Category: Envision Math

Help kindergarten students to become proficient in maths by downloading enVision Math Common Core Kindergarten Answer Key. This answer key contains the answers for all the questions and all the chapters. It helps to enhance your math skills and score good marks in the exam. Practising envision math answers grade K develops problem-solving ability among elementary school students right from a young age.

Get volume-wise and topic-wise enVision Math Answer Key Common Core Kindergarten Volume 1 & Volume 2 Pdf from the quick links available and complete your assignments with ease.

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

Preschool students can get enVision Math Common Core K Grade Solution Key in the following sections. It has the answers with detailed explanations for topics such as numbers 0 to 5, compare numbers 0 to 5, numbers 6 to 10, compare numbers 0 to 10, classify and count data, understand addition, understand subtraction, more addition and subtraction.

Teachers and students can access envision math common core grade K volume 1 answer key for all the chapters via quick links without paying any amount. You can find them during your test preparation. All you need to do is click on the respective chapter link and start preparation accordingly. Practice with the help of enVision Common Core Math Grade K 1st Volume Answers regularly and improve your accuracy in solving questions.

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

• Topic 1 Numbers 0 to 5
• Topic 2 Compare Numbers 0 to 5
• Topic 3 Numbers 6 to 10
• Topic 4 Compare Numbers 0 to 10
• Topic 5 Classify and Count Data
• Topic 6 Understand Addition
• Topic 7 Understand Subtraction
• Topic 8 More Addition and Subtraction

enVision Math Common Core Grade K Volume 2 Answer Key | enVision Math Common Core Grade K Volume 2 Answers

• Topic 9 Count Numbers to 20
• Topic 10 Compose and Decompose Numbers 11 to 19
• Topic 11 Count Numbers to 100
• Topic 12 Identify and Describe Shapes
• Topic 13 Analyze, Compare, and Create Shapes
• Topic 14 Describe and Compare Measurable Attributes

Benefits of referring to Grade K enVision Math Common Core Answer Key

We have several advantages of solving enVision Math Common Core Grade K Volume 1 Answer Key. Some of them are listed here.

• These preschool enVision Math Common Core Grade K Solutions are given as per the enVision Common Core Textbooks.
• Elaborate answers provided by subject experts that are useful to gain deeper level knowledge on the Grade K concepts.
• You can assess preparation standards using enVision Math Common Core Kindergarten Book Questions.
• Improve your proficiency by practising from the enVision Common Core Math Grade K Solution Key PDF on a regular basis.
• It acts as a one-stop destination to resolve all your queries regarding the Common Core enVision Math Book concepts.

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1. How to download enVision Math Common Core Grade K Answer Key PDF?

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3. What are the benefits of enVision Math Common Core Grade K Volume 2 Solution Key?

It helps you to score the highest marks in the exam also improve your knowledge by referring to enVision Math Common Core Grade K Volume 1 Answer Key from envisionmathanswerkey.com

Go through the enVision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Value regularly and improve your accuracy in solving questions.

enVision Math Common Core 5th Grade Answers Key Topic 1 Understand Place Value

Envision Math Common Core 5th Grade Answers Key Topic 1 Understand Place Value has all the topics covered , which are related to the basics of Math by giving live examples of our daily life. This chapter is loaded with Decimals, Fractions, and how to do rounding to the nearest numbers, which are quite helpful for students to deal with Math basics. Experience the most satisfying and understandable answers and easy methods with Envision Math Common Core 5th Grade Answers Key Topic 1 Understand Place Value.

Envision Math Common Core 5th Grade Answers Key Topic 1 Understand Place Value

Are you facing difficulties to understand the simple Math problems or numbers , that we have to use in day to day life. Then you have come to the right place to grasp the simple tricks of small calculations. Envision Math Common Core 5th Grade Answers Key Topic 1 Understand Place Value. will help you with rounding the numbers , understand the place values , converting decimal into fractions .These are the major topics covered, get on the track with your kids to know them a better way of learning.

Envision STEM Project: Pollinating Insects

• Envision STEM Project Pollinating Insects

Lesson 1 Patterns with exponents and Powers of 10

• Lesson 1.1 Patterns with exponents and Powers of 10

Lesson 2 Understand Whole Number Place Value

• Lesson 1.2 Understand Whole Number Place Value

Lesson 3 Decimals to Thousandths

• Lesson 1.3 Decimals to Thousandths

Lesson 4 Understand Decimal Place Value

• Lesson 1.4 Understand Decimal Place Value

Lesson 5 Compare Decimals

• Lesson 1.5 Compare Decimals

Lesson 6 Round Decimals

• Lesson 1.6 Round Decimals

Lesson 7 Look For and Use Structure

• Lesson 1.7 Look For and Use Structure

 Performance Task

• Topic 1 Fluency Review
• Topic 1 Assessment Practice
• Topic 1 Performance Task

Essential Question: How are whole numbers and decimals written, compared, and ordered?

Envision STEM Project Pollinating Insects
Do Research Use the Internet or other sources to find out more about pollinating insects in the United States. What types of insects are they? How many are there of each type? How many crops and flowering plants depend on pollinating insects in order to produce the foods we eat?
Journal: Write a Report Include what you found. Also in your report:

• Choose two of the pollinating insects. Estimate how many crop plants each type of insect pollinates.
• Estimate how many of your favorite foods and beverages come from pollinated plants.
• Make up and solve ways to compare and order your data.

Answer: Pollinated insects are nothing but, the insects which are helpful to carry the pollinated grains along with them with the help of their legs or wings from the flower to promote the vast growth of new plants and are commonly known as insect pollinators.

Report for the project :

•  Insect pollinators include bees, flies, butterflies, beetles, wasps, moths, midges and ants, among others. Of these, bees are the most important group, with both wild and managed species acting as pollinators.
• Pollinators are essential for continued plant growth in the wild. There are seven insect pollinators other than bees and butterflies that also help spread plant seeds and enable plant growth.
• Three-fourths of the world’s flowering plants and about 35 percent of the world’s food crops depend on animal pollinators to reproduce. More than 3,500 species of native bees help increase crop yields.
• More than 75 percent of the world’s food crops depend on pollination . When a seed forms in flowering plants, a fruit is able to grow to protect the seed.

Among these two major pollinators are honey bees and butterflies,
Honey bees are the commonly known pollinators which are helpful for the crops like Okra, kiwifruit, Onion, cashew, strawberry, Broccoli, Cauliflower , Cabbage etc.
Butterflies are mostly the pollinators of vegetables and herbs and are helpful for the crops like especially those in the carrot family (dill, fennel, celery, cilantro, parsnip), sunflower family (artichokes, lettuce, chicory, chamomile), legume family (peas, beans), mint family (lavender, basil).

Review What You Know

Vocabulary

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

• digits
• place value
• period
• whole numbers

Question 1.
____ are the symbols used to show numbers.
Answer: Digits are the symbols used to show numbers.

Explanation:
Because, A digit is a single symbol used to make numerals or numbers ,
That is 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are the ten digits we use in everyday numerals.
For example, the number 78 is a two digit number which is made up of two digits that is ‘7’ and ‘8’.

Question 2.
A group of 3 digits in a number is a ____
Answer: A group of 3 digits in a number is a Period.

Explanation:
Every number consists of digits. The place value is the position of each digit in a number.
Digits are separated into groups of three by commas. So, a period is a group of three digits,
For example, The number 13,456 is a 5 digit number and the ending 3 digits are called a period, because it is separated by the commas in the number which is referred to a period.

Question 3.
____ is the position of a digit in a number that is used to determine the value of the digit.
Answer: Place value is the position of a digit in a number that is used to determine the value of the digit.

Explanation:
The value of a digit depends on its place, or position, in the number.
Place value is the value of a digit according to its position in the number such as ones, tens, hundreds, and so on.
For example, In the number 3,548
3 is in thousands place and its place value is 3,000,
5 is in hundreds place and its place value is 500,
4 is in tens place and its place value is 40,
8 is in ones place and its place value is 8.

Comparing

Compare. Use <,>, or = for each .

Question 4.
869 912
Answer: 869 < 912.

Explanation:
Because 869 is less than 912.
For the number 869
8 is in hundreds place and its place value is 800,
6 is in tens place and its place value is 60,
9 is in ones place and its place value is 9.

And For the number 912
9 is in hundreds place and its place value is 900,
1 is in tens place and its place value is 10,
2 is in ones place and its place value is 2.

So, 869 is less than 912.

Question 5.
9,033 9,133
Answer: 9,033 < 9,133

Explanation:
Because 9,033 is less than 9,133
For the number 9,033
9 is in thousands place and its place value is 9,000,
0 is in hundreds place and its place value is 000,
3 is in tens place and its place value is 30,
3 is in ones place and its place value is 3.

And For the number 9,133
9 is in thousands place and its place value is 9,000,
1 is in hundreds place and its place value is 100,
3 is in tens place and its place value is 30,
3 is in ones place and its place value is 3.

So, 9,033 is less than 9,133

Question 6.
1,338 1,388
Answer:  1,338 < 1,388

Explanation:
Because 1,338 is less than 1,388
For the number 1,338
1 is in thousands place and its place value is 1,000,
3 is in hundreds place and its place value is 300,
3 is in tens place and its place value is 30,
8 is in ones place and its place value is 8.

And For the number 1,388
1 is in thousands place and its place value is 1,000,
3 is in hundreds place and its place value is 300,
8 is in tens place and its place value is 80,
8 is in ones place and its place value is 8.

So, 1,338 is less than 1,388

Question 7.
417,986 417,986
Answer: 417,986 = 417,986

Explanation:
Because, Numbers given are the same digits having the same place value .

So, 417,986 Equal to 417,986

Question 8.
0.25 0.3
Answer:  0.25 < 0.3

Explanation:
Because, regarding with its place value the number  0.25 is less than 0.3

Question 9.
0.5 0.50
Answer:  0.5 = 0.50

Explanation:
Because the 0 after the number next to decimal is exactly the same number with or without 0
Both the given numbers are in their same places with respect to each other
So, 0.5 is equal to 0.50.

Question 10.
Kamal has 7,325 songs on his computer. Benito has 7,321 songs on his computer. Who has more songs?
Answer:  Kamal has more songs than Benito.

Explanation:
Given , Kamal has 7,325 songs on his computer.
Benito has 7,321 songs on his computer.
Then 7,325 > 7,321 or 7,325 is greater than 7,321

So, Kamal has more songs than Benito.

