Category: Envision Math

Go through the enVision Math Common Core Grade 8 Answer Key Topic 1 Real Numbers regularly and improve your accuracy in solving questions.

enVision Math Common Core 8th Grade Answers Key Topic 1 Real Numbers

enVision STEM Project

Did You Know?
Natural resources are materials that occur in nature, such as water, fossil fuels, wood, and minerals. Natural resources not only meet basic human needs, but also support industry and economy.

Minerals are used in the manufacturing of all types of common objects, including cell phones, computers, light bulbs, and medicines.

Water, oil, and forests are some of the natural resources that are in danger of someday being depleted.
70% of available fresh water is used in agriculture…

…and 10% for human consumption.

Each person in the United States needs over 48,000 pounds of minerals each year.

About 18 million acres of forest are lost to deforestation each year.

Solar power, wind power and other renewable energy sources are helping to lessen the dependency on oil and fossil fuels.

Fossil fuels are expected to supply almost 80% of world energy use through 2040.

Your Task: Going, Going, Gone?

Natural resource depletion is an important issue facing the world. Suppose a natural resource is being depleted at the rate of 1.333% per year. If there were 300 million tons of this resource in 2005, and there are no new discoveries, how much will be left in the year 2045? You and your classmates will explore the depletion of this resource over time.

Answer:
It is given that
A natural resource is being depleted at the rate of 1.333% per year and there were 300 million tons of this resource in 2005, and there are no new discoveries
So,
The number of resources left in 2045 = The number of resources present in 2005 – The decrease of the number of resources from 2005 to 2045
So,
The number of resources left in 2045 = 300 million – 1.33% of 300 million × (2045 – 2005)
The number of resources left in 2045 = 300 million – 159.6 million
The number of resources left in 2045 = 140.4 million
Hence, from the above,
We can conclude that the number of resources left in 2045 is: 140.4 million

Topic 1 GET READY!

Review What You Know!

Vocabulary

Choose the best term from the box. Write it on the blank.

Question 1.
A(n) ____ is a decimal that ends in repeating zeros.
Answer:
A “Terminating decimal” is a decimal that ends in repeating zeros
Hence, from the above,
We can conclude that the best term from the box for this definition is “Terminating decimal”

Question 2.
A(n) ____ is a decimal in which a digit or digits repeat
Answer:
A “Repeating decimal” is a decimal in which a digit or digits repeat
Hence, from the above,
We can conclude that the best term from the box for this definition is “Repeating decimal”

Question 3.
A(n) ____ is either a counting number, the opposite of a counting number, or zero
Answer:
An “Integer” is either a counting number, the opposite of a counting number, or zero
Hence, from the above,
We can conclude that the best term from the box for this definition is “Integer”

Question 4.
A(n) ___ is a number that can be used to describe a part of a whole, a part of a set, a location on a number line, or a division of whole numbers.
Answer:
A “Fraction” is a number that can be used to describe a part of a whole, a part of a set, a location on a number line, or a division of whole numbers.
Hence, from the above,
We can conclude that the best term from the box for this definition is “Fraction”

Terminating and Repeating Decimals

Determine whether each decimal is terminating or repeating.

Question 5.
5.692
Answer:
The given decimal is: 5.692
We know that,
A terminating decimal is usually defined as a decimal number that contains a finite number of digits after the decimal point
Hence, from the above,
We can conclude that the given decimal is a terminating decimal

Question 6.
-0.222222…
Answer:
The given decimal is -0.222222….
We know that,
A repeating decimal or recurring decimal is the decimal representation of a number whose digits are periodic and the infinitely repeated portion is not zero
Hence, from the above,
We can conclude that the given decimal is a repeating decimal

Question 7.
7.0001
Answer:
The given decimal is: 7.0001
We know that,
A terminating decimal is usually defined as a decimal number that contains a finite number of digits after the decimal point
Hence, from the above,
We can conclude that the given decimal is a terminating decimal

Question 8.
7.2\(\overline{8}\)
Answer:
The given decimal is: 7.2\(\overline{8}\)
We know that,
A repeating decimal or recurring decimal is the decimal representation of a number whose digits are periodic and the infinitely repeated portion is not zero
Hence, from the above,
We can conclude that the given decimal is a repeating decimal

Question 9.
1.\(\overline{178}\)
Answer:
The given decimal is: 1.\(\overline{178}\)
We know that,
A repeating decimal or recurring decimal is the decimal representation of a number whose digits are periodic and the infinitely repeated portion is not zero
Hence, from the above,
We can conclude that the given decimal is a repeating decimal

Question 10.
-4.03479
Answer:
The given decimal is: -4.03479
We know that,
A terminating decimal is usually defined as a decimal number that contains a finite number of digits after the decimal point
Hence, from the above,
We can conclude that the given decimal is a terminating decimal

Multiplying Integers

Find each product.

Question 11.
2.2
Answer:
The given expression is: 2 × 2
Hence,
2 × 2 = 4

Question 12.
-5. (-5)
Answer:
The given expression is: -5 × -5
We know that,
– × – = +
Hence,
-5 × -5 = 25

Question 13.
7.7
Answer:
The given expression is: 7 × 7
Hence,
7 × 7 = 49

Question 14.
-6 ∙ (-6) ∙ (-6)
Answer:
The given expression is: -6 × -6 × -6
We know that,
– × – × –
= + × –
= –
Hence,
-6 × -6 × -6
= 36 × -6
= -216

Question 15.
10 ∙ 10 ∙ 10
Answer:
The given expression is: 10 × 10 × 10
Hence,
10 × 10 × 10 = 1,000

Question 16.
-9 ∙ (-9) ∙ (-9)
Answer:
The given expression is: -9 × -9 × -9
We know that,
– × – × –
= + × –
= –
Hence,
-9 × -9 × -9
= 81 × -9
= -729

Simplifying Expressions
Simplify each expression.

Question 17.
(4 ∙ 10) + (5 ∙ 100)
Answer:
The given expression is:
(4 × 10) + (5 × 100)
So,
(4 × 10) + (5 × 100)
= 40 + 500
= 540
Hence, from the above,
We can conclude that the value of the expression is: 540

Question 18.
(2100) + (7.10)
Answer:
The given expression is:
2100 + (7 × 10)
So,
2100 + (7 × 10)
= 2100 + 70
= 2170
Hence, from the above,
We can conclude that the value of the given expression is: 2170

Question 19.
(6 · 100) – (1 · 10)
Answer:
The given expression is:
(6 × 100) – (1 × 10)
So,
(6 × 100) – (1 × 10)
= 600 – 10
= 590
Hence, from the above,
We can conclude that the value of the given expression is: 590

Question 20.
(9 ∙ 1,000) + (4 ∙ 10)
Answer:
The given expression is:
(9 × 1,000) + (4 × 10)
So,
(9 × 1,000) + (4 × 10)
= 9,000 + 40
= 9,040
Hence, from the above,
We can conclude that the value of the given expression is: 9,040

Question 21.
(3 · 1,000) – (2 ∙ 100)
Answer:
The given expression is:
(3 × 1,000) – (2 × 100)
So,
(3 × 1,000) – (2 × 100)
= 3,000 – 200
= 2,800
Hence, from the above,
We can conclude that the value of the given expression is: 2,800

Question 22.
(2 ∙ 10) + (7 · 100)
Answer:
The given expression is:
(2 × 10) + (7 × 100)
So,
(2 × 10) + (7 × 100)
= 20 + 700
= 720
Hence, from the above,
we can conclude that the value of the given expression is: 720

Language Development

Fill in the word map with new terms, definitions, and supporting examples or illustrations.

Topic 1 PICK A PROJECT

PROJECT 1A
Who is your favorite poet, and why?
PROJECT: WRITE A POEM

PROJECT 1B
If you moved to a tiny house, what would you bring with you?
PROJECT: DESIGN A TINY HOUSE

PROJECT 1C
If you could travel anywhere in space, where would you go?
PROJECT: PLAN A TOUR OF THE MILKY WAY

PROJECT 1D
Why do you think people tell stories around a campfire?
PROJECT: TELL A FOLK STORY

Lesson 1.1 Rational Numbers as Decimals

Solve & Discuss It!

Jaylon has a wrench labeled 0.1875 inches and bolts labeled in fractions of an inch. Which size bolt will fit best with the wrench? Explain.

Answer:
It is given that
Jaylon has a wrench labeled 0.1875 inches and bolts labeled in fractions of an inch.
Now,
We know that,
The bolt will be fit in a wrench only when
The size of the bolt (inches) = The size of the wrench (inches)
Now,
The representation of the bolts in the decimal numbers is:
\(\frac{3}{8}\) = 0.375 inches
\(\frac{1}{8}\) = 0.046 inches
\(\frac{3}{16}\) = 0.1875 inches
\(\frac{1}{4}\) = 0.25 inches
Hence, from the above,
We can conclude that the bolt which has the size \(\frac{3}{16}\) inches will fit best with the wrench

Reasoning
How can you write these numbers in the same form?
Answer:
The representation of the sizes of bolts in the decimal form is by using the properties of place values
We know that,
A terminating decimal can be written as a fraction by using properties of place value.
Example:
3.75 = three and seventy-five hundredths or \(\frac{375}{100}\), which is equal to the improper fraction

Focus on math practices
Reasoning Why is it useful to write a rational number as a fraction or as a decimal?
Answer:
Rational numbers are whole numbers, fractions, and decimals – the numbers we use in our daily lives. They can be written as a ratio of two integers. … The definition says that a number is rational if you can write it in the form \(\frac{a}{b}\) where a and b are integers, and b is not zero.

? Essential Question
How can you write repeating decimals as fractions?
Answer:
Because repeating decimals are rational numbers, you can write them in fraction form.
STEP 1
Assign a variable to represent the repeating decimal.
STEP 2
Write an equation: variable = decimal.
STEP 3
Multiply each side of the equation by 10^d, where d is the number of repeating digits in the repeating decimal.
STEP 4
Subtract equivalent expressions of the variable and the repeating decimal from each side of the equation.
STEP 5
Solve for the variable. Write an equivalent fraction so that the numerator and denominator are integers, if necessary.

Try It!

In another baseball division, one team had a winning percentage of 0.444…. What fraction of their games did this team win?
The team won their games.

Answer:
It is given that
In another baseball division, one team had a winning percentage of 0.444…
Since only 1 number is repeated,
The repeating decimal can be written as 0.\(\overline{4}\)
Now,
According to the steps of converting a repeating decimal to a fraction (Essential Question),
Step 1:
Let x = 0.\(\overline{4}\)
Step 2:
Multiply with 10 on both sides since only 1 number is repeating
So,
10x = 10 (0.\(\overline{4}\))
10x = 4.\(\overline{4}\)
Step 3:
10x – x = 4.\(\overline{4}\) – 0.\(\overline{4}\)
9x = 4
x = \(\frac{4}{9}\)
Hence, from the above,
We can conclude that the team won \(\frac{4}{9}\) of their games

Convince Me!
How do you know what power of ten to multiply by in the second step at the right?
Answer:
Let x be the repeating portion.
Multiply this equation by a power of 10 to move the repeating digits immediately to the left of the decimal point (in other words, to eliminate any zeros preceding the repeating digits).

Try It!
Write the repeating decimal 0.63333… as a fraction.
Answer:
The given repeating decimal is: 0.6333333
Now,
According to the steps of converting a repeating decimal to a fraction (Essential Question),
Step 1:
Let x = 0.6\(\overline{3}\)
Step 2:
Multiply with 10 on both sides since only 1 number is repeating
So,
10x = 10 (0.6\(\overline{3}\))
10x = 6.\(\overline{3}\)
Step 3:
10x – x = 6.\(\overline{3}\) – 0.6\(\overline{3}\)
9x = 6.33 – 0.63
9x = 5.7
Divide by 9 into both sides
So,
\(\frac{9}{9}\)x = \(\frac{5.7}{9}\)
x = \(\frac{57}{90}\)
Hence, from the above,
We can conclude that the value of the given repeating decimal in the fraction form is: \(\frac{57}{90}\)

Try It!
Write the repeating decimal 4.1363636… as a fraction.
Answer:
The given repeating decimal is 4.1363636…..
Now,
According to the steps of converting a repeating decimal to a fraction (Essential Question),
Step 1:
Let x = 4.1\(\overline{36}\)
Step 2:
Multiply with 100 on both sides since 2 numbers are repeating
So,
100x = 100 (4.1\(\overline{36}\))
100x = 41.\(\overline{36}\)
Step 3:
100x – x = 41.\(\overline{36}\) – 4.1\(\overline{36}\)
99x = 413.636 – 4.136
99x = 409.5
Divide by 99 into both sides
So,
\(\frac{99}{99}\)x = \(\frac{409.5}{99}\)
x = \(\frac{4095}{990}\)
Hence, from the above,
We can conclude that the value of the given repeating decimal in the fraction form is: \(\frac{4095}{990}\)

KEY CONCEPT

Because repeating decimals are rational numbers, you can write them in fraction form.
STEP 1
Assign a variable to represent the repeating decimal.
STEP 2
Write an equation: variable = decimal.
STEP 3
Multiply each side of the equation by 10^d, where d is the number of repeating digits in the repeating decimal.
STEP 4
Subtract equivalent expressions of the variable and the repeating decimal from each side of the equation.
STEP 5
Solve for the variable. Write an equivalent fraction so that the numerator and denominator are integers, if necessary.

Do You Understand?

Question 1.
? Essential Question
How can you write repeating decimals as fractions?
Answer:
Because repeating decimals are rational numbers, you can write them in fraction form.
STEP 1
Assign a variable to represent the repeating decimal.
STEP 2
Write an equation: variable = decimal.
STEP 3
Multiply each side of the equation by 10^d, where d is the number of repeating digits in the repeating decimal.
STEP 4
Subtract equivalent expressions of the variable and the repeating decimal from each side of the equation.
STEP 5
Solve for the variable. Write an equivalent fraction so that the numerator and denominator are integers, if necessary.

Question 2.
Use Structure Why do you multiply by a power of 10 when writing a repeating decimal as a rational number?
Answer:
The idea is to multiply by some number (10, 100, 1000, etc.) so that when we subtract the original number from the multiple, the repeating part cancels out.

Question 3.
Be Precise How do you decide by which power of 10 to multiply an equation when writing a decimal with repeating digits as a fraction?
Answer:
The idea is to multiply by some number (10, 100, 1000, etc.) so that when we subtract the original number from the multiple, the repeating part cancels out.

Do You Know How?

Question 4.
A survey reported that 63.63% of moviegoers prefer action films. This percent represents a repeating decimal. Write it as a fraction.
Answer:
It is given that
A survey reported that 63.63% of moviegoers prefer action films. This percent represents a repeating decimal
So,
The given repeating decimal is 63.6363…..%
Now,
According to the steps of converting a repeating decimal to a fraction (Essential Question),
Step 1:
Let x = 63.\(\overline{63}\)
Step 2:
Multiply with 100 on both sides since 2 numbers are repeating
So,
100x = 100 (63.\(\overline{63}\))
100x = 6363.\(\overline{63}\)
Step 3:
100x – x = 6363.\(\overline{63}\) – 63.\(\overline{63}\)
99x = 6,300
Divide by 99 into both sides
So,
\(\frac{99}{99}\)x = \(\frac{6,300}{99}\)
x = \(\frac{6,300}{99}\)
Hence, from the above,
We can conclude that the value of the given repeating decimal in the fraction form is: \(\frac{6,300}{99}\)

Question 5.
A student estimates the weight of astronauts on the Moon by multiplying their weight by the decimal 0.16666… What fraction can be used for the same estimation?

Answer:
It is given that
A student estimates the weight of astronauts on the Moon by multiplying their weight by the decimal 0.16666…
Now,

Hence, from the above,
We can conclude that the fraction that can be used for the same estimation is: \(\frac{1}{6}\)

Question 6.
Write 2.3181818… as a mixed number.
Answer:
The given repeating decimal is 2.3181818…
Now,
According to the steps of converting a repeating decimal to a fraction (Essential Question),
Step 1:
Let x = 2.3\(\overline{18}\)
Step 2:
Multiply with 100 on both sides since 2 numbers are repeating
So,
100x = 100 (2.3\(\overline{18}\))
100x = 23.\(\overline{18}\)
Step 3:
100x – x = 23.\(\overline{18}\) – 2.3\(\overline{18}\)
99x = 231.818 – 2.318
99x = 229.5
Divide by 99 into both sides
So,
\(\frac{99}{99}\)x = \(\frac{229.5}{99}\)
x = \(\frac{2295}{990}\)
x = \(\frac{51}{22}\)
So,
The representation of the above fraction in the mixed form is: 2\(\frac{7}{22}\)
Hence, from the above,
We can conclude that the value of the given repeating decimal in the mixed fraction form is: 2\(\frac{7}{22}\)

Practice & Problem Solving

Leveled Practice In 7 and 8, write the decimal as a fraction or mixed number.

Question 7.
Write the number 0.21212121… as a fraction.
Let x =
100x =
100x – x = –
99x =
x =
So 0.2121… is equal to
Answer:
The given repeating decimal is: 0.212121….
Now,

Hence, from the above,
We can conclude that the value of the repeating decimal in the fraction form is: \(\frac{21}{99}\)

Question 8.
Write 3.7 as a mixed number.
Let x =
10x =
9x =
x =
So 3.\(\overline{7}\) is equal to
Answer:
The given repeating decimal is: 3.\(\overline{7}\)
Now,

Hence, from the above,
We can conclude that the representation of the given repeating decimal in the mixed fraction form is: 3\(\frac{7}{9}\)

Question 9.
Write the number shown on the scale as a fraction.

Answer
The number that is shown on the scale is: 0.233333
Now,
From the: above number,
We can observe that the number is a repeating decimal
Now,

So,
The simplified form of \(\frac{2.1}{9}\) is: \(\frac{7}{30}\)
Hence, from the above,
We can conclude that the representation of the number that is shown on the scale as a fraction is: \(\frac{7}{30}\)

Question 10.
Tomas asked 15 students whether summer break should be longer. He used his calculator to divide the number of students who said yes by the total number of students. His calculator showed the result as 0.9333…
a. Write this number as a fraction.
Answer:
The given repeating decimal is 0.93333…
Now,

So,
The simplified form of \(\frac{8.4}{9}\) is: \(\frac{14}{15}\)
Hence, from the above,
We can conclude that the representation of the repeating number in the form of the fraction is: \(\frac{14}{15}\)

b. How many students said that summer break should be longer?
Answer:
It is given that
Tomas asked 15 students whether summer break should be longer. He used his calculator to divide the number of students who said yes by the total number of students.
So,
The number of students that said summer break should be longer = \(\frac{The number of students that said yes that summer break is longer}{The total number of students}
Now,
From part (a),
The fraction form of his calculated result from part (a) is: 14 / 15
Hence, from the above,
We can conclude that the number of students that said the summer break should be longer is: 14 students

Question 11.
Write 0.[latex]\overline{87}\) as a fraction.
Answer:
The given repeating decimal is 0.\(\overline{87}\)
Now,

Hence, from the above,
We can conclude that the representation of the repeating decimal in the fraction form in the simplest form is: \(\frac{29}{33}\)

Question 12.
Write 0.\(\overline{8}\) as a fraction.
Answer:
The given repeating decimal is 0.\(\overline{8}\)
Now,

Hence, from the above,
We can conclude that the representation of the given repeating decimal in the fraction form is: \(\frac{8}{9}\)

Question 13.
Write 1.\(\overline{48}\) as a mixed number.
Answer:
The given repeating decimal number is 1.\(\overline{48}\)
Now,

Hence, from the above,
We can conclude that the representation of the given repeating decimal in the mixed fraction form is: 1\(\frac{16}{33}\)

Question 14.
Write 0.\(\overline{6}\) as a fraction.
Answer:
The given repeating decimal is 0.\(\overline{6}\)
Now,

Hence, from the above,
We can conclude that the representation of the given repeating decimal in the fraction form is: \(\frac{2}{3}\)

Question 15.
A manufacturer determines that the cost of making a computer component is $2.161616. Write the cost as a fraction and as a mixed number.

Answer:
It is given that
A manufacturer determines that the cost of making a computer component is $2.161616
So,
The given repeating decimal is 2.161616…
Now,

Hence, from the above,
We can conclude that
The cost of a computer component in a fraction form is: \(\frac{214}{99}\)
The cost of a computer component in a mixed fraction form is: 2\(\frac{16}{99}\)

Question 16.
Reasoning When writing a repeating decimal as a fraction, does the number of repeating digits you use matter? Explain.
Answer:
No. Even if the number of different digits in the cycle is 1 or 1 million, the method of finding the fraction is the same

Question 17.
Higher Order Thinking When writing a repeating decimal as a fraction, why does the fraction always have only 9s or 9s and 0s as digits in the denominator?
Answer:
When writing a repeating decimal as a fraction, the fraction always has only 9s or 9s and 0s as digits in the denominator because we are talking here about a geometric series and they are decimals, so the right side i.e., after the decimal point, the digits are in tenths, hundredths and so on

Assessment Practice

Question 18.
Which decimal is equivalent to \(\frac{188}{11}\)?
A. 17.\(\overline{09}\)
B. 17.0\(\overline{09}\)
C. 17.\(\overline{1709}\)
D. 17.\(\overline{1709}\)0
Answer:
The given fraction is: \(\frac{188}{11}\)
So,
The representation of the given fraction in the decimal form is:

we know that,
\(\frac{1}{11}\) = 0.090909…..
So,
\(\frac{188}{11}\) = 17.090909….
= 17.\(\overline{09}\)
Hence, from the above,
We can conclude that option A matches the representation of the repeating decimal for the given fraction

Question 19.
Choose the repeating decimal that is equal to the fraction on the left.

Answer:
Follow the process that is mentioned below to solve the given repeating decimals in the fraction form
Now,
STEP 1
Assign a variable to represent the repeating decimal.
STEP 2
Write an equation: variable = decimal.
STEP 3
Multiply each side of the equation by 10^d, where d is the number of repeating digits in the repeating decimal.
STEP 4
Subtract equivalent expressions of the variable and the repeating decimal from each side of the equation.
STEP 5
Solve for the variable. Write an equivalent fraction so that the numerator and denominator are integers, if necessary.
Hence,

Lesson 1.2 Understand Irrational Numbers

Explain It!
Sofia wrote a decimal as a fraction. Her classmate Nora says that her method and answer are not correct. Sofia disagrees and says that this is the method she learned.

A. Construct Arguments Is Nora or Sofia correct? Explain your reasoning.
Answer:
The given number is 0.121121112111112…..
Now,
From the given number,
We can observe that the given decimal is not a repeating decimal because there are other numbers other than the repeating numbers in the given decimal or a terminating decimal because the decimal is not finite
So,
Since the given decimal is not a repeating decimal,
The method that we used to convert the repeating decimal into a fraction is not applicable
Hence, from the above,
We can conclude that Nora is correct

B. Use Structure What is a nonterminating decimal number that can not be written as a fraction.
Answer:
A non-terminating, non-repeating decimal is a decimal number that continues endlessly, with no group of digits repeating endlessly. Decimals of this type cannot be represented as fractions, and as a result, are irrational numbers

Focus on math practices
Construct Arguments is 0.12112111211112… a rational number? Explain.
Answer:
0.12112111211112… can’t be represented in the form of \(\frac{p}{q}\) and it has non terminating non-repeating decimal expansion
Hence, from the above,
We can conclude that 0.12112111211112… is not a rational number

? Essential Question
How is an irrational number different from a rational number?
Answer:
Numbers that can be expressed in \(\frac{a}{b}\) or fraction form are rational numbers where a is an integer and b is a non-zero integer and the irrational numbers are the numbers that cannot be written in \(\frac{a}{b}\) form

Try It!
Classify each number as rational or irrational.


Answer:
The given numbers are:

Now,
We know that,
Numbers that can be expressed in \(\frac{a}{b}\) or fraction form are rational numbers where a is an integer and b is a non-zero integer and the irrational numbers are the numbers that cannot be written in \(\frac{a}{b}\) form
Hence,
The representation of the rational and irrational numbers are:

Convince Me!
Construct Arguments Jen classifies the number 4.567 as irrational because it does not repeat. Is Jen correct? Explain.
Answer:
The given decimal is: 4.567
We know that,
A rational number is a number that can be written in the form of \(\frac{a}{b}\)
A terminating decimal has the finite number of digits without repeating and it is also a rational number
So,
We can observe that we can write 4.567 as a rational number
Hence, from the above,
We can conclude that Jen is not correct

Try It!
Classify each number as rational or irrational and explain.
A) \(\frac{2}{3}\)
B) \(\sqrt{25}\)
C) -0.7\(\overline{5}\)
D) \(\sqrt{2}\)
E) 7,548,123
Answer:
The given numbers are:
A) \(\frac{2}{3}\)
B) \(\sqrt{25}\)
C) -0.7\(\overline{5}\)
D) \(\sqrt{2}\)
E) 7,548,123
Now,
We know that,
A “Rational number” is a number that can be written in the form of \(\frac{a}{b}\)
Ex: Terminating decimals, perfect squares, etc
An “Irrational number” is a number that can not be written in the form of \(\frac{a}{b}\)
Ex: Non-terminating decimals
So,
From the given numbers,
Rational numbers ——> A, B, E
Irrational numbers ——> C, D

KEY CONCEPT
Numbers that are not rational are called irrational numbers.

Do You Understand?

Question 1.
? Essential Question How is an irrational number different from a rational number?
Answer:
Numbers that can be expressed in \(\frac{a}{b}\) or fraction form are rational numbers where a is an integer and b is a non-zero integer and the irrational numbers are the numbers that cannot be written in \(\frac{a}{b}\) form

Question 2.
Reasoning How can you tell whether a square root of a whole number is rational or irrational?
Answer:
If the square root of an integer is itself an integer (Ex: √4 = 2), then by definition it is rational – If the square root is not an integer (Ex: √2 = 1.41414…), then it must be irrational. Put another way the only integers for which the square root of an integer can be rational is if is a perfect square – that is where x is an integer

Question 3.
Construct Arguments Could a number ever be both rational and irrational? Explain.
Answer:
No. A rational number is a number that can be expressed as the quotient of two integers. An irrational number is a number that cannot be expressed as a quotient of two integers. So if a number is either rational or irrational, it cannot also be the other.

Do You Know How?

Question 4.
Is the number 65.4349224… rational or irrational? Explain.
Answer:
The given number is 65.4349224…
From the given number,
We can observe that the given number is a non-repeating and non-terminating decimal number
Hence, from the above,
We can conclude that the given number is an irrational number

Question 5.
Is the number \(\sqrt{2,500}\) rational or irrational? Explain.
Answer:
The given number is: \(\sqrt{2,500}\)
We know that,
A perfect square number is a rational number
So,
\(\sqrt{2,500}\) = 50
Hence, from the above,
We can conclude that the given number is a rational number

Question 6.
Classify each number as rational or irrational.

