was the dollar amount of sales of the mystery novel on Friday? Write an equation to model your work.

Answer:
A. The dollar amount of sales of the mystery novel on Saturday = 2472.
B. The dollar amount of sales of the mystery novel on Friday = 3768.

Explanation:
In the above-given question,
given that,
The latest mystery novel costs $24.
98 books were sold on Thursday.
103 books were sold on Friday.
157 books were sold on Saturday.
116 books were sold on Sunday.
103 x 24 = 2472.
157 x 24 = 3768.

Question 6.
There are 45 cans of mi×ed nuts. Each can has 338 nuts. Below is Mary’s work to find the total number of nuts. What is the missing number? Enter your answer in the box.
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 99.65

Answer:
The missing number is 5.

Explanation:
In the above-given question,
given that,
There are 45 cans of mi×ed nuts.
Each can have 338 nuts.
338 x 45 = 15210.
so the missing number is 5.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-68

Question 7.
There are 36 large fish tanks at the zoo. Each tank holds 205 gallons of water. How many gallons of water would it take to fill all of the tanks?

Answer:
The number of gallons of water would it take to fill all of the tanks = 7380 gallons.

Explanation:
In the above-given question,
given that,
There are 36 large fish tanks at the zoo.
Each tank holds 205 gallons of water.
205 x 36 = 7380.
so the number of gallons of water would it take to fill all of the tanks = 7380 gallons.

Question 8.
Kai ordered 1,012 baseball cards. Sharon ordered 5 times as many cards as Kai. Write and solve an equation to find b, the number of baseball cards Sharon ordered.
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 99.70

Answer:
The number of baseball cards Sharon ordered = 5060 cards.

Explanation:
In the above-given question,
given that,
Kai ordered 1,012 baseball cards.
Sharon ordered 5 times as many cards as Kai.
1012 x 5 = 5060.
so the number of baseball cards Sharon ordered = 5060 cards.

Question 9.
Multiply
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 99.71

Answer:
289 x 16 = 4624.

Explanation:
In the above-given question,
given that,
the two numbers are 289 x 16.
multiply the two numbers.
289 x 16 = 4624.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-69

Question 10.
Match each number on the left with an equivalent expression.
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 99.72

Answer:
12 x 100 = 1200.
120 = 12 x 10.
12 = 12 x 10.
12000 = 12 x 1000.

Explanation:
In the above-given question,
given that,
12 x 100 = 1200.
120 = 12 x 10.
12 = 12 x 10.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-3- Fluently Multiply Multi-Digit Whole Numbers-70

Question 11.
Select all the expressions that are equal to 3 × 103.
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 108 3 × 1,000
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 108 3 × 100
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 108 30 × 100
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 108 300 × 100
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 108 300 × 10

Answer:
3 x 1000, 30 x 100, 300 x 10.

Explanation:
In the above-given question,
given that,
3 x 1000 = 3000.
30 x 100 = 3000.
300 x 10 = 3000.

Question 12.
Rosanne has 142 songs on her MP3 player. Teresa has 11 times as many songs as Rosanne. How many songs does Teresa have?

Answer:
The number of songs does Teresa has = 1562 songs.

Explanation:
In the above-given question,
given that,
Rosanne has 142 songs on her MP3 player.
Teresa has 11 times as many songs as Rosanne.
142 x 11 = 1562 songs.
so the number of songs does Teresa has = 1562 songs.

Topic 3 Performance Task

Baseball Apparel
Coach Sandberg wants to buy items for the baseball league. The league already has caps with the league logo on them, but the coach would like to offer the option of purchasing a T-shirt, sweatshirt, sweatpants, or jacket with the logo. Use the information in the table to answer the questions.

Question 1.
The players asked their families and friends if they want to buy T-shirts with the league logo. If 254 people want T-shirts, what would be the total cost? Write an equation to model your work.

Answer:
The total cost is $3556.