Adding Whole Numbers

Find each sum.

Question 11.
10,000 + 2,000 + 60 + 1
Answer: 12,061

Explanation:

Adding all the given numbers by placing them in order of their place value of the digit
So, the total sum is 12,061.

Question 12.
20,000 + 5,000 + 400 + 3
Answer: 25,403

Explanation:

Adding all the given numbers by placing them in order of their place value of the digit
So, the total sum is 25,403.

Question 13.
900,000 + 8,000 + 200 + 70 + 6
Answer: 9,08,276

Explanation:

Adding all the given numbers by placing them in order of their place value of the digit
So, the total sum is 9,08,276.

Question 14.
7,000,000 + 50,000 + 900 + 4
Answer: 70,50,904

Explanation:

Adding all the given numbers by placing them in order of their place value of the digit
So, the total sum is 70,50,904.

Place Value

Question 15.
The largest playing card structure was made of 218,792 cards. What is the value of the digit 8 in 218,792?
A. 80
B. 800
C. 8,000
D. 80,000
Answer: C

Explanation:
Given, The largest playing card structure was made of 218,792 cards.
According to the place value method, the value of the digit 8 in 218,792 is 8,000

Question 16.
Construct Arguments In the number 767, does the first 7 have the same value as the final 7? Why or why not?
Answer: In the number  767, The first 7 does not have the same value as the final 7.

Explanation:
Given , number is 767 , The first 7 does not have the same value as the final 7.
Because, the place value is counted from right to left by ones, tens, hundreds, thousand and so on.
So, in this number 767, the first  number 7 holds the hundreds place and the final number 7 holds the ones place, giving it a different value for both the numbers in their place respectively.

Pick a Project

PROJECT 1A
Manatees or sea cows?
Project: Create a Manatee Poster

Answer:

PROJECT 1B
What makes a game fun?
Project: Design a Game with Place-Value Blocks

Answer:

PROJECT 1С
How far are we from the sun?
Project: Research Measurements in Our Solar System

Answer: The Sun is at an average distance of about 93,000,000 miles (150 million kilometers) away from Earth. It is so far away that light from the Sun, traveling at a speed of 186,000 miles (300,000 kilometers) per second, takes about 8 minutes to reach us.

Lesson 1.1 Patterns with exponents and Powers of 10

Solve & share
A store sells AA batteries in packages of 10 batteries. They also sell boxes of 10 packages, cases of 10 boxes, and cartons of 10 cases. How many AA batteries are in one case? One carton? 10 cartons? Solve these problems any way you choose.

You can use appropriate tools, such as place-value blocks, to help solve the problems. However you choose to solve it, show your work!

Look Back! How many 10s are in 100? How many 10s are in 1,000? Write equations to show your work.

Answer: There are 1000 AA batteries in one case , 10,000 batteries in one carton and 1,00,000 batteries in 10 cartons.

Explanation:
Given, A store sells AA batteries in packages of 10 batteries, boxes of 10 packages, cases of 10 boxes, and cartons of 10 cases.
Then we have 10 batteries for one package and given that 1 box contains 10 packages ,
10 × 10 = 100 , that is 100 batteries for each box.
Next for 10 boxes we have a case which means 100 × 10 = 1000 batteries ,
That is each case contains 1000 batteries.
Now for 10 cases we have a carton which means 1000 × 10 = 10,000 batteries ,
That is each carton contains 10,000 batteries.
And now they asked for 10 cartons ,
So, for 10 cartons we have 10,000 × 10 = 1,00,000 batteries .

Visual Learning Bridge

Essential Question
How Can You Explain Patterns in the Number of Zeros in a Product?
Answer: An exponent identifies a quantity representing the power to which a given number or expression is to be raised.

Explanation:
For example , 4 × 60
= 4 x (6 x 10)
= (4 x 6) x 10
= 24 x 10
=240
or
50 x 700
= (5 x 10) + (7 x 100)
= (5 x 7) x (10 x 100)
= 35 x 1,000
= 35,000.

A.
Tamara’s new horse weighs about 1,000 pounds. How can you show 1,000 as a power of 10 using an exponent?

The exponent is the number that tells how many times a base number is used as a factor.

Answer: 10³

Explanation:
Given, horse weighs about 1,000 pounds. then show 1,000 as a power of 10 using an exponent
So we can write 1000 as 10 × 10 × 10 = 10³.

B.
Write 1,000 as a product using 10 as a factor.

The exponent, 3, shows that the base number, 10, is multiplied 3 times.
So, 1,000 is written as 10³ using exponents.

C.
Tamara estimates that her horse will eat about 5,000 pounds of hay each year. How can you write 5,000 using exponents? 5 × 10¹ = 5 × 10 = 50
5 × 10² = 5 × 10 × 10 = 500
5 × 10³ = 5 × 10 × 10 × 10 = 5,000
The number of zeros in the product is the same as the exponent.
So, 5,000 is written as 5 × 10³ using exponents.

Convince Me! Look for Relationships What pattern do you notice in the number of zeros in the products in box C above?

Guided Practice

Do You Understand?

Question 1.
Why are there three zeros in the product of 6 × 10³?
Answer: The product of 6 × 10³ is 6,000 so it contains three zeros.

Explanation:
The product of 6 × 10³ is 6,000,
Because, 6 × 10 × 10 × 10 = 6,000.
So it contains three zeros.

Question 2.
Susan said that 10⁵ is 50. What mistake did Susan make? What is the correct answer?
Answer: The correct answer is 1,00,000.

Explanation:
10 × 10 × 10 × 10 × 10 = 1,00,000.
The mistake Susan made is he multiplied the exponent with the base.

Do You Know How?

In 3 and 4, complete the pattern.

Question 3.
10¹ =
10² =
10³ =
10⁴ =
Answer:
10¹ = 10
10² = 100
10³ = 1000
10⁴ =  10,000

Explanation:
10  = 10
10 × 10  =  100
10 × 10 × 10  =  1000
10 × 10 × 10 × 10  = 10,000.

Question 4.
= 7 × 10¹
= 7 × 10²
= 7 × 10³
= 7 × 10⁴
Answer:
70 = 7 × 10¹
700 = 7 × 10²
7,000 = 7 × 10³
70,000  = 7 × 10⁴

Explanation:
Given, 7 ×10  = 70
7 × 10 × 10  =  700
7 ×10 × 10 × 10  =  7,000
7 × 10 × 10 × 10 × 10  = 70,000.

Independent Practice

In 5-15, find each product. Use patterns to help.

Question 5.
3 × 10¹ =
3 × 10² =
3 × 10³ =
3 × 10⁴ =
Answer:
3 × 10¹ = 30
3 × 10² = 300
3 × 10³ = 3,000
3 × 10⁴ = 30,000

Explanation:
3 ×10  = 30
3 × 10 × 10  =  300
3 ×10 × 10 × 10  =  3,000
3 × 10 × 10 × 10 × 10  = 30,000.

Question 6.
2 × 10 =
2 × 100 =
2 × 1,000 =
2 × 10,000 =
Answer:
2 × 10 =  20
2 × 100 = 200
2 × 1,000 = 2,000
2 × 10,000 = 20,000

Explanation:
2 × 10 =  2 × 10¹ = 20
2 × 100 = 2 ×  10² =  2 × 10 × 10 = 200
2 × 1,000 = 2 × 10³  =  2 × 10 × 10 × 10 = 2,000
2 × 10,000 = 2 × 10⁴  = 2 × 10 × 10 × 10  × 10 = 20,000

Question 7.
= 9 × 10¹
= 9 × 10²
= 9 × 10³
= 9 × 10⁴
Answer:

90 = 9 × 10¹
900 = 9 × 10²
9,000 = 9 × 10³
90,000  = 9 × 10⁴

Explanation:
Given, 7 ×10  = 70
7 × 10 × 10  =  700
7 ×10 × 10 × 10  =  7,000
7 × 10 × 10 × 10 × 10  = 70,000.

Question 8.
8 × 10⁴
Answer:  8 × 10⁴ = 80,000

Explanation:
8 × 10⁴ = 8 × 10 × 10 × 10 × 10  = 80,000

Question 9.
4 × 1,000
Answer:  4 × 1,000 = 4,000

Explanation:
4 × 10³  = 4 ×10 × 10 × 10  =  4,000

Question 10.
5 × 10²
Answer:  5 × 10² = 500

Explanation:
5 × 10² =  5 × 10 × 10  = 500

Question 11.
6 × 10,000
Answer:  6 × 10,000 = 60,000

Explanation:
6 × 10,000 =  6 × 10 × 10 × 10 × 10  = 6 × 10⁴ = 60,000

Question 12.
4 × 10¹
Answer:  4 × 10¹  =  40

Explanation:
4 × 10¹  =  40

Question 13.
100 × 9
Answer: 100 × 9 = 900

Explanation:
100 × 9  = 10 × 10  × 9 = 900

Question 14.
10³ × 6
Answer:  10³ × 6 = 6,000

Explanation:
10³ × 6 = 10 × 10 × 10 × 6  = 6,000

Question 15.
8 × 10⁵
Answer: 8 × 10⁵  = 8,00,000

Explanation:
8 × 10⁵  = 8 × 10 × 10 × 10 × 10 × 10 = 8,00,000

Question 16.
Write 10 × 10 × 10 × 10 × 10 × 10 with an exponent. Explain how you decided what exponent to write.
Answer: 10⁶

Explanation:
As Given there are 6 tens together
So, 10 × 10 × 10 × 10 × 10 × 10  = 10⁶

Problem Solving

Question 17.
One box of printer paper has 3 × 10² sheets of paper. Another box has 103 sheets of paper. What is the total number of sheets in both boxes?
Answer: The total number of sheets in both boxes is 403

Explanation:
Given, One box of printer paper has 3 × 10² sheets of paper. then , 3 × 10² = 300
Another box has 103 sheets of paper.
So, 300 + 103  = 403
The total number of sheets in both boxes is 403

Question 18.
A post is put every 6 feet along a fence around a rectangular field that is 42 ft long and 36 ft wide. How many posts are needed?
Answer: 26 posts are needed.

Explanation:
42 feet for one side, 42 feet for the parallel side
36 feet for one side, 36 feet for the parallel side
42 ÷ 6 = 7 ,One of these posts is a corner, so make it 6 because we do not want to count a corner post more than once.
36 ÷ 6 = 6 ,One of these posts is a corner, so make it 5 because we do not want to count a corner post more than once
We now add these: 2 x 6 + 2 x 5 = 12 + 10 = 22
Add in the four corner posts: 22 + 4 = 26 posts.
So , 26 posts are needed.