Answer:
We know that,
A “Rational number” is a number that can be written in the form of \(\frac{a}{b}\)
Ex: Perfect squares
An “Irrational number” is a number that can not be written in the form of \(\frac{a}{b}\)
Ex: Non-terminating decimal numbers
Hence,
The representation of the rational and irrational numbers from the given numbers are:

Practice & Problem Solving

Question 7.
Is 5.787787778… a rational or irrational number? Explain.
Answer:
The given number is 5.787787778…
From the given number,
We can observe that the given number is a non-repeating and non-terminating decimal number
Hence, from the above,
We can conclude that the given number is an irrational number

Question 8.
Is \(\sqrt{42}\) rational or irrational? Explain.
Answer:
The given number is \(\sqrt{42}\)
From the given number,
We can observe that the given number is not a perfect square
Hence, from the above,
We can conclude that the given number is an irrational number

Question 9.
A teacher places seven cards, lettered A-G, on a table. Which cards show irrational numbers?

Answer:
The given cards are:

We know that,
An “Irrational number” is a number that can not be written in the form of \(\frac{a}{b}\)
Hence, from the above,
We can conclude that from the given cards,
The irrational numbers are:
A) π
B) 8.25635…,
C) 6.\(\overline{31}\)

Question 10.
Circle the irrational number in the list below.
A) 7.\(\overline{27}\)
B) \(\frac{5}{9}\)
C) \(\sqrt{15}\)
D) \(\sqrt{196}\)
Answer:
The given numbers are:
A) 7.\(\overline{27}\)
B) \(\frac{5}{9}\)
C) \(\sqrt{15}\)
D) \(\sqrt{196}\)
Now,
We know that,
A “Rational number” is a number that can be written in the form of \(\frac{a}{b}\)
Ex: Perfect squares
An “Irrational number” is a number that can not be written in the form of \(\frac{a}{b}\)
Ex: Non-terminating decimal numbers
Hence, from the above,
We can conclude that
From the given numbers,
The irrational numbers are A) and C)

Question 11.
Lisa writes the following list of numbers.
5.737737773…, 26, \(\sqrt{45}\), –\(\frac{3}{2}\), 0, 9
Answer:
The given numbers are:
A) 5.7377377737… B) 26  C) \(\sqrt{45}\)
D) –\(\frac{3}{2}\) E) 0 F) 9
Now,
We know that,
A “Rational number” is a number that can be written in the form of \(\frac{a}{b}\)
Ex: Perfect squares
An “Irrational number” is a number that can not be written in the form of \(\frac{a}{b}\)
Ex: Non-terminating decimal numbers

a. Which numbers are rational?
Answer:
From the given numbers,
The rational numbers are: B, D, E, and F

b. Which numbers are irrational?
Answer:
From the given numbers,
The irrational numbers are: A and C

Question 12.
Construct Arguments Deena says that 9.565565556… is a rational number because it has a repeating pattern. Do you agree? Explain.
Answer:
The given number is 9.565565556…
From the given number,
We can observe that the number is a non-repeating and a non-terminating decimal
So,
The given number is an irrational number
Hence, from the above,
We can conclude that we don’t have to agree with Deena

Question 13.
Is \(\sqrt{1,815}\) rational? Explain.
Answer:
The given number is: \(\sqrt{1,815}\)
We know that,
A “Rational number” is a number that can be written in the form of \(\frac{a}{b}\)
Ex: Perfect squares
Now,
From the given square root,
We can observe that it won’t form a perfect square
Hence, from the above,
We can conclude that the given number is an irrational number

Question 14.
Is the decimal form of \(\frac{13}{3}\) Explain.
Answer:
The given number is: \(\frac{13}{3}\)
We know that,
\(\frac{13}{3}\) = 4.3333…..
We know that,
An “Irrational number” is a number that can not be written in the form of \(\frac{a}{b}\)
Hence, from the above,
We can conclude that the decimal form of \(\frac{13}{3}\) is an irrational number

Question 15.
Write the side length of the square rug as a square root. Is the side length a rational or irrational number? Explain.

Answer:
The given figure is:

From the given figure,
We can observe that the given figure depicts the shape of a square
Now,
Let the side length of a square be x
We know that,
Area = (Side length)²
x² = 100
x = \(\sqrt{100}\)
We know that,
A “Perfect square” is a rational number
Hence, from the above,
We can conclude that a side length is a rational number

Question 16.
Reasoning The numbers 2.888… and 2.999… are both rational numbers. What is an irrational number that is between the two rational numbers?
Answer:
A rational number is a number which is can be represented as the quotient of two numbers without having any remainder i.e., having remainder 0. For example 2.45, 2, 3 etc.
An irrational number has a non-zero remainder and has a nonterminating quotient.
Hence,
The numbers between 2.888… and 2.999… are 2.8889………, 2.8890…….., 2.8891…… etc

Question 17.
Higher Order Thinking You are given the expressions \(\sqrt{76+n}\) and \(\sqrt{2 n+26}\). What is the smallest value of n that will make each number rational?
Answer:
The given expressions are: \(\sqrt{76+n}\) and \(\sqrt{2 n+26}\)
Now,
To find the smallest value of n so that each expression will be a rational number,
\(\sqrt{76+n}\) = \(\sqrt{2 n+26}\)
Squaring on both sides
So,
76 + n = 2n + 26
2n – n = 76 – 26
n = 50
Hence, from the above,
We can conclude that the smallest value of n so that the given expressions will become a rational number is: 5

Assessment Practice

Question 18.
Which numbers are rational?
I. 1.1111111…
II. 1.567
III. 1.101101110…
A. II and III
B. III only
C. II only
D. I and II
E I only
F. None of the above
Answer:
The given numbers are:
I. 1.1111111…
II. 1.567
III. 1.101101110…
We know that,
A “Rational number” is a number that can be written in the form of \(\frac{a}{b}\)
So,
From the given numbers,
1 and 2 are the rational numbers
Hence, from the above,
we can conclude that option D matches with the given situation

Question 19.
Determine whether the following numbers are rational or irrational.

Answer:
The representation of the given numbers as rational and irrational numbers is:

Lesson 1.3 Compare and Order Real Numbers

Solve & Discuss It!

Courtney and Malik are buying a rug to fit in a 50-square-foot space. Which rug should they purchase? Explain.

Answer:
It is given that
Courtney and Malik are buying a rug to fit in a 50-square-foot space
Now,
The given figure is:

Now,
From the above figure,
We can observe that
The rugs are in different shapes i.e., a square, a circle, and a rectangle
Now,
The area of a square rug = The side length of a square rug × The side length of a square rug
= 7 × 7
= 49 ft²
Now,
The area of a circular rug = π × \(\frac{Diameter of a circular rug²}{4}\)
= 3.14 × \(\frac{8 ×8}{4}\)
= 3.14 ×16
= 50.24 ft²
Now,
The area of a rectangular rug = Length × Width
= 6 × 8\(\frac{1}{2}\)
= 6 × \(\frac{17}{2}\)
= \(\frac{6 ×17}{2}\)
= 51 ft²
Now,
When we compare the area of the rugs,
The area of the square rug is less than 50 ft²
Hence, from the above,
We can conclude that
Courtney and Malik should buy the square rug

Focus on math practices
Make Sense and Persevere How did you decide which rug Courtney and Malik should purchase?
Answer:
It is given that
Courtney and Malik are buying a rug to fit in a 50 ft² space
So,
To fit in a 50 ft² space,
The area of any type of rug should be less than 50 ft²
Now,
From the above problem,
We can observe that
The area of the square rug is the only area that is less than 50 ft²
Hence, from the above,
We can conclude that
Courtney and Mali should purchase the rugs based on the areas of the rugs

?Essential Question
How can you compare and order rational and irrational numbers?
Answer:
In the given numbers, one of them is rational while other one is irrational. To make the comparison, let us first make the given irrational number into rational number and then carry out the comparison. So, let us square both the given numbers

Try It!
Between which two whole numbers is \(\sqrt{12}\)?

Answer:
The given number is: \(\sqrt{12}\)
Now,

Hence, from the above,
We can conclude that
\(\sqrt{12}\) is in between 3 and 4

Convince Me!
Which of the two numbers is a better estimate for \(\sqrt{12}\)? Explain.
Answer:
The given number is: \(\sqrt{12}\)
Now,
We know that,
12 will be 3² and 4²
So,
\(\sqrt{12}\) will be between 3 and 4
Now,
When we observe the numbers between 3 and 4
The value of \(\frac{12}\) will be near to 3.4
Hence, from the above,
We can conclude that
The two numbers that are better estimate for \(\sqrt{12}\) is: 3 and 4

Try It!
Compare and order the following numbers:
\(\sqrt{11}\), 2\(\frac{1}{4}\), -2.5, 3.\(\overline{6}\), -3.97621 …
Answer:
The given numbers are:
\(\sqrt{11}\), 2\(\frac{1}{4}\), -2.5, 3.\(\overline{6}\), -3.97621 …
So,
\(\sqrt{11}\) ≅ 3.3
2\(\frac{1}{4}\) = 2.25
3.\(\overline{6}\) = 3.666……
So,
The order of the numbers from the least to the greatest is:
-3.97621…….. < -2.5 < 2.25 < \(\sqrt{11}\) < 3.\(\overline{6}\)

KEY CONCEPT
To compare rational and irrational numbers, you must first find rational approximations of the irrational numbers. You can approximate irrational numbers using perfect squares or by rounding.

Do You Understand?

Question 1.
? Essential Question How can you compare and order rational and irrational numbers?
Answer:
In the given numbers, one of them is rational while other one is irrational. To make the comparison, let us first make the given irrational number into rational number and then carry out the comparison. So, let us square both the given numbers

Question 2.
Reasoning
The “leech” is a technical term for the slanted edge of a sail. Is the length of the leech shown closer to 5 meters or 6 meters? Explain.

Answer:
It is given that
The “leech” is a technical term for the slanted edge of a sail
Now,
The given figure is:

Now,
From the given figure,
We can observe that
The length of the leech is: \(\sqrt{30}\) meters
Now,
We know that,
5² < 30 < 6²
5 < \(\sqrt{30}\) < 6
Now,
We know that,
\(\sqrt{30}\) ≅ 5.4
So,
\(\sqrt{30}\) is close to 5
Hence, from the above,
We can conclude that
The length of the leech shown above is close to 5 meters

Question 3.
Construct Arguments which is a better approximation of \(\sqrt{20}\), 4.5 or 4.47? Explain.
Answer:
The given number is: \(\sqrt{20}\)
Now,
We know that,
4² < 20 < 5²
4 < \(\sqrt{20}\) < 5
Now,
We know that,
4.5² = 20.25
So,
The value of \(\sqrt{20}\) is close to 4.4
Hence, from the above,
We can conclude that
The better approximation of \(\sqrt{20}\) is: 4.4

Do You Know How?

Question 4.
Approximate \(\sqrt{39}\) to the nearest whole number.
Answer:
The given number is: \(\sqrt{39}\)
Now,
We know that,
6² < 39 < 7²
6 < \(\sqrt{39}\) < 7
Now,
We know that,
6.5² = 42.25
So,
The value of \(\sqrt{39}\) is close to 6.2
So,
The value of \(\sqrt{39}\) is closes to 6 that is the nearest whole number
Hence, from the above,
We can conclude that
The better approximation of \(\sqrt{39}\) that is the closest to the whole number is: 6

Question 5.
Approximate \(\sqrt{18}\) to the nearest tenth and plot the number on a number line.

Answer:
The given number is: \(\sqrt{18}\)
Now,
We know that,
4² < 18 < 5²
4 < \(\sqrt{18}\) < 5
Now,
We know that,
4.5² = 20.25
4.2² = 17.69
So,
\(\sqrt{18}\) is close to 4.2
Hence, from the above,
We can conclude that
The representation of the approximate value of \(\sqrt{18}\) on the given number line is:

The approximate value of \(\sqrt{18}\) is: 4.2

Question 6.
Compare 5.7145… and \(\sqrt{29}\). Show your work.
Answer:
The given numbers are: 5.7145…… and \(\sqrt{29}\)
Now,
We know that,
5² < 29 < 6²
So,
5 < \(\sqrt{29}\) < 6
Now,
We know that,
5.5² = 30.25
5.3² = 28.09
So,
The approximate value of \(\sqrt{29}\) is: 5.3
Now,
When we compare the given numbers,
We can observ ethat
5.7145….. > 5.3
Hence, from the above,
We can conclude that
The order of the given numbers is:
5.7145……….. > \(\sqrt{29}\)

Question 7.
Compare and order the following numbers
5.2, -5.\(\overline{6}\), 3\(\frac{9}{10}\), \(\sqrt{21}\)
Answer:
The given numbers are:
5.2, -5.\(\overline{6}\), 3\(\frac{9}{10}\), \(\sqrt{21}\)
Now,
We know that,
-5.\(\overline{6}\) = -5.666666…..
3\(\frac{9}{10}\) = 3.9
\(\sqrt{21}\) ≅ 4.58
So,
The order of the given numbers from the least to the greatest is:
-5.\(\overline{6}\) < 3\(\frac{9}{10}\) < \(\sqrt{21}\) < 5.2
Hence, from the above,
We can conclude that
The order of the given numbers from the least to the greatest is:
-5.\(\overline{6}\) < 3\(\frac{9}{10}\) < \(\sqrt{21}\) < 5.2

Practice & Problem Solving

Question 8.
Leveled Practice Find the rational approximation of \(\sqrt{15}\).
a. Approximate using perfect squares.
< 15 <
= < \(\sqrt{15}\) <
< \(\sqrt{15}\) <
Answer:
The given number is: \(\sqrt{15}\)
Now,
We know that,
By using the approximation using the perfect squares,

Hence, from the above,
We can conclude that
The approximate numbers that are between \(\sqrt{15}\) are: 3 and 4

b. Locate and plot \(\sqrt{15}\) on a number line. Find a better approximation using decimals.

Answer:
From Part (a),
We know that,
The approximate numbers that are between \(\sqrt{15}\) are: 3 and 4
Now,
We know that,
3.5 ² = 12.25
Now,

So,
The approximate number that is the closest to \(\sqrt{15}\) is: 3.8
Hence, from the above,
We can conclude that
The representation of the approximation of \(\sqrt{15}\) in the given number line is:

The approximate number that is close to \(\sqrt{15}\) is: 3.8

Question 9.
Compare – 1.96312… and –\(\sqrt{5}\). Show your work.
Answer:
The given numbers are: -1.96312…… and –\(\sqrt{5}\)
Now,
We know that,
2² < 5 < 3²
2 < \(\sqrt{5}\) < 3
Now,
We know that,
2.5² = 6.25
2.2² = 4.84
So,
The approximate value of –\(\sqrt{5}\) is: -2.2
So,
The order of the given numbers from the least to the greatest is:
-1.96312……. > –\(\sqrt{5}\)
Hence, from the above,
We can conclude that
The order of the given numbers from the least to the greatest is:
-1.96312……. > –\(\sqrt{5}\)

Question 10.
Does \(\frac{1}{6}\), -3, \(\sqrt{7}\), –\(\frac{6}{5}\), or 4.5 come first when the numbers are listed from least to greatest? Explain.
Answer:
The given numbers are: \(\frac{1}{6}\), -3, \(\sqrt{7}\), –\(\frac{6}{5}\), and 4.5
Now,
\(\frac{1}{6}\) = 0.166
\(\sqrt{7}\) = 2.64
–\(\frac{6}{5}\) = -1.2
So,
The order of the given numbers from the least to the greatest is:
-3 < –\(\frac{6}{5}\) < \(\frac{1}{6}\) < \(\sqrt{7}\) < 4.5
Hence, from the above,
We can conclude that
“-3” will come first when the given numbers will be arranged from the least to the greatest

Question 11.
A museum director wants to hang the painting on a wall. To the nearest foot, how tall does the wall need to be?

Answer:
It is given that
A museum director wants to hang the painting on a wall
Now,
The given figure is:

Now,
From the given figure,
We can obsere that
The painting on a wall is about \(\sqrt{90}\) ft
Now,
We know that,
9² < 90 < 10²
9 < \(\sqrt{90}\) < 10
Now,
We know that,
9.5² = 90.25
So,
The approximate value of \(\sqrt{90}\) is: 9.4 ft
Hence, from the above,
We can conclude that
The height of the wall needed to hang a painting is about 9.4 ft

Question 12.
Dina has several small clay pots. She wants to display them in order of height, from shortest to tallest. What will be the order of the pots?

Answer:
It is given that
Dina has several small clay pots. She wants to display them in order of height, from shortest to tallest
Now,
The given heights are:
\(\sqrt{8}\), 2\(\frac{1}{3}\), \(\sqrt{5}\), and 2.5
Now,
We now that,
\(\sqrt{8}\) ≅ 2.82 in.
2\(\frac{1}{3}\) = 2.033 in.
\(\sqrt{5}\) ≅ 2.23 in.
So,
The order of the heights from the shortest to the tallest is:
2\(\frac{1}{3}\) in. < \(\sqrt{5}\) in. < 2.5 in. < \(\sqrt{8}\) in.
Hence, from the above,
We can conclude that
The order of the pots is:
2\(\frac{1}{3}\) in. < \(\sqrt{5}\) in. < 2.5 in. < \(\sqrt{8}\) in.

Question 13.
Rosie is comparing \(\sqrt{7}\) and 3.44444…. She says that \(\sqrt{7}\) > 3.44444… because \(\sqrt{7}\) = 3.5.
a. What is the correct comparison?
Answer:
It is given that
Rosie is comparing \(\sqrt{7}\) and 3.44444…. She says that \(\sqrt{7}\) > 3.44444… because \(\sqrt{7}\) = 3.5.
Now,
We know that,
2² < 7 < 3²
2 < \(\sqrt{7}\) < 3
Now,
We know that,
2.6² = 6.76
2.7² = 7.29
So,
The approximate value of \(\sqrt{7}\) is: 2.6
So,
The order of the given numbers is:
\(\sqrt{7}\) < 3.44444…..
Hence, from the above,
We can conclude that
The correct comparison of the given numbers is:
\(\sqrt{7}\) < 3.44444…..

b. Critique Reasoning What mistake did Rosie likely make?
Answer:
The given numbers are: \(\sqrt{7}\) and 3.44444…..
Now,
It is also given that
\(\sqrt{7}\) = 3.5
But, 3.5 = \(\frac{7}{2}\)
Hence, from the above,
We can conclude that
The mistake Rosie likely make is:
She considered \(\sqrt{7}\) = \(\frac{7}{2}\)

Question 14.
Model with Math Approximate – √23 to the nearest tenth. Draw the point on the number line.

Answer:
The given number is: –\(\sqrt{23}\)
Now,
We know that,
4² < 23 < 5²
4 < \(\sqrt{23}\) < 5
Now,
We know that,
4.5² = 20.25
4.7² = 22.09
So,
The approximate value of –\(\sqrt{23}\) is: -4.7
Hence,
The representtaion of the approximate value of –\(\sqrt{23}\) on the given number line is:

Question 15.
Higher Order Thinking The length of a rectangle is twice the width. The area of the rectangle is 90 square units. Note that you can divide the rectangle into two squares.

a. Which irrational number represents the length of each side of the squares?
Answer:
It is given that
The length of a rectangle is twice the width. The area of the rectangle is 90 square units. Note that you can divide the rectangle into two squares.
Now,
The given figure is:

Now,
According to the given information,
The area of each square = \(\frac{90}{2}\)
= 45 square units
Now,
We know that,
The area of a square = Side²
So,
Side of a squre = \(\sqrt{The area of a square}\)
So,
The side length of each square = \(\sqrt{45}\) units
Hence, from the above,
We can conclude that
The irrational number that represents the length of each side of the squares is: \(\sqrt{45}\) units

b. Estimate the length and width of the rectangle.
Answer:
It is given that
The length of a rectangle is twice the width. The area of the rectangle is 90 square units. Note that you can divide the rectangle into two squares.
Now,
The given figure is:

Now,
Let the width of the rectangle be x units
So,
The length of the rectangle = 2 (Width) = 2x units
Now,
We know that,
The length of a rectangle = Length × Width
So,
According to the given information,
90 = 2x × x
90 = 2x²
x² = \(\frac{90}{2}\)
x = \(\sqrt{45}\)
Hence, from the above,
We can conclude that
The length of the rectangle is: 2\(\sqrt{45}\) units
The width of the rectangle is: \(\sqrt{45}\) units

Assessment Practice

Question 16.
Which list shows the numbers in order from least to greatest?
A. -4, –\(\frac{9}{4}\), \(\frac{1}{2}\), 3.7, \(\sqrt{5}\)
B. -4, –\(\frac{9}{4}\), \(\frac{1}{2}\), \(\sqrt{5}\), 3.7
C. –\(\frac{9}{4}\), \(\frac{1}{2}\), 3.7, \(\sqrt{5}\), -4
D. –\(\frac{9}{4}\), -4, \(\frac{1}{2}\), 3.7, \(\sqrt{5}\)
Answer:
The given list of numbers are:
-4, –\(\frac{9}{4}\), \(\frac{1}{2}\), 3.7, \(\sqrt{5}\)
Now,
We know that,
–\(\frac{9}{4}\) = -2.25
\(\frac{1}{2}\) = 0.5
\(\sqrt{5}\) ≅ 2.23
So,
The order of the given list of numbers from the least to the greatest is:
-4 < –\(\frac{9}{4}\) < \(\frac{1}{2}\) < \(\sqrt{5}\) < 3.7
Hence, from the above,
We can conclude that
The list that shows the numbers from the least to the greatest is:

Question 17.
The area of a square poster is 31 square inches. Find the length of one side of the poster. Explain.

PART A
To the nearest whole inch
Answer:
It is given that
The area of a square poster is 31 square inches
Now,
We know that,
The area of a square = Side²
Side = \(\sqrt{Area of a square}\)
So,
According to the given information,
The length of one side of the poster = \(\sqrt{31}\) inches
Now,
We know that,
5² < 31 < 6²
5 < \(\sqrt{31}\) < 6
Now,
We know that,
5.5² = 30.25
5.6² = 31.36
So,
The approximate value of \(\sqrt{31}\) is: 5.5
Hence, from the above,
We can conclude that
The length of one side of the poster to the nearest whole inch is: 6 inches

PART B
To the nearest tenth of an inch
Answer:
It is given that
The area of a square poster is 31 square inches
Now,
We know that,
The area of a square = Side²
Side = \(\sqrt{Area of a square}\)
So,
According to the given information,
The length of one side of the poster = \(\sqrt{31}\) inches
Now,
We know that,
5² < 31 < 6²
5 < \(\sqrt{31}\) < 6
Now,
We know that,
5.5² = 30.25
5.6² = 31.36
So,
The approximate value of \(\sqrt{31}\) is: 5.5
Hence, from the above,
We can conclude that
The length of one side of the poster to the nearest tenth of an inch is: 5.5 inches

Lesson 1.4 Evaluate Square Roots and Cube Roots

Solve & Discuss It!

ACTIVITY

Matt and his dad are building a tree house. They buy enough flooring material to cover an area of 36 square feet. What are all possible dimensions of the floor?

Answer:
It is given that
Matt and his dad are building a tree house. They buy enough flooring material to cover an area of 36 square feet
Now,
To find the length and width of the floor, find the multiples of 36
So,
The multiples of 36 are:
36 = 1 × 36, 2 × 18, 3 × 12, 4 × 9, 9 ×4, 12 × 3, 18 × 2, 36 × 1
So,
All the possible dimensions of the floor are:
1 × 36, 2 × 18, 3 × 12, 4 × 9, 9 ×4, 12 × 3, 18 × 2, 36 × 1
Hence, from the above,
We can conclude that
All the possible dimensions of the floor are:
1 × 36, 2 × 18, 3 × 12, 4 × 9, 9 ×4, 12 × 3, 18 × 2, 36 × 1

Look for Relationships
Can different floor dimensions result in the same area?
Answer:
From the above problem,
We can observe that
All the possible dimensions of the floor are:
1 × 36, 2 × 18, 3 × 12, 4 × 9, 9 ×4, 12 × 3, 18 × 2, 36 × 1
Now,
When we find the area by using all the different dimensions of the floor,
We can observe that the area of the floor is the same
Hence, from the above
We can conclude that
The different floor dimensions result in the same area

Focus on math practices
Reasoning Why is there only one set of dimensions for a square floor when there are more sets for a rectangular floor? Are all the dimensions reasonable? Explain.
Answer:
From the above problem,
We can observe that
All the possible dimensions of the floor are:
1 × 36, 2 × 18, 3 × 12, 4 × 9, 9 ×4, 12 × 3, 18 × 2, 36 × 1
Now,
We know that,
A square has the same side lengths
A square has the same parallel side lengths
Hence,
The square floor has only one set of dimensions whereas the rectangular floor has more sets and all the dimensions will be reasonable

? Essential Question
How do you evaluate cube roots and square roots?
Answer:
Let the number be: p
Now,
The square of a number is: p²
The cube of a number is: p³
The square root of a number is: \(\sqrt{p}\)
The cube root of a number is: \(\sqrt[3]{p}\)

Try It!
A cube-shaped art sculpture has a volume of 64 cubic feet. What is the length of each edge of the cube?
The length of each edge is feet.

Answer:
It is given that
A cube-shaped art sculpture has a volume of 64 cubic feet.
Now,
We know that,
A cube has all the same side lengths
Now,
Let the side length of a cube be: s
So,
The volume of a cube (V) = s³
So,
Side = \(\sqrt[3]{V}\)
Now,

So,

Hence, from the above,
We can conclude that
The length of each edge of the cube is: 4 feet

Convince Me!
How can you find the cube root of 64?
Answer:
Let the number be: p
Now,
The cube root of a number is: \(\sqrt[3]{p}\)
So,
The cube root of 64 = \(\sqrt[3]{64}\)
= \(\sqrt[3]{4 × 4 × 4}\)
= 4
Hence, from the above,
We can conclude that
The cube root of 64 is: 4

Try It!

Evaluate.
a. \(\sqrt[3]{27}\)
Answer:
The given number is: \(\sqrt[3]{27}\)
Now,

Hence, from the above,
We can conclude that
The cube root of the given number is: 3

b. \(\sqrt{25}\)
Answer:
The given number is: \(\sqrt{25}\)
Now,

Hence, from the above,
We can conclude that
The square root of the given number is: 5

c. \(\sqrt{81}\)
Answer:
The given number is: \(\sqrt{81}\)
Now,

Hence, from the above,
We can conclude that
The square root of the given number is: 9

d. \(\sqrt[3]{1}\)
Answer:
The given number is: \(\sqrt[3]{1}\)
Now,

Hence, from the above,
We can conclude that
The cube root of the given number is: 1

Try It!