Explanation:
In the above-given question,
given that,
The players asked their families and friends if they want to buy T-shirts with the league logo.
If 254 people want T-shirts,
254 x $14 = 3556.
so the total cost is $3556.

Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 110

Question 2.
Coach Sandberg wants to order 127 sweatshirts.
Part A
Will the total cost of the sweatshirts be greater than or less than $3,000? Use estimation to decide. Explain your reasoning.
Part B
What is the total cost of 127 sweatshirts?

Answer:
The total cost of 127 sweatshirts = $4064.

Explanation:
In the above-given question,
given that,
the cost of sweatshirts = $32.
127 x $32 = $4064.
so the total cost of 127 sweatshirts = $4064.

Question 3.
Which would cost more, 32 T-shirts or 14 sweatshirts? How can you tell without multiplying?

Answer:
The two items cost the same.

Explanation:
In the above-given question,
given that,
the cost of the T-shirts = $14.
cost of  sweatshirts = $32.
32 x 14 = $448.
14 x 32 = $448.
so the two items cost the same.

Question 4.
There are 18 × 101 players in the league.
Part A
The league raised $1,560 through fundraisers. Trenton estimates the cost of buying jackets for each player in the league. He concludes that the league has raised enough money. Do you agree with Trenton? Explain.
Envision Math Common Core Grade 5 Answers Topic 3 Fluently Multiply Multi-Digit Whole Numbers 102

Answer:
Yes, I agree with it.

Explanation:
In the above-given question,
given that,
The league raised $1,560 through fundraisers.
Trenton estimates the cost of buying jackets for each player in the league.
200 x 50 = 1000.
so I agree with it.

Part B
How much would it cost to order sweatpants for each player? Write and solve an equation with a variable to show your work.

Answer:
The cost to order sweatpants for each player =

Explanation:
In the above-given question,
given that,
$24

Question 5.
Which costs more: 136 sweatpants or 103 sweatshirts? How much more?

Answer:
The more is 32.

Explanation:
In the above-given question,
given that,
136 x $24 = $3264.
103 x $32 = 3296.
3296 – 3264 = 32.

Question 6.
Coach Sandberg wants to order 115 jackets and 27 caps for $12 each.
Part A
Estimate the total cost for his order. Show your work.
Part B
What is his total cost? Compare your answer to your estimate.

Answer:
The total cost is $439.

Explanation:
In the above-given question,
given that,
Coach Sandberg wants to order 115 jackets and 27 caps for $12 each.
27 x 12 = $324.
$324 + 115 = 439.

enVision Math Common Core Grade 5 Answer Key Topic 4 Use Models and Strategies to Multiply Decimals

enVision Math Common Core Grade 5 Answer Key Topic 4 Use Models and Strategies to Multiply Decimals

Go through the enVision Math Common Core Grade 5 Answer Key Topic 4 Use Models and Strategies to Multiply Decimals regularly and improve your accuracy in solving questions.

enVision Math Common Core 5th Grade Answers Key Topic 4 Use Models and Strategies to Multiply Decimals

Essential Question: What are some common procedures for estimating and finding products involving decimals?

Envision Math Common Core Grade 5 Answer Key Topic 4 Use Models and Strategies to Multiply Decimals 1
enVision STEM Project: Solar Energy
Do Research Use the Internet or other sources to learn about solar energy. Find at least five ways that we use the Sun’s energy today.

Answer:
To dry your clothes.
To grow your food.
To heat your water.
To power your car.
To generate your electricity.

Explanation:
In the above-given question,
given that,
The sun has gone to a lot of trouble to send us its energy.
we can use solar energy in many ways.
they are:
to dry your clothes.
to grow your food.
to heat your water.
to power your car.
to generate your electricity.

Journal: Write a Report Include what you found. Also in your report:
• Describe at least one way that you could use solar energy. Could it save you money?
• Estimate how much your family pays for energy costs such as lights, gasoline, heating, and cooling.
• Make up and solve problems by multiplying whole numbers and decimals.