Question 19.
Number Sense A company had 9 × 10⁶ dollars in sales last year. Explain how to find the product 9 × 10⁶.
Answer:  9 × 10⁶   = 9,000,000.

Explanation:
9 × 10⁶  =  9 ×10 × 10 × 10 × 10 × 10 × 10  = 9,000,000.

Question 20.
An aquarium has the same shape as the solid figure shown below. What is the name of this solid figure?

Answer: A Rectangular prism.

Explanation:
a solid figure that has six sides, called faces, that are rectangles.
So, they are known as Rectangular prism.

Question 21.
Model with Math Isaac takes 5 minutes to ride his bike down the hill to school and 10 minutes to ride up the hill from school. He attends school Monday through Friday. How many minutes does he spend biking to and from school in two weeks? Write an equation to model your work.
Answer:  It takes 150 minutes

Explanation:
Given, Issac take 5 minutes to ride his bike down the hill to school
And 10 minutes to ride up the hill from school .
So, it takes 15 min each day to and from school He attends school Monday through Friday,
So, 5 days in a week for 5 days it takes  15 × 5 = 75 minutes
For 2 weeks , 75 × 2 = 150 minutes.

Question 22.
Higher Order Thinking Santiago hopes to buy a 4-horse trailer for about $12,000. Describe all the numbers that when rounded to the nearest hundred are 12,000.
Answer: Any number from 11,950 to 12,049, will result to 12,000 when rounded to the nearest hundred. 

Explanation:
Since we have given that,
Number of horse trailer = 4 Cost of 4 horse trailer = $12000 ,
As we know about the “Estimation”, if there is a number greater or equal to 5 in tens place then it will be rounded off to the nearest next greatest integer.
So, if this number is $11950 to 12049.
So, the possible numbers that when rounded to the nearest hundred are 12000 are from 11500 to 12049.

Assessment Practice

Question 23.
Choose all the equations that are true.
10 × 10 × 10 × 10 = 40
10 × 10 × 10 × 10 = 10⁴
10 × 10 × 10 × 10 = 1,000,
10 × 10 × 10 × 10 = 10,000
10 × 10 × 10 × 10 = 4 × 10⁴
Answer:  10 × 10 × 10 × 10 = 10⁴  and  10 × 10 × 10 × 10 = 10,000

Explanation:
There are 4 tens in 10 × 10 × 10 × 10 = 10⁴
So, 10 × 10 × 10 × 10 = 10,000.

Question 24.
Choose all the equations that are true.
6 × 10⁵ = 6 × 100,000
6 × 10⁵ = 6 × 10,000
6 × 10⁵ = 600,000
6 × 10⁵ = 60,000
6 × 10⁵ = 650,000
Answer: 6 × 10⁵ = 6 × 100,000 and  6 × 10⁵ = 600,000

Explanation:
6 × 10⁵ = 6 × 10 × 10 × 10 × 10  × 10 = 6,00,000. or
6 × 10⁵ = 6 × 10 × 10 × 10 × 10  × 10 = 6 × 100,000.

Lesson 1.2 Understand Whole Number Place Value

Activity

Solve & Share

The population of a city is 1,880,000. What is the value of each of the two 8s in this number? How are the two values related? Use tools like this place-value chart to help solve the problem.

Answer: These are the values of each number

Explanation:
Add up all these values and we will have 1,880,000.

Use Structure You can use place value to analyze the relationship between the digits of a number. Show your work!

Look Back! Is the relationship between the value of the two 85 in 1,088,000 the same as the relationship between the value of the two 8s in the problem above? Explain.

Visual Learning Bridge

Essential Question
How Are Place-Value Positions Related?

A.
According to the 2010 U.S. Census, the population of Phoeni×, Arizona is about 1,440,000. What is the relationship between the value of the two 4s in this number?

Writing the number in expanded form can help.

C.
Look at the expanded form of 1,440,000. The value of the 4 in the hundred thousands place is 400,000. The value of the 4 in the ten thousands place is 40,000.
400,000 is 10 times as great as 40,000.
40,000 is \(\frac{1}{10}\) of 400,000.

Sometimes word form is used instead of number name.

Standard form
1,440,000
Expanded form:
1 × 1,000,000 + 4 × 100,000 + 4 × 10,000
Using exponents, this can be written as:
(1 × 10⁶) + (4 × 10⁵) + (4 × 10⁴)
Number name:
one million, four hundred forty thousand

Convince Me! Construct Arguments is the value of the 1 in 1,440,000 10 times as great as the value of the 4 in the hundred thousands place? Explain.

Another example
When two digits next to each other in a number are the same, the digit on the left has 10 times the value of the digit to its right.
When two digits next to each other are the same, the digit on the right has to the value of the digit to its left.

Guided Practice

Do You Understand?

Question 1.
In 9,290, is the value of the first 9 ten times as great as the value of the second 9? Explain.
Answer:  No

Explanation:
The first 9 is on the place of the thousands, while the second 9 is on the place of the tens. so the first 9 is a hundred times as great as the second 9

Do You Know How?

Question 2.
Write 4,050 in expanded form.
Answer:  4,000 + 50

Explanation:
4,000 + 50  = 4,050.

In 3 and 4, write the values of the given digits.

Question 3.
the 7s in 7,700
Answer:  7,000 + 700

Explanation:
The first 7 is on the place of the thousands, while the second 7 is on the place of the hundreds.

Question 4.
the 2s in 522
Answer:  500 + 22

Explanation:
The first 2 is on the place of the tens, while the second 2 is on the place of the ones.

Independent Practice

In 5-7, write each number in standard form.

Question 5.
8,000,000 + 300 + 9
Answer: 8,000,309.

Explanation:
8,000,000 + 300 + 9 = 8,000,309.

Question 6.
(4 × 10⁴) + (6 × 10²)
Answer: 40,600

Explanation:
4 × 10⁴ =4 × 10 × 10 × 10 × 10  =  40,000
6 × 10²  = 6 × 10 × 10 =  600
So, 40,000 + 600 = 40,600

Question 7.
10,000 + 20 + 3
Answer:   10023

Explanation:
10,000 + 20 + 3 = 10023.

In 8-10, write each number in expanded form.

Question 8.
5,360
Answer: (5 × 10³) + (3 × 10²) + (6 × 10)

Explanation:
5,360 =  5000 + 300 + 60  = (5 × 10³) + (3 × 10²) + (6 × 10)

Question 9.
102,200
Answer:  10⁵ + (2 × 10³) + ( 2 × 10²)

Explanation:
102,200 = 1,00,000 + 2000 + 200 =  10⁵ + (2 × 10³) + ( 2 × 10²)

Question 10.
85,000,011
Answer:  (85 ×  10⁶) + 10 + 1

Explanation:
85,000,011 = 85,000,000 + 10 + 1 = (85 ×  10⁶) + 10 + 1

In 11-13, write the values of the given digits.

Question 11.
the 7s in 6,778
Answer: 6,000 + 700 + 70 + 8

Explanation:
The first 7 is on the place of the hundreds, while the second 7 is on the place of the tens.

Question 12.
the 9s in 990,250
Answer: 9,00,000 + 90,000 + 200 + 50

Explanation:
The first 9 is on the place of the hundred thousands, while the second 9 is on the place of the thousands.

Question 13.
the 1s in 2,011,168
Answer: 2,000,000 + 11,000 + 100 + 60 + 8

Explanation:
The first 1 is on the place of the ten thousands, while the second 1 is on the place of the thousands and the third 1 is on the place of hundreds.

Problem Solving

Question 14.
Write the number name and expanded form for the number of driver ants that could be in two colonies.

Up to 22,000,000 driver ants can live in a single colony.

Answer:  For 2 colonies  44,000,000
The number name is Forty four million
The expanded form is (4 ×10⁶) + (4 ×10⁶)

Explanation:
Given , Up to 22,000,000 driver ants can live in a single colony.
For 2 colonies 22,000,000 + 22,000,000 = 44,000,000
The expanded form is 40,000,000 + 4,000,000 = (4 ×10⁶) + (4 ×10⁶)

Question 15.
enVision® STEM A queen ant can produce about nine million ants in her lifetime. Write this number in standard form.
Answer: The number in standard form is 9,000,000

Explanation:
Given, A queen ant can produce about nine million ants in her lifetime.
The number in standard form is 9,000,000.

Question 16.
Critique Reasoning Paul says that in the number 6,367, one 6 is 10 times as great as the other 6. Is he correct? Explain why or why not.
Answer: No

Explanation:
Given in the number 6,367, one 6 is 10 times as great as the other 6.
The first 6 is on the place of the thousands, while the second 6 is on the place of the tens. so the first 6 is a hundred times as great as the second 6.

Question 17.
Jorge drew a square that had a side length of 8 inches. What is the perimeter of Jorge’s square?
Remember, the perimeter of a shape is the distance around it.

Answer: The perimeter of Jorge’s square is 32 inches.

Explanation:
Given, Jorge drew a square that had a side length of 8 inches.
Square has all equal sides, so perimeter of the square P = 4a = 4 × 8 = 32 inches

Question 18.
Higher Order Thinking
Dan wrote (2 × 10⁶) + (3 × 10⁴) + (5 × 10³) + 4 for the expanded form of two million, three hundred fifty thousand, four. What error did he make in the expanded form? What is the standard form of the number?

Answer: The standard form of the number is 20,35,004.
The number name is two million, thirty five thousand and four.

Explanation:
(2 × 10⁶) + (3 × 10⁴) + (5 × 10³) + 4
= 2,000,000 + 30,000 + 5,000 + 4
= 20,35,004.
The standard form of the number is 20,35,004.
The number name is two million, thirty five thousand and four.
He made mistake with the three hundred fifty thousand, instead of  thirty five thousand

Assessment Practice

Question 19.
Colleen says she is thinking of a 4-digit number in which all the digits are the same. The value of the digit in the hundreds place is 200.
Part A
What is the number? Explain.

Part B
Describe the relationship between the values of the digits in the number.
Answer:  The number is 2,222.
The first 2 is on the place of the thousands, while the second 2 is on the place of the hundreds, The third 2 is on the place of the tens, and the last 2 is on the place of the ones.