Emily wants to buy a tablecloth to cover a square card table. She knows the tabletop has an area of 9 square feet. What are the minimum dimensions of the tablecloth Emily needs?

Emily should buy a tablecloth that measures at least
feet by feet.
Answer:
It is given that
Emily wants to buy a tablecloth to cover a square card table. She knows the tabletop has an area of 9 square feet.
Now,
We know that,
The area of a square = Side²
So,
According to the given information,
The area of a square table = Side²
Side² = 9
Now,

So,

Hence, from the above,
We can conclude that
The minimum dimensions of the table cloth Emily needs is: 3 feet × 3 feet

KEY CONCEPT

The cube root of a number is a number whose cube is equal to that number.

Cubing a number and taking the cube root of the number are inverse operations.

The square root of a number is a number whose square is equal to that number.

Squaring a number and taking the square root of the number are inverse operations.

Do You Understand?

Question 1.
? Essential Question How do you evaluate cube roots and square roots?
Answer:
Let the number be: p
Now,
The square of a number is: p²
The cube of a number is: p³
The square root of a number is: \(\sqrt{p}\)
The cube root of a number is: \(\sqrt[3]{p}\)

Question 2.
Generalize A certain number is both a perfect square and a perfect cube. Will its square root and its cube root always be different numbers? Explain.
Answer:
We know that,
A perfect square is a number whose square root is an integer; and a perfect cube is a number whose cube root is an integer.
A number that is a perfect square and perfect cube will not always have different numbers as its square root and cube root.

Question 3.
Critique Reasoning A cube-shaped box has a volume of 27 cubic inches. Bethany says each side of the cube measures 9 inches because 9 × 3 = 27. Is Bethany correct? Explain your reasoning.
Answer:
It is given that
A cube-shaped box has a volume of 27 cubic inches. Bethany says each side of the cube measures 9 inches because 9 × 3 = 27
Now,
We know that,
A “Cube” has the equal side lengths
Now,
We know that,
The volume of a cube (V) = Side³
So,
Side = \(\sqrt[3]{V}\)
Now,
For the volume of 27 cubic inches,
Side = \(\sqrt[3]{27}\)
= \(\sqrt[3]{3 × 3 ×3}\)
= 3 inches
So,
Each side of the cube measures 3 inches
Hence, from the above,
We can conclude that
Bethany is not correct

Do You Know How?

Question 4.
A cube has a volume of 8 cubic inches. What is the length of each edge of the cube?
Answer:
It is given that
A cube has a volume of 8 cubic inches
Now,
We know that,
A “Cube” has the equal side lengths
Now,
We know that,
The volume of a cube (V) = Side³
So,
Side = \(\sqrt[3]{V}\)
Now,
For the volume of 8 cubic inches,
Side = \(\sqrt[3]{8}\)
= \(\sqrt[3]{2 × 2 ×2}\)
= 2 inches
Hence, from the above,
We can conclude that
The length of each edge of the given cube is: 2 inches

Question 5.
Below is a model of the infield of a baseball stadium. How long is each side of the infield?

Answer:
It is given that
Below is a model of the infield of a baseball stadium
Now,
The given figure is:

Now,
From the given figure,
We can observe that
The infield is in the form of a square
Now,
We know that,
The area of a square = Side²
So,
Each Side of the infield =\(\sqrt{The area of the infield}\)
= \(\sqrt{81}\)
Now,

Hence, from the above,
We can conclude that
The length of each side of the infield is: 9 inches

Question 6.
Julio cubes a number and then takes the cube root of the result. He ends up with 20. What number did Julio start with?
Answer:
It is given that
Julio cubes a number and then takes the cube root of the result. He ends up with 20
Now,
Let us say that the number is x.
So
The first step that Julio do is to cube the number so that the number will become: x³
Then,
The next he did was take cube root of the number, so that the result will become: \(\sqrt[3]{x³}\)
Now,
By solving the above expression,
\(\sqrt[3]{x³}\) = x
Now,
It is given that
The end result is 20
So,
x = 20
So,
Julio started and ended with the same number which is 20.
Hence, from the above,
We can conclude that
The number did Julio start with is: 20

Practice & Problem Solving

Leveled Practice
In 7 and 8, evaluate the cube root or square root.

Question 7.
Relate the volume of the cube to the length

Answer:
The given figure is:

Now,
We know that,
The volume of a cube (V) = Side³
So,
Side = \(\sqrt[3]{V}\)
So,
Side = \(\sqrt[3]{8}\)
Now,

Hence, from the above,
We can conclude that
The length of each edge of the cube is: 2 cm

Question 8.
Relate the area of the square to the length of each edge.

Answer:
The given figure is:

Now,
We know that,
The area of a square (V) = Side²
So,
Side = \(\sqrt{V}\)
So,
Side = \(\sqrt{16}\)
Now,

Hence, from the above,
We can conclude that
The length of each side of the square is: 4 cm

Question 9.
Would you classify the number 169 as a perfect square, a perfect cube, both, or neither? Explain.
Answer:
The given number is: 169
Now,
We know that,
A perfect cube is a number that can be expressed as the product of three equal integers
A perfect square is a number that can be expressed as the product of two equal integers
Now,
169 can be written as:
169 = 13 × 13
Hence, from the above,
We can conclude that
169 would be classified as a perfect square

Question 10.
The volume of a cube is 512 cubic inches. What is the length of each side of the cube?
Answer:
It is given that
The volume of a cube is 512 cubic inches
Now,
We know that,
The volume of a cube (V) = Side³
So,
Side = \(\sqrt[3]{V}\)
So,
Side = \(\sqrt[3]{512}\)
So,

Hence, from the above,
We can conclude that
The length of each edge of the cube is: 8 inches

Question 11.
A square technology chip has an area of 25 square cm. How long is each side of the chip?
Answer:
It is given that
A square technology chip has an area of 25 square cm.
Now,
We know that,
The area of a square = Side²
So,
Side = \(\sqrt{The area of a square}\)
So,
Side = \(\sqrt{25}\)
So,

Hence, from the above,
We can conclude that
The length of each side of the chip is: 5 cm

Question 12.
Would you classify the number 200 as a perfect square, a perfect cube, both, or neither? Explain.
Answer:
The given number is: 200
Now,
We know that,
A perfect cube is a number that can be expressed as the product of three equal integers
A perfect square is a number that can be expressed as the product of two equal integers
Now,
200 can be written as:
200 = 100 × 2
= 10 × 10 × 2
Hence, from the above,
We can conclude that
200 would not be classified neither as a perfect square nor a perfect cube

Question 13.
A company is making building blocks. What is the length of each side of the block?

Answer:
It is given that
A company is making building blocks
Now,
The given figure is:

Now,
From the given figure,
We can observe that
The building blocks is in the form of a cube
Now,
We know that,
The volume of a cube (V) = Side³
So,
Side = \(\sqrt[3]{V}\)
So,
Side = \(\sqrt[3]{1}\)
Now,

Hence, from the above,
We can conclude that
The length of each side of the block is: 1 ft

Question 14.
Mrs. Drew wants to build a square sandbox with an area of 121 square feet. What is the total length of wood Mrs. Drew needs to make the sides of the sandbox?
Answer:
It is given that
Mrs. Drew wants to build a square sandbox with an area of 121 square feet
Now,
We know that,
The area of a square (A) = Side²
So,
Side = \(\sqrt{A}\)
So,
Side = \(\sqrt{121}\)
Now,

So,
The side of the sandbox is: 11 feet
Now,
To find the total length of wood Mrs. Drew needs to make the sides of the sandbox = 4 × (The length of the side of the sandbox)
= 4 × 11
= 44 feet
Hence, from the above,
We can conclude that
The total length of wood Mrs. Drew needs to make the sides of the sandbox is: 44 feet

Question 15.
Construct Arguments Diego says that if you cube the number 4 and then take the cube root of the result, you end up with 8. Is Diego correct? Explain.
Answer:
It is given that
Diego says that if you cube the number 4 and then take the cube root of the result, you end up with 8
Now,
According to the given information,
Step 1:
4³ = 4 × 4 × 4
= 64
Step 2:
\(\sqrt[3]{64}\)
= \(\sqrt[3]{4 × 4 ×4}\)
= 4
But,
It is given that
The end result is 8 and we got 4
Hence,f rom the above,
We can conclude that
Diego is not correct

Question 16.
Higher Order Thinking Talia is packing a moving box. She has a square-framed poster with an area of 9 square feet. The cube-shaped box has a volume of 30 cubic feet. Will the poster lie flat in the box? Explain.

Answer:
It is given that
Talia is packing a moving box. She has a square-framed poster with an area of 9 square feet. The cube-shaped box has a volume of 30 cubic feet
Now,
The given figure is:

Now,
To make the square-framed poster fit into a cube-shaped box,
The side of square-framed poster < The side of each edge of the cube-shaped box
Now,
We know that,
The area of a square = Side²
The volume of a cube = Side³
So,
\(\sqrt{9}\) = 3 feet
\(\sqrt[3]{30}\) = 3.10 feet
So,
3 < 3.10
So,
The side of square-framed poster < The side of each edge of the cube-shaped box
Hence, from the above,
We can conclude that
The poster lie flat in the box

Assessment Practice

Question 17.
Which expression has the greatest value?
A. \(\sqrt{49}\) . 2
B. \(\sqrt{49}\) – \(\sqrt{16}\)
C. \(\sqrt{25}\) + \(\sqrt{16}\)
D. \(\sqrt{25}\).3
Answer:
The given expressions are:
a.
The given expression is: \(\sqrt{49}\) . 2
Now,
We know that,
\(\sqrt{49}\) = 7
So,
The value of the given expression is: 14
b.
The given expression is:
\(\sqrt{49}\) – \(\sqrt{16}\)
Now,
We know that,
\(\sqrt{49}\) = 7
\(\sqrt{16}\) = 4
So,
The value of the given expression is: 3
c.
The given expression is:
\(\sqrt{25}\) + \(\sqrt{16}\)
Now,
We know that,
\(\sqrt{25}\) = 5
\(\sqrt{16}\) = 4
So,
The value of the given expression is: 9
d.
The given expression is:
\(\sqrt{25}\).3
Now,
We know that,
\(\sqrt{25}\) = 5
So,
The value of the given expression is: 15
Hence, from the above,
We can conclude that
The expression that has the greatest value is:

Question 18.
A toy has various shaped objects that a child can push through matching holes. The area of the square hole is 8 square cm. The volume of a cube-shaped block is 64 cubic cm
PART A
Which edge length can you find? Explain.
Answer:
It is given that
A toy has various shaped objects that a child can push through matching holes. The area of the square hole is 8 square cm. The volume of a cube-shaped block is 64 cubic cm
Now,
We know that,
The area of a square (A) = Side²
The volume of a cube (V) = Side³
So,
\(\sqrt{8}\) = 2.82 cm
\(\sqrt[3]{64}\) = 4 cm
Hence, from the above,
We can conclude that
The value of the edge lengths you found are:
The side of a square-shaped hole is: 2.82 cm
The side of a cube-shaped block is: 4 cm

PART B
Will the block fit in the square hole? Explain.
Answer:
Now,
From Part A,
We can observe that
The side of a square-shaped hole is: 2.82 cm
The side of a cube-shaped block is: 4 cm
Now,
For the block to fit in the square hole,
The side of the block < The side of the hole
But,
4 cm > 2.82 cm
Hence, from the above,
We can conclude that
The block will not fit in the square hole

Lesson 1.5 Solve Equations Using Square Roots and Cube Roots

Solve & Discuss It!

Janine can use up to 150 one-inch blocks to build a solid, cube-shaped model. What are the dimensions of the possible models that she can build? How many blocks would Janine use for each model? Explain.

Answer:
It is given that
Janine can use up to 150 one-inch blocks to build a solid, cube-shaped model
Now,
The given figure is:

Now,
We know that,
The volume of a cube = Length × Width × Height
Now,
To find the dimensions of the possible models that Janine can model,
We have to find the multiples of 150 in terms of three
So,
150 = 25 × 6
150 = 5 × 5 × 6
So,
The total number of blocks Janine would use for each model = The sum of the above three multiples of 150
= 5 + 5 + 6
= 16 blocks
Hence, from the above,
We can conclude that
The dimensions of the possible model that Janine can build is: 5 × 5 × 6
The total number of blocks Janine would use for each model is: 16 blocks

Look for Relationships
How are the dimensions of a solid related to its volume?
Answer:
The volume, V , of any rectangular solid is the product of the length, width, and height. We could also write the formula for volume of a rectangular solid in terms of the area of the base. The area of the base, B , is equal to length × Width.

Focus on math practices
Reasoning Janine wants to build a model using \(\frac{1}{2}\)-inch cubes. How many \(\frac{1}{2}\)-inch cubes would she use to build a solid, cube-shaped model with side lengths of 4 inches? Show your work.
Answer:
It is given that
Janine wants to build a model using \(\frac{1}{2}\)-inch cubes and a cube-shaped model with side lengths of 4 inches
Now,
According to the given information,
The number of \(\frac{1}{2}\)-inch cubes would Janine used to build a solid = \(\frac{1}{2}\) × 8 blocks
Hence, from the above,
We can conclude that
The number of \(\frac{1}{2}\)-inch cubes would she use to build a solid, cube-shaped model with side lengths of 4 inches is: 8 blocks

? Essential Question
How can you solve equations with squares and cubes?
Answer:
The steps to solve equations with squares are:
Step 1:
Divide all terms by a (the coefficient of x²).
Step 2:
Move the number term (c/a) to the right side of the equation.
Step 3:
Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.
The steps to solve equations with cubes are:
Step 1:
Divide all terms by a (the coefficient of x^³).
Step 2:
Move the number term (\(\frac{d}{a}\)) to the right side of the equation.
Step 3:
Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.

Try It!

What is the side length, s, of the square below?

Each side of the square measures meters.
Answer:
The given figure is:

Now,
From the given figure,
We can observe that
It is a square
Now,
We know that,
The area of a square = Side²
Now,

So,

Hence, from the above,
We can conclude that
Each side of the square measures 10 meters

Convince Me!
Why are there two possible solutions to the equation s² = 100? Explain why only one of the solutions is valid in this situation.
Answer:
The given equation is: s²
Now,
We know that,
The square of a positive number or a negative number is always positive and the square root of a number must always be positive
So,
s² = 100
s = \(\sqrt{100}\)
s = ± 10
Now,
We know that,
The side of any figure will always be positive
Hence, from the above,
We can conclude that
Only one of the solutions is valid in this situation because of the property of sides of the geometrical figures

Try It!

Solve x³ = 64.
Answer:
The given equation is:
x³ = 64
So,
x = \(\sqrt[3]{64}\)
x = \(\sqrt[3]{4 × 4 × 4}\)
x = 4
Hence, from the above,
We can conclude that
The value of x for the given equation is: 4

Try It!

a. Solve a³ = 11.
Answer:
The given equation is:
a³ = 11
So,
a = \(\sqrt[3]{11}\)
Now,
We know that,
The cube of a number will always be positive
Hence, from the above,
We can conclude that
The possible solution for the given equation is: \(\sqrt[3]{11}\)

b. Solve c² = 27.
Answer:
The given equation is:
c² = 27
So,
c = ±\(\sqrt{27}\)
Hence, from the above,
We can conclude that
The possible solutions for the given equation is: \(\sqrt{27}\), –\(\sqrt{27}\)

KEY CONCEPT
You can use square roots to solve equations involving squares.
x² = a
\(\sqrt{x^{2}}\) = \(\sqrt{a}\)
x = + \(\sqrt{a}\), –\(\sqrt{a}\)

You can use cube roots to solve equations involving cubes. x2 = b Vx3 = xb
x³ = b
\(\sqrt[3]{x^{3}}\) = \(\sqrt[3]{b}\)
x = \(\sqrt[3]{b}\)

Do You Understand?

Question 1.
? Essential Question
How can you solve equations with squares and cubes?
Answer:
The steps to solve equations with squares are:
Step 1:
Divide all terms by a (the coefficient of x²).
Step 2:
Move the number term (c/a) to the right side of the equation.
Step 3:
Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.
The steps to solve equations with cubes are:
Step 1:
Divide all terms by a (the coefficient of x^³).
Step 2:
Move the number term (\(\frac{d}{a}\)) to the right side of the equation.
Step 3:
Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.

Question 2.
Be Precise Suri solved the equation x² = 49 and found that x = 7. What error did Suri make?
Answer:
It is given that
Suri solved the equation x² = 49 and found that x = 7.
Now,
The given equation is:
x² = 49
Now,
We know that,
The square of a number is always positive but the square root of a number can either be positive or negative
So,
x = \(\sqrt{49}\)
x = ±7
x = 7, -7
Hence, from the above,
We can conclude that
The error did Suri made is that he did not consider the negative square root of 49

Question 3.
Construct Arguments There is an error in the work shown below. Explain the error and provide a correct solution.
x³ = 125
\(\sqrt[3]{x^{3}}\) = \(\sqrt[3]{125}\)
x = 5 and x = -5
Answer:
The given equation is:
x³ = 125
Now,
We know that,
The cube root of a number will always be positive
So,
\(\sqrt[3]{x^{3}}\) = \(\sqrt[3]{125}\)
x = \(\sqrt[3]{x^{3}}\) = \(\sqrt[3]{5 × 5 × 5}\)
x = 5
Hence, from the above,
We can conclude that
The error is not considering that the cube root of a number will always be positive
The correct solution is:
x = 5

Question 4.
Why are the solutions to x² = 17 irrational?
Answer:
The given equation is:
x² = 17
So,
x = ±\(\sqrt{17}\)
Now,
We know that,
The perfect square will be an integer and we know that an integer is a rational number
Now,
When we look at 17,
It is not a perfect square
Hence, from the above,
We can conclude that
The solutions of the given equation are irrational

Do You Know How?

Question 5.
If a cube has a volume of 27 cubic cm, what is the length of each edge? Use the volume formula, V = s³, and show your work.
Answer:
It is given that
A cube has a volume of 27 cubic cm
Now,
We know that,
The volume of a cube (V) = Side³
So,
Side = \(\sqrt[3]{V}\)
Now,
According to the given information,
Side = \(\sqrt[3]{27}\)
Side = \(\sqrt[3]{3 × 3 × 3}\)
Side = 3 cm
Hence, from the above,
We can conclude that
The length of each edge of the give cube is: 3 cm

Question 6.
Darius is building a square launch pad for a rocket project. If the area of the launch pad is 121 square cm, what is its side length? Use the area formula, A = s², and show your work.

Answer:
It is given that
Darius is building a square launch pad for a rocket project. If the area of the launch pad is 121 square cm,
Now,
We know that,
The area of a square (A) = Side²
So,
Side = \(\sqrt{A}\)
Now,
According to the given information,
Side = \(\sqrt{121}\)
Side = \(\sqrt{11 × 11}\)
Side = 11 cm
Hence, from the above,
We can conclude that
The side length of the launch pad is: 1 cm

Question 7.
Solve the equation x³ = -215.
Answer:
The given equation is:
x³ = -215
Now,
We know that,
The square root won’t accept negative values but the cube root will accept both positive and negative values
So,
x = \(\sqrt[3]{-215}\)
Now,

Hence, from the above,
We can conclude that
The solution for the given equation is: -5.99

Practice & Problem Solving

Leveled Practice
In 8 and 9, solve.

Question 8.
z² = 1

Answer:
The given equation is:
z² = 1
Now,

Hence, from the above,
We can conclude that

Question 9.
a³ = 216

Answer:
The given equation is:
a³ = 216
Now,

Hence, from the above,
We can conclude that
The solution for the given equation is: 6

Question 10.
Solve v² = 47.
Answer:
The give equation is:
v² = 47
Now,
\(\sqrt{v²}\) = \(\sqrt{47}\)
v = ±\(\sqrt{47}\)
Now,

Hence, from the above,
We can conclude that
The solutions for the given equation are: 6.85, and -6.85

Question 11.
The area of a square photo is 9 square inches. How long is each side of the photo?
Answer:
It is given that
The area of a square photo is 9 square inches
Now,
We know that,
The area of a square (A) = Side² (s)
So,
s² = 9
\(\sqrt{s²}\) = \(\sqrt{9}\)
s = 3 inches [Because the length of the side will never be negative]
Hence, from the above,
We can conclude that
The length of each side of the photo is: 3 inches

Question 12.
Solve the equation y² = 81.
Answer:
The given equation is:
y² = 81
Now,
\(\sqrt{y²}\) = ±\(\sqrt{81}\)
y = ±9
Hence, from the above,
We can conclude that
The solutions for the given equation are: 9, and -9

Question 13.
Solve the equation w³ = 1,000.
Answer:
The given equation is:
w³ = 1,000
Now,
\(\sqrt[3]{w³}\) = \(\sqrt[3]{1,000}\)
w = \(\sqrt[3]{10 × 10 × 10}\)
w = 10
Hence, from the above,
We can conclude that
The solution for the given equation is: 10

Question 14.
The area of a square garden is shown. How long is each side of the garden?

Answer:
It is given that
The area of a square garden is shown.
Now,
The given figure is:

Now,
From the given figure,
We can observe that
The area of the square garden is: 121 ft²
Now,
We know that,
The area of a square (A) = s²
So,
According to the given information,
s² = 121
\(\sqrt{s²}\) = \(\sqrt{121}\)
s = 11 ft [Since the length of the side will never be negative]
Hence, from the above,
We can conclude that
The length of each side of the garden is: 11 ft

Question 15.
Solve b² = 77.
Answer:
The given equation is:
b² = 77
Now,
\(\sqrt{b²}\) = ±\(\sqrt{77}\)
Now,

Hence, from the above,
We can conclude that
The solutions for the givene quation are: 8.77, and -8.77

Question 16.
Find the value of c in the equation c³ = 1,728.
Answer:
The given equation is:
c³ = 1,728
Now,
\(\sqrt[3]{c³}\) = \(\sqrt[3]{1,728}\)
c = 12
Hence, from the above,
We can conclude that
The value of c for the given equation is: 12

Question 17.
Solve the equation v³ = 12.
Answer:
The given equation is:
v³ = 12
Now,
\(\sqrt[3]{v³}\) = \(\sqrt[3]{12}\)
Now,

Hence, from the above,
We can conclude that
The solution for the given equation is: 2.28

Question 18.
Higher Order Thinking Explain why
\(\sqrt[3]{-\frac{8}{27}}\) is –\(\frac{2}{3}\)
Answer:
The given equation is:
\(\sqrt[3]{-\frac{8}{27}}\)
Now,
We know that,
\(\sqrt[3]{\frac{a}{b}}\) = \(\frac{\sqrt[3]{a}}{\sqrt[3]{b}}\)
So,
\(\sqrt[3]{-\frac{8}{27}}\) = –\(\frac{\sqrt[8]{a}}{\sqrt[3]{27}}\)
= –\(\frac{\sqrt[2 × 2 ×2]{a}}{\sqrt[3]{3 ×3 × 3}}\)
= –\(\frac{2}{3}\)
Hence, from the above,
We can conclude that
\(\sqrt[3]{-\frac{8}{27}}\) is –\(\frac{2}{3}\) due to the below property of Exponents:
\(\sqrt[3]{\frac{a}{b}}\) = \(\frac{\sqrt[3]{a}}{\sqrt[3]{b}}\)

Question 19.
Critique Reasoning Manolo says that the solution of the equation g² = 36 is g = 6 because 6 × 6 = 36. Is Manolo’s reasoning complete? Explain.
Answer:
It is given that
Manolo says that the solution of the equation g² = 36 is g = 6 because 6 × 6 = 36
Now,
The given equation is:
g² = 36
Now,
We know that,
The square of a number will always be positive but the square root of a number will either be positive or negative
So,
\(\sqrt{g²}\) = ±\(\sqrt{36}\)
g = ±6
So,
The solutions for the given equation are: 6, -6
Hence, from the above,
We can conclude that
Manolo’s reasoning is not complete

Question 20.
Evaluate \(\sqrt[3]{-512}\).
a. Write your answer as an integer.
Answer:
The given equation is:
\(\sqrt[3]{-512}\)
Now,
\(\sqrt[3]{-512}\) = \(\sqrt[3]{(-8) × (-8) × (-8)}\)
= -8
Hence, from the above,
We can conclude that
The value of \(\sqrt[3]{-512}\) as an integer is: -8

b. Explain how you can check that your result is correct.
Answer:
The given equation is:
\(\sqrt[3]{-512}\)
Now,
From part (a),
We get the value of the given equation is: 8
Now,
\(\sqrt[3]{(-8) × (-8) × (-8)}\)
= \(\sqrt[3]{64 × (-8)}\)
= \(\sqrt[3]{-512}\)
Hence, from the above,
We can conclude that
The result is correct because the givene quation and the result are the same

Question 21.
Yael has a square-shaped garage with 228 square feet of floor space. She plans to build an addition that will increase the floor space by 50%. What will be the length, to the nearest tenth, of one side of the new garage?