Review What You Know

Vocabulary

Choose the best term from the box. Write it on the blank.
• exponent
• hundredths
• overestimate
• partial products
• power
• round
• tenths
• thousandths
• underestimate

Use Models and Strategies to Multiply Decimals 1

Question 1.
One way to estimate a number is to ____ the number.

Answer:
One way to estimate a number is to round the number.

Explanation:
In the above-given question,
given that,
one way to estimate a number is to round the number.
for example:
2.456 round to tenths.
2.556.

Question 2.
Using 50 for the number of weeks in a year is a(n) _____.

Answer:
Using 50 for the number of weeks in a year is a(n) is exponents.

Explanation:
In the above-given question,
given that,
using 50 for the number of weeks in a year is a(n) is exponents.
for example:
a(n) = 50.
a = 5.
50/5 = 10.

Question 3.
In the number 3.072, the digit 7 is in the ___ place and the digit 2 is in the ____ place.

Answer:
In the number 3.072, the digit 7 is in the hundredths place and the digit 2 is in the thousandths place.

Explanation:
In the above-given question,
given that,
In the number 3.072, the digit 7 is in the hundredths place and the digit 2 is in the thousandths place.
for example:
3.072.
7 is in the hundredths place.
2 is in the thousands place.

Question 4.
10,000 is a(n) ____ of 10 because 10 × 10 × 10 × 10 = 10,000.

Answer:
10,000 is a(n) power of 10 because 10 x 10 x 10 x 10 = 10,000.

Explanation:
In the above-given question,
given that,
for example:
10 x 10 x 10 x 10.
100 x 100 = 10,000.

Whole Number Multiplication Find each product.

Use Models and Strategies to Multiply Decimals 2

Question 5.
64 × 100

Answer:
The product is 6400.

Explanation:
In the above-given question,
given that,
the two numbers are 64 and 100.
multiply the two numbers.
64 x 100 = 6400.
so the product is 6400.

Question 6.
7,823 × 103

Answer:
The product is 7823000.

Explanation:
In the above-given question,
given that,
the two numbers are 7823 and 1000.
multiply the two numbers.
7823 x 1000 = 7823000.
so the product is 7823000.

Question 7.
10 × 1,405

Answer:
The product is 14050.

Explanation:
In the above-given question,
given that,
the two numbers are 10 and 1405.
multiply the two numbers.
10 x 1405 = 14050.
so the product is 14050.

Question 8.
53 × 413

Answer:
The product is 21889.

Explanation:
In the above-given question,
given that,
the two numbers are 53 and 413.
multiply the two numbers.
53 x 413 = 21889.
so the product is 21889.

Question 9.
906 × 57

Answer:
The product is 51,642.

Explanation:
In the above-given question,
given that,
the two numbers are 906 and 57.
multiply the two numbers.
906 x 57 = 51,642.
so the product is 51,642.

Use Models and Strategies to Multiply Decimals 3

Question 10.
1,037 × 80

Answer:
The product is 82,960.

Explanation:
In the above-given question,
given that,
the two numbers are 1037 and 80.
multiply the two numbers.
1037 x 80 = 82,960.
so the product is 82,960.

Round Decimals

Round each number to the nearest tenth.

Question 11.
842.121

Answer:
The number to the nearest tenth = 842.10.

Explanation:
In the above-given question,
given that,
the number is 842.121.
to round a number to the nearest tenth look at the number of ones.
if this is 5 or more round up.
if it is 4 or less round down.
842.10.
so the number to the nearest tenth = 842.10.

Question 12.
10,386.145

Answer:
The number to the nearest tenth = 10386.10.

Explanation:
In the above-given question,
given that,
the number is 10,386.145.
to round a number to the nearest tenth look at the number of ones.
if this is 5 or more round up.
if it is 4 or less round down.
10386.10.
so the number to the nearest tenth = 10386.10.

Question 13.
585.055

Answer:
The number to the nearest tenth = 585.155.