Explanation:
Given,  Colleen says she is thinking of a 4-digit number in which all the digits are the same. The value of the digit in the hundreds place is 200.
The number is 2,222.
The first 2 is on the place of the thousands, while the second 2 is on the place of the hundreds, The third 2 is on the place of the tens, and the last 2 is on the place of the ones.

Lesson 1.3 Decimals to Thousandths

Solve & Share
At Suzie’s Sticker City, customers can buy a book of stickers, a page, a strip, or a single sticker. Provide the missing fractions in the boxes below.

How can you use what you know about powers of 10 to help you fill in the boxes?

Look Back! Use Structure Describe any patterns you notice in the fractions.

Visual Learning Bridge

Essential Question
How Can You Read and Write Question Decimals to the Thousandths?

A.
A box is filled with 1,000 cubes. Amy picks out 4 cubes. How can you represent 4 out of 1,000 cubes as a decimal?

You can write 4 out of 1,00 as the fraction \(\frac{4}{1,000}\).

B.
The number name for \(\frac{4}{1,000}\) is four thousandths. A decimal place-value chart can help you determine the decimal. Notice that the thousandths place is three places to the right of the decimal point.

So, \(\frac{4}{1,000}\) can be represented by the decimal 0.004.

C.
How can \(\frac{444}{1,000}\) be represented by a decimal? \(\frac{444}{1,000}\) is read as four hundred forty-four thousandths and represented by the decimal 0.444.

The value of the digit 4 in the hundredths place has 10 times the value of the digit 4 in the thousandths place and \(\frac{1}{10}\) the value of the digit 4 in the tenths place.

Convince Me! Reasoning How is 0.004 the same as and different from 0.444?

Answer: 0.004 and 0.444 both have 4 in the thousandth place

Explanation:
0.004 and 0.444 both have 4 in the thousandth place,
both are in thousandth place
different: 0.444 is greater than 0.004 when divide by 1000
444/1000 is greater than 4/1000
if convert to percentage : 0.444 is 44.4% and 0.004 is 0.4%

Guided Practice

Do You Understand?

Question 1.
If four cubes are pulled from the box on the previous page, how would you write the fraction representing the cubes that are left? the decimal representing the cubes that are left?
Answer:  In decimal form we can write it as 0.996.

Explanation:
Given, A box is filled with 1,000 cubes, picking out 4 cubes,
1000 – 4 = 996,
Then \(\frac{996}{1000}\) are the remaining of the cubes which are left in the box
In decimal form we can write it as 0.996.

Do You Know How?

Question 2.
0.3 is 10 times as great as what decimal? 0.003 is \(\frac{1}{10}\) of what decimal?
Answer: 0.3 is 10 times as great as 0.03 and 0.003 is \(\frac{1}{10}\) of 0.03

Explanation:
0.3 × \(\frac{1}{10}\) = 0.03
0.03 = 0.003 × \(\frac{1}{10}\)
So, 0.3 is 10 times as great as 0.03 and 0.003 is \(\frac{1}{10}\) of 0.03

In 3-6, write each decimal as a fraction.

Question 3.
0.001 =
Answer: \(\frac{1}{1000}\)

Explanation:
Given 0.001 ,
In fraction form we can write it in to \(\frac{1}{1000}\)

Question 4.
0.05 =
Answer: \(\frac{5}{100}\)

Explanation:
Given 0.05 ,
In fraction form we can write it in to \(\frac{5}{100}\)

Question 5.
0.512 =
Answer: \(\frac{512}{1000}\)

Explanation:
Given 0.512 ,
In fraction form we can write it in to \(\frac{512}{1000}\)

Question 6.
0.309 =
Answer: \(\frac{309}{1000}\)

Explanation:
Given 0.309 ,
In fraction form we can write it in to \(\frac{309}{1000}\)

In 7-10, write each fraction as a decimal.

Question 7.
\(\frac{2}{1,000}\) =
Answer: 0.002

Explanation:
Given , \(\frac{2}{1,000}\)
In decimal form we can  write it into 0.002

Question 8.
\(\frac{34}{100}\) =
Answer: 0.34

Explanation:
Given , \(\frac{34}{100}\)
In decimal form we can  write it into 0.34

Question 9.
\(\frac{508}{1,000}\) =
Answer: 0.508

Explanation:
Given , \(\frac{508}{1,000}\)
In decimal form we can  write it into 0.508

Question 10.
\(\frac{99}{1,000}\) = =
Answer:

Explanation:
Given , \(\frac{99}{1,000}\)
In decimal form we can  write it into 0.099

Independent Practice

In 11-18, write each decimal as a fraction.

Question 11.
0.007
Answer: \(\frac{7}{1000}\)

Explanation:
Given 0.007 ,
In fraction form we can write it in to \(\frac{7}{1000}\)

Question 12.
0.08
Answer: \(\frac{8}{100}\)

Explanation:
Given 0.08 ,
In fraction form we can write it in to \(\frac{8}{100}\)

Question 13.
0.065
Answer: \(\frac{65}{1000}\)

Explanation:
Given 0.08 ,
In fraction form we can write it in to \(\frac{65}{1000}\)

Question 14.
0.9
Answer: \(\frac{9}{10}\)

Explanation:
Given 0.9 ,
In fraction form we can write it in to \(\frac{9}{10}\)

Question 15.
0.832
Answer: \(\frac{832}{1000}\)

Explanation:
Given 0.832 ,
In fraction form we can write it in to \(\frac{832}{1000}\)

Question 16.
0.203
Answer: \(\frac{203}{1000}\)

Explanation:
Given 0.203 ,
In fraction form we can write it in to \(\frac{203}{1000}\)

Question 17.
0.78
Answer:  \(\frac{78}{100}\)

Explanation:
Given 0.78 ,
In fraction form we can write it in to \(\frac{78}{100}\)

Question 18.
0.999
Answer: \(\frac{999}{1000}\)

Explanation:
Given 0.999 ,
In fraction form we can write it in to \(\frac{999}{1000}\)

In 19-26, write each fraction as a decimal.

Question 19.
\(\frac{434}{1,000}\) =
Answer:  0.434

Explanation:
Given, \(\frac{434}{1,000}\)
In decimal form we can  write it into 0.0.434

Question 20.
\(\frac{3}{10}\) =
Answer: 0.3

Explanation:
Given, \(\frac{3}{10}\)
In decimal form we can  write it into 0.3

Question 21.
\(\frac{873}{1,000}\) =
Answer: 0.873

Explanation:
Given, \(\frac{873}{1,000}\)
In decimal form we can  write it into 0.873

Question 22.
\(\frac{17}{1,000}\) =
Answer:0.017

Explanation:
Given, \(\frac{17}{1,000}\)
In decimal form we can  write it into 0.017

Question 23.
\(\frac{309}{1,000}\) =
Answer: 0.309

Explanation:
Given, \(\frac{309}{1,000}\)
In decimal form we can  write it into 0.309

Question 24.
\(\frac{5}{1,000}\) =
Answer: 0.005

Explanation:
Given, \(\frac{5}{1,000}\)
In decimal form we can  write it into 0.005

Question 25.
\(\frac{6}{100}\) =
Answer: 0.06

Explanation:
Given, \(\frac{6}{100}\)
In decimal form we can  write it into 0.06

Question 26.
\(\frac{999}{1,000}\) =
Answer: 0.999

Explanation:
Given, \(\frac{999}{1,000}\)
In decimal form we can  write it into 0.999

Question 27.
Look at the middle 9 in exercise 18. How is its value related to the value of the 9 to its left? to the value of the 9 to its right?
Answer: The value of the digit 9 in the hundredths place has 10 times the value of the digit 9 in the thousandths place and \(\frac{1}{10}\) the value of the digit 9 in the tenths place.

Explanation:
The value of the number given in exercise 18 is 0.999,
The first 9 is on the place of the tenths, while the second 9 is on the place of the hundredths and the second 9 is  on the place of thousandths.
So, The value of the digit 9 in the hundredths place has 10 times the value of the digit 9 in the thousandths place and \(\frac{1}{10}\) the value of the digit 9 in the tenths place.

Problem Solving

Question 28.
The Palmers’ property tax bill for the year is $3,513. In their first installment, they paid $1,757. How much do they still owe on their bill? Write an equation to model your work.
Answer:  They still owe $1,756 on their bill .

Explanation:
Given, The Palmers’ property tax bill for the year is $3,513.
In their first installment, they paid $1,757.
$3,513 – $1,757 = $1,756.
So, they still have to pay the half amount that is $1,756
Then $1,757 × 2 = $3,513.
They still owe $1,756 on their bill .

Question 29.
Write the fractions \(\frac{22}{100}\) and \(\frac{22}{1,000}\) as decimals. How are the values of the digit 2 related in each of the decimals?
Answer: 0.22 and 0.022

Explanation:
Given, \(\frac{22}{100}\)
In decimal form we can write it as 0.22,
The value of the digit 2 in the tenths place has 10 times the value of the digit 4 in the hundredths place .

And \(\frac{22}{1,000}\)
In decimal form we can write it as 0.022,
The value of the digit 2 in the hundredths place has 10 times the value of the digit 4 in the thousandths place.

Question 30.
Simon scored 4 × 10² points in a game. Joe scored 2 × 10³ points in the same game. Whose score is higher? How much higher?
Answer: Joe scored 5 times as many as Simon.

Explanation:
Given, Simon scored 4 × 10² points in a game.
Joe scored 2 × 10³ points in the same game.
Now we have  4 × 10² =  4 × 10 × 10 = 400
2 × 10³ =  2 × 10 × 10 × 10 = 2000
Finally, Simon scored 400 point and Joe scored 2000 points
So, Joe scored 5 times as many as Simon.

Question 31.
Higher Order Thinking Kelly said that \(\frac{97}{1,000}\) can be written as 0.97. Is she correct? Explain.
Answer: No , \(\frac{97}{1,000}\)  should be 0.097 not 0.97

Explanation:
Given, Kelly said that \(\frac{97}{1,000}\) can be written as 0.97
The decimal form can written according to the number of zeros in the denominator
So , \(\frac{97}{1,000}\)  should be 0.097 not 0.97

Question 32.
Critique Reasoning Frank reasoned that in the number 0.555, the value of the 5 in the thousandths place is ten times as great as the 5 in the hundredths place. Is he correct? Explain.
Answer: NO

Explanation:
Given, the number 0.555
The value of the digit 5 in the hundredths place has 10 times the value of the digit 5 in the thousandths place

Question 33.
How many cubes are in the box? What fraction of the entire box do the 7 cubes represent? Explain your answer.