Answer:
It is given that
Yael has a square-shaped garage with 228 square feet of floor space. She plans to build an addition that will increase the floor space by 50%.
Now,
According to the given information,
50% of 228 = \(\frac{50}{100}\) × 228
= \(\frac{50 × 228}{100}\)
= 114 square feet
So,
The area of the new garage = 228 + 114
= 342 square feet
Now,
We know that,
The area of a square (A) = Side (s)²
So,
s² = 342
\(\sqrt{s²}\) = \(\sqrt{342}\)
Now,

Hence, from the above,
We can conclude that
The length of one side of the new garage is: 18.5 feet

Assessment Practice

Question 22.
The Traverses are adding a new room to their house. The room will be a cube with a volume of 6,859 cubic feet. They are going to put in hardwood floors, which costs $10 per square foot. How much will the hardwood floors cost?
Answer:
It is given that
The Traverses are adding a new room to their house. The room will be a cube with a volume of 6,859 cubic feet. They are going to put in hardwood floors, which costs $10 per square foot
Now,
We know that,
The volume of a cube (V) = s³
So,
According to the given information,
s³ = 6,859
\(\sqrt[3]{s³}\) = \(\sqrt[3]{6,859}\)
s = 19 feet
So,
The length of each edge of the new room is: 19 feet
Now,
To find the total cost of hardwood floors, find the perimeter of the room and multiply the result with the cost per square foot
Now,
We know that,
The perimeter of a cube = 6s
So,
The perimeter of the cube (p) = 6 × 19
= 114 feet
Now,
The total cost of hardwood floors = 114 × $10
= $1,140
Hence, from the above,
We can conclude that
The total cost of hardwood floors is: $1,140

Question 23.
While packing for their cross-country move, the Chen family uses a crate that has the shape of a cube.
PART A
If the crate has the volume V = 64 cubic feet, what is the length of one edge?
It is given that
While packing for their cross-country move, the Chen family uses a crate that has the shape of a cube and the crate has the volume V = 64 cubic feet
Now,
We know that,
The volume of a cube (V) = s³
So,
According to the given information,
s³ = 64
\(\sqrt[3]{s³}\) = \(\sqrt[3]{64}\)
s = 4 feet
Hence, from the above,
We can conclude that
The length of each edge of the crate is: 4 feet

PART B
The Chens want to pack a large, framed painting. If the framed painting has the shape of a square with an area of 12 square feet, will the painting fit flat against a side of the crate? Explain.
Answer:
It is given that
The Chens want to pack a large, framed painting. If the framed painting has the shape of a square with an area of 12 square feet
Now,
We know that,
The area of a square (A) = s²
So,
According to the given information,
s² = 12
Now,
\(\sqrt{s²}\) = \(\sqrt{12}\)
Now,

So,
According to the given information,
The side of the crate > The side of the painting
4 > 2.28
Hence, from the above,
We can conclude that
The painting will fit flat against a side of the crate

Topic 1 MID-TOPIC CHECKPOINT

Question 1.
Vocabulary How can you show that a number is a rational number? Lesson 1.2
Answer:
A rational number is a number that can be written as a ratio. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers

Question 2.
Which shows 0.2\(\overline{3}\) as a fraction? Lesson 1.1
A. \(\frac{2}{33}\)
B. \(\frac{7}{33}\)
C. \(\frac{23}{99}\)
D. \(\frac{7}{30}\)
Answer:
The given expression is: 0.2\(\overline{3}\)
Hence, from the above,
We can conclude that
The options that show 0.2\(\overline{3}[/latex as a fraction is:

Question 3.
Approximate [latex]\sqrt{8}\) to the nearest hundredth. Show your work. Lesson 1.3
Answer:
The given expression is: \(\sqrt{8}\)
Now,

Hence, from the above,
We can conclude that
The approximate value of the given expression to the nearest hundredth is: 2.82

Question 4.
Solve the equation m² = 14. Lesson 1.5
Answer:
The given equation is:
m² = 14
Now,
\(\sqrt{m²}\) = ±\(\sqrt{14}\)
Now,

Hence, from the above,
We can conclude that

The solutions for the given equation are: 3.74, -3.74

Question 5.
A fish tank is in the shape of a cube. Its volume is 125 ft³. What is the area of one face of the tank? Lessons 1.4 and 1.5
Answer:
It is given that
A fish tank is in the shape of a cube. Its volume is 125 ft³.
Now,
We know that,
The volume of a cube (V) = s³
So,
s³= 125
Now,
\(\sqrt[3]{s³}\) = \(\sqrt[3]{125}\)
s = \(\sqrt[3]{5 × 5 × 5}\)
s = 5 ft
Now,
We know that,
The surface area of a cube = 4s²
So,
The area of one face of the tank = 4 × 5²
= 4 × 25
= 100 ft²
Hence, from the above,
We can conclude that
The area of one face of the tank is: 100 ft²

Question 6.
Write 1.\(\overline{12}\) as a mixed number. Show your work. Lesson 1.1
Answer:
The given expression is: 1.\(\overline{12}\)
Now,
The representation of the given expression in the form of a fraction is: \(\frac{28}{25}\)
Now,
The representation of \(\frac{28}{25}\) into a mixed number is: 1\(\frac{3}{25}\)
Hence, from the above,
We can conclude that
The representation of 1.\(\overline{12}\) as a mixed number is: 1\(\frac{3}{25}\)
How well did you do on the mid-topic checkpoint? Fill in the stars.

Topic 1 MID-TOPIC PERFORMANCE TASK

Six members of the math club are forming two teams for a contest. The teams will be determined by having each student draw a number from a box.

PART A

The table shows the results of the draw. The students who drew rational numbers will form the team called the Tigers. The students who drew irrational numbers will form the team called the Lions. List the members of each team.
Answer:
It is given that
Six members of the math club are forming two teams for a contest. The teams will be determined by having each student draw a number from a box.
Now,
The given table is:

Now,
We know that,
The numbers that can be written in the form of \(\frac{p}{q}\) are “Rational numbers”
The numbers that can not be written in the form of \(\frac{p}{q}\) are “Irrational numbers”
Now,
From the given table,
The list of rational numbers are:
6.\(\overline{34}\), \(\sqrt{36}\), 6.3\(\overline{4}\)
The list of Irrational numbers are:
\(\sqrt{38}\), 6.343443444….,, \(\sqrt{34}\)
Hence, from the above,
We can conclude that
The students present in the Tigers Team are:
6.\(\overline{34}\), \(\sqrt{36}\), 6.3\(\overline{4}\)
The students present in the Lions Team are:
\(\sqrt{38}\), 6.343443444….,, \(\sqrt{34}\)

PART B

The student on each team who drew the greatest number will be the captain of that team. Who will be the captain of the Tigers? Show your work.
Answer:
From Part A,
We can observe that
The students present in the Tigers Team are:
6.\(\overline{34}\), \(\sqrt{36}\), 6.3\(\overline{4}\)
Now,
From the above list of numbers,
We can observe that
6.\(\overline{34}\) is the greater number
Hence, from the above,
We can conclude that
The captain of the Tigers Team is: Anya

PART C

Who will be the captain of the Lions? Show your work.
Answer:
The students present in the Lions Team are:
\(\sqrt{38}\), 6.343443444….,, \(\sqrt{34}\)
Now,
From the above list of numbers,
We can observe that
6.343443444…., is the greater number
Hence, from the above,
We can conclude that
The captain of the Lions Team is: Ryan

Lesson 1.6 Use Properties of Integer Exponents

Solve & Discuss It!

One band’s streaming video concert to benefit a global charity costs $1.00 to view.
The first day, the concert got 2,187 views. The second day, it got about three times as many views. On the third day, it got 3 times as many views as on the second day. If the trend continues, how much money will the band raise on Day 7?

Answer:
It is given that
One band’s streaming video concert to benefit a global charity costs $1.00 to view.
The first day, the concert got 2,187 views. The second day, it got about three times as many views. On the third day, it got 3 times as many views as on the second day
So,
According to the given information,
The amount of money got on the first day of concert = (The total number of views) × $1
= 2,187 × $1
= $2,187
The amount of money got on the second day of concert = (The amount of money got on the first day of concert) × 3
= $2,187 × 3
= $6,561
The amount of money got on the third day of concert = (The amount of money got on the second day of concert) × 3
= $6,561 × 3
= $19,683
The amount of money got on the fourth day of concert = (The amount of money got on the third day of concert) × 3
= $19,683 × 3
= $59,049
The amount of money got on the fifth day of concert = (The amount of money got on the fourth day of concert) × 3
= $59,049 × 3
= $1,77,147
The amount of money got on the sixth day of concert = (The amount of money got on the fifth day of concert) × 3
= $1,77,147 × 3
= $5,31,441
The amount of money got on the seventh day of concert = (The amount of money got on the sixth day of concert) × 3
= $5,31,441 × 3
= $15,94,323
Hence, from the above,
We can conclude that
The amount of money the Band raise on Day 7 is: $15,94,323

Focus on math practices
Use Structure Use prime factorization to write an expression equivalent to the amount of money raised by the band on the last day of the week.
Answer:
From the above,
We can observe that
The amount of money got on the last day of the wee is: $15,94,323
Now,
By using the Prime factorisation method,

Hence, from the above,
We can conclude that
The prime factorisation of 15,94,323 is: 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3

? Essential Question
How do properties of integer exponents help you write equivalent expressions?
Answer:
The properties of integer exponents can be used to write equivalent expressions by combining numeric or algebraic expressions that have a common base, distributing exponents to products and quotients, and simplifying powers of powers.

Convince Me!
Explain why the Product of Powers Property makes mathematical sense.
Answer:
The Power of a Product rule states that a term raised to a power is equal to the product of its factors raised to the same power

Try It!
Write equivalent expressions using the properties of exponents.

a. (7³)²
Answer:
The given expression is: (7³)²
Now,
We know that,
The “Power of Powers Property” states that to find the power of a power, multiply the exponents
So,
(7³)² = 7^{3 × 2}
= 7⁶
Hence, from the above,
We can conclude that
The value of the given expression is: 7⁶

b. (4⁵)³
Answer:
The given expression is: (4⁵)³
Now,
We know that,
The “Power of Powers Property” states that to find the power of a power, multiply the exponents
So,
(4⁵)³ = 4^{3 × 5}
= 4¹⁵
Hence, from the above,
We can conclude that
The value of the given expression is: 4¹⁵

c. 9⁴ × 8⁴
Answer:
The given expression is: 9⁴ × 8⁴
Now,
We know that,
By using the “Power of Products Property”, when multiplying two exponential expressions with the same exponent and different powers, multiply the bases and keep the exponents the same
So,
9⁴ × 8⁴ = (9 × 8)⁴
= 72⁴
Hence, from the above,
We can conclude that
The value of the given expression is: 72⁴

d. 8⁹ ÷ 8³
Answer:
The given expression is: 8⁹ ÷ 8³
Now,
We know that,
The “Quotient of powers Property” states that when dividing two exponential expressions with the same base, subtract the exponents
So,
8⁹ ÷ 8³ = 8^{9 – 3}
= 8⁶
Hence, from the above,
We can conclude that
The value of the given expression is: 8⁶

KEY CONCEPT

Use these properties when simplifying expressions with exponents (when a, m, and n ≠ 0).

Do You Understand?

Question 1.
Essential Question How do properties of integer exponents help you write equivalent expressions?
Answer:
The properties of integer exponents can be used to write equivalent expressions by combining numeric or algebraic expressions that have a common base, distributing exponents to products and quotients, and simplifying powers of powers.

Question 2.
Look for Relationships If you are writing an equivalent expression for 2³ × 2⁴, how many times would you write 2 as a factor?
Answer:
It is given that
you are writing an equivalent expression for 2³ × 2⁴
Now,
The given expression is: 2³ × 2⁴
Now,
We know that,
The “Power of Powers Property” states that to find the power of a power, multiply the exponents
So,
2³ × 2⁴ = 2^{3 + 4}
= 2⁷
= 2 × 2 × 2 × 2 × 2 × 2 × 2
Hence, from the above,
We can conclude that
We would write 2 as a factor 7 times

Question 3.
Construct Arguments Kristen wrote 5⁸ as an expression equivalent to (5²)⁴. Her math partner writes 5⁶. Who is correct?
Answer:
It is given that
Kristen wrote 5⁸ as an expression equivalent to (5²)⁴. Her math partner writes 5⁶
Now,
The given expression is: 5⁸
Now,
We know that,
(a^m)ⁿ = a^mn
Now,
According to the above Property,
5⁸ = 5^{4 × 2}
= (5⁴)²
= (5²)⁴
Hence, from the above,
We can conclude that
Kristen is correct

Question 4.
Critique Reasoning Tyler says that an equivalent expression for 2³ × 5³ is 10⁹. Is he correct? Explain.
Answer:
It is given that
Tyler says that an equivalent expression for 2³ × 5³ is 10⁹
Now,
The given expression is: 2³ × 5³
Now,
We know that,
a^m × b^m = (a × b)^m
So,
2³ × 5³ = (2 × 5)³
= 10³
So,
The equivalent expression for 2³ × 5³ is: 10³
Hence, from the above,
We can conclude that
Tyler is not correct

Do You Know How?

Question 5.
Write an equivalent expression for 7¹² × 7⁴.
Answer:
The given expression is: 7¹² × 7⁴
Now,
We know that,
a^m × aⁿ = a^{m + n}
So,
7¹² × 7⁴ = (7)^{12 + 4}
= 7¹⁶
Hence, from the above,
We can conclude that
The equivalent expression for the given expression is: 7¹⁶

Question 6.
Write an equivalent expression for (8²)⁴.
Answer:
The given expression is: (8²)⁴
Now,
We know that,
(a^m)ⁿ = a^mn
So,
(8²)⁴ = 8^{2 × 4}
= 8⁸
Hence, from the above,
We can conclude that
The equivalent expression for the given expression is: 8⁸

Question 7.
A billboard has the given dimensions.

Using exponents, write two equivalent expressions for the area of the rectangle.
Answer:
It is given that
A billboard has the given dimensions.
Now,
The given figure is:

Now,
From the given figure,
We can observe that
The billboard is in the form of a rectangle.
Now,
From the given figure,
We can observe that
The length of the billboard is: 7² ft
The width of the billboard is: 4² ft
Now,
We know that,
We know that,
a^m × b^m = (a × b)^m
So,
The area of the rectangle (A) = 7² × 4²
= (7 × 4)²
= 28² ft²
Hence, from the above,
We can conclude that
The two equivalent expressions for the area of the rectangle is: 7² × 4² and 28² ft²

Question 8.
Write an equivalent expression for 18⁹ – 18⁴.
Answer:
The given expression is: 18⁹ – 18⁴
Now,
18⁹ – 18⁴ = 18⁴ (18⁵ – 1)
Hence, from the above,
We can conclude that
The equivalent expression for the given expression is: 18⁴ (18⁵ – 1)

Practice & Problem Solving

Leveled Practice
In 9-12, use the properties of exponents to write an equivalent expression for each given expression.

Question 9.
2⁸ × 2⁴

Answer:
The given expression is: 2⁸ × 2⁴
Now,

Hence, from the above,
We can conclude that
The equivalent expression for the given expression is: 2¹²

Question 10.
\(\frac{8^{7}}{8^{3}}\)

Answer:
The given expression is: \(\frac{8⁷}{8³}\)
Now,
We know that,
\(\frac{a^m}{aⁿ}\) = a^{m – n}
So,

Hence, from the above,
We can conclude that
The equivalent expression for the given expression is: 8⁴

Question 11.
(3⁴)⁵

Answer:
The given expression is: (3⁴)⁵
Now,
We know that,
(a^m)ⁿ = a^mn
Now,

Hence, from the above,
We can conclude that
The equivalent expression for thegiven expression is: 3²⁰

Question 12.
3⁹ × 2⁹

Answer:
The given expression is: 3⁹ × 2⁹
Now,
We know that,
a^m × b^m = (a × b)^m
Now,

Hence, from the above,
We can conclude that
The equivalent expression for the given expression is: (3 ×  2)⁹

Question 13.
a. How do you multiply powers that have the same base?
Answer:
If the two exponential expressions have the same base and different exponents and both are multiplying each other, then add the exponents by keeping the bases same

b. How do you divide powers that have the same base?
Answer:
If the two exponential expressions have the same base and different exponents and both are dividing each other, then subtract the exponents by keeping the bases same

c. How do you find the power of a power?
Answer:
When an exponential expression contains the power of a power, we will multiply both the powers by keeping the base constant

d. How do you multiply powers with different bases but the same exponent?
Answer:
If the two exponential expressions have the same exponent and different bases and both are multiplying each other, then multiply the bases by keeping the exponents same

Question 14.
Which expressions are equivalent to 2¹¹?
Select all that apply.
\(\frac{2^{23}}{2^{12}}\)
2⁷ ∙ 2⁴
\(\frac{2^{9}}{2^{2}}\)
2² ∙ 2⁹
Answer:
The given expression is: 2¹¹
Hence, from the above,
We can conclude that
The expressions that are equivalent to 2¹¹ are:

In 15-18, use the properties of exponents to write an equivalent expression for each given expression.

Question 15.
(4⁴)³
Answer:
The given expression is: (4⁴)³
Now,
We know that,
(a^m)ⁿ = a^{m × n}
So,
(4⁴)³ = 4^{4 × 3}
= 4¹²
Hence, from the above,
We can conclude that
(4⁴)³ = 4¹²

Question 16.
\(\frac{3^{12}}{3^{3}}\)
Answer:
The given expression is: \(\frac{3^{12}}{3^{3}}\)
Now,
We know that,
\(\frac{a^m}{aⁿ}\) = a^{m – n}
So,
\(\frac{3^{12}}{3^{3}}\) = 3^{12 – 3}
= 3⁹
Hence, from the above,
We can conclude that
\(\frac{3^{12}}{3^{3}}\) = 3⁹

Question 17.
4⁵ × 4²
Answer:
The given expression is: 4⁵ × 4²
Now,
We know that,
a^m × aⁿ = a^{m + n}
So,
4⁵ × 4² = 4^{5 + 2}
= 4⁷
Hence, from the above,
We can conclude that
4⁵ × 4² = 4⁷

Question 18.
6⁴ × 2⁴
Answer:
The given expression is: 6⁴ × 2⁴
Now,
We know that,
a^m × b^m = (a × b)^m
So,
6⁴ × 2⁴ = (6 × 2)⁴
= 12⁴
Hence, from the above,
We can conclude that
6⁴ × 2⁴ = 12⁴

Question 19.
Critique Reasoning Alberto incorrectly stated that \(\frac{5^{7}}{5^{4}}\) = 1³. What was Alberto’s error? Explain your reasoning and find the correct answer.
Answer:
It is given that
Alberto incorrectly stated that \(\frac{5^{7}}{5^{4}}\) = 1³
Now,
The given expression is: \(\frac{5^{7}}{5^{4}}\)
Now,
We know that,
\(\frac{a^m}{aⁿ}\) = a^{m – n>}
So,
\(\frac{5^{7}}{5^{4}}\) = 5 ^{7 – 4}
= 5³
So,
From the above,
We can observe that
Alberto applied the “Quotient of Powers Property” incorrectly
Hence, from the above,
We can conclude that
Alberto’s error is: Alberto applied the “Quotient of Powers Property” incorrectly

Question 20.
Is the expression 8 × 8⁵ equivalent to (8 × 8)⁵? Explain.
Answer:

Question 21.
Is the expression (3²)^-3 equivalent to (3³)^-2? Explain.
Answer:

Question 22.
Is the expression 3² ∙ 3^-3 equivalent to 3³ ∙ 3^-2? Explain.
Answer:

Question 23.
Model with Math What is the width of the rectangle written as an exponential expression?

Answer:

Question 24.
Simplify the expression \(\left(\left(\frac{1}{2}\right)^{3}\right)^{3}\).
Answer:

Question 25.
Higher Order Thinking Use a property of exponents to write (3b)⁵ as a product of powers.
Answer:

Assessment Practice

Question 26.
Select all the expressions equivalent to 4⁵ ∙ 4¹⁰
4⁵ + 4¹⁰
4³ ∙ 4⁵
4³ ∙ 4¹²
4³ + 4¹²
4¹⁸ – 4³
4¹⁵

Question 27.
Your teacher asks the class to evaluate the expression (2³)¹. Your classmate gives an incorrect answer of 16.
PART A
Evaluate the expression.
PART B
What was the likely error?
A. Your classmate divided the exponents.
B. Your classmate multiplied the exponents.
C. Your classmate added the exponents.
D. Your classmate subtracted the exponents.
Answer:

Lesson 1.7 More Properties of Integer Exponents

ACTIVITY

Explore It!

Calvin and Mike do sit-ups when they work out. They start with 64 sit-ups for the first set and do half as many each subsequent set.

Look for Relationships
Determine whether the relationship shown for Set 1 is also true for Sets 2-5.

A. What representation can you use to show the relationship between the set number and the number of sit-ups?

B. What conclusion can you make about the relationship between the number of sit-ups in each set?

Focus on math practices
Use Structure How could you determine the number of sit-up sets Calvin and Mike do?

? Essential Question
What do the Zero Exponent and Negative Exponent Properties mean?

Try It!

Evaluate
a. (-7)⁰
b. (43)⁰
c. 1⁰
d. (0.5⁰)

Convince Me!
Why is 2(7⁰) = 2?

Try It!

Write each expression using positive exponents.
a. 8^-2
b. 2^-4
c. 3^-5
Answer:

Try It!

Write each expression using positive exponents.

a. \(\frac{1}{5^{-3}}\)

b. \(\frac{1}{2^{-6}}\)

KEY CONCEPT
Use these additional properties when simplifying or generating equivalent expressions with exponents (when a ≠ 0 and n ≠ 0).
Zero Exponent Property
a⁰ = 1

Negative Exponent Property
a^-n = \(\frac{1}{a^{n}}\)

Do You Understand?

Question 1.
Essential Question What do the Zero Exponent and Negative Exponent Properties mean?

Question 2.
Reasoning In the expression 9^-12, what does the negative exponent mean?

Question 3.
Reasoning in the expression 3(2⁰), what is the order of operations? Explain how you would evaluate the expression.

Do You Know How?

Question 4.
Simplify 1,999,999⁰.
Answer:

Question 5.
a. Write 7^-6 using a positive exponent.
b. Rewrite \(\frac{1}{10^{-3}}\) using a positive exponent.
Answer:

Question 6.
Evaluate 27x^oy^-2 for x = 4 and y = 3.
Answer:

Practice & Problem Solving

Leveled Practice In 7-8, complete each table to find the value of a nonzero number raised to the power of 0.

Question 7.

Answer:

Question 8.

Answer:

Question 9.
Given: (-3.2)⁰
a. Simplify the given expression.
b. Write two expressions equivalent to the given expression. Explain why the three expressions are equivalent.
Answer:

Question 10.
Simplify each expression for x = 6.
a. 12x⁰(x^-4)
b. 14(x^-2)

In 11 and 12, compare the values using >,<, or =.

Question 11.
3^-2 1
Answer:

Question 12.
\(\left(\frac{1}{4}\right)^{0}\) 1

In 13 and 14, rewrite each expression using a positive exponent.

Question 13.
9^-4
Answer:

Question 14.
\(\frac{1}{2^{-6}}\)
Answer:

Question 15.
Given: 9y⁰
a. Simplify the expression for y = 3.
b. Construct Arguments Will the value of the given expression vary depending on y? Explain.
Answer:

Question 16.
Simplify each expression for x = 4.
a. -5x^-4
b. 7x^-3
Answer:

Question 17.
Evaluate each pair of expressions.
a. (-3)^-8 and -3^-8
b. (-3)^-9 and -3^-9
Answer:

Question 18.
Be Precise To win a math game, Lamar has to pick a card with an expression that has a value greater than 1. The card Lamar chooses reads \(\left(\frac{1}{2}\right)^{-4}\). Does Lamar win the game? Explain.

Answer:

Question 19.
Simplify the expression. Assume that x is nonzero. Your answer should have only positive exponents. x^-10 ∙ x⁶
Answer:

Question 20.
Higher Order Thinking
a. Is the value of the expression \(\left(\frac{1}{4^{-3}}\right)^{-2}\) greater than 1, equal to 1, or less than 1?
b. If the value of the expression is greater than 1, show how you can change one sign to make the value less than 1. If the value is less than 1, show how you can change one sign to make the value greater than 1. If the value is equal to 1, show how you can make one change to make the value not equal to 1.

Assessment Practice

Question 21.
Which expressions are equal to 5^-3? Select all
that apply.
125
125^-1
5³
\(\frac{1}{5^{3}}\)
\(\frac{1}{125}\)
Answer:

Question 22.
Which expressions have a value less than 1 when x = 4? Select all that apply.
\(\left(\frac{3}{x^{2}}\right)^{0}\)
\(\frac{x^{0}}{3^{2}}\)
\(\frac{1}{6^{-x}}\)
\(\frac{1}{x^{-3}}\)
3x^-4

Lesson 1.8 Use Powers of 10 to Estimate Quantities

ACTIVITY

Explain It!

Keegan and Jeff did some research and found that there are approximately 7,492,000,000,000,000,000 grains of sand on Earth. Jeff says that it is about 7 × 10¹⁵ grains of sand. Keegan says that this is about 7 × 10¹⁸ grains of sand.

A. How might Jeff have determined his estimate? How might Keegan have determined his estimate?

B. Whose estimate, Jeff’s or Keegan’s, is more logical? Explain.

Focus on math practices
Be Precise Do you think the two estimates are close in value? Explain your reasoning.

? Essential Question
when would you use a power of 10 to estimate a quantity?

Try It!

Light travels 299,792,458 meters per second. Sound travels at 332 meters per second. Use a power of 10 to compare the speed of light to the speed of sound.
299,792,458 rounded to the greatest place value is
There are zeros in the rounded number.
The estimated speed of light is × 10 meters per second.
3 × 10 > 3 × 10, so the speed of light is faster than the speed of sound.

322 rounded to the greatest place value is
There are zeros in the rounded number. The estimated speed of sound is × 10 meters per second.

Convince Me!
Country A has a population of 1,238,682,005 and Country B has a population of 1,106,487,394. How would you compare these populations?

Try It!

There are approximately 1,020,000,000 cars in the world. The number of cars in the United States is approximately 239,800,000.
Compare the number of cars in the world to that in the United States.

KEY CONCEPT

You can estimate a very large or very small number by rounding the number to its greatest place value, and then writing that number as a single digit times a power of 10.

Do You Understand?

Question 1.
? Essential Question when would you use powers of 10 to estimate a quantity?
Answer:

Question 2.
Construct Arguments Kim writes an estimate for the number 0.00436 as 4 × 10³. Explain why this cannot be correct.
Answer:

Question 3.
Be Precise Raquel estimated 304,900,000,000 as 3 × 10⁸. What error did she make?
Answer:

Do You Know How?

Question 4.
Use a single digit times a power of 10 to estimate the height of Mt. Everest to the nearest ten thousand feet.

Answer:

Question 5.
A scientist records the mass of a proton as 0.0000000000000000000000016726231 gram. Use a single digit times a power of 10 to estimate the mass.
Answer:

Question 6.
The tanks at the Georgia Aquarium hold approximately 8.4 × 10⁶ gallons of water. The tanks at the Audubon Aquarium of the Americas hold about 400,000 gallons of water. Use a single digit times a power of 10 to estimate how many times greater the amount of water is at the Georgia Aquarium.
Answer:

Practice & Problem Solving

Leveled Practice in 7-9, use powers of 10 to estimate quantities.

Question 7.
A city has a population of 2,549,786 people. Estimate this population to the nearest million. Express your answer as the product of a single digit and a power of 10. Rounded to the nearest million, the population is about
Written as the product of a single digit and a power of ten, this number is
Answer:

Question 8.
Use a single digit times a power of 10 to estimate the number 0.00002468. Rounded to the nearest hundred thousandth, the number is about
Written as a single digit times a power of ten, the estimate is
Answer:

Question 9.
The approximate circumferences of Earth and Saturn are shown. How many times greater is the circumference of Saturn than the circumference of Earth?
The circumference of Saturn is


Saturn’s circumference is about times greater than the circumference of Earth.
Answer:

Question 10.
Estimate 0.037854921 to the nearest hundredth. Express your answer as a single digit times a power of ten.
Answer:

Question 11.
Compare the numbers 6 × 10^-6 and 2 × 10^-8.
a. Which number has the greater value?
b. Which number has the lesser value?
c. How many times greater is the greater number?
Answer:

Question 12.
Taylor made $43,785 last year. Use a single digit times a power of ten to express this value rounded to the nearest ten thousand.
Answer:

Question 13.
The length of plant cell A is 8 × 10^-5 meter. The length of plant cell B is 0.000004 meter. How many times greater is plant cell A’s length than plant cell B’s length?
Answer:

Question 14.
Critique Reasoning The diameter of one species of bacteria is shown. Bonnie approximates this measure as 3 × 10^-11 meter. Is she correct? Explain.