Explanation:
In the above-given question,
given that,
the number is 585.055.
to round a number to the nearest tenth look at the number of ones.
if this is 5 or more round up.
if it is 4 or less round down.
585.055.
so the number to the nearest tenth = 585.155.

Properties of Multiplication
Use the Commutative and Associative Properties of Multiplication to complete each multiplication.

Use Models and Strategies to Multiply Decimals 4

Question 14.
96 × 42 = 4,032 so 42 × 96 = ___

Answer:
42 x 96 = 4032.

Explanation:
In the above-given question,
given that,
the two numbers are 96 and 42.
multiply the two numbers.
96 x 42 = 4032.
42 x 96 = 4032.

Question 15.
4 (58 × 25) = 4 × (25 × ___) = (___ × ___) × 58 = ___

Answer:
4(58 x 25) = 4 x (25 x 58) = (25 x 4) x 58 = 5800.

Explanation:
In the above-given question,
given that,
the two numbers are 58, 4, and 25.
multiply the two numbers.
4 x 25 x 58.
25 x 4 x 8 = 5800.

Question 16.
(293 × 50) × 20 = 293 × (50 × ___) = ___

Answer:
293 x 50 x 20 = 293 x 50 x 20 = 2,93,000.

Explanation:
In the above-given question,
given that,
the two numbers are 293, 50, and 20.
multiply the two numbers.
293 x 50 x 20.
50 x 293 x 20 = 2,93,000.

pick a Project

PROJECT 4A

How can you set up an exercise plan?
Project: Plan an exercise Program
Envision Math Common Core Grade 5 Answer Key Topic 4 Use Models and Strategies to Multiply Decimals 3.1

PROJECT 4B
How much does it cost to dress a team?
Project: Budget a Team
Envision Math Common Core Grade 5 Answer Key Topic 4 Use Models and Strategies to Multiply Decimals 3.2

PROJECT 4C
How far can a rocket go in 100 seconds?
Project: Make a Poster
Envision Math Common Core Grade 5 Answer Key Topic 4 Use Models and Strategies to Multiply Decimals 3.3

Answer:
The rocket can go in 100 sec = 790 km.

Explanation:
The above-given question,
given that,
the rocket can go in 100 sec is:
1 minute = 60 sec.
1 sec = 7.9 km.
100sec = 7.9 x 100.
790 km.

PROJECT 4D
How much extra do you have to pay?
Project: Make a Data Display
Envision Math Common Core Grade 5 Answer Key Topic 4 Use Models and Strategies to Multiply Decimals 3.4

Lesson 4.1 Multiply Decimals by Powers of 10

Activity

Solve & Share

Javier is helping his parents put up posters in their movie theater. Each poster has a thickness of 0.012 inch. How thick is a stack of 10 posters? 100 posters? 1,000 posters? Solve this problem any way you choose.

Answer:
The thick is a stack of 10 posters = 0.12.
the thick is a stack of 100 posters = 1.2.
the thick is a stack of 1000 posters = 12.

Explanation:
In the above-given question,
given that,
Javier is helping his parents put up posters in their movie theater.
Each poster has a thickness of 0.012.
0.012 x 10 = 0.12.
0.012 x 100 = 1.2.
0.012 x 1000 = 12.

You can use the structure of our number system and mental math to help you.
Envision Math Common Core Grade 5 Answer Key Topic 4 Use Models and Strategies to Multiply Decimals 3.5

Look Back! Use Structure How is your answer for 1,000 posters similar to 0.012? How is it different?

Visual Learning Bridge

Essential Question
What Patterns Can Help You Multiply Decimals by Powers of 10?

A.
You can use place value and what you know about whole numbers to multiply decimals by powers of 10. What patterns can you find?