Answer: \(\frac{7}{1000}\)

Explanation:
Given, 10 × 10 × 10  = 1000
And we have the 7 cubes
So, the fraction will be  \(\frac{7}{1000}\)

Assessment Practice

Question 34.
0.04 is 10 times as great as which decimal?
A. 0.4
B. 0.1
C. 0.004
D. 0.001
Answer: C , 0.04 is 10 times as great as 0.004

Explanation:
0.04 × \(\frac{1}{10}\) = 0.004
So, 0.04 is 10 times as great as 0.004

Question 35.
0.009 is o of which decimal?
A. 0.01
B. 0.09
C. 0.1
D. 0.9
Answer:  B, 0.009 is \(\frac{1}{10}\) of 0.09

Explanation:
0.09 × \(\frac{1}{10}\) = 0.009
So, 0.009 is \(\frac{1}{10}\) of 0.09

Lesson 1.4 Understand Decimal Place Value

Solve & Share

A runner won a 100-meter race with a time of 9.85 seconds. How can you use place value to Explain this time? Complete a place-value chart to show this time.

Generalize You can use what you know about whole-number place value to help you think about place value of decimal numbers.

Look Back! In the decimal 9.85, what is the value of the 8? What is the value of the 5?

Visual Learning Bridge

Essential Question How Can You Represent Decimals?

A.
Jo picked a seed from her flower. The seed has a mass of 0.245 gram. What are some different ways you can represent 0.245?

You can write the standard form, expanded form, and number name for a decimal just like you can for a whole number.

B.

Number Name: two hundred forty-five

A place-value chart can help you identify the tenths, hundredths, and thousandths places in a decimal.

Convince Me! Use Structure How many hundredths are in one tenth? How many thousandths are in one hundredth? Tell how you know.

Another example
Equivalent decimals name the same amount.
What are two other decimals equivalent to 1.4?
One and four tenths is the same as one and forty hundredths.
1.4 = 1.40
One and four tenths is the same as one and four hundred thousandths.
1.4 = 1.400.

So, 1.4 = 1.40 = 1.400.

Guided Practice

Do You Understand?

Question 1.
The number 3.453 has two 3s. Why does each 3 have a different value?
Answer: These are different due to the place values. 3 and 0.003

Explanation:
The given number is 3.453.
The first 3 that is before the decimal is at ones place, that is 3 × 1 = 3
But the second 3 which is after the decimal is at thousandth place, \(\frac{3}{1000}\) = 0.003
Hence, you can clearly see the different values of both three’s. These are different due to the place values.

Do You Know How?

In 2 and 3, write each number in standard form.

Question 2.
4 × 100 + 7 × 10 + 6 × 1 + 6 × \(\left(\frac{1}{10}\right)\) + 3 × \(\left(\frac{1}{100}\right)\) + 7 × \(\left(\frac{1}{1,000}\right)\)
Answer:  The standard form is 476.637.

Explanation:
Given,
(4 × 100)+ (7 × 10) + (6 × 1) + {6 × \(\left(\frac{1}{10}\right)\) + 3 × \(\left(\frac{1}{100}\right)\) + 7 × \(\left(\frac{1}{1,000}\right)\)}
Solve for the brackets,
we have 400 + 70 + 6 + [ 0.6 + 0.03 + 0.007]
= 476 + 0.637
= 476.637
The standard form is 476.637.

Question 3.
four and sixty-eight thousandths
Answer: The standard form is 0.468.

Explanation:
Given, four and sixty-eight thousandths
The expanded form is 4 × \(\left(\frac{1}{10}\right)\) + 6 × \(\left(\frac{1}{100}\right)\) + 8 × \(\left(\frac{1}{1,000}\right)\)
Then, 0.4 + 0.06 + 0.008 = 0.468
So, the standard form is 0.468

Independent Practice

In 4-6, write each number in standard form.

Question 4.
(2 × 1) + (6 × \(\frac{1}{1,000}\))
Answer: The standard form is 2.006

Explanation:
Given, (2 × 1) + (6 × \(\frac{1}{1,000}\))
= 2 + 0.006
= 2.006
The standard form is 2.006

Question 5.
(3 × 1) + (3 × \(\frac{1}{10}\)) + (9 × \(\frac{1}{1,000}\))
Answer: The standard form is 3.309

Explanation:
Given, (3 × 1) + (3 × \(\frac{1}{10}\)) + (9 × \(\frac{1}{1,000}\))
= 3 + 0.3 + 0.009
= 3.309
So, The standard form is 3.309

Question 6.
nine and twenty hundredths
Answer: The standard form is 0.920.

Explanation:
Given, nine and twenty hundredths
The expanded form is (9 × \(\frac{1}{10}\)) + (20 × \(\frac{1}{100}\))
= 0.9 + 0.020
= 0.920
The standard form is 0.920

In 7-10, write two decimals that are equivalent to the given decimal.

Question 7.
2.200
Answer: The two decimals that are equivalent to the given decimal are 2.2 and 2.20

Explanation:
Given, 2.200
Two and two tenths is the same as two and twenty hundredths.
2.2 = 2.20
Two and Two tenths is the same as two and twenty tenths.
2.2 = 2.2.
So, The two decimals that are equivalent to the given decimal are 2.2 and 2.20

Question 8.
8.1
Answer: The two decimals that are equivalent to the given decimal are 8.10 and 8.100

Explanation:
Given, 8.1
Eight and one tenths is the same as eight and one hundredths.
8.1 = 8.10
Eight and one tenths is the same as eight and one hundred thousandths.
8.1 = 8.100.
So, The two decimals that are equivalent to the given decimal are 8.10 and 8.100

Question 9.
9.50
Answer: The two decimals that are equivalent to the given decimal are 9.5 and 9.500

Explanation:
Given, 9.50
Nine and five tenths is the same as nine and  fifty tenths.
9.50 = 9.5.
Nine and five tenths is the same as nine and fifty hundred thousandths.
9.50 = 9.500.
So, The two decimals that are equivalent to the given decimal are 9.5 and 9.500

Question 10.
4.200
Answer: The two decimals that are equivalent to the given decimal are 4.2 and 4.20

Explanation:
Given, 4.200
Four and two tenths is the same as four and twenty hundredths.
4.200 = 4.20
Four and two tenths is the same as two and twenty tenths.
4.200 = 4.2.
So, The two decimals that are equivalent to the given decimal are 4.2 and 4.20

Problem Solving

Question 11.
The annual fundraising goal of a charity is $100,000. So far $63,482 has been raised. How much more money is needed to reach the goal?

Answer: To reach the goal they need $36,518.

Explanation:
Given, The annual fundraising goal of a charity is $100,000.
So far $63,482 has been raised.
Then, $100,000 – $63,482 = $36,518.
So, To reach the goal they need $36,518.

Question 12.
Santiago has a rope that measures 205.95 centimeters. Write this number in expanded form.
Answer:

Explanation:
Given, Santiago has a rope that measures 205.95 centimeters
The standard form is 205.95
Then, The expanded form is (2 × 100) + (5 × 1) + (9 × \(\frac{1}{10}\)) + (5 × \(\frac{1}{100}\))
= 200 + 5 +[ 0.9 + 0.05 ]
= 205 + 0.95
= 205.95
So, The expanded form is (2 × 100) + (5 × 1) + (9 × \(\frac{1}{10}\)) + (5 × \(\frac{1}{100}\))

Question 13.
How can you tell that 7.630 and 7.63 are equivalent decimals?
Answer: The two decimals that are equivalent to both the numbers

Explanation:
Given, 7.630 and 7.63
Seven and sixty three tenths is the same as Seven and sixty three hundredths.
7.63 = 7.630
So, The two decimals that are equivalent to both the numbers

Question 14.
In Justin’s school, 0.825 of the students participate in a sport. If there are one thousand students in Justin’s school, how many participate in a sport?
Answer: Totally 825 students are participating in the school.

Explanation:
Given, In Justin’s school, 0.825 of the students participate in a sport.
Since, there are 1 thousand students, and the decimal is 0.825,
So, there are 825 students participating.

Question 15.
Be Precise Maria incorrectly placed the decimal point when she wrote 0.65 inch for the width of her tablet computer. What is the correct decimal number for the width?

Answer: The correct decimal number for the width is 6.5.

Explanation:
Given, Maria incorrectly placed the decimal point when she wrote 0.65 inch for the width of her tablet computer.
As shown in the figure, the scale placed there to measure the width , as per the scale the width is 6.5.
So, The correct decimal number for the width is 6.5.

Question 16.
Higher Order Thinking Three boys cut out hundredths decimal models. Derrick does not shade any of his models. Ari shades half of one model. Wesley shades two models and one tenth of another model. What decimal represents the amount each boy shades?

Answer: Derrick’s value = 0,
Ari’s value = 0.5,
Wesley’s value = 2.1 ,

Explanation:
Given, Derrick does not shade any of his models. Ari shades half of one model. Wesley shades two models and one tenth of another model.
Derrick doesn’t shade anything, so his value is 0.
Ari shades half of one model, so 1/2 = 0.5 is his value Wesley shades 2 full models, plus 1/10 = 0.1 of another one, leading to 2+0.1 = 2.1 as his value. 
Derrick’s value = 0,
Ari’s value = 0.5,
Wesley’s value = 2.1 ,

Assessment Practice

Question 17.
Find two decimals that are equivalent to (4 × 10) + (7 × \(\frac{1}{100}\)). Write the decimals in the box.

Answer:  The two decimals that are equivalent to the 40.07 are 40.070 and 40.07

Explanation:
Given, (4 × 10) + (7 × \(\frac{1}{100}\)).
= 40 + 0.07
= 40.07
The standard form is 40.07
The two decimals that are equivalent to the 40.07 are 40.070 and 40.07 , from the given box of decimals.

Lesson 1.5 Compare Decimals

Activity

Solve & Share

The lengths of three ants were measured in a laboratory. The lengths were 0.521 centimeter, 0.498 centimeter, and 0.550 centimeter. Which ant was the longest? Which ant was the shortest?

How can you use the math you know to compare and order the decimals? Tell how you decided.

Look Back! Be Precise What are the lengths of the ants in order from least to greatest?

Visual Learning Bridge

Essential Question How Can You Compare Decimals?

A.
Scientists collected and measured the lengths of different cockroach species. Which cockroach had the greater length, the American or the Oriental cockroach?

Comparing decimals is like comparing whole numbers!