Answer:

Question 15.
The populations of Cities A and B are 2.6 × 10⁵ and 1,560,000, respectively. The population of City C is twice the population of City B. The population of City C is how many times the population of City A?
Answer:

Assessment Practice

Question 16.
Earth is approximately 5 × 10⁹ years old. For which of these ages could this be an approximation?
A. 4,762,100,000 years
B. 48,000,000,000 years
C. 4.45 × 10⁹ years
D. 4.249999999 × 10⁹ years
Answer:

Question 17.
PART A
Express 0.000000298 as a single digit times a power of ten rounded to the nearest ten millionth.

PART B
Explain how negative powers of 10 can be helpful when writing and comparing small numbers.
Answer:

Lesson 1.9 Understand Scientific Notation

ACTIVITY

Solve & Discuss It!

Scientists often write very large or very small numbers using exponents. How might a scientist write the number shown using exponents?

Use Structure
How can you use your knowledge of powers of 10 to rewrite the number?

Focus on math practices
Look for Relationships What does the exponent in 1015 tell you about the value of the number?

? Essential Question
What is scientific notation and why is it used?

Try It!
The height of Angel Falls, the tallest waterfall in the world, is 3,212 feet. How do you write this number in scientific notation?

Convince Me!
Why do very large numbers have positive exponents when written in scientific notation? Explain.

Try It!
A common mechanical pencil lead measures about 0.005 meter in diameter. How can you express this measurement using scientific notation ?

Try It!

Write the numbers in standard form.
a. 9.225 × 10¹⁸
b. 6.3 × 10^-8
Answer:

KEY CONCEPT
Scientific notation is a way to write very large numbers or very small numbers. Scientists use scientific notation as a more efficient and convenient way of writing such numbers.
A number in scientific notation is the product of two factors. The first factor must be greater than or equal to 1 and less than 10. The second factor is a power of 10.

To write a number in scientific notation in standard form, multiply the decimal number by the power of 10.

Do You Understand?

Question 1.
?Essential Question What is scientific notation and why is it used?
Answer:

Question 2.
Critique Reasoning Taylor states that 2,800,000 in scientific notation is 2.8 × 10^-6 because the number has six places to the right of the 2. Is Taylor’s reasoning correct?
Answer:

Question 3.
Construct Arguments Sam will write 0.000032 in scientific notation. Sam thinks that the exponent of 10 will be positive. Do you agree? Construct an argument to support your response.
Answer:

Do You Know How?

Question 4.
Express 586,400,000 in scientific notation.
Answer:

Question 5.
The genetic information of almost every living thing is stored in a tiny strand called DNA. Human DNA is 3.4 × 10^-8 meter long. Write the length in standard form.

Answer:

Question 6.
The largest virus known to man is the Megavirus, which measures 0.00000044 meter across. Express this number in scientific notation.
Answer:

Question 7.
How would you write the number displayed on the calculator screen in standard form?

Answer:

Practice & Problem Solving

Leveled Practice In 8 and 9, write the numbers in the correct format.

Question 8.
The Sun is 1.5 × 10⁸ kilometers from Earth. 1.5 × 10⁸ is written as in standard form.
Answer:

Question 9.
Brenna wants an easier way to write 0.0000000000000000587.
0.0000000000000000587 is written as × 10 in scientific notation.
Answer:

Question 10.
Is 23 × 10^-8 written in scientific notation? Justify your response.
Answer:

Question 11.
Is 8.6 × 10⁷ written in scientific notation? Justify your response.
Answer:

Question 12.
Simone evaluates an expression using her calculator. The calculator display is shown at the right. Express the number in standard form.

Answer:

Question 13.
Express the number 0.00001038 in scientific notation.
Answer:

Question 14.
Express the number 80,000 in scientific notation.
Answer:

Question 15.
Peter evaluates an expression using his calculator. The calculator display is shown at the right. Express the number in standard form.

Answer:

Question 16.
a. What should you do first to write 5.871 × 10^-7 in standard form?
b. Express the number in standard form
Answer:

Question 17.
Express 2.58 × 10^-2 in standard form.
Answer:

Question 18.
At a certain point, the Grand Canyon is approximately 1,600,000 centimeters across. Express this number in scientific notation.

Answer:

Question 19.
The length of a bacterial cell is 5.2 × 10^-6 meter. Express the length of the cell in standard form.
Answer:

Question 20.
Higher Order Thinking Express the distance 4,300,000 meters using scientific notation in meters, and then in millimeters.
Answer:

Assessment Practice

Question 21.
Which of the following numbers are written in scientific notation?
12 × 10⁶
12
6.89 × 10⁶
6.89
0.4
4 × 10^-1
Answer:

Question 22.
Jeana’s calculator display shows the number to the right.
PART A
Express this number in scientific notation.

PART B

Express this number in standard form.

3-Act Mathematical Modeling: Hard-Working Organs

3-ACT MATH

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.
Construct Arguments Predict an answer to this Main Question. Explain your prediction.

Answer:

Question 4.
On the number line below, write a number that is too small to be the answer. Write a number that is too large.

Answer:

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

АСТ 2

Question 6.
What information in this situation would be helpful to know? How would you use that information?
Answer:

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 greater or less than your prediction? Explain why.

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?
Answer:

Question 12.
Make Sense and Persevere Would you change your model now that you know the answer? Explain.

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?
Answer:

Question 14.
Generalize What pattern did you notice in your calculations? How did that pattern help you solve the problem?
Answer:

SEQUEL

Question 15.
Use Structure How many times does a heart beat in a lifetime? Use your solution to the Main Question to help you solve.

Answer:

Lesson 1.10 Operations with Numbers in Scientific Notation

Solve & Discuss It!

The homecoming committee wants to fly an aerial banner over the football game. The banner is 1,280 inches long and 780 inches tall. How many different ways can the area of the banner be expressed?

Focus on math practices
Be Precise Which of the solutions is easiest to manipulate?

? Essential Question
How does using scientific notation help when computing with very large or very small numbers?

Try It!

The planet Venus is on average 2.5 × 10⁷ kilometers from Earth. The planet Mars is on average 2.25 × 10⁸ kilometers from Earth. When Venus, Earth, and Mars are aligned, what is the average distance from Venus to Mars?

Answer:

Convince Me!
In Example 1 and the Try it, why did you move the decimal point to get the final answer?

Try It!

There are 1 × 10¹⁴ good bacteria in the human body. There are 2.6 x 10¹⁸ good bacteria among the spectators in a soccer stadium. About how many spectators are in the stadium? Express your answer in scientific notation.

KEY CONCEPT
Operations with very large or very small numbers can be carried out more efficiently using scientific notation. The properties of exponents apply when carrying out operations.

Do You Understand?

Question 1.
? Essential Question
How does using scientific notation help when computing with very small or very large numbers?

Question 2.
Use Structure When multiplying and dividing two numbers in scientific notation, why do you sometimes have to rewrite one factor?
Answer:

Question 3.
Use Structure For the sum of (5.2 × 10⁴) and (6.95 × 10⁴) in scientific notation, why will the power of 10 be 10⁵?
Answer:

Do You Know How?

Question 4.
A bacteriologist estimates that there are 5.2 × 10⁴ bacteria growing in each of 20 petri dishes. About how many bacteria in total are growing in the petri dishes? Express your answer in scientific notation.
Answer:

Question 5.
The distance from Earth to the Moon is approximately 1.2 × 10⁹ feet. The Apollo 11 spacecraft was approximately 360 feet long. About how many spacecraft of that length would fit end to end from Earth to the Moon? Express your answer in scientific notation.

Answer:

Question 6.
The mass of Mars is 6.42 × 10²³ kilograms. The mass of Mercury is 3.3 × 10²³ kilograms.
a. What is the combined mass of Mars and Mercury expressed in scientific notation?
b. What is the difference in the mass of the two planets expressed in scientific notation?
Answer:

Practice & Problem Solving

Leveled Practice In 7 and 8, perform the operation and express your answer in scientific notation.

Question 7.
(7 × 10^-6)(7 × 10^-6)

Answer:

Question 8.
(3.76 × 10⁵) + (7.44 × 10⁵)

Answer:

Question 9.
What is the value of n in the equation
1.9 × 10⁷ = (1 × 10⁵)(1.9 × 10ⁿ)?
Answer:

Question 10.
Find (5.3 × 10³) – (8 × 10²).
Express your answer in scientific notation.
Answer:

Question 11.
What is the mass of 30,000 molecules? Express your answer in scientific notation.

Answer:

Question 12.
Critique Reasoning Your friend says that the product of 4.8 × 10⁸ and 2 × 10^-3 is 9.6 × 10^-5. Is this answer correct? Explain.
Answer:

Question 13.
Find \(\frac{7.2 \times 10^{-8}}{3 \times 10^{-2}}\). Write your answer in scientific white notation.
Answer:

Question 14.
A certain star is 4.3 × 10² light years from Earth. One light year is about 5.9 × 10¹² miles. How far from Earth (in miles) is the star? Express your answer in scientific notation.

Question 15.
The total consumption of fruit juice in a particular country in 2006 was about 2.28 × 10⁹ gallons. The population of that country that year was 3 × 10⁸. What was the average number of gallons consumed per person in the country in 2006?
Answer:

Question 16.
The greatest distance between the Sun and Jupiter is about 8.166 × 10⁸ kilometers. The greatest distance between the Sun and Saturn is about 1.515 × 10⁹ kilometers. What is the difference between these two distances?
Answer:

Question 17.
What was the approximate number of pounds of garbage produced per person in the country in one year? Express your answer in scientific notation.

Answer:

Question 18.
Higher Order Thinking
a. What is the value of n in the equation
1.5 × 10¹² = (5 × 10⁵)(3 × 10ⁿ)?
b. Explain why the exponent on the left side of the equation is not equal to the sum of the exponents on the right side.

Assessment Practice

Question 19.
Find (2.2 × 10⁵) ÷ (4.4 × 10^-3). When you regroup the factors, what do you notice about the quotient of the decimal factors? How does this affect the exponent of the quotient?
Answer:

Question 20.
Which expression has the least value?
A. (4.7 × 10⁴) + (8 × 10⁴)
B. (7.08 × 10³) + (2.21 × 10³)
C. (5.43 × 10⁸) – (2.33 × 10⁸)
D. (9.35 × 10⁶) – (6.7 × 10⁶)
Answer:

Topic 1 Review

? Topic Essential Question
What are real numbers? How are real numbers used to solve problems?

Vocabulary Review

Use Vocabulary in Writing
Use vocabulary words to explain how to find the length of each side of a square garden with an area of 196 square inches.

Concepts and Skills Review

LESSON 1.1 Rational Numbers as Decimals

Quick Review

You can write repeating decimals in fraction form by writing two equations. You multiply each side of one equation by a power of 10. Then you subtract the equations to eliminate the repeating decimal.

Practice
Write each number as a fraction or a mixed number.

Question 1.
0.\(\overline{7}\)
Answer:

Question 2.
0.0\(\overline{4}\)
Answer:

Question 3.
4.\(\overline{45}\)

Question 4.
2.191919….
Answer:

LESSON 1.2 Understand Irrational Numbers

Quick Review
An irrational number is a number that cannot be written in the form \(\frac{a}{b}\), where a and b are integers and b ≠ 0. Rational and irrational numbers together make up the real number system.

Practice

Question 1.
Determine which numbers are irrational. Select all that apply.
\(\sqrt{36}\)
\(\sqrt{23}\)
-4.232323….
0.151551555….
0.3\(\overline{5}\)
π

Question 2.
Classify -0.\(\overline{25}\) as rational or irrational. Explain.
Answer:

LESSON 1.3 Compare and Order Real Numbers

Quick Review
To compare and order real numbers, it helps to first write each number in decimal form.

Practice

Question 1.
Between which two whole numbers does \(\sqrt{89}\) lie?
\(\sqrt{89}\) is between and
Answer:

Question 2.
Compare and order the following numbers. Locate each number on a number line. 2.\(\overline{3}\), \(\sqrt{8}\), 2.5, 2\(\frac{1}{4}\)
Answer:

LESSON 1.4 Evaluate Square Roots and Cube Roots

Quick Review
Remember that a perfect square is the square of an integer. A square root of a number is a number that when multiplied by itself is equal to the original number. Similarly, a perfect cube is the cube of an integer. A cube root of a number is a number that when cubed is equal to the original number.

Practice
Classify each number as a perfect square, a perfect cube, both, or neither.

Question 1.
27
Answer:

Question 2.
100
Answer:

Question 3.
64
Answer:

Question 4.
24
Answer:

Question 5.
A gift box is a cube with a volume of 512 cubic inches. What is the length of each edge of the box?
Answer:

LESSON 1.5 Solve Equations Using Square Roots and Cube Roots

Quick Review

You can use square roots to solve equations involving squares. You can use cube roots to solve equations involving cubes. Equations with square roots often have two solutions. Look at the context to see whether both solutions are valid.

Practice Solve for x.

Question 1.
x³ = 64
Answer:

Question 2.
x² = 49
Answer:

Question 3.
x³ = 25
Answer:

Question 4.
x² = 125
Answer:

Question 5.
A container has a cube shape. It has a volume of 216 cubic inches. What are the dimensions of one face of the container?

LESSON 1.6 Use Properties of Integer Exponents

Quick Review
These properties can help you write equivalent expressions that contain exponents.
Product of Powers Property
a^m.aⁿ = a^m+n
Power of Powers Property
(a^m)ⁿ = a^mn
Power of Products Property
aⁿ ∙ bⁿ = (a ∙ b)ⁿ
Quotient of Powers Property
a^m ÷ aⁿ = a^m-n, when a ≠ 0

Practice
Use the properties of exponents to write an equivalent expression for each given expression.

Question 1.
6⁴ ∙ 6³
Answer:

Question 2.
(3⁶)^-2
Answer:

Question 3.
7³ ∙ 2³
Answer:

Question 4.
4¹⁰ ÷ 4⁴
Answer:

LESSON 1.7 More Properties of Integer Exponents

Quick Review
The Zero Exponent Property states that any nonzero number raised to the power of 0 is equal to 1. The Negative Exponent Property states that for any nonzero rational number a and integer n, a^-n = \(\frac{1}{a^{n}}\)

Practice
Write each expression using positive exponents.

Question 1.
9^-4
Answer:

Question 2.
\(\frac{1}{3^{-5}}\)
Answer:

Evaluate each expression for x = 2 and y = 5

Question 3.
-4x^-2 + 3y⁰
Answer:

Question 4.
2x⁰y^-2
Answer:

LESSON 1.8 Use Powers of 10 to Estimate Quantities

Quick Review
You can estimate very large and very small quantities by writing the number as a single digit times a power of 10.

Practice

Question 1.
In the year 2013 the population of California was about 38,332,521 people. Write the estimated population as a single digit times a power of 10.
Answer:

Question 2.
The wavelength of green light is about 0.00000051 meter. What is this estimated wavelength as a single digit times a power of 10?
Answer:

Question 3.
The land area of Connecticut is about 12,549,000,000 square meters. The land area of Rhode Island is about 2,707,000,000 square meters. How many times greater is the land area of Connecticut than the land area of Rhode Island?
Answer:

LESSON 1.9 Understand Scientific Notation

Quick Review
A number in scientific notation is written as a product of two factors, one greater than or equal to 1 and less than 10, and the other a power of 10.

Practice

Question 1.
Write 803,000,000 in scientific notation.
Answer:

Question 2.
Write 0.0000000068 in scientific notation.
Answer:

Question 3.
Write 1.359 × 10⁵ in standard form.
Answer:

Question 4.
The radius of a hydrogen atom is 0.000000000025 meter. How would you express this radius in scientific notation?

LESSON 1.10 Operations with Numbers in Scientific Notation

Quick Review
When multiplying and dividing numbers in scientific notation, multiply or divide the first factors. Then multiply or divide the powers of 10. When adding and subtracting numbers in scientific notation, first write the numbers with the same power of 10. Then add or subtract the first factors, and keep the same power of 10.
If the decimal part of the result is not greater than or equal to 1 and less than 10, move the decimal point and adjust the exponent.

Practice
Perform each operation. Express your answers in scientific notation.

Question 1.
(2.8 × 10⁴) × (4 × 10⁵)
Answer:

Question 2.
(6 × 10⁹) ÷ (2.4 × 10³)

Question 3.
(4.1 × 10⁴) + (5.6 × 10⁶)
Answer:

Question 4.
The population of Town A is 1.26 × 10⁵ people. The population of Town B is 2.8 × 10⁴ people. How many times greater is the population of Town A than the population of Town B?
Answer:

Topic 1 Fluency Practice

Crisscrossed

Solve each equation. Write your answers in the cross-number puzzle below. Each digit, negative sign, and decimal point of
your answer goes in its own box.

Across
A -377 = x – 1,000
B x³ = 1,000
C x³ = -8
D x + 7 = -209
F x + 19 = -9
J 14 + x = -9
L m – 2.02 = -0.58
M -3.09 + x = -0.7
N -2.49 = -5 + x
Q x – 3.5 = -3.1
T q – 0.63 = 1.16
V 8.3 + x = 12.1

Down
A y – 11 = 49
B x + 8 = 20
C z³ = -1,331
D 11 + x = 3
E x – 14 -7.96
F 14 + x = -19
G d + 200 = 95
H x² = 144
K -12 = t – 15.95
P 0.3 + x = 11
R x – 3 = -21
S – 7 = -70 + y

Go through the enVision Math Common Core Grade 3 Answer Key Topic 4 Use Multiplication to Divide: Division Facts regularly and improve your accuracy in solving questions.

enVision Math Common Core 3rd Grade Answers Key Topic 4 Use Multiplication to Divide: Division Facts

Essential Questions:
How can you use known multiplication facts to find unknown division facts? How are multiplication and division related?

enVision STEM Project: Testing Models
Do Research Tests can be done to see if a model works or if a change makes it better. Use the Internet or other sources to find information about a model or prototype that was tested. Identify how the testing was done.

Journal: Write a Report Include what you found. Also in your report:

• Make a chart that includes the model, what changed in the test, and what stayed the same.
• Explain the results of the test.
• Write an equation to show one of the relationships in the test. Explain what the numbers represent.

Review What You Know

Vocabulary
Choose the best term from the box. Write it on the blank.

• division
• factors
• multiplication
• equation

Question 1.
_________ are multiplied together to give a product.
Answer:
__Factors _______ are multiplied together to give a product.

Question 2.
Use _________ to find how many equal groups or how many are in each group.
Answer:
Use __division_______ to find how many equal groups or how many are in each group.

Question 3.
___________ is an operation that gives the total number when you put together equal groups.
Answer:
__Multiplication_________ is an operation that gives the total number when you put together equal groups.

Division
Solve each problem. You can use bar diagrams, counters, or draw a picture to help.
Question 4.
Stuart has 15 stickers to give to his 3 friends. How many stickers can each friend have?

Answer:
Number of stickers each friend gives = 5.

Explanation:
Number of stickers Stuart has = 15.
Number of friends to give = 3.
Number of stickers each friend gives = Number of stickers Stuart has ÷ Number of friends to give
= 15 ÷ 3
= 5.

Question 5.
There are 32 muffins. Eight people share them equally. How many muffins does each person get?
Answer:
Number of muffins each person get = 4.

Explanation:
Number of muffins = 32.
Number of people share them equally = 8.
Number of muffins each person get = Number of muffins ÷ Number of people share them equally
= 32 ÷ 8
= 4.

Question 6.
Suzy has 12 granola bars. There are 2 granola bars in each package. How many packages of granola bars are there?
Answer:
Number of packages of granola bars there = 6.

Explanation:
Number of granola bars Suzy has = 12.
Number of granola bars in each package = 2.
Number of packages of granola bars there = Number of granola bars Suzy has ÷ Number of granola bars in each package
= 12 ÷ 2
= 6.

Equations
Question 7.
Brian has 5 boxes. He puts 8 markers in each box. Which equation shows the total number of markers?
A. 5 + 8 = 13
B. 5 × 8 = 40
C. 40 ÷ 5 = 8
D. 40 ÷ 8 = 5
Answer:
Total number of markers = 40.
B. 5 × 8 = 40.

Explanation:
Number of boxes Brian has = 5.
Number of markers in each box = 8.
Total number of markers = Number of boxes Brian has × Number of markers in each box
= 5 × 8
= 40.

Pick a Project

PROJECT 4A
Who are your favorite athletes?
Project: Make a Poster of Your Favorite Athletes

Answer:

PROJECT 4B
Who is on our money?
Project: Write a Report About Money

Answer:

PROJECT 4C
How do you score in horseshoes?
Project: Create a Score Sheet

Answer:

PROJECT 4D
What kind of game would you create?
Project: Develop a Game

Answer:

Lesson 4.1 Relate Multiplication and Division

Solve & Share
Use 24 counters to make arrays with equal rows. Write multiplication and division equations to describe your arrays.
I can … use fact families to see how multiplication and division are related.

Look Back! What relationships do you see between the multiplication and division equations for each of your arrays?
Answer:
The relationships  between the multiplication and division equations for each of the arrays may be different in process yet the result value is same.

Explanation:
Process different to multiplication and division equations yet both are done to find the solution to the problem.

Essential Question
How Can Multiplication Question Facts Help You Divide?
Answer:
In multiplication, we find the product of two factors. In division, we find the missing factor if the other factor and the product are known.

Explanation:
In multiplication, we find the product of two factors. In division, we find the missing factor if the other factor and the product are known. This is how Multiplication Question Facts Helps in Divide.

Visual Learning Bridge
This array can show the relationship between multiplication and division.

Multiplication 3 rows of 10 drums
3 × 10 = 30
30 drums
Division 30 drums in 3 equal rows
30 ÷ 3 = 10
10 drums in each row

A fact family shows how multiplication and division are related.
Fact family for 3, 10, and 30:

The dividend is the number of objects to be divided.
The divisor is the number by which another number is divided.
The quotient is the answer to a division problem.

Convince Me! Reasoning 4 × 7 = 28 is one fact in a fact family. Draw an array for this fact. Write the other three facts in the fact family.
Answer:
4 × 7 = 28.
28 = dividend.
7 = divisor
4 = quotient.

Explanation:

Guided Practice
Do You Understand?
Question 1.
Look at the fact family for 3, 10, and 30 on the previous page. What do you notice about the products and the dividends?
Answer:
It is noticed that the product in mathematics means that the two factors should be multiplied together.
In division, the amount or number to be divided is called the dividend.

Explanation:
Multiplication 3 rows of 10 drums
3 × 10 = 30 drums.
Division 30 drums in 3 equal rows
30 ÷ 3 = 10 drums in each row.
It is noticed that the product in mathematics means that the two factors should be multiplied together.
In division, the amount or number to be divided is called the dividend. Dividend is the whole that is to be divided into parts.

Question 2.
Is 4 × 6 = 24 part of the fact family for 3, 8, and 24? Explain.
Answer:
Yes, 4 × 6 = 24 is a part of the fact family for 3, 8, and 24.

Explanation:
Is 4 × 6 = 24 part of the fact family for 3, 8, and 24.
4 × 6 = 24.
24 ÷ 4 = 6.
Dividend = 24.
Divisor = 4
Quotient = 6
.
3 × 8 = 24.
24 ÷ 3 = 8.
Dividend = 24.
Divisor = 3
Quotient = 8.

Do You Know How?
In 3-5, use the relationship between multiplication and division to complete each equation.
Question 3.
3 × 7 = 21
21 ÷ 3 = ________
Answer:
3 × 7 = 21.
21 ÷ 3 = __7__.

Explanation:
3 × 7 = 21.
21 ÷ 3 = 7.
Dividend ÷ Quotient = Divisor.

Question 4.
18 ÷ 2 = 9
2 × ______ = 18
Answer:
18 ÷ 2 = 9.
2 × __9____ = 18.

Explanation:
18 ÷ 2 = 9.
2 × 9 = 18.
Quotient × divisor = dividend.

Question 5.
2 × 10 = 20
20 ÷ 2 = _________
Answer:
2 × 10 = 20.
20 ÷ 2 = ____10_____.

Explanation:
2 × 10 = 20.
20 ÷ 2 = 10.
Dividend ÷ quotient = Divisor.

Independent Practice
In 6 and 7, use the relationship between multiplication and division to complete each equation.

Question 6.
2 × ______ = 16
16 ÷ 2 = _______
Answer:
2 × __8____ = 16.
16 ÷ 2 = __8_____.

Explanation:
2 × 8 = 16
16 ÷ 2 = 8.
Dividend ÷ quotient = Divisor.

Question 7.
56 ÷ 8 = 7
8 × _____ = 56
Answer:
56 ÷ 8 = 7.
8 × __7___ = 56.

Explanation:
56 ÷ 8 = 7.
8 × 7 = 56.
Dividend = Divisor ÷ quotient.

In 8-13, write the fact family.
Question 8.
Write the fact family for 6, 7, and 42.
Answer:
6 × 7 = 42.
42 ÷ 6 = 7.

Explanation:
6 × 7 = 42.
42 ÷ 6 = 7.
Dividend ÷ quotient = Divisor.

Question 9.
Write the fact family for 9, 10, and 90.
Answer:
9 × 10 = 90.
90 ÷ 9 = 10.

Explanation:
9 × 10 = 90.
90 ÷ 9 = 10.
Dividend ÷ quotient = Divisor.

Question 10.
Write the fact family for 2, 3, and 6.
Answer:
2 × 3 = 6.
6 ÷ 2 = 3.

Explanation:
2 × 3 = 6.
6 ÷ 2 = 3.
Dividend ÷ quotient = Divisor.

Question 11.
Write the fact family for 1, 5, and 5.
Answer:
1 × 5 = 5.
5 ÷ 1 = 5.

Explanation:
1 × 5 = 5.
5 ÷ 1 = 5.
Dividend ÷ quotient = Divisor.

Question 12.
Write the fact family for 3, 8, and 24.
Answer:
3 × 8 = 24.
24 ÷ 3 = 8.

Explanation:
3 × 8 = 24.
24 ÷ 3 = 8.
Dividend ÷ quotient = Divisor.