We already know what happens when a whole number is multiplied by powers of 10.
Envision Math Common Core Grade 5 Answer Key Topic 4 Use Models and Strategies to Multiply Decimals 3.6
Envision Math Common Core Grade 5 Answer Key Topic 4 Use Models and Strategies to Multiply Decimals 3.7

B.
In a place-value chart, the same pattern appears when a decimal is multiplied by powers of 10.
Envision Math Common Core Grade 5 Answer Key Topic 4 Use Models and Strategies to Multiply Decimals 3.8
A digit in one place is worth 10 times more when moved to the place on its left. Every time a number is multiplied by 10, the digits of the number shift to the left.

C.
Holding the numbers still, another pattern appears.
3.63 × 1 = 3.63
3.63 × 101 = 36.3
3.63 × 102 = 363.0
3.63 × 103 = 3630.0
It looks like the decimal point moves to the right each time.
Envision Math Common Core Grade 5 Answer Key Topic 4 Use Models and Strategies to Multiply Decimals 3.9

Convince Me! Use Structure Complete the chart. What patterns can you use to place the decimal point?
Envision Math Common Core Grade 5 Answer Key Topic 4 Use Models and Strategies to Multiply Decimals 3.10

Answer:
1.275 x 10 = 12.75, 1.275 x 100 = 127.5, 1.275 x 1000 = 1275.
26.014 x 10 = 260.14, 26.014 x 100 = 2601.4, 26.014x 1000 = 26014.
0.4 x 10 = 4, 0.4 x 100 = 40, 0.4 x 1000 = 400.

Explanation:
In the above-given question,
given that,
the numbers are 1.275, 26.014, and 0.4.
multiply the numbers by 10, 100, and 1000.
1.275 x 10 = 12.75, 1.275 x 100 = 127.5, 1.275 x 1000 = 1275.
26.014 x 10 = 260.14, 26.014 x 100 = 2601.4, 26.014x 1000 = 26014.
0.4 x 10 = 4, 0.4 x 100 = 40, 0.4 x 1000 = 400.
Envision-Math-Common-Core-5th-Grade-Answer-Key-Topic-4- Use Models and Strategies to Multiply Decimals-1

Guided Practice

Do You Understand?

Question 1.
When multiplying by a power of 10, like 4.58 × 103, how do you know you are moving the decimal in the correct direction?

Answer:
The number is 4580.

Explanation:
In the above-given question,
given that,
4.58 × 103.
4.58 x 10 x 10 x 10.
4.58 x 1000 = 4580.
so the number is 4580.

Do You Know How?

In 2-5, find each product.

Question 2.
0.009 × 10

Answer:
The product is 0.09.

Explanation:
In the above-given question,
given that,
the numbers are 0.009  and 10.
multiply the numbers.
0.009 x 10 = 0.09.
so the product is 0.09.

Question 3.
3.1 × 103

Answer:
The product is 3100.

Explanation:
In the above-given question,
given that,
the numbers are 3.1 and 1000.
multiply the numbers.
3.1 x 1000 = 3100.
so the product is 3100.

Question 4.
0.062 × 102

Answer:
The product is 6.2.

Explanation:
In the above-given question,
given that,
the numbers are 0.062  and 100.
multiply the numbers.
0.062 x 100 = 6.2.
so the product is 6.2.

Question 5.
1.24 × 104

Answer:
The product is 1240.

Explanation:
In the above-given question,
given that,
the numbers are 1.24  and 10000.
multiply the numbers.
1.24 x 10000 = 1240.
so the product is 1240.

Independent Practice

Leveled Practice in 6 and 7, find each product.

Place-value patterns can help you solve these problems.
Envision Math Common Core Grade 5 Answer Key Topic 4 Use Models and Strategies to Multiply Decimals 4.2

Question 6.
42.3 ×1 = ___
42.3 × 10 = ___
42.3 × 102 = ___

Answer:
42.3 x 1 = 42.3.
42.3 x 10 = 423.
42.3 x 100 = 4230.

Explanation:
In the above-given question,
given that,
the numbers are 42.3, 42.2 x 10, and 42.3 x 100.
42.3 x 1 = 42.3.
42.3 x 10 = 423.
42.3 x 100 = 4230.