B.
Step 1
Line up the decimal points.
Start at the left.
Compare digits of the same place value.
3.576
3.432

C.
Step 2
Find the first place where the digits are different.
3.576
3.432

D.
Step 3
Compare.
5 > 4
0.5 > 0.4
So, 3.576 > 3.432.
The American cockroach is longer than the Oriental cockroach.

Convince Me! Critique Reasoning Valerie said, “12.68 is greater than 12.8 because 68 is greater than 8.” Is she correct? Explain.

Answer: No, 12.8 is greater than 12.68

Explanation:
Given  : 12.68 is greater than 12.8
We have given 12.68 and 12.8
In 12.68,
Tens place is 1,
Ones place is 2,
Tenth place is 6,
Hundredth place is 8.
In 12.8,
Tens place is 1 .
Ones place is 2,
Tenth place is 8,
So, We can see Tens place and ones place of both number are same but tenth place of 12.8 is greater than 12.68.
Therefore, 12.8 is greater than 12.68 . 

Another example
Order the cockroaches from least to greatest length.
Step 1
Write the numbers, lining up the decimal points. Start at the left. Compare digits of the same place value.
3.576
3.582
3.432 is the least.

Step 2
Write the remaining numbers, lining up the decimal points. Start at the left. Compare.
3.576
3.582
3.582 is greater than 3.576.

Step 3
Write the numbers from least to greatest.
3.432 3.576 3.582
From least to greatest lengths are the Oriental, the American, and the Australian.

Guided Practice

Do You Understand?

Question 1.
Scientists measured a Madeira cockroach and found it to be 3.44 centimeters long. Toby says that the Madeira is shorter than the Oriental because 3.44 has fewer digits than 3.432. Is he correct? Explain.
Answer: 3.44cm cockroach is the longest.

Explanation:
Given,  3.44  and  3.432
3.44
3.432
Comparing the numbers after decimals,
4 > 3
So, 3.44cm cockroach is the longest.

Do You Know How?

In 2 and 3, write >, <, or = for each .

Question 2.
3.692 3.697
Answer: 3.692 < 3.697

Explanation:
Given,  3.692  and  3.697
3.692
3.697
Comparing the numbers after decimals, we have
2 < 7
So, 3.692 < 3.697.

Question 3.
7.216 7.203
Answer:  7.216 > 7.203.

Explanation:
Given,  7.216  and  7.203
7.216
7.203
Comparing the numbers after decimals, we have
1 > 0
So, 7.216 > 7.203.

In 4 and 5, order the decimals from least to greatest.

Question 4.
5.540, 5.631, 5.625
Answer: The order from least to greatest is  5.540, 5.625, 5.631.

Explanation:
Given, 5.540, 5.631, 5.625
lining up the decimal points.
5.540, is the least
5.631,
5.625,
Comparing the numbers after decimals, we have
5.631,
5.625,
Compare digits of the same place value.
3 > 2
5.631 is the greatest
So, The order from least to greatest is  5.540, 5.625, 5.631.

Question 5.
0.675, 1.529, 1.35, 0.693
Answer: The order from least to greatest is 0.675, 0.693, 1.35, 1.529.

Explanation:
Given, 0.675, 1.529, 1.35, 0.693
lining up the decimal points.
0.675,
1.529,
1.35,
0.693
Comparing the numbers after decimals, we have
0.675,
0.693,
Compare digits of the same place value.
7 < 9
0.675 is the less than 0.693
Then 1.529,
1.35,
Compare digits of the same place value.
5 > 3
1.529 is greater than 1.35
So, The order from least to greatest is 0.675, 0.693, 1.35, 1.529.

Independent Practice

In 6-8, compare the two numbers. Write >, <, or = for each .

Question 6.
0.890 0.890
Answer: 0.890 = 0.890

Explanation:
Given,  0.890  and 0.890
0.890
0.890
Comparing the numbers after decimals, we have
All the numbers are equal
So, 0.890 = 0.890.

Question 7.
5.733 5.693
Answer: 5.733 > 5.693.

Explanation:
Given,  5.733  and  5.693
5.733
5.693
Comparing the numbers after decimals, we have
7 > 6
So, 5.733 > 5.693.

Question 8.
9.707 9.717
Answer: 9.707 < 9.717

Explanation:
Given,  9.707  and  9.717
lining up the decimal points
9.707
9.717
Comparing the numbers after decimals, we have, Compare digits of the same place value.
0 < 1
So, 9.707 < 9.717.

In 9 and 10, order the decimals from greatest to least.

Question 9.
878.403, 887.304,887.043
Answer: The order from least to greatest is 878.403 ,887.043 , 887.304.

Explanation:
Given, 878.403, 887.304,887.043
lining up the decimal points.
878.403, is the least
887.304,
887.043
Comparing the numbers after decimals, we have
887.304,
887.043
Compare digits of the same place value.
3 > 0
887.304 is greater than 887.043
So, The order from least to greatest is 878.403 ,887.043 , 887.304.

Question 10.
435.566, 436.565, 435.665
Answer: The order from least to greatest is 435.566, 435.665 , 436.565.

Explanation:
Given, 435.566, 436.565, 435.665
lining up the decimal points.
435.566,
436.565, is the greatest
435.665
Comparing the numbers after decimals, we have
435.566,
435.665
Compare digits of the same place value.
5 < 6
435.566 is less than 435.665
So, The order from least to greatest is 435.566, 435.665 , 436.565.

Problem Solving

Question 11.
Critique Reasoning Explain why it is not reasonable to say that 4.23 is less than 4.135 because 4.23 has fewer digits after the decimal point than 4.135.
Answer: 4.23 is greater than 4.135.

Explanation:
Given, 4.23 and 4.135
lining up the decimal points.
4.23,
4.135
Comparing the numbers after decimals, we have ,Compare digits of the same place value.
2 < 1
So, 4.23 is greater than 4.135.

Question 12.
Number Sense Carlos wrote three numbers between 0.33 and 0.34. What numbers could Carlos have written?
Answer: there can be any three numbers mentioned Below between 0.33 and 0.34.

Explanation:
Given, three numbers between 0.33 and 0.34
The numbers between the 0.33 and 0.34 are 0.331, 0.332, 0.333, 0.334, 0.335, 0.336, 0.337, 0.338 and 0.339,
Then the next number will be 0.340 that is same as 0.34
So, there can be any three numbers mentioned above between 0.33 and 0.34.

Question 13.
Vocabulary Draw lines to match each decimal on the left to its equivalent decimal on the right.

Answer: The given decimals are compared to its nearest decimals by place value method.

Explanation:

Question 14.
Is 0.5 greater than or less than \(\frac{6}{10}\)? Draw a number line to show your answer.
Answer: 0.5 is < 0.6

Explanation:

Given,  0.5 and \(\frac{6}{10}\)
We can write \(\frac{6}{10}\) as 0.6 in decimal form,
Compare digits of the same place value.
5 < 6
So, 0.5 is < 0.6

Question 15.
Higher Order Thinking Ana’s gymnastics scores were posted on the scoreboard in order from highest to lowest score. One digit in her floor score is not visible. List all the possible digits for the missing number.

Answer: The possible digits are 1 , 2 , 3 , 4

Explanation:
Given,
The floor score must be higher than 15.133 and lower than 15.500
Then all the possible scores are: 15.166 , 15.266 , 15.366 and 15.466
Because 15.166 > 15.133 and 15.466 < 15.500
Then all the possible digits are 1 , 2 , 3 , 4 

Question 16.
Marcia’s vault score is 15.050. How does it compare to Ana’s vault score?
Answer: Marcia’s vault score is less than Ana’s vault score

Explanation:
Given, Marcia’s vault score is 15.050
Ana’s vault score is 15.500
Lining up the decimal points.
15.050
15.500
Comparing the numbers after decimals, we have ,Compare digits of the same place value.
0 < 5
So, 15.050 is less than 15.500.
Finally, Marcia’s vault score is less than Ana’s vault score

Assessment Practice

Question 17.
Which statements correctly compare two numbers?
0.1 <0.125
0.2 < 0.125
125 > 0.13
0.125 > 0.12
0.126 < 0.125
Answer: 0.1 < 0.125 , 125 > 0.13 , 0.125 > 0.12 .

Explanation:
Compare digits of the same place value.
0.1 < 0.125 , because 0 < 2
125 > 0.13 , because .0 > 0.
0.125 > 0.12 . because 5 > 0

Question 18.
Cara weighed 4.16 pounds of apples at the grocery store. Which numbers make the statement true? > 4.16
04.15
04.19
4.2
4.09
4.1
Answer: 04.15

Explanation:
Given, Cara weighed 4.16 pounds of apples at the grocery store
4.16 and 04.15 are the closest decimals
Compare digits of the same place value.
5 < 6
So, the immediate number of 4.16 is 04.15.

Lesson 1.6 Round Decimals

Activity

Solve & Share

In science class, Marci recorded numbers from an experiment as 12.87, 12.13, 12.5, and 12.08. Which numbers are closer to 12? Which are closer to 13? How can you tell?

You can use structure to help determine what number is halfway between two whole numbers. Show your work!

Look Back! What is the halfway point between 12 and 13? Is that point closer to 12 or 13?

Visual Learning Bridge

Essential Question How Can You Round Decimals?

A.
Rounding replaces one number with another number that tells about how many or about how much. Round 2.36 to the nearest tenth. Is 2.36 closer to 2.3 or 2.4?

A number line can help you round a decimal.

B.
Step 1
Find the rounding place. Look at the digit to the right of the rounding place.
2.36

C.
Step 2
If the digit is 5 or greater, add 1 to the rounding digit. If the digit is less than 5, leave the rounding digit alone.
Since 6 >5, add 1 to the 3.

D.
Step 3
Drop the digits to the right of the rounding digit.
2.36 rounds to 2.4

Rounding can help (you find which tenth or hundredth a decimal is closest to.

Convince Me! Critique Reasoning Carrie said, “448 rounds to 500 because 448 rounds to 450 and 450 rounds to 500.” Is she correct? Explain. Use the number line in your explanation.

Answer: No

Explanation:
Given, 448 rounds to 500 because 448 rounds to 450 and 450 rounds to 500.”
The nearest number to 448 is 450 and that is correct but,
450 is not near the number 500 ,
The difference between these numbers is 50 ,
So, Carrie is wrong.

Another example
Round 3.2 to the nearest whole number. Is 3.2 closer to 3 or 4?