Question 13.
Write the fact family for 5, 6, and 30.
Answer:
5 × 6 = 30.
30 ÷ 5 = 6.

Explanation:
5 × 6 = 30.
30 ÷ 5 = 6.
Dividend ÷ quotient = Divisor.

Problem Solving
Question 14.
Write a multiplication equation and a division equation for the array.
4 × ______ = 20
20 ÷ ______ = 5

Answer:
4 × _5____ = 20.
20 ÷ __4____ = 5.

Explanation:
4 × 5 = 20.
20 ÷ 4 = 5.
Dividend ÷ quotient = Divisor.

Question 15.
Make Sense and Persevere How many inches shorter is the red fabric than the green and yellow fabrics combined?

Answer:
Length of  the red fabric shorter than the green and yellow fabrics combined = 18 inches.

Explanation:
Length of the red fabric = 72 inches.
Length of the green fabric = 36 inches.
Length of the yellow fabric = 54 inches.
Length of  the red fabric shorter than the green and yellow fabrics combined = (Length of the green fabric + Length of the yellow fabric) – Length of the red fabric
= (36 + 54) – 72
= 90 – 72
= 18 inches.

Question 16.
Higher Order Thinking
Anya says that with 24 counters she can make only 6 possible arrays. Todd says he can make 8 arrays. Who is correct? Explain.
Answer:
Anya says with 24 counters she can make only 6 possible arrays is incorrect because we can make 8 arrays out of 24 counters as Todd says.

Explanation:
Anya says that with 24 counters she can make only 6 possible arrays.
=> 24 ÷ 6 = 4.

Todd says that with 24 counters he can make 8 arrays.
=> 24 ÷ 8 = 3.

Question 17.
Algebra Carla picked 9 apples a day for three days. Which number tells you how many apples she picked in three days and makes this equation true? ☐ ÷ 3 = 9
Answer:
Number of apples she picked in three days = 27.
Equation: 27 ÷ 3 = 9.

Explanation:
Number of apples Algebra Carla picked a day = 9.
Number of days Algebra Carla picked  apples = 3.
Number of apples she picked in three days = Number of apples Algebra Carla picked a day  × Number of days Algebra Carla picked  apples
= 9 × 3
= 27.

Question 18.
Vocabulary
Can you write a fact family for 3, 5, and 7? Explain.
Answer:
3, 5 , and 7 are nor fact family because there is no product number among this three numbers.

Explanation:
No, you cannot write a fact family for 3, 5 , and 7 because they are odd numbers and there is no product among these three numbers.

Question 19.
Lisa, Bret, and Gary harvested apples. Lisa filled 3 carts with apples. Bret also filled 3 carts with apples. Gary filled another 3 carts with apples. Write a multiplication equation and a division equation for this story.
Answer:
Number of carts each person filled with apples = 9 ÷ 3 = 3.
Total number of carts of apples these three filled = 3 × 3 = 9.

Explanation:
Number of carts Lisa filled with apples = 3.
Number of carts Bret filled with apples = 3.
Number of carts Gary filled with apples = 3.
Total number of carts these three filled = Number of carts Lisa filled with apples + Number of carts Bret filled with apples + Number of carts Gary filled with apples
= 3 + 3 + 3
= 6 + 3
= 9.
Number of people filled carts with apples = 3.
Number of carts each person filled with apples = Total number of carts of apples these three filled ÷ Number of people filled carts with apples
= 9 ÷ 3 = 3.
Total number of carts of apples these three filled = Number of people filled carts with apples × Number of carts each person filled with apples
= 3 × 3 = 9.

Assessment Practice
Question 20.
Select numbers to create a multiplication equation that could be used to solve 20 ÷ 5 = ☐.

Answer:
Numbers to create a multiplication equation that could be used to solve 20 ÷ 5 = ☐.
20 ÷ 5 = 4.
4 × 5 = 20.

Explanation:
Numbers to create a multiplication equation that could be used to solve 20 ÷ 5 = ☐.
20 ÷ 5 = 4.
5 × 4 = 20.
4 × 5 = 20.

Question 21.
Select numbers to create a multiplication equation that could be used to solve 24 ÷ 8 = ☐0

Answer:
Numbers to create a multiplication equation that could be used to solve 24 ÷ 8 = ☐.
24 ÷ 8 = 3.
3 × 8 = 24.

Explanation:
Numbers to create a multiplication equation that could be used to solve 24 ÷ 8 = ☐.
24 ÷ 8 = 3.
8 × 3 = 24.
3 × 8 = 24.

Lesson 4.2 Use Multiplication to Divide with 2, 3, 4, and 5

Solve & Share
Kara puts 30 toys into 5 party bags. She puts the same number of toys into each bag. How many toys are in each bag?
I can … divide by 2, 3, 4, and 5 by thinking about how I multiply with those numbers.

Look Back! Show two pictures you might draw to represent 30 ÷ 5.
Answer:
Number of toys are in each bag = 6.
30 ÷ 5 = 6.
5 × 6 = 30.
6 × 5 = 30.

Explanation:
Number of toys Kara has = 30.
Number of party bags Kara has = 5.
Number of toys are in each bag = Number of toys Kara has ÷ Number of party bags Kara has
= 30 ÷ 5
= 6.
5 × 6 = 30.
6 × 5 = 30.

Essentials Question
What Multiplication Fact Can You Use?
Answer:
Some basic multiplication facts are needed to follow for multiplying numbers. The repeated addition of the same number is expressed by multiplication in short.

Explanation:
Multiplication facts which we commonly use is multiplying two numbers together to get the product.

Visual Learning Bridge
Dee has 14 noisemakers. She puts the same number on each of 2 tables. How many noisemakers are on each table?

Dee has 40 stickers. If she puts 5 stickers on each bag, how many bags can Dee decorate?

Find 40 ÷ 5.
What You Think
What number times 5 is 40?
8 × 5 = 40

What You Write
40 ÷ 5 = 8
Dee can decorate 8 bags.
You can use multiplication to help divide.

Dee wants to put 15 cups in 3 equal stacks on the table. How many cups will Dee put in each stack?

Find 15 ÷ 3.
What You Think
3 times what number is 15?
3 × 5 = 15

What You Write
15 ÷ 3 = 5
Dee will put 5 cups in each stack.
Multiplication and division facts form relationships.

Convince Me! Reasoning How can you use multiplication to help solve 20 ÷ 4? Write the related multiplication fact you use to help solve the problem.
Answer:
20 ÷ 4 = 5.
5 × 4 = 20.

Explanation:
Dee wants to put 20cups in 4 equal stacks on the table. How many cups will Dee put in each stack?
20 ÷ 4 = 5.
5 × 4 = 20.
4 × 5 = 20.

Another Example!
Here are two ways to write a division problem.

Guided Practice
Do You Understand?
Question 1.
How can 5 × 3 = 15 help you divide 15 by 3?
Answer:
5 × 3 = 15.
15 ÷ 3 = 5.

Explanation:
5 × 3 = 15.
15 ÷ 3 = 5.

Question 2.
Mr. Dean has 3 children. He buys 30 pencils to share equally among his children for the school year. How many pencils will each child get? Write the answer and the fact family you used.
Answer:
Number of pencils each child gets = 10.

Explanation:
Number of children Mr. Dean has = 3.
Number of pencils he buys to share equally among his children for the school year = 30.
Number of pencils each child gets = Number of pencils he buys to share equally among his children for the school year ÷ Number of children Mr. Dean has
= 30 ÷ 3
= 10.

Do You Know How?
In 3 and 4, complete each fact family.
Question 3.
3 × 6 = 18 _________
18 ÷ 3 = 6 _________
Answer:
3 × 6 = 18 __Dividend_______.
18 ÷ 3 = 6 ____quotient___.

Explanation:
3 × 6 = 18 (Dividend)
18 ÷ 3 = 6 (quotient).

Question 4.
9 × 4 = 36 _________
36 ÷ 4 = 9 _________
Answer:
9 × 4 = 36 __Dividend_______.
36 ÷ 4 = 9 ___quotient______.

Explanation:
9 × 4 = 36 (Dividend).
36 ÷ 4 = 9 (quotient).

In 5-8, find each quotient.
Question 5.
36 ÷ 4 = ________
Answer:
36 ÷ 4 = ____9____.

Explanation:
36 ÷ 4 = ___9_____.

Question 6.
_______ = 15 ÷ 5
Answer:
___3____ = 15 ÷ 5.

Explanation:
___3____ = 15 ÷ 5.

Question 7.

Answer:

= 9.

Explanation:

= 9.

Question 8.

Answer:
= 10.

Explanation:
= 10.

Independent Practice
In 9-20, find each quotient.
Question 9.
12 ÷ 2 = _______
Answer:
12 ÷ 2 = __6_____.

Explanation:
12 ÷ 2 = 6.

Question 10.
_______ = 12 ÷ 3
Answer:
___4____ = 12 ÷ 3.

Explanation:
4 = 12 ÷ 3.

Question 11.
16 ÷ 4 = _______
Answer:
16 ÷ 4 = ___4____.

Explanation:
16 ÷ 4 = 4.

Question 12.
35 ÷ 5 = _______
Answer:
35 ÷ 5 = ___7____.

Explanation:
35 ÷ 5 = 7.

Question 13.
14 ÷ 2 = _______
Answer:
14 ÷ 2 = ___7___.

Explanation:
14 ÷ 2 = 7.

Question 14.
20 ÷ 4 = ______
Answer:
20 ÷ 4 = __5____.

Explanation:
20 ÷ 4 = 5.

Question 15.
_______ = 24 ÷ 4
Answer:
___6____ = 24 ÷ 4.

Explanation:
6= 24 ÷ 4.

Question 16.
45 ÷ 5 = ______
Answer:
45 ÷ 5 = ___9___.

Explanation:
45 ÷ 5 = 9.

Question 17.

Answer:

= 9.

Explanation:

= 9.

Question 18.

Answer:

= 10.

Explanation:

= 10.

Question 19.

Answer:

= 8.

Explanation:

= 8.

Question 20.

Answer:

= 7.

Explanation:

= 7.

Problem Solving
In 21 and 22, use the rectangle at the right.
Question 21.
How many individual squares are inside the rectangle? Write a division equation in which the quotient represents the number of rows.

Answer:
Total number of squares in the rectangle = 20.
Division Equation:
20 ÷ 4 = 5.

Explanation:
Number of squares rows = 5.
Number of squares columns = 4.
Total number of squares in the rectangle = Number of squares rows × Number of squares columns
= 5 × 4
= 20.
Division Equation:
Total number of squares in the rectangle ÷ Number of squares columns = Number of squares rows.
20 ÷ 4 = 5.

Question 22.
Make Sense and Persevere If Anna arranges the squares into an array with 2 columns, how many rows will there be?
Answer:
Number of rows = 10.

Explanation:
Number of squares = 20.
Number of columns = 2
Number of rows = Number of squares ÷ Number of columns
= 20 ÷ 2
= 10.

Question 23.
Number Sense Joey says, “I cannot solve 8 ÷ 2 by using the fact 2 × 8 = 16.” Do you agree or disagree? Explain.
Answer:
Yes, I agree with Joey. We cannot solve 8 ÷ 2 by using the fact 2 × 8 = 16.

Explanation:
2 × 8 = 16.
=> 16 ÷ 8 = 2.
=> 16 ÷ 2 = 8.
8 ÷ 2 = 4.

Question 24.
Miko has 8 counters to arrange in an array. Write multiplication and division equations to represent all the ways Miko might arrange her counters.
Answer:
Way:1:  Total number of counters = Number of rows × Number of columns
= 4 × 2
= 8.
Way:2: Total number of counters = Number of rows × Number of columns
= 2 × 4
= 8.
Way:3: Total number of counters = Number of rows × Number of columns
= 8 × 1
= 8.
Way:4: Total number of counters = Number of rows × Number of columns
= 1 × 8
= 8.

Explanation:
Number of counters Miko has = 8.
Way:1:
Number of rows = 4.
Number of columns = 2.
Total number of counters = Number of rows × Number of columns
= 4 × 2
= 8.

Way:2:
Number of rows = 2.
Number of columns = 4.
Total number of counters = Number of rows × Number of columns
= 2 × 4
= 8.

Way:3:
Number of rows = 8.
Number of columns = 1.
Total number of counters = Number of rows × Number of columns
= 8 × 1
= 8.

Way:4:
Number of rows = 1.
Number of columns = 8.
Total number of counters = Number of rows × Number of columns
= 1 × 8
= 8.

Question 25.
Vocabulary Write a division equation. Tell which number is the quotient, the dividend, and the divisor.
Answer:
Division Equation:
Total number of counters ÷ Number of columns = Number of rows
8 ÷ 8 = 1.
Dividend = 8.
Divisor = 8.
Quotient = 1.

Explanation:
Number of rows = 1.
Number of columns = 8.
Total number of counters = Number of rows × Number of columns
= 1 × 8
= 8.
Division Equation:
Total number of counters ÷ Number of columns = Number of rows
8 ÷ 8 = 1.
Dividend = 8.
Divisor = 8.
Quotient = 1.

Question 26.
Higher Order Thinking Chris gives 18 pretzels equally to 3 friends. Martha gives 20 pretzels equally to 4 friends. Whose friends got more pretzels? Use equations to justify your answer.
Answer:
Chris’s friends gets each 6 pretzels more than Martha’s friends.

Explanation:
Number of pretzels Chris gives = 18.
Number of friends Chris gives equally = 3.
Number of pretzels Chris’s friends gets each = Number of pretzels Chris gives ÷ Number of friends Chris gives equally
= 18 ÷ 3
= 6.

Number of pretzels Martha gives = 20.
Number of friends Martha gives equally = 4.
Number of pretzels Martha’s friends gets each = Number of pretzels Martha gives ÷ Number of friends Martha gives equally
= 20 ÷ 4
= 5.

Assessment Practice
Question 27.
Which expression can help you divide 12 ÷ 3?
A. 2 × 3
B. 3 × 3
C. 4 × 3
D. 5 × 3
Answer:
C. 4 × 3 is the expression which helps to divide 12 ÷ 3 = ??.

Explanation:
Expression can help you divide 12 ÷ 3 = ??
12 ÷ 3 = 4.
=> 4 × 3 = 12.

Question 28.
Which expression can help you divide 28 ÷ 4?
A. 7 × 4
B. 6 × 4
C. 5 × 4
D. 4 × 4
Answer:
A. 7 × 4 is the expression which helps to divide 28 ÷ 4 = ??.

Explanation:
Expression can help you divide 28 ÷ 4 = ??
28 ÷ 4 = 7.
=> 7 × 4 = 28.

Lesson 4.3 Use Multiplication to Divide with 6 and 7

Solve & Share
There are 18 children in a ballet class. They are standing in rows of 6 for a dance recital. How many rows of children are there?
I can … divide by 6 and 7 by thinking about how I multiply with those numbers.

Look Back! Draw a bar diagram to represent the problem.
Answer:
Number of children in each row = 3.

Explanation:
Number of children in a ballet class = 18
Number of rows children are standing for a dance recital = 6.
Number of children in each row = Number of children in a ballet class ÷ Number of rows children are standing for a dance recital
= 18 ÷ 6
= 3.

Essential Question
How Do You Divide with 6 and 7?
Answer:
The dividend to be divided with the number 6 or 7 to get the answer which is called quotient.

Explanation:
What is being divided is called the dividend, which is divided by the divisor, and the result is called the quotient. In the example, 24 is the dividend, 6 is the divisor, and 4 is the quotient

Visual Learning Bridge
There are 48 dogs entered in a dog show. The judge wants 6 dogs in each group. How many groups will there be?

Find 48 ÷ 6.
What You Think
What number times 6 is 48?
8 × 6 = 48

What You Write
48 ÷ 6 = 8
There will be 8 groups.
Use a multiplication problem to make sense of a division problem.

Another dog was entered in the show. There will now be 7 dogs in each group. How many groups will there be?
Find 49 ÷ 7.
What You Think
What number times 7 is 49?
7 × 7 = 49

What You Write
49 ÷ 7 = 7
There will be 7 groups.

Convince Me! Model with Math Draw a bar diagram using the numbers 36, 6, and 6. Write the division fact and the related multiplication fact that your bar diagram shows.
Answer:
36 ÷ 6 = 6.
6 × 6 = 36.

Explanation:
Division Equation:
36 ÷ 6 = 6.
Multiplication Equation:
6 × 6 = 36.

Guided Practice
Do You Understand?
Question 1.
How can you tell without dividing that 42 ÷ 6 will be greater than 42 ÷ 7?
Answer:
If the divisor is small, the quotient in division is going to be greater number and vice versa.

Explanation:
42 ÷ 6  will be greater than 42 ÷ 7  you can say without doing division by seeing the divisor. If the divisor is small the quotient in division is going to be greater number and vice versa.

Question 2.
How can 8 × 6 = 48 help you divide 48 by 6?
Answer:
8 × 6 = 48 helps in dividing 48 by 6.
48 ÷ 6 = 8.

Explanation:
8 × 6 = 48 helps in dividing 48 by 6.
48 ÷ 6 = 8.

Do You Know How?
In 3-8, write the related multiplication fact, and then find each quotient.
Question 3.
36 ÷ 6 = ________
Answer:
36 ÷ 6 = ____6____.

Explanation:
36 ÷ 6 = 6 (quotient).
6 × 6 = 36.

Question 4.
______ = 42 ÷ 6
Answer:
___7___ = 42 ÷ 6.

Explanation:
42 ÷ 6 = 7 (quotient).
6 × 7 = 42.

Question 5.
42 ÷ 7 = _______
Answer:
42 ÷ 7 = __6_____.

Explanation:
42 ÷ 7 = 6 (quotient).
7 × 6 = 42.

Question 6.
______ = 18 ÷ 6
Answer:
__3____ = 18 ÷ 6.

Explanation:
18 ÷ 6 = 3 (quotient).
6 × 3 = 18.

Question 7.

Answer:
= 4.

Explanation:
= 4(quotient).
6 × 4 = 24.

Question 8.

Answer:

= 5.

Explanation:

= 5 (quotient).
6 × 5 = 30.

Independent Practice
Leveled Practice In 9-20, use related multiplication and division facts to find the quotient.
Question 9.
12 ÷ 6 = ?
What number times 6 is 12?
6 × ☐ = 12
12 ÷ 6 = ☐
Answer:
12 ÷ 6 = 2.

Explanation:
12 ÷ 6 = ??
6 × 2 = 12.
12 ÷ 6 = 2.

Question 10.
21 ÷ 3 = ?
What number times 3 is 21?
3 × ☐ = 21
21 ÷ 3 = ☐
Answer:
21 ÷ 3 = 7.

Explanation:
21 ÷ 3 = ??
3 × 7 = 21.
21 ÷ 3 = 7.

Question 11.
30 ÷ 6 = ?
What number times 6 is 30?
6 × ☐ = 30
30 ÷ 6 = ☐
Answer:
30 ÷ 6 = 5.

Explanation:
30 ÷ 6 = ??
6 × 5 = 30.
30 ÷ 6 = 5.

Question 12.

Answer:
= 7.

Explanation:
= ??
= 7.
2 × 7 = 14.

Question 13.

Answer:
= 7.

Explanation:
= ??
= 7.
7 × 7 = 49.

Question 14.

Answer:
= 10.

Explanation:
= ??
= 10.
6 × 10 = 60.

Question 15.

Answer:
= 9.

Explanation:
= ??
= 9.
6 × 9 = 54.

Question 16.

Answer:
= 1.

Explanation:
=??
= 1.
6 × 1 = 6.

Question 17.

Answer:
= 4.

Explanation:
= ??
= 4.
7 × 4 = 28.

Question 18.
Find 49 divided by 7.

Answer:
49 ÷ 7 = 7.

Explanation:
49 ÷ 7 = 7.
7 × 7 = 49.

Question 19.
Divide 54 by 6.
Answer:
54 ÷ 6 = 9.

Explanation:
54 ÷ 6 = 9.
6 × 9 = 54.

Question 20.
Find 35 divided by 7.
Answer:
35 ÷ 7 = 5.

Explanation:
35 ÷ 7 = 5.
7 × 5 = 35.

Problem Solving
Question 21.
A pizza parlor made 88 deep-dish pizzas. It made 10 more thin-crust pizzas than deep-dish pizzas. How many thin-crust pizzas did the parlor make?
Answer:
10 more thin-crust pizzas the parlor makes than deep-dish pizzas.

Explanation:
Number of deep-dish pizzas pizza parlor made = 88.
It made 10 more thin-crust pizzas than deep-dish pizzas.
=> Number of thin-crust pizzas pizza parlor made = 88 + 10 = 98.
Difference:
Number of thin-crust pizzas pizza parlor made – Number of deep-dish pizzas pizza parlor made
= 98 – 88
= 10.

Question 22.
Higher Order Thinking There are 35 new tires. Each truck will get 6 tires plus 1 tire for a spare. How many trucks will get new tires?
Answer:
Number of trucks gets new tires = 5.

Explanation:
Number of new tires = 35.
Each truck will get 6 tires plus 1 tire for a spare.
=> Number of tires each truck gets = 6 + 1 = 7.
Number of trucks gets new tires = Number of new tires ÷ Number of tires each truck gets
= 35 ÷ 7
= 5.

Question 23.
Make Sense and Persevere Explain the mistake in the fact family below. Give the correct fact.
4 × 7 = 28
7 × 4 = 28
7 ÷ 4= 28
28 ÷ 7 = 4
Answer:
4 × 7 = 28. Correct.
7 × 4 = 28. Correct.
7 ÷ 4= 28. Incorrect.
28 ÷ 7 = 4. Correct.

Explanation:
4 × 7 = 28. Correct.
7 × 4 = 28. Correct.
7 ÷ 4 = 28. Division is not correct. Incorrect.
28 ÷ 7 = 4. Correct.

Question 24.
Gloria mowed 7 lawns and earned $56. She was paid the same amount for each lawn. How much money did Gloria earn for mowing each lawn? Write an equation to represent this problem.
Answer:
Amount Gloria earn for mowing each lawn = $392.

Explanation:
Number of lawns Gloria mowed = 7.
Money Gloria earned = $56.
She was paid the same amount for each lawn.
Amount Gloria earn for mowing each lawn = Number of lawns Gloria mowed × Money Gloria earned
= 7 × $56.
= $392.

Question 25.
Andy bought 35 beads. He bought only one color of beads. Which beads could Andy have bought? How many packages of that color bead did he buy?
Answer:
Andy would have bought red or gold beads.
Number of packets red beads Andy bought = 5.
Number of packets gold beads Andy bought = 7.

Explanation:
Number of beads Andy bought = 35.
Number of beads in green package = 6.
Number of beads in red package = 7.
Number of beads in gold package = 5.
Number of packets red beads Andy bought = Number of beads Andy bought ÷ Number of beads in red package
= 35 ÷ 7
= 5.
Number of packets gold beads Andy bought = Number of beads Andy bought ÷ Number of beads in gold package
= 35 ÷ 5
= 7.

Question 26.
Cassidy bought 48 beads. She bought only one color of beads. Which beads could Cassidy have bought? How many packages of that color bead did she buy?
Answer:
Number of packets green beads Cassidy bought = 8.

Explanation:
Number of beads Cassidy bought = 48.
Number of beads in green package = 6.
Number of beads in red package = 7.
Number of beads in gold package = 5
Number of packets green beads Cassidy bought = Number of beads Cassidy bought ÷ Number of beads in green package
= 48 ÷ 6
= 8.

Assessment Practice
Question 27.
Which multiplication fact can you use to help find the value of the unknown number in the equation 42 ÷ 7 = ☐?
A. 5 × 7
B. 6 × 7
C. 7 × 7
D. 8 × 7
Answer:
Equation 42 ÷ 7 = ☐?
B. 6 × 7 multiplication fact can be used to help, find the value of the unknown number in the equation 42 ÷ 7 = ??.

Explanation:
Equation 42 ÷ 7 = ☐?
42 ÷ 7 = 6.
7 × 6 = 42.
6 × 7 = 42.

Question 28.
Which multiplication fact can you use to help find the value of the unknown number in the equation 36 ÷ 6 = ☐?
A. 5 × 6
B. 6 × 6
C. 7 × 6
D. 8 × 6
Answer:
Equation 36 ÷ 6 = ☐?
B. 6 × 6 multiplication fact can be used to help, find the value of the unknown number in the equation 36 ÷ 6 = ☐?

Explanation:
Equation 36 ÷ 6 = ☐?
36 ÷ 6 = 6.
6 × 6 = 36.

Lesson‌ ‌4.4‌ ‌Use‌ ‌Multiplication‌ ‌to‌ ‌Divide‌ ‌with‌ ‌8‌ ‌and‌ ‌9‌ ‌

Solve & Share‌ ‌
An‌ ‌art‌ ‌teacher‌ ‌has‌ ‌72‌ ‌crayons.‌ ‌The‌ ‌crayons‌ ‌came‌ ‌in‌ ‌boxes‌ ‌with‌ ‌8‌ ‌crayons‌ ‌ in‌ ‌each‌ ‌box.‌ ‌How‌ ‌many‌ ‌boxes‌ ‌of‌ ‌crayons‌ ‌were‌ ‌there?‌ ‌
I‌ ‌can‌ ‌…‌ ‌ divide‌ ‌by‌ ‌8‌ ‌and‌ ‌9‌ ‌by‌ ‌thinking‌ ‌about‌ ‌how‌ ‌I‌ ‌multiply‌ ‌with‌ ‌those‌ ‌numbers.

Look‌ ‌Back!‌ ‌Draw‌ ‌a‌ ‌picture‌ ‌you‌ ‌could‌ ‌use‌ ‌to‌ ‌help‌ ‌solve‌ ‌the‌ ‌problem‌ ‌above.‌ ‌

Answer:
Number of boxes crayons are there = 9.

Explanation:
Number of crayons art teacher has = 72.
Number of crayons in each box = 8
Number of boxes crayons are there = Number of crayons art teacher has ÷ Number of crayons in each box
= 72 ÷ 8
= 9.

Essential Question
What Multiplication Fact Can You Use?

Answer:
Some basic multiplication facts are needed to follow for multiplying numbers. The repeated addition of the same number is expressed by multiplication in short.

Explanation:
Basic multiplication facts are needed to follow for multiplying numbers. The repeated addition of the same number is expressed by multiplication in short.
For example: 36 ÷ 6 = 6.
6 × 6 = 36.

Visual Learning Bridge
John has 56 straws. He needs 8 straws to make a spider. How many spiders can John make? Find 56 ÷ 8.

Luz made 9 animals. She used 54 straws. She used the same number of straws for each animal. How many straws did Luz use

Find 54 ÷ 9.