Question 7.
____ = 0.086 × 101
___ = 0.086 × 100
____ = 0.086 × 1,000

Answer:
0.086 x 10 = 0.86.
0.086 x 100 = 8.6.
0.086 x 1000 = 86.

Explanation:
In the above-given question,
given that,
the numbers are 0.086 x 10, 0.086 x 100, and 0.086 x 1000.
0.086 x 10 = 0.86.
0.086 x 100 = 8.6.
0.086 x 1000 = 86.

In 8-15, find each product.

Question 8.
63.7 × 10

Answer:
The product is 637.

Explanation:
In the above-given question,
given that,
the numbers are 63.7  and 10.
multiply the numbers.
63.7 x 10 = 637.
so the product is 637.

Question 9.
563.7 × 102

Answer:
The product is 56370.

Explanation:
In the above-given question,
given that,
the numbers are 563.7  and 100.
multiply the numbers.
563.7 x 100 = 56370.
so the product is 56370.

Question 10.
0.365 × 104

Answer:
The product is 3650.

Explanation:
In the above-given question,
given that,
the numbers are 0.365  and 10000.
multiply the numbers.
0.365 x 10000 = 3650.
so the product is 3650.

Question 11.
5.02 × 100

Answer:
The product is 502.

Explanation:
In the above-given question,
given that,
the numbers are 5.02  and 100.
multiply the numbers.
5.02 x 100 = 502.
so the product is 502.

Question 12.
94.6 × 103

Answer:
The product is 94600.

Explanation:
In the above-given question,
given that,
the numbers are 94.6  and 1000.
multiply the numbers.
94.6 x 1000 = 94600.
so the product is 94600.

Question 13.
0.9463 × 102

Answer:
The product is 94.63.

Explanation:
In the above-given question,
given that,
the numbers are 0.9463  and 100.
multiply the numbers.
0.9463 x 100 = 94.63.
so the product is 94.63.

Question 14.
0.678 × 100

Answer:
The product is 67.8.

Explanation:
In the above-given question,
given that,
the numbers are 0.678  and 100.
multiply the numbers.
0.678 x 100 = 67.8.
so the product is 67.8.

Question 15.
681.7 × 104

Answer:
The product is 6817000.

Explanation:
In the above-given question,
given that,
the numbers are 681.7 and 10000.
multiply the numbers.
681.7 x 10000 = 6817000.
so the product is 6817000.

In 16-18, find the missing exponent.

Question 16.
0.629 × Envision Math Common Core Grade 5 Answer Key Topic 4 Use Models and Strategies to Multiply Decimals 6 = 62.9

Answer:
The missing exponent is 2.

Explanation:
In the above-given question,
given that,
0.629 x 10 x 10.
0.629 x 100 = 62.9.
0.629

Question 17.
Envision Math Common Core Grade 5 Answer Key Topic 4 Use Models and Strategies to Multiply Decimals 6 × 0.056 = 560

Answer:
The missing exponent is 4.

Explanation:
In the above-given question,
given that,
10 x 10 x 10 x 10 x 0.056.
100 x 100 x 0.056.
10000 x 0.056.
560.

Question 18.
1.23 = Envision Math Common Core Grade 5 Answer Key Topic 4 Use Models and Strategies to Multiply Decimals 6 × 0.123

Answer:
The missing exponent is 0.

Explanation:
In the above-given question,
given that,
1.23 x 10.
1.23.

Problem Solving

In 19-21, use the table to find the answers.

Question 19.
Monroe uses a microscope to observe specimens in science class. The microscope enlarges objects to 100 times their actual size. Find the size of each specimen as seen in the microscope.

Answer:
The size of each specimen as seen in the microscope = 0.8, 11, 0.25, and 0.4.

Explanation:
In the above-given question,
given that,
Monroe uses a microscope to observe specimens in science class.
The microscope enlarges objects to 100 times their actual size.
0.008 x 100 = 0.8.
0.011