Step 1
Find the rounding place. Look at the digit to the right of the rounding place.
3.2
Step 2
If the digit is 5 or greater, add 1 to the rounding digit. If the digit is less than 5, leave the rounding digit alone. Since 2 < 5, leave 3 the same.
Step 3
Drop the digits to the right of the decimal point. Drop the decimal point.
3.2 rounds to 3.

Guided Practice

Do You Understand?

Question 1.
To round 74.58 to the nearest tenth, which digit do you look at? What is 74.58 rounded to the nearest tenth?
Answer: the nearest tenth to 74.58 is 74.6

Explanation:
Given, To round 74.58 to the nearest tenth,
Find the rounding place. Look at the digit to the right of the rounding place.
8 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point.
74.6
So, the nearest tenth to 74.58 is 74.6

Question 2.
A car-rental service charges customers for the number of miles they travel, rounded to the nearest whole mile. George travels 40.8 miles. For how many miles will he be charged? Explain.
Answer: He will be charged for 41 miles

Explanation:
Given, George travels 40.8 miles.
Find the rounding place. Look at the digit to the right of the rounding place.
8 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point.
The nearest to 40.8 is 41
So, He will be charged for 41 miles

Do You Know How?

In 3-10, round each number to the place of the underlined digit.

Question 3.
16.5
Answer: 17

Explanation:
Given, 16.5
6 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 17
The rounding number is 17.

Question 4.
56.1
Answer: 56

Explanation:
Given, 56.1
6 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 56
The rounding number is 56.

Question 5.
1.32
Answer: 1.3

Explanation:
Given, 1.32
3 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 1.3
The rounding number is 1.3.

Question 6.
42.78
Answer: 42.8

Explanation:
Given, 42.78
7 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 42.8
The rounding number is 42.8.

Question 7.
1.652
Answer: 1.65

Explanation:
Given, 1.652
5 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 1.65
The rounding number is 1.65.

Question 8.
582.04
Answer: 582.0

Explanation:
Given, 582.04
4 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 582.0
The rounding number is 582.0.

Question 9.
80,547.645
Answer: 80,547.65

Explanation:
Given, 80,547.645
4 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 80,547.65
The rounding number is 80,547.65.

Question 10.
135,701.949
Answer: 135,701.9

Explanation:
Given, 135,701.949
9 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 135,701.9
The rounding number is 135,701.9.

Independent Practice

In 11-14, round each decimal to the nearest whole number.

Question 11.
4.5
Answer: 5

Explanation:
Given, 4.5
4 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 5
The rounding number is 5.

Question 12.
57.3
Answer: 57

Explanation:
Given, 57.3
7 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 57
The rounding number is 57.

Question 13.
34.731
Answer: 34.73

Explanation:
Given, 34.731
3 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 34.73
The rounding number is 34.73.

Question 14.
215.39
Answer: 215.4

Explanation:
Given,215.39
3 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 215.4
The rounding number is 215.4.

In 15-18, round each number to the place of the underlined digit.

Question 15.
7.158
Answer: 7.2

Explanation:
Given, 7.158
1 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 7.2
The rounding number is 7.2.

Question 16.
0.758
Answer: 0.76

Explanation:
Given, 0.758
5 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 0.76
The rounding number is 0.76.

Question 17.
6.4382
Answer: 6.44

Explanation:
Given, 6.4382
3 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 6.44
The rounding number is 6.44.

Question 18.
84.732
Answer: 84.7

Explanation:
Given, 84.732
7 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 84.7
The rounding number is 84.7.

Problem Solving

Question 19.
The picture at the right shows the length of an average American alligator. What is the length of the alligator rounded to the nearest tenth?

Answer: 4.4 is the length of the alligator rounded to the nearest tenth.

Explanation:
Given, 4.39
3 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 4.4
The rounding number is 4.4.
So,4.4  is the length of the alligator rounded to the nearest tenth

Question 20.
Name two different numbers that round to 8.21 when rounded to the nearest hundredth.
Answer: 8.211 and 8.212

Explanation:
Given, 8.21 rounded to the nearest hundredth,
2 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 8.211 and 8.212
The rounding number is 8.211 and 8.212.

Question 21.
Number Sense To the nearest hundred, what is the greatest whole number that rounds to 2,500? the least whole number?
Answer: 2,499 and 2,491.

Explanation:
Given, 2,500
The greatest whole number to nearest hundred of 2,500 is 2,499
And, The less whole number to nearest hundred of 2,500 is 2,491

Question 22.
Draw all of the lines of symmetry in the figure shown below.

Answer: There are 2 line lines of symmetry.

Explanation:

The given figure is in the form of a Rectangle
So, it has 2 lines of symmetry.

Question 23.
Higher Order Thinking Emma needs 2 pounds of ground meat to make a meatloaf. She has one package with 2.36 pounds of ground meat and another package with 2.09 pounds of ground meat. She uses rounding and finds that both packages are close to 2 pounds. Explain how Emma can choose the package closer to 2 pounds.
Answer: She can use the package of 2.09 because it is close to 2 pounds.

Explanation:
Given, 2.09 and 2.36
For 2.09 , 0 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point. 2.1
For 2.36, 3 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point. 2.4
So, She can use the package of 2.09 because it is close to 2 pounds.

Question 24.
Make Sense and Persevere Robert slices a large loaf of bread to make 12 sandwiches. He makes 3 turkey sandwiches and 5 veggie sandwiches. The rest are ham sandwiches. What fraction of the sandwiches Robert makes are ham?
Answer: \(\frac{3}{12}\), \(\frac{5}{12}\), \(\frac{4}{12}\).

Explanation:
Given, Robert slices a large loaf of bread to make 12 sandwiches,
He makes 3 turkey sandwiches and 5 veggie sandwiches
The rest are ham sandwiches.
The number of turkey sandwiches are \(\frac{3}{12}\),
The number of veggie sandwiches are \(\frac{5}{12}\),
The number of ham sandwiches are \(\frac{4}{12}\),
Total 12 sandwiches are there made by Robert.

Question 25.
Algebra After buying school supplies, Ruby had $32 left over. She spent $4 on notebooks, $18 on a backpack, and $30 on a new calculator. How much money, m, did Ruby start with? Write an equation to show your work.
Answer: Ruby started with $84.

Explanation:
Given, Ruby had $32 left over.
She spent $4 on notebooks, $18 on a backpack, and $30 on a new calculator.
The total money spent =  $4 + $18  +  $30  = $52
Add the money she had left over and money she had spent
we have, $52 + $32 = $84
So, Ruby started with $84.

Assessment Practice

Question 26.
Find two numbers that round to 35.4 when rounded to the nearest tenth. Write the numbers in the box.

Answer: 35.45 and 35.391

Explanation:
Given, 35.40
4 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 35.45
if the digit is 5 or greater, add 1 to the rounding digit. 35.391
The rounding number is 35.45 and 35.391.

Lesson 1.7 Look For and Use Structure

Problem Solving

Activity

Solve & Share
Angie volunteers in the school library after school. The librarian gave her a stack of books and told her to use the number on each book to shelve it where it belongs.
How can Angie arrange the books in order from least to greatest to make shelving them easier?

Thinking Habits
Be a good thinker! These questions can help you.
• What patterns can I see and describe?
• How can I use the patterns to solve the problem?
• Can I see expressions and objects in different ways?
• What equivalent expressions can I use?

Look Back! Use Structure Explain why 323.202 is less than 323.21 even though 202 is greater than 21.

Answer: Because , when rounding the decimals we have to look for the place values to decide which is greater or least.

Explanation:
Given, 323.202
0 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 323.21
The rounding number is 323.21.

Visual Learning Bridge

Essential Question
How Can You Use Structure to Solve Problems?

A.
Analyze the chart. What do you notice that can help you complete the chart?

What do I need to do to solve this problem?

I can use the structure of the decimal place-value system to complete the chart.

You can look for patterns to find the missing numbers.

B.
How can I make use of structure to solve this problem?
I can
• find and describe patterns.
• use the patterns to see how the numbers are organized.
• analyze patterns to see the structure in the table.
• break the problem into simpler parts.

C.
Solve
Here’s my thinking…

As you move down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same, except for the last number, while the hundredths increase by 1.

Convince Me! Use Structure Write the missing numbers. Explain how you can use structure to find the last number in the bottom row.

Answer:   

Explanation:
Because, down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same,
except for the last number, while the hundredths increase by 1.

Guided Practice
Use Structure Each of these grids is a part of a decimal number chart similar to the one on page 30.

You can use what you know about place value when you look for patterns with decimals.

Question 1.
Describe the pattern for moving from a pink square to a green square. Then write the missing numbers.

Answer:

Explanation:
Because, down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same,
except for the last number, while the hundredths increase by 1.

Question 2.
How can you use patterns to find the number that would be in the box below 0.52?
Answer: the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same,
except for the last number, while the hundredths increase by 1. and completing the pattern.

Independent Practice

Use Structure Pamela is hiking. When she returns to camp, she passes the mile markers shown at the right.

Question 3.
Explain how you can use structure to find the decimal numbers that will be shown on the next four mile markers.

Answer:  the next four mile markers will be 2.2 , 2.1 , 2.0 , 1.9

Explanation:

Moving from left to right in the rows, tenths stay the same,
except for the last number, while the hundredths increase by 1
So, the next four mile markers will be 2.2 , 2.1 , 2.0 , 1.9

Question 4.
Pamela stops at the 1.8 mile marker. Where will she be if she walks one tenth of a mile towards camp? one mile towards camp? Explain.
Answer: she will be on 0.18 if she walks one tenth of a mile towards camp and 1.7 if she walks one mile towards the camp.

Explanation:
Given, 1.8
8 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 1.7
The reducing mile as reaching out for camp is 1.7
If she will be on 0.18 if she walks one tenth of a mile towards camp, That is 1.8 × \(\frac{1}{10}\)
Finally, she will be on 0.18 if she walks one tenth of a mile towards camp and 1.7 if she walks one mile towards the camp.

Problem Solving

Performance Task

Thousandths Chart
The students in Ms. Lowell’s class wrote a thousandths decimal chart on the board. Some of the numbers got erased.

Answer:

Explanation:
Because, down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same,
except for the last number, while the hundredths increase by 1.

Question 5.
Use Structure Describe the pattern for moving across a row from left to right.
Answer:

Explanation:
Moving from left to right in the rows, tenths stay the same, except for the last number, while the hundredths increase by 1.