What You Think
9 times what number is 54?
9 × 6 = 54

What You Write
54 ÷ 9 = 6
Luz used 6 straws for each animal.

Convince Me! Look for Relationships Write the related multiplication fact that can be used to complete each division fact.

Answer:
Division Fact    Related Multiplication Fact.
72 ÷ 8 = 9.               9 × 8 = 72.
48 ÷ 8 = 6.               8 × 6 = 48.
63 ÷ 9 = 7.               9 × 7 = 63.

Explanation:
Related multiplication fact that can be used to complete each division fact is as below:
Division Fact    Related Multiplication Fact.
72 ÷ 8 = 9.               9 × 8 = 72.
48 ÷ 8 = 6.               8 × 6 = 48.
63 ÷ 9 = 7.               9 × 7 = 63.

Guided Practice

Do You Understand?
Question 1.
What multiplication fact can you use to find 18 ÷ 9?
Answer:
2 × 9 = 18 multiplication fact can be used to find 18 ÷ 9.

Explanation:
18 ÷ 9 = ??
18 ÷ 9 = 2.
2 × 9 = 18.

Question 2.
Carla and Jeff each use 72 straws. Carla makes animals with 9 legs. Jeff makes animals with 8 legs. Who makes more animals? Explain.

Answer:
Jeff makes 1 more animal with legs than Carla makes.

Explanation:
Number of straws Carla and Jeff each use = 72
Number of animals with legs Carla makes = 9.
Number of animals with legs Jeff makes = 8.
Total number of animals with legs Carla makes = Number of straws Carla and Jeff each use ÷ Number of animals with legs Carla makes
= 72 ÷ 9
= 8.
Total number of animals with legs Jeff makes = Number of straws Carla and Jeff each use ÷ Number of animals with legs Jeff makes
= 72 ÷ 8
= 9.

Do You Know How?
In 3 and 4, use the multiplication equation to help find each quotient.
Question 3.
16 ÷ 8 = ?
What number times 8 is 16?
______ × 8 = 16
So, 16 ÷ 8 = _______

Answer:
16 ÷ 8 = 2.
___2___ × 8 = 16.
So, 16 ÷ 8 = __2_____.

Explanation:
16 ÷ 8 = ?
16 ÷ 8 = 2.
=> 2 × 8 = 16.
So, 16 ÷ 8 =2.

Question 4.
64 ÷ 8 = ?
What number times 8 is 64?
______ × 8 = 64
So, 64 ÷ 8 = _______

Answer:
64 ÷ 8 = 8.
__8____ × 8 = 64.
So, 64 ÷ 8 = __8_____.

Explanation:
64 ÷ 8 = ?
64 ÷ 8 = 8.
8 × 8 = 64
So, 64 ÷ 8 = 8.

Independent Practice
Leveled Practice In 5-7, use the multiplication equation to help find each quotient.

Question 5.
24 ÷ 8= ?
What number times 8 is 24?
______ × 8 = 24
24 ÷ 8 = ______

Answer:
24 ÷ 8= 3.
___3___ × 8 = 24.
24 ÷ 8 = ___3___.

Explanation:
24 ÷ 8= ?
24 ÷ 8= 3.
3 × 8 = 24.
24 ÷ 8 = 3.

Question 6.
45 ÷ 9 = ?
What number times 9 is 45?
______ × 9 = 45
45 ÷ 9 = ______
Answer:
45 ÷ 9 = 5.
___5___ × 9 = 45.
45 ÷ 9 = ___5___.

Explanation:
45 ÷ 9 = ?
45 ÷ 9 = 5.
5 × 9 = 45.
45 ÷ 9 = 5.

Question 7.
27 ÷ 9 = ?
What number times 9 is 27?
______ × 9 = 27
27 ÷ 9 = _______

Answer:
27 ÷ 9 = 3.
___3___ × 9 = 27.
27 ÷ 9 = ___3____.

Explanation:
27 ÷ 9 = ?
27 ÷ 9 = 3.
3 × 9 = 27.
27 ÷ 9 = 3.

In 8-16, find each quotient.
Question 8.
48 ÷ 8 = _______

Answer:
48 ÷ 8 = 8.

Explanation:
48 ÷ 8 = 8. (quotient)

Question 9.
72 ÷ 9 = ______

Answer:
72 ÷ 9 = ___8___.

Explanation:
72 ÷ 9 = 8. (quotient)

Question 10.
______ = 8 ÷ 8

Answer:
__1____ = 8 ÷ 8.

Explanation:
8 ÷ 8 = 1. (quotient)

Question 11.
_____ = 54 ÷ 9

Answer:
__6___ = 54 ÷ 9.

Explanation:
54 ÷ 9 = 6. (quotient)

Question 12.
72 ÷ 8 = ______

Answer:
72 ÷ 8 = ___9___.

Explanation:
72 ÷ 8 = 9. (quotient)

Question 13.
90 ÷ 9 = ________

Answer:
90 ÷ 9 = __10______.

Explanation:
90 ÷ 9 = 10. (quotient)

Question 14.

Answer:
= 10.

Explanation:
= 10. (quotient)

Question 15.

Answer:
= 4.

Explanation:
= 4. (quotient)

Question 16.

Answer:
= 1.

Explanation:
= 1. (quotient)

Problem Solving
Question 17.
Callie biked 27 miles on Saturday. She biked 9 miles every hour. How many hours did Callie bike? Draw a picture to represent the problem.

Answer:
Total number of hours she biked = 3.

Explanation:
Number of miles Callie biked  on Saturday = 27.
Number of miles Callie biked  every hour = 9.
Total number of hours she biked = Number of miles Callie biked  on Saturday ÷ Number of miles Callie biked  every hour
= 27 ÷ 9
= 3.

Question 18.
en Vision® STEM Eight friends decide to test how far 40 paper airplanes with different shapes will fly. If each friend tests the same number of airplanes, how many airplanes does each friend test?

Answer:
Number of airplanes each friend gets to test = 5.

Explanation:
Number of friends of Stem = 8.
Number of paper airplanes with different shapes  = 40.
Number of airplanes each friend gets to test = Number of paper airplanes with different shapes ÷ Number of friends of Stem
= 40 ÷ 8
= 5.

Question 19.
Reasoning What other equations are in the same fact family as 18 ÷ 9 = 2?

Answer:
Other equations which are in the same fact family as 18 ÷ 9 = 2:
18 ÷ 2 = 9.
2 × 8 = 18.
9 × 2 = 18.

Explanation:
18 ÷ 9 = 2.
Other equations which are in the same fact family as 18 ÷ 9 = 2:
18 ÷ 2 = 9.
2 × 8 = 18.
9 × 2 = 18.

Question 20.
Higher Order Thinking Jeremy had 30 gummy bears. He ate 6, and then gave the rest to 8 friends. Each friend got the same number of gummy bears. How many did each friend get?

Answer:
Number of gummy bears each friend gets = 3.

Explanation:
Number of gummy bears Jeremy had = 30.
Number of gummy bears Jeremy ate = 6.
Number of friends he gave the rest gummy bears = 8.
Number of gummy bears each friend gets = (Number of gummy bears Jeremy had – Number of gummy bears Jeremy ate) ÷ Number of friends he gave the rest gummy bears
= (30 – 6) ÷ 8
= 24 ÷ 8
= 3.

Question 21.
Mr. Stern spends $36 on tickets. He buys only one type of ticket.

a. Which types of ticket could he buy?
Answer:
Types of tickets he could buy:
Number of child tickets he buys = 9.
Number of adult tickets he buys = 4.

Explanation:
Mr. Stern spends $36 on tickets.
Cost of the child ticket = $4
Amount Mr. Stern spends on child tickets = $36
Number of child tickets he buys = Amount Mr. Stern spends on child tickets ÷ Cost of the child ticket
= $36 ÷ $4
= 9.

Cost of the adult ticket = $9.
Amount Mr. Stern spends on adult tickets = $36
Number of adult tickets he buys = Amount Mr. Stern spends on adult tickets ÷ Cost of the adult ticket
= $36 ÷ $9
= 4.

b. Which type of ticket could Mr. Stern NOT buy? Explain why not.
Answer:
Cost the youth ticket = $8 because the amount cannot be divided exactly by the amount he spent on tickets purchase.

Explanation:
Cost the youth ticket = $8.
This ticket may not be purchased by Mr. Stern because the amount cannot be divided exactly by the amount he spent on tickets purchase.

Assessment Practice

Question 22.
Find 32 ÷ 8 by selecting numbers to complete the following equations. Numbers may be selected more than once.

Answer:
8 × 9 = 32.
32 ÷ 8 = 4.

Explanation:
32 ÷ 8 = 4.
8 × 9 = 32.
Number 4 is selected from the given numbers to complete the following equations.

Question 23.
Find 54 ÷ 9 by selecting numbers to complete the following equations. Numbers may be selected more than once.

Answer:
9 × 6 = 54.
54 ÷ 9 = 6.

Explanation:
54 ÷ 9 = 6.
9 × 6 = 54.
6 number is selected from the given numbers to complete the following equations.

Lesson 4.5 Multiplication Patterns: Even and Odd Numbers

Solve & Share
Prizes for a school fair are packaged with 2 of the same prizes in each package. Which of the prizes listed below can be packaged with none left over? Tell how you decided.
I can … find and explain patterns for even and odd numbers.

You can use reasoning. Think about numbers that can be separated into two equal groups.

Look Back! What do you notice about the numbers for the prizes that can be packaged in 2s with none left over? What do you notice about the numbers for the other prizes?

Answer:
The prizes listed can be packaged with none left over are: car prizes, boats prizes and books prizes.
It is noticed about the numbers for the prizes that can be packaged in 2s with none left over that they are the multiples of 2. It is noticed about the numbers for the other prizes that they are prime numbers and are not multiples of 2.

Explanation:
Number of car prizes = 6.
Number of hats prizes = 15.
Number of balls prizes = 23.
Number of boats prizes = 18.
Number of books prizes = 36.
Number of groups the prizes are packed = 2.
Numbers for the prizes that can be packaged in 2s with none left over:
= Number of car prizes ÷ Number of groups the prizes are packed
= 6 ÷ 2
= 3.
Number of boats prizes ÷ Number of groups the prizes are packed
= 18 ÷ 2
= 9.
Number of books prizes ÷ Number of groups the prizes are packed
= 18 ÷ 2
= 9.

Essential Question
How Can You Explain Multiplication Question Patterns for Even and Odd Numbers?
Answer:
An  even number can only be formed by multiplication in three ways:
even × odd, odd × even and even × even.
An odd number can only be formed by multiplication in one way:
odd × odd = odd.

Explanation:
Multiplication Question Patterns for Even and Odd Numbers can be explained as below:
An  even number can only be formed by multiplication in three ways:
even × odd, odd × even and even × even.
An odd number can only be formed by multiplication in one way:
odd × odd = odd.

Visual Learning Bridge
Nita says that the product of an even number and an odd number is always even. Is she correct?
Even numbers are whole numbers that can be divided by 2 with none left over.
Odd numbers are whole numbers that cannot be divided by 2 with none left over.

Even numbers greater than 0 can be shown as two equal groups.
Think about 2 × 3 and 2 × 5.
2 is an even number.
2 × 3 means 2 equal groups of 3.
2 × 3 = 6
2 × 5 means 2 equal groups of 5.
2 × 5 = 10
There are always 2 equal groups, so the product of 2 times any number is even.

You can generalize.
All even numbers are multiples of 2.
Think about 4 × 3.
You can think of 4 as 2 groups of 2.
Using properties you can write
4 × 3 = (2 × 2) > 3 as
4 × 3 = 2 × (2 × 3).
So, 4 × 3 = 2 × 6.
There are 2 equal groups of 6. So, the product will be even.

You can write any even number as 2 equal groups. So, Nita is correct: me even X odd = even.

Convince Me! Generalize Does multiplying by 8 also always result in an even product? Explain.

Answer:
Multiplying by 8 also always result in an even product is true because even numbers are whole numbers that can be divided by 2 with none left over. Odd numbers are whole numbers that cannot be divided by 2 with none left over.

Explanation:
Multiplying by 8 also always result in an even product is true because even numbers are whole numbers that can be divided by 2 with none left over. Odd numbers are whole numbers that cannot be divided by 2 with none left over.
2 × 8 = 16.
3 × 8 = 24.

Another Example!
An odd number cannot be divided by 2 with none left over.

Think about 3 × 5.
3 cannot be divided by 2 with none left over.
5 cannot be divided by 2 with none left over.
3 × 5 = 15
15 is odd.

Guided Practice
Do You Understand?
Question 1.
If you multiply two even numbers, will the product be even or odd? Explain with an example.

Answer:
If you multiply two even numbers, will the product be even not odd.

Explanation:
If you multiply two even numbers, will the product be even not odd.
For example: 2 × 6 = 12.

Do You Know How?
Write or circle to complete the sentences. Then solve.
Question 2.
4 × 6 = ?
Can 4 be divided by 2? _______
Can 6 be divided by 2? _______
So 4 × 6 is even, odd
4 × 6 = _______

Answer:
4 × 6 = 24.
Can 4 be divided by 2? __Yes_____.
Can 6 be divided by 2? ____Yes___.
So 4 × 6 is even.
4 × 6 = _24______.

Explanation:
4 × 6 = 24.
Can 4 be divided by 2? 4 ÷ 2 = 2 yes.
Can 6 be divided by 2? 6 ÷ 2 = 3 yes.
So 4 × 6 is even.
4 × 6 = 24.

Independent Practice
In 3-5, circle the factors that can be divided by 2. Then write even or odd to describe the product and solve the equation.
Question 3.
9 × 5 = ?
9 × 5 is _______
9 × 5 = _______

Answer:
9 × 5 = ?
9 × 5 is __odd_____.
9 × 5 = __45_____.

Explanation:
9 × 5 = ??
9 × 5 is odd.
9 × 5 = 45.

Question 4.
8 × 7 = ?
8 × 7 is ______
8 × 7 = ______

Answer:
8 × 7 = ?
8 × 7 is __even____.
8 × 7 = __56____.

Explanation:
8 × 7 = ?
8 × 7 is even.
8 × 7 = 56.

Question 5.
4 × 8 =?
4 × 8 is ______
4 × 8 = ______

Answer:
4 × 8 =?
4 × 8 is _Even_____.
4 × 8 = __32____.

Explanation:
4 × 8 =?
4 × 8 is even.
4 × 8 = 32.

Problem Solving
In 6-8, use the table at the right. Look at the factors. Write even or odd to describe the product. Then solve.

Question 6.
How many miles did Joseph bicycle in 6 days?
The product is ________.

Answer:
Number of miles Joseph bicycled in 6 days = 18.
The product is ___18(Even)_____.

Explanation:
Number of miles Joseph bicycled per day = 3.
Number of miles Joseph bicycled in 6 days = Number of miles Joseph bicycled per day  × 6
= 3 × 6
= 18.

Question 7.
How many miles did Caitlin bicycle in 8 days?
The product is ________.

Answer:
Number of miles Caitlin bicycled in 8 days = 48.
The product is ___48 (Even)_____.

Explanation:
Number of miles Caitlin bicycled per day = 6.
Number of miles Caitlin bicycled in 8 days = Number of miles Caitlin bicycled per day  × 8
= 6 × 8
= 48.

Question 8.
How many miles did Maria and Riaz bicycle in 3 days?
The total is _________

Answer:
Total number of miles Maria and Riaz bicycled in 3 days = 27.
The total is ___27 (Odd)______.

Explanation:
Number of miles Maria bicycled per day = 4.
Number of miles Maria bicycled in 3 days = Number of miles Maria bicycled per day  × 3
= 4 × 3
= 12.
Number of miles Riaz bicycled per day = 5.
Number of miles Riaz bicycled in 3 days = Number of miles Riaz bicycled per day  × 3
= 5 × 3
= 15.

Total number of miles Maria and Riaz bicycled in 3 days = Number of miles Maria bicycled in 3 days + Number of miles Riaz bicycled in 3 days
= 12 + 15
= 27.

Question 9.
Critique Reasoning Ryan says that the following patterns are true:
even × odd = even
odd × even = odd
Is he correct? Explain.

Answer:
(even × odd = even) pattern Ryan’s says is correct yet not odd × even = odd pattern.

Explanation:
even × odd = even.
=> 2 × 5 = 10.
odd × even = odd
=> 5 × 4 = 20.

Question 10.
Draw a shape with an odd number of sides. Then write the name of the shape.

Answer:

Explanation:
Number of sides in the shape = 5.
Name of the shape = Pentagon.

Question 11.
Higher Order Thinking The bakery had 84 muffins. Ms. Craig bought 5 packs of 6 muffins. Did she purchase an even or an odd number of muffins? Is the number of muffins left even or odd? Explain.

Answer:
Total number of muffins Ms. Craig bought = 30. (Even)
Number of muffins left = 54. (Even)

Explanation:
Number of muffins bakery had = 84.
Number of muffins Ms. Craig bought = 6.
Number of packs Ms. Craig bought = 5.
Total number of muffins Ms. Craig bought = Number of muffins Ms. Craig bought  × Number of packs Ms. Craig bought
= 6 × 5
= 30. (Even)
Number of muffins left = Number of muffins bakery had – Total number of muffins Ms. Craig bought
= 84 – 30
= 54. (Even)

Assessment Practice
Question 12.
Select all of the equations where you can use properties of operations to show that the product will be even.
☐ 7 × 9 = ?
☐ 9 × 2 = ?
☐ 5 × 3 = ?
☐ 1 × 6 = ?
☐ 7 × 5 = ?

Answer:
9 × 2 = 18. => odd × even = even.
1 × 6 = 6. => odd × even = even.

Explanation:
☐ 7 × 9 = 63.
=> odd × odd = odd.
☐ 9 × 2 = 18.
=> odd × even = even.
☐ 5 × 3 = 15.
=> odd × odd = odd.
☐ 1 × 6 = 6.
=> odd × even = even.
☐ 7 × 5 = 35.
=> odd × odd = odd.

Question 13.
Select all of the equations that do NOT have even products.
☐ 5 × 1 = ?
☐ 2 × 7 = ?
☐ 3 × 9 = ?
☐ 8 × 8 = ?
☐ 6 × 4 = ?

Answer:
5 × 1 = 5. => odd × odd = odd.
3 × 9 = 27. => odd × odd = odd.

Explanation:
☐ 5 × 1 = 5.
=> odd × odd = odd.
☐ 2 × 7 = 14.
=> even × odd = even.
☐ 3 × 9 = 27.
=> odd × odd = odd.
☐ 8 × 8 = 64.
=> even × even = even.
☐ 6 × 4 = 24.
=> even × even = even.

Lesson 4.6 Division Involving 0 and 1

Solve & Share
Find 5 ÷ 1,0 ÷ 5, and 5 ÷ 5. Explain how you found each quotient. You can use counters to help.
I can … understand the patterns of division with 0 and 1.

Look Back! Use your understanding of multiplying by 0 to find 0 ÷ 7,0 ÷ 4, and 0 ÷ 10. Describe the patterns you see.

Answer:
0 ÷ 4 = 0.
0 ÷ 7 = 0.
0 ÷ 10 = 0.
It is observed that any number multiplied to zero or zero divided by any number the product of it is same that is zero.

Explanation:
0 × 4 = 0.
0 ÷ 4 = 0.
0 × 7 = 0.
0 ÷ 7 = 0.
0 × 10 = 0.
0 ÷ 10 = 0.

Essential Question
How Do You Divide with 1 or 0?
Answer:
Any number divided by the given number to be divided.
2 ÷ 1 = 2.
2 ÷ 0 = 0.

Explanation:
We divided the number with one or 0 by the following:
2 ÷ 1 = 2.
2 ÷ 0 = 0.

Visual Learning Bridge
Neil has 3 goldfish. He puts 1 goldfish in each bowl. How many bowls did Neil use? Find 3 ÷ 1.

What number times 1 is 3?
3 × 1 = 3
So, 3 ÷ 1 = 3.
Neil used 3 bowls.

1 as a Quotient
Find 3 ÷ 3.
3 times what number equals 3?
3 × 1 = 3
S0, 3 ÷ 3 = 1.
Rule: Any number (except 0) divided by itself is 1.

Dividing O by a Number
Find 0 ÷ 3.
3 times what number equals 0?
3 × 0 = 0
So, 0 ÷ 3 = 0.
Rule: 0 divided by any number (except 0) is 0.

Dividing by 0
Find 3 ÷ 0.
0 times what number equals 3?
There is no such number. So, 3 ÷ 0 can’t be done.
Rule: You cannot divide any number by 0.

Convince Me! Be Precise Sue wrote 9 invitations. She put 1 invitation in each mailbox on her street. How many mailboxes got invitations? Which equation shows the problem and the solution? Explain your thinking.
0 ÷ 9 = 0
9 ÷ 1 = 9

Answer:
Number of mailboxes got invitations = Number of invitations Sue wrote ÷ Number of invitation she puts in each mailbox on her street
= 9 ÷ 1
= 9.

Explanation:
Number of invitations Sue wrote = 9.
Number of invitation she puts in each mailbox on her street = 1.
Number of mailboxes got invitations = Number of invitations Sue wrote ÷ Number of invitation she puts in each mailbox on her street
= 9 ÷ 1
= 9.

Guided Practice
Do You Understand?
Question 1.
How can you tell, without dividing, that 375 ÷ 375 = 1?

Answer:
Any number divided by itself, the product is always 1.
375 ÷ 375 = 1

Explanation:
We can tell without dividing, that 375 ÷ 375 = 1 because any number divided by itself, the product is always 1.

Question 2.
Use a representation to explain why zero divided by any number except zero is zero.
Answer:
In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as  where a is the dividend. Any number dividend by zero is zero yet not by itself divided.

Explanation:
In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as  where a is the dividend.
Any number multiplied by zero is zero, the expression is also undefined.

Do You Know How?
In 3 and 4, solve the multiplication equation to find each quotient.

Question 3.
Find 8 ÷ 8.
8 × _____ = 8
So, 8 ÷ 8 = _______

Answer:
8 × __1___ = 8
So, 8 ÷ 8 = ___1____. (quotient)

Explanation:
8 × 1 = 8
So, 8 ÷ 8 = 1.

Question 4.
Find 0 ÷ 9.
9 × ______ = 0
So, 0 ÷ 9 = ______

Answer:
9 × __0____ = 0
So, 0 ÷ 9 = __0____. (quotient)

Explanation:
9 × 0 = 0
So, 0 ÷ 9 = 0.

Independent Practice
Leveled Practice In 5-7, solve the multiplication equation to find each quotient.
Question 5.
Find 0 ÷ 7.
7 × _______ = 0
So, 0 ÷ 7 = _______

Answer:
7 × ___0____ = 0.
So, 0 ÷ 7 = ___0____. (quotient)

Explanation:
7 × 0 = 0
So, 0 ÷ 7 = 0.

Question 6.
Find 4 ÷ 4.
4 × _______ = 4
So, 4 ÷ 4 = ________

Answer:
4 × ___1____ = 4
So, 4 ÷ 4 = ___1_____. (quotient)

Explanation:
4 × 1 = 4
So, 4 ÷ 4 = 1.

Question 7.
Find 6 ÷ 1.
1 × _______ = 6
So, 6 ÷ 1 = _______

Answer:
1 × ___6____ = 6
So, 6 ÷ 1 = ___6____. (quotient)

Explanation:
1 × 6 = 6
So, 6 ÷ 1 = 6.

In 8-18, find each quotient.
Question 8.
3 ÷ 3 = __________

Answer:
3 ÷ 3 = ____1______.(quotient)

Explanation:
3 ÷ 3 = 1.

Question 9.
______ = 0 ÷ 8

Answer:
___0___ = 0 ÷ 8. (quotient)

Explanation:
0 = 0 ÷ 8.

Question 10.
______ = 5 ÷ 5

Answer:
___1___ = 5 ÷ 5. (quotient)

Explanation:
1 = 5 ÷ 5.

Question 11.
7 ÷ 1 = _______

Answer:
7 ÷ 1 = __7_____. (quotient)

Explanation:
7 ÷ 1 = 7.

Question 12.

Answer:
= 1. (quotient)

Explanation:
= 1.

Question 13.

Answer:
= 5. (quotient)

Explanation:
= 5.

Question 14.

Answer:
= 1. (quotient)

Explanation:
= 1.

Question 15.

Answer:
= 13. (quotient)

Explanation:
= 13.

Question 16.
Find 0 divided by 8.

Answer:
0 ÷ 8 = 0.

Explanation:
0 divided by 8.
=> 0 ÷ 8 = 0.

Question 17.
Find 9 divided by 1.
Answer:
9 ÷ 1 = 9.

Explanation:
9 divided by 1.
9 ÷ 1 = 9.

Question 18.
Find 10 divided by 10.

Answer:
10 ÷ 10 = 1.

Explanation:
10 divided by 10.
=> 10 ÷ 10 = 1.

Problem Solving
In 19-22, use the picture at the right.

Question 19.
Addie hiked 3 different trails for a total distance of 11 miles. Which trails did Addie hike?
Answer:
Trails Addie hiked = Number of miles of White hike + Number of miles of Red hike + Number of miles of Green hike
= 5 + 2 + 4
= 11.

Explanation:
Number of miles of Blue hike = 3.
Number of miles of White hike = 5.
Number of miles of Red hike = 2.
Number of miles of Green hike = 4.
Number of different trails Addie hiked = 3.
Number of total miles distance Addie hiked = 11.
Trails Addie hiked = Number of miles of White hike + Number of miles of Red hike + Number of miles of Green hike
= 5 + 2 + 4
= 7 + 4
= 11.

Question 20.
Marty hikes one of the trails 4 times. In all, he hikes more than 10 miles but less than 16 miles. Which trail does Marty hike? Explain your answer.
Answer:
Blue Trail Marty hiked 4times hiking of 12 miles.

Explanation:
Number of different trails Marty hiked = 1.
Number of times Marty hiked same trail = 4.
Number of miles of Blue hike = 3.
Trail Marty hiked = Number of miles of Blue hike  × 4
= 3 × 4
= 12 > 10 < 16.

Question 21.
Four teams are tidying the Green trail. They will each tidy an equal distance. How many miles does each team tidy?
Answer:
Miles each team tidy = 1.

Explanation:
Number of miles of Green hike = 4.
Number of teams hiking the green trial = 4.
Miles each team tidy = Number of miles of Green hike  ÷ Number of teams hiking the green trial
= 4 ÷ 4
= 1.

Question 22.
Fiona hiked on Wednesday and Sunday. Each day she hiked all of the trails. How many miles did Fiona hike?
Answer:
Total number of Miles Fiona hiked = 28.

Explanation:
Number of miles of Blue hike = 3.
Number of miles of White hike = 5.
Number of miles of Red hike = 2.
Number of miles of Green hike = 4.
Fiona hiked on Wednesday and Sunday.
=> Number of days Fiona hiked  = 2.
Total number of Miles Fiona hiked = Number of miles of Blue hike + Number of miles of White hike + Number of miles of Red hike + Number of miles of Green hike × Number of days Fiona hiked
= (3 + 5 + 2 + 4) × 2
= ( 8 + 2 + 4)
= ( 10 + 4)
= 14 × 2
= 28.

Question 23.
Model with Math Use a representation to explain why any number divided by 1 is itself.
Answer:
Same number divided by itself, always gives the product of 1.

Explanation:
1 ÷ 1 = 1 because same number divided by itself, always gives the product of 1.
2 ÷ 2 = 1.

Question 24.
Higher Order Thinking Yvonne says that 0 ÷ 21 and 21 ÷ 0 both have a quotient of 0. Is Yvonne correct? Explain.

Answer:
Yes, Yvonne is correct 0 ÷ 21 and 21 ÷ 0 both have a quotient of 0 because any number divided by zero or zero divided by any number the quotient is always zero.

Explanation:
0 ÷ 21 = 0.
21 ÷ 0 = 0.

Assessment Practice
Question 25.
Use division properties to match each equation to its quotient.

Answer:

Explanation:
8 ÷ 8 = 1. Any number divided by itself, quotient is always 1.
0 ÷ 4 = 0. Zero divided by any number, quotient is always 0.
3 ÷ 3 = 1. Any number divided by itself, quotient is always 1.

Question 26.
Use division properties to match each equation to its quotient.

Answer:

Explanation:
4 ÷ 4 = 1. Any number divided by itself, quotient is always 1.
0 ÷ 5 = 0. Zero divided by any number, quotient is always 0.
0 ÷ 7 = 0. Zero divided by any number, quotient is always 0.

Lesson 4.7 Practice Multiplication and Division Facts

Solve & Share
A tour bus to a national park holds 56 people. There are 7 tour guides at the park to lead equal groups of people from the bus. How many people are in each tour group? Each person in a group pays a $2 entrance fee to a tour guide. How much does 1 tour guide collect?
I can .. use patterns and related facts to solve multiplication and division problems.

Look Back! How can 7 × ? = 56 help you find 56 ÷ 7 = ?
Answer:
Number of people in each tour group = 8.
Amount of each tour guide collects = $16.

Explanation:
Number of people A tour bus to a national park holds = 56.
Number of tour guides at the park to lead equal groups of people from the bus = 7.
Number of people in each tour group = Number of people A tour bus to a national park holds ÷ Number of tour guides at the park to lead equal groups of people from the bus
= 56 ÷ 7
= 8.
Amount of entrance fee each person in a group to pay tour guide = $2.
Amount of each tour guide collects = Number of people in each tour group × Amount of entrance fee each person in a group to pay tour guide
= 8 × $2
= $16.

Essential Question
What Fact Can You Use?
Answer:
To solve the problem we can use multiplication facts or division facts according to the requirment of the question asked.

Explanation:
We can use multiplication facts 1s, 2s, 3s, 4s, 5s…… and division facts to solve the problems to find the asked solution.

Visual Learning Bridge
Sabrina has 28 quarters in her bank. She wants to trade all of them for one-dollar bills. How many one-dollar bills will Sabrina get?
There are 4 quarters in one dollar.

One Way
How many groups of 4 are in 28?
You can draw a bar diagram to help solve the problem

28 ÷ 4 = 7
There are 7 groups of 4 in 28. Sabrina can trade 28 quarters for 7 one-dollar bills.

Another Way
What number times 4 equals 28?
You can use multiplication facts to help solve the problem.
28 quarters
? × 4 = 28
7 × 4= 28
Sabrina can trade 28 quarters for 7 one-dollar bills.

Convince Me! Construct Arguments Why can both 28 ÷ 7 = ? and ? × 7 = 28 be used to solve the problem above?
Answer:
Number of one-dollar bills Sabrina gets = 7.
4 × 7 = 28.
28 ÷ 7 = 4.
We can use both 28 ÷ 7 = ? and ? × 7 = 28  to solve the problem above explained as below:
Number of quarters in one dollar × Number of one-dollar bills Sabrina gets = Number of quarters Sabrina has in her bank.
=> 4 × 7 = 28.
Number of quarters Sabrina has in her bank ÷ Number of one-dollar bills Sabrina gets = Number of quarters in one dollar.
28 ÷ 7 = 4.

Explanation:
There are 4 quarters in one dollar.
Number of quarters Sabrina has in her bank = 28.
Number of quarters in one dollar = 4.
Number of one-dollar bills Sabrina gets = Number of quarters Sabrina has in her bank  ÷ Number of quarters in one dollar
= 28 ÷ 4
= 7.
We can use both 28 ÷ 7 = ? and ? × 7 = 28  to solve the problem above explained as below:
Number of quarters in one dollar × Number of one-dollar bills Sabrina gets = Number of quarters Sabrina has in her bank.
=> 4 × 7 = 28.
Number of quarters Sabrina has in her bank ÷ Number of one-dollar bills Sabrina gets = Number of quarters in one dollar.
=> 28 ÷ 7 = 4.

Guided Practice
Do You Understand?
Question 1.
Look back at the problem on the previous page. Suppose Sabrina put 8 more quarters in her bank. How many one-dollar bills can she trade for the quarters in her bank now?
Answer:
Number of one-dollar bills Sabrina gets = 10.

Explanation:
There are 4 quarters in one dollar.
Number of quarters Sabrina has in her bank = 28 + 12 = 40.
Number of quarters in one dollar = 4.
Number of one-dollar bills Sabrina gets = Number of quarters Sabrina has in her bank  ÷ Number of quarters in one dollar
= 40 ÷ 4
= 10.

Question 2.
Calvin solves the equation 49 ÷ 7 = ☐. How does this help him complete the equation 7 × ☐ = 49?
Answer:
Yes, this helps Calvin to complete the equation 7 × ☐ = 49.

Explanation:
Equation:
49 ÷ 7 = ☐.
49 ÷ 7 = 7.
=> 7 × 7= 49.

Do You Know How?
In 3-7, use a multiplication or a division fact to complete the equations.
You can use multiplication to help divide.
Question 3.
______ = 45 ÷ 5
45 = 5 × ______
Answer:
___9___ = 45 ÷ 5.
45 = 5 × ___9___.

Explanation:
9 = 45 ÷ 5
45 = 5 × 9.

Question 4.
_____ × 7 = 21
21 ÷ ______ = 7
Answer:
__3___ × 7 = 21.
21 ÷ __3____ = 7.

Explanation:
3 × 7 = 21.
21 ÷ 3 = 7.

Question 5.
6 × _____ = 30
30 ÷ 6 = ______
Answer:
6 × __5___ = 30.
30 ÷ 6 = ___5___.

Explanation:
6 × 5 = 30.
30 ÷ 6 = 5.

Question 6.
4 = 24 ÷ _______
24 = ______ × 4
Answer:
4 = 24 ÷ ___6____.
24 = __6____ × 4.

Explanation:
4 = 24 ÷ 6.
24 = 6 × 4.

Question 7.
6 × _____ = 12
12 ÷ 6 = _______
Answer:
6 × __2___ = 12.
12 ÷ 6 = __2_____.

Explanation:
6 × 2 = 12.
12 ÷ 6 = 2.

Independent Practice
Leveled Practice In 8-10, use fact families to complete the equations.
Question 8.
42 ÷ 7 = ______
7 × _____ = 42
Answer:
42 ÷ 7 = ___6___.
7 × __6___ = 42.

Explanation:
42 ÷ 7 = 6.
7 × 6 = 42.

Question 9.
18 = 6 × ______
_____ = 18 ÷ 6
Answer:
18 = 6 × ___3___.
__3___ = 18 ÷ 6.

Explanation:
18 = 6 × 3.
3 = 18 ÷ 6.

Question 10.
9 = _____ ÷ 8
9 × 8 = ______
Answer:
9 = __2___ ÷ 8.
9 × 8 = ___2___.

Explanation:
9 = 2 ÷ 8.
9 × 8 = 2.

In 11-19, find the product or quotient.
Question 11.
36 ÷ 4 = ______
Answer:
36 ÷ 4 = __9____.

Explanation:
36 ÷ 4 = 9. (quotient)

Question 12.
_____ = 8 × 8
Answer:
__1___ = 8 × 8.

Explanation:
1= 8 × 8. (quotient)

Question 13.
15 ÷ 3 = ________
Answer:
15 ÷ 3 = ___5_____.

Explanation:
15 ÷ 3 = 5. (quotient)

Question 14.

Answer:
= 6.

Explanation:
= 6. (quotient)

Question 15.

Answer:
= 7.

Explanation:
= 7. (quotient)

Question 16.

Answer:
= 6.

Explanation:
= 6. (quotient)

Question 17.
Multiply 8 times 5.
Answer:
8 × 5 = 40.

Explanation:
Multiply 8 times 5.
8 × 5 = 40. (product)

Question 18.
Divide 18 by 9.
Answer:
18 ÷ 9 = 2.

Explanation:
18 ÷ 9 = 2. (product)

Question 19.
Divide 27 by 3.
Answer:
27 ÷ 3 = 9.

Explanation:
Divide 27 by 3.
27 ÷ 3 = 9.(product)

Problem Solving
In 20-22, use the recipe at the right.

Question 20.
How many cups of peanuts would Eric need to make 5 batches of trail mix? Write an equation to show your thinking.
Answer:
Number of cups of peanuts would Eric need to make of 5 batches trail mix = 5 × 4 = 20.

Explanation:
Number of cups of peanuts would Eric need to make of trail mix = 4.
Number of batches to make trail mix = 5.
Number of cups of peanuts would Eric need to make of 5 batches trail mix = Number of cups of peanuts would Eric need to make of trail mix × Number of batches to make trail mix
= 5 × 4
= 20.

Question 21.
How many batches of trail mix can Eric make with 16 cups of peanuts, 15 cups of raisins, and 8 cups of walnuts?
Answer:
Number of batches of trail mix can Eric make with 16 cups of peanuts = 16 ÷ 4 = 4.
Number of batches of trail mix can Eric make with 8 cups of walnuts = 8 ÷ 2 = 4.
Number of batches of trail mix can Eric make with 15 cups of raisins = 15 ÷ 3 = 5.

Explanation:
Number of cups of peanuts would Eric need to make of trail mix = 4.
Number of cups of peanuts Eric want the trail mix = 16
Number of batches of trail mix can Eric make with 16 cups of peanuts = Number of cups of peanuts Eric want the trail mix ÷ Number of cups of peanuts would Eric need to make of trail mix
= 16 ÷ 4
= 4.
Number of  cups of raisins would Eric need to make of trail mix = 3.
Number of  cups of raisins Eric want the trail mix = 15.
Number of batches of trail mix can Eric make with 15 cups of raisins = Number of cups of raisins Eric want the trail mix ÷ Number of cups of raisins would Eric need to make of trail mix
= 15 ÷ 3
= 5.
Number of cups of walnuts would Eric need to make of trail mix = 2.
Number of cups of walnuts Eric want the trail mix = 8.
Number of batches of trail mix can Eric make with 8 cups of walnuts = Number of cups of walnuts Eric want the trail mix ÷ Number of cups of walnuts would Eric need to make of trail mix
= 8 ÷ 2
= 4.

Question 22.
Reasoning Eric spends $30 to buy the ingredients for 5 batches of trail mix. Find the cost of the ingredients Eric needs for one batch. How much would Eric need for 2 batches?
Answer:
Cost of the ingredients Eric needs for one batch = $6.
Amount of money Eric need for buying 2 batches = $12.

Explanation:
Amount of money Eric spends to buy the ingredients for 5 batches of trail mix = $30.
Number of batches of trail mix Eric buys = 5.
Cost of the ingredients Eric needs for one batch = Amount of money Eric spends to buy the ingredients for 5 batches of trail mix ÷ Number of batches of trail mix Eric buys
= $30 ÷ 5
= $6.
Amount of money Eric need for buying 2 batches = Cost of the ingredients Eric needs for one batch × 2
= $6 × 2
= $12.

Question 23.
Emilia drew lines to divide these squares into parts. What is one way to name these parts?

Answer:
One way to name these parts:
Square.
Triangle.

Explanation:
one way to name these parts:
Square.
Triangle.

Question 24.
Higher Order Thinking Wilson is thinking of 2 one-digit numbers. When he multiplies them, the product is 27. What is the sum of the two numbers? Explain your answer.
Answer:
2 one-digit numbers: 3 and 9.
Product: 3 × 9 = 27. or 9 × 3 = 27.
Sum: 3 + 9 = 12. or 9 + 3 = 12.

Explanation:
2 one-digit numbers: 3 and 9.
Product: 3 × 9 = 27.
or 9 × 3 = 27.
Sum: 3 + 9 = 12.
or 9 + 3 = 12.

Assessment Practice
Question 25.
Use the relationship between multiplication and division to find the value of each unknown.

Answer:

Explanation:
42 ÷ 7 = ?   => 42 ÷ 7 = 6.
7 × ? = 42.  => 7 × 6 = 42.
36 ÷ 4 = ?   => 36 ÷ 4 = 9.
9 × 4 = 36.   => 9 × 4 = 36.

Question 26.
Use properties of operations to find the value of each unknown.

Answer:

Explanation:
8 ÷ 1 = ?    => 8 ÷ 1 = 8.
? = 9 ÷ 9.    =>1 = 9 ÷ 9.
? = 0 ÷ 3.    => 0 = 0 ÷ 3.
6 × 0 = ?     => 6 × 0 = 0.

Lesson 4.8 Solve Multiplication and Division Equations

Solve & Share
The expression 24 ÷ 4 is on the right side of the balance below. What can you write on the left side that will have the same value as the right side? Write 5 different multiplication or division problems that will keep the pans balanced.
I can… use multiplication and division facts to find unknown values in an equation.

Look Back! Would the problem 1 × 3 × 2 × 1 keep the pans above balanced? Explain.
Answer:
Yes, the problem 1 × 3 × 2 × 1 keep the pans above balanced because 24 ÷ 4 = 6.
(1 × 3) × (2 × 1) = 3 × (2 × 1) = 3 × 2 = 6.

Explanation:
Problem 1 × 3 × 2 × 1 keep the pans above balanced.
24 ÷ 4 = 6.
(1 × 3) × (2 × 1) = 3 × (2 × 1) = 3 × 2 = 6.

Essential Question
How Do Multiplication and Division Equations Work?
Answer:
Multiplication works by multiplying the numbers to find the answers and Division works by dividing the numbers to find the answers.

Explanation:
Multiplication is the process of multiplying the numbers and Division  means the process of divinding the numbers to find the answers.

Visual Learning Bridge
The pan balance shows 35 ÷ 7 = 5.

Remember, in an equation, the symbol = means “is equal to.” It tells you the value on the left is the same as the value on the right.

These are other examples of equations.

Frank has some tubes of tennis balls. Each tube has 4 tennis balls. Frank has 8 tennis balls in all. How many tubes of tennis balls does he have?
You can write an equation to represent the problem.
8 = ? × 4

Some equations have symbols to represent unknowns. The ? represents the number of tubes of tennis balls Frank has.
8 = ? × 4
A multiplication fact that matches this is 8 = 2 × 4.

The value of ? is 2. So, ? = 2.

Convince Me! Reasoning Use the value of ? in the multiplication equation to write and solve two division equations.
7 × ? = 42
42 ÷ ? = ☐
42 ÷ ☐ = ?
Answer:
7 × ? = 42     => 7 × 6 = 42.
42 ÷ ? = ☐    => 42 ÷ 6 = 7.
42 ÷ ☐ = ?    => 42 ÷ 7 = 6.

Explanation:
7 × 6 = 42.
42 ÷ 6 = 7.
42 ÷ 7 = 6.

Guided Practice
Do You Understand?
Question 1.
Write an equation that represents the following problem: Walt makes some sandwiches. Each sandwich uses 2 slices of bread. He uses 16 slices of bread. How many sandwiches does Walt make? Use ? to represent the number of sandwiches.
Answer:
Number of sandwiches Walt makes = 8.

Explanation:
Number of slices of bread each sandwich uses = 2.
Number of slices of bread Walt uses = 16.
Number of sandwiches Walt makes = ??
=> Number of slices of bread Walt uses ÷ Number of slices of bread each sandwich uses
=> 16 ÷ 2
=> 8.

Do You Know How?
In 2-5, find the value for ? that makes the equation true.
Question 2.
9 × ? = 27
Answer:
9 × 3 = 27.

Explanation:
9 × ? = 27.
9 × 3 = 27.

Question 3.
8 = 40 ÷ ?
Answer:
8 = 40 ÷ 5.

Explanation:
8 = 40 ÷ ?
8 = 40 ÷ 5.

Question 4.
32 = ? × 8
Answer:
32 = 4 × 8.

Explanation:
32 = ? × 8.
32 = 4 × 8.

Question 5.
? ÷ 3 = 6
Answer:
18 ÷ 3 = 6.

Explanation:
? ÷ 3 = 6.
18 ÷ 3 = 6.

Independent Practice
In 6-9, find the value for ? that makes the equation true.
Question 6.
? ÷ 4 = 7
Answer:
28 ÷ 4 = 7.

Explanation:
? ÷ 4 = 7.
28 ÷ 4 = 7.

Question 7.
25 = 5 × ?
Answer:
25 = 5 × 5.

Explanation:
25 = 5 × ?
25 = 5 × 5.

Question 8.
72 = ? × 9
Answer:
72 = 8 × 9.

Explanation:
72 = ? × 9.
72 = 8 × 9.

Question 9.
4 = 20 ÷ ?
Answer:
4 = 20 ÷ 5.

Explanation:
4 = 20 ÷ ?.
4 = 20 ÷ 5.

In 10-13, write and solve an equation that represents the problem.
Question 10.
Sasha has 21 dimes. She puts them in stacks with the same number of dimes in each stack. In all, she has 3 stacks. How many dimes are in each stack? Use? to represent the number of dimes in each stack.
Answer:
Number of dimes in each stack = 7.

Explanation:
Number of dimes Sasha has = 21.
She puts them in stacks with the same number of dimes in each stack.
Number of stacks Sasha has = 3.
Number of dimes in each stack = Number of dimes Sasha has ÷ Number of stacks Sasha has
= 21 ÷ 3
= 7.

Question 11.
There were some sheep in a barnyard. Each sheep had 4 legs. There were 24 legs in the barnyard. How many sheep were in the barnyard? Use ? to represent the number of sheep in the barnyard.
Answer:
Number of sheep in the barnyard = 6.

Explanation:
Number of legs each sheep has = 4.
Total number of legs in the barnyard = 24.
Number of sheep in the barnyard = ??
=> Total number of legs in the barnyard  ÷ Number of legs each sheep has
=> 24 ÷ 4
=> 6.

Question 12.
A football team scored 48 points. The team only scored on touchdowns, worth 6 points each. How many touchdowns did the team score? Use ? to represent the number of touchdowns.
Answer:
Number of touchdowns the team scored = 8.

Explanation:
Number of points football team scored = 48.
Number of points team scored on touchdowns each = 6.
Number of touchdowns the team scored = ??
=> Number of points football team scored ÷ Number of points team scored on touchdowns each
=> 48 ÷ 6
=> 8.

Question 13.
There were 6 ladybugs on a leaf. Each ladybug had the same number of spots. There were 36 spots. How many spots were on each ladybug? Use ? to represent the number of spots on each ladybug.
Answer:
Number of spots on each ladybug = 6.

Explanation:
Number of ladybugs on a leaf = 6.
Each ladybug had the same number of spots.
Total number of spots = 36.
Number of spots on each ladybug =??
=> Total number of spots ÷ Number of ladybugs on a leaf
=> 36 ÷ 6
=> 6.

Problem Solving
Question 14.
A baker is decorating 5 cakes. He uses 9 chocolate flowers to decorate each cake. How many flowers will he need to decorate all the cakes? Write an equation to represent the problem, using ? to represent the missing information. Then solve your equation.

Answer:
Number of flowers he needs to decorate all the cakes = 45.

Explanation:
Number of cakes baker decorating = 5.
Number of chocolate flowers  he uses to decorate each cake = 9.
Number of flowers he needs to decorate all the cakes = ??
=> Number of cakes baker decorating × Number of chocolate flowers  he uses to decorate each cake
=> 5 × 9
=> 45.

Question 15.
Be Precise Hal asked 20 people to name their favorite sport. The tally chart shows how many people answered baseball and swimming. All the other people he asked said their favorite sport was football. Complete the tally chart to show how many people chose football.

Answer:
Number of people choose Football as their favorite sport = 8.

Explanation:
Total number of people Hal asked to name their favorite sport = 20.
Number of people choose Baseball as their favorite sport = 3.
Number of people choose Swimming as their favorite sport = 5 + 4 = 9.
Number of people choose Football as their favorite sport = ??
=> Total number of people Hal asked to name their favorite sport – ( Number of people choose Baseball as their favorite sport + Number of people choose Swimming as their favorite sport )
=> 20 – ( 3 + 9 )
=> 20 – 12
=> 8.

Question 16.
Higher Order Thinking A building has more than 2 stories but fewer than 10 stories. Each story of the building has the same number of windows. The building has 25 windows. Complete the sentence. Then explain how you found your answer.
The building has each story has ________stories, and each story has ______ windows.
Answer:
The building has each story has ___25_____stories, and each story has ___25___ windows.

Explanation:
A building has more than 2 stories but fewer than 10 stories.
Number of windows building has = 25.
Each story of the building has the same number of windows.
=> Number of stories each building has = 25.

Question 17.
Rosi and Karen are trying to solve 4= ? ÷ 8. Rosi says the value of the unknown is 32. Karen says the value of the unknown is 2. Is Rosi or Karen correct? Explain.
Answer:
Rosi is correct because 32 ÷ 8 = 4 not 2 ÷ 8 = 4.

Explanation:
Equation:
4 = ? ÷ 8.
=> 4 = 32 ÷ 8.

Assessment Practice
Question 18.
What is the value of the unknown in the equation 30 ÷ ? = 6?
A. 4
B. 5
C. 6
D. 7
Answer:
30 ÷ 5 = 6.
C. 6

Explanation:
Equation :
30 ÷ ? = 6
=> 30 ÷ 5 = 6.

Question 19.
What is the value of the unknown in the equation 8 × ? = 64?
A. 5
B. 6
C. 7
D. 8
Answer:
8 × 8 = 64.
D. 8

Explanation:
Equation:
8 × ? = 64
8 × 8 = 64.

Lesson 4.9 Problem Solving

Make Sense and Persevere
Solve & Share
Natalie prepared a crate of oranges for shipment. She packed the oranges in 2 layers. In each layer, she packed 6 rows with 7 oranges in each row. How many oranges did Natalie pack? Use equations to represent your work.
I can… make sense of problems and keep working if I get stuck.

Thinking Habits
Be a good thinker! These questions can help you.

• What do I need to find?
• What do I know?
• What’s my plan for solving the problem?
• What else can I try if get stuck?
• How can I check that my solution makes sense?
  

Look Back! Make Sense and Persevere How did you find how many oranges Natalie packed in each layer? How did this plan help you solve the problem?
Answer:
Total number of oranges Natalie packed = 84. This plan helped to solve the problem by step by step process and its easy to understand.

Explanation:
Natalie prepared a crate of oranges for shipment. She packed the oranges in 2 layers. In each layer, she packed 6 rows with 7 oranges in each row. How many oranges did Natalie pack?
Number of layers of oranges she packed = 2.
Number of rows of oranges she packed = 6.
Number of oranges in each row she kept = 7.
Total number of oranges Natalie packed = ??
=> Number of layers of oranges she packed ( Number of rows of oranges she packed × Number of oranges in each row she kept )
=> 2 ( 6 × 7)
=> 2 × 42
=> 84.

Essential Question
How Can You Make Sense of a Problem and Persevere in Solving It?
Answer:
According to the  Common Core State Standards, mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. You should understand the problem step by step then should try to find out the solve the problem.

Explanation:
According to the  Common Core State Standards, mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals.

Visual Learning Bridge
A store has boxes of video games for sale. In each box, the video games are in 2 rows with 3 video games in each row. Each video game costs the same amount. What is the cost of each video game?
To persevere you can check your strategy and your work.

What do I need to do?
I need to make sense of the problem before I can solve it. If I get stuck, I need to persevere until I find the cost of each video game.

How can I make sense of and solve this problem?
I can

• identify what is known from the problem.
• look for and answer hidden questions in the problem.
• make a plan to solve the problem.
• check to make sure my work and answer make sense.

Here’s my thinking…

I know a box costs $54.
There are 2 rows of 3 games in a box.
First, I need to find the total number of games in a box.
I will multiply the number of rows by the number of games in each row.
2 × 3 = 6
6 games are in a box.
Then I will divide the cost of a box by the total number of games to find the cost of each game.
$54 ÷ 6 = $
9 Each game costs $9.

Convince Me! Make Sense and Persevere How can you check to make sure the work and answer given above make sense?
Answer:
CHECK:
Cost of video game box = Total number of video games in a box × Cost of each game
= 6 × $9
= $54.

Explanation:
Number of rows of video games = 2.
Number of video games in each row = 3.
Cost of video game box= $54.
Total number of video games in a box = Number of rows of video games × Number of video games in each row
= 2 × 3
= 6.
Cost of each game = Cost of video game box ÷ Total number of video games in a box
= $54 ÷ 6
= $9.

Guided Practice
Make Sense and Persevere Twelve friends went camping. All except 4 of them went on a hike. The hikers carried 32 water bottles. Each hiker carried the same number of water bottles. How many water bottles did each hiker carry?

Question 1.
Tell what you know. Then explain what you can persevere. Think: need to find first to solve the problem. Does the strategy I am sing make sense?
Answer:
Number of water bottles each hiker carried = 4. Yes, the strategy makes sense which you are singing.

Explanation:
Number of friends went camping = 12.
All except 4 of them went on a hike.
=> Number of friends went on a hike = 12 – 4 = 8.
Number of water bottles the hikers carried = 32.
Number of water bottles each hiker carried = Number of water bottles the hikers carried  ÷ Number of friends went on a hike
= 32 ÷ 8
= 4.

Question 2.
Tell which operations you will use. Then solve the problem.
Answer:
Multiplication and Division are the operations used and solved the problem.

Explanation:
The operations used re multiplication and division to solve the problem.

Independent Practice
Make Sense and Persevere Four students went bowling. They bowled 2 games each. The cost was $5 per game. How much money did the students spend on bowling? Explain.
Question 3.
Tell what you know. Then explain what you need to find first to solve the problem.
Answer:
Amount of money the students spend on bowling = $40.