You can use structure to decide if decimal numbers are following a pattern.

Answer:

Explanation:
Above question has the figure, we have the decimal numbers following a pattern of  tenths stay the same, except for the last number, while the hundredths increase by 1.

Question 6.
Be Precise How does the pattern change in the last square of each row?
Answer: As you move down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.

Explanation:

As you move down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.

Question 7.
Use Structure Describe the pattern for moving down a column.
Answer: As you move down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.

Explanation:

As you move down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.

Question 8.
Use Repeated Reasoning Write the missing numbers in the decimal chart above.
Answer:

Explanation:
Because, down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same,
except for the last number, while the hundredths increase by 1.

Question 9.
Use Structure Suppose the students add to the chart. Write the missing numbers in the row and the column below.

Answer:  The complete chart is as follows,

Explanation:
Because, down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same,
except for the last number, while the hundredths increase by 1.

Topic 1 Fluency Review

Activity

Find a Match

Work with a partner. Point to a clue. Read the clue. Look below the clues to find a match. Write the clue letter in the box above the match. Find a match for every clue.

Clues
A The sum is between 15,000 and 20,000.
B The difference is less than 10,000.
C The difference is between 41,000 and 42,000.
D The sum is exactly 52,397.
E The difference is between 82,000 and 84,000.
F The sum is greater than 79,000.
G The sum is exactly 52,407,
H The difference is exactly 42,024.

Answer:

Topic 1 Vocabulary Review

Understand Vocabulary

Glossary

Word List
• base
• equivalent decimals
• expanded form
• exponent
• power
• thousandths
• value

Choose the best term from the Word List. Write it on the blank.

Question 1.
Decimal numbers that name the same part of a whole or the same point on a number line are called ____
Answer: Decimal numbers that name the same part of a whole or the same point on a number line are called Equivalent decimals

Question 2.
The ____ of a digit in a number depends on its place in the number.
Answer: The value of a digit in a number depends on its place in the number.

Question 3.
The product that results from multiplying the same number over and over is a(n) ____ of that number.
Answer:  The product that results from multiplying the same number over and over is a(n) POWER of that number.

Question 4.
A digit in the hundredths place has ten times the value of the same digit in the ___ place.
Answer: A digit in the hundredths place has ten times the value of the same digit in the thousandths place.

Question 5.
In 10⁵, the number 10 is the _____.
Answer: In 10⁵, the number 10 is the BASE.

Draw a line from each number in Column A to the same number in Column B.

Answer:

Use Vocabulary in Writing

Question 10.
Explain why each 8 in the number 8.888 has a different value. Use one or more terms from the Word List in your explanation.
Answer:  Because of its place value.

Explanation:
Given, 8.888
The first 8 has the place value of tenths, The second 8 has the place value of hundredths and the final 8 has the place value of thousandths.

Set A
pages 5-8
How can you write 7,000 using exponents?
7,000 = 7 × 10 × 10 × 10 = 7 × 10³
So, using exponents, you can write 7,000 as 7 × 10³

Remember the number of zeros in the product is the same as the exponent.

Find each product.

Question 1.
9 × 10¹
Answer: 90

Explanation:
9 × 10¹ = 9 × 10 = 90

Question 2.
8 × 1,000
Answer: 8,000

Explanation:
8 × 10 × 10 × 10 = 8,000

Question 3.
5 × 10²
Answer: 5 × 10² = 500

Explanation:
5 × 10² =  5 × 10 × 10  = 500

Question 4.
2 × 10⁵
Answer: 2 × 10⁵  = 2,00,000

Explanation:
2 × 10⁵  = 2 × 10 × 10 × 10 × 10 × 10 = 2,00,000

Set B
pages 9-12
Write the number name and tell the value of the underlined digit for 930,365.
Nine hundred thirty thousand, three hundred sixty-five
Since the 0 is in the thousands place, its value is 0 thousands, or 0.
Use digital tools to solve these and other problems.

Remember you can find the value of a digit by its place in a number.

Write the number name and tell the value of the underlined digit.

Question 1.
9,000,000
Answer: Nine million

Explanation:
the value of the underlined digit is in millions place.

Question 2.
485,002,000
Answer: four hundred and eighty five million and two thousands

Explanation:
the value of the underlined digit is in millions place.

Question 3.
25,678
Answer: Twenty five thousand and six hundred and seventy eight

Explanation:
the value of the underlined digit is in thousands place.

Question 4.
17,874,000
Answer: seventeen million and eight hundred and seventy four thousand

Explanation:
the value of the underlined digit is in ten millions place.

Set C
pages 13-16, 17-20
A place-value chart can help you write the standard form, expanded form, and number name for a decimal.

Standard form: 8.026
expanded form: 8 + 2 × \(\frac{1}{100}\) + 6 × \(\frac{1}{1,000}\)
Number name: Eight and twenty-six thousandths

Remember the word and is written for the decimal point.

Question 1.
How can you write 0.044 as a fraction?
How are the values of the two 4s related in 0.044?
Answer: \(\frac{44}{1,000}\) and The first 4 is in hundredths place and last 4 is in thousandths place

Explanation:
\(\frac{44}{1,000}\) = 0.044,
The first 4 is in hundredths place and last 4 is in thousandths place

Write each number in standard form.

Question 2.
eight and fifty-nine hundredths
Answer: 8.59

Question 3.
seven and three thousandths + 4. 3 + 2 × \(\frac{1}{10}\) + 4 × \(\frac{1}{1000}\)
Answer: 11.507

Explanation:
Given, seven and three thousandths + 4. 3 + 2 × \(\frac{1}{10}\) + 4 × \(\frac{1}{1000}\)
7.003 + 4.3 + [ 0.2 + 0.004]
= 11.303 + 0.204
= 11.507

Set D
pages 21-24 Compare. Write >, <, or =.

8. 45 8.47
Line up the decimal points. Start at the left to compare. Find the first place where the digits are different.
8.45
8.47 0.05 < 0.07
So, 8.45 < 8.47.

Remember that equivalent decimals, such as 0.45 and 0.450, can help you compare numbers.
Compare. Write >, <, or =.

Question 1.
0.584 0.58
Answer: 0.584 > 0.58

Explanation:
Given,  0.584 and 0.58
0.584
0.58
Comparing the numbers after decimals,
4 > 0
So, 0.584 > 0.58

Question 2.
9.327 9.236
Answer: 9.327 > 9.236

Explanation:
Given, 9.327 and 9.236
9.327
9.236
Comparing the numbers after decimals,
3 > 2
So, 9.327 > 9.236

Question 3.
5.2 5.20
Answer: 5.2 = 5.20

Explanation:
Given, 5.2 and 5.20
5.2
5.20
Comparing the numbers after decimals, we have
All the numbers are equal
So, 5.2 = 5.20

Question 4.
5.643 5.675
Answer: 5.643 < 5.675

Explanation:
Given, 5.643 and 5.675
5.643
5.675
Comparing the numbers after decimals, we have
4 < 7
So, 5.643 < 5.675

Question 5.
0.07 0.08
Answer: 0.07 < 0.08.

Explanation:
Given,  0.07 and 0.08
0.07
0.08
Comparing the numbers after decimals, we have
7 < 8
So, 0.07 < 0.08.

Set E
pages 25-28
Round 12.087 to the place of the underlined digit.
12.087 Look at the digit following the underlined digit. Look at 7.
Round to the next greater number of hundredths because 7 > 5.
12.087 rounded to the nearest hundredth is 12.09.

Remember that rounding a number means replacing it with a number that tells about how many or how much.
Round each number to the place of the underlined digit.

Question 1.
10.245
Answer: 10.2

Explanation:
Given, 10.245
2 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 10.2
The rounding number is 10.2.

Question 2.
73.4
Answer: 73

Explanation:
Given, 73.4
7 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 73
The rounding number is 73.

Question 3.
0.145
Answer: 0.1

Explanation:
Given, 0.145
1 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 0.1
The rounding number is 0.1.

Question 4.
3.999
Answer: 4

Explanation:
Given, 3.999
9 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 4
The rounding number is 4.

Question 5.
13.023
Answer: 13

Explanation:
2 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 13
The rounding number is 13.

Question 6.
45.398
Answer: 45

Explanation:
3 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 45
The rounding number is 45.

Set F
pages 29-32
Think about these questions to help you look for and use structure to understand and Explain patterns with decimal numbers.

Thinking Habits
• What patterns can I see and describe?
• How can I use the patterns to solve the problem?
• Can I see expressions and objects in different ways?
• What equivalent expressions can I use?

Remember to check that all of your answers follow a pattern.
Each grid is part of a decimal number chart. Write the missing numbers to complete the grids.

Question 1.

Answer:

Explanation:
As you move down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same, except for the last number, while the hundredths increase by 1.

Question 2.

Answer:

Explanation:
As you move down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same, except for the last number, while the hundredths increase by 1.

Topic 1 Assessment Practice

Question 1.
Complete the sentences to make true statements.
6 is 100 times ____.
0.06 is 10 times ______.
60 is \(\frac{1}{100}\) of ____.
Answer: 600, 0.6, 0.6

Explanation:
6 × 100 = 600
0.06 × 10 = 0.6
60 × \(\frac{1}{100}\) = 0.6

Question 2.
A national park has eighty thousand, nine-hundred twenty-three and eighty-six hundredths acres of land. Which shows this in standard form?
A. 80,923.086
B. 80,923.68
C. 80,923.806
D. 80,923.86
Answer: A , that is 80,923.086

Explanation:
Because, the number name of 80,923.086 is eighty thousand, nine-hundred twenty-three and eighty-six hundredths

Question 3.
Which numbers have a digit in the ones place that is \(\frac{1}{10}\) the value of the digit in the tens place? Select all that apply.
9,077
9,884
1,303
1,055
3,222
Answer: 9,884 and  3,222

Explanation:
These numbers 9,884 and  3,222 have a digit in the ones place that is \(\frac{1}{10}\) the value of the digit in the tens place

Question 4.
Mrs. Martin has $7,000 in her savings account. Alonzo has \(\frac{1}{10}\) to as much money in his account as Mrs. Martin. How much money does Alonzo have in his account?
Answer: Amount Alonzo has is $ 700

Explanation:
he amount Martin has in her account is $ 7000
Alonzo has 1/10th the amount Martin has
Therefore Alonzo has \(\frac{1}{10}\) of 7000 ;
= 7000/10 = $ 700
Amount Alonzo has is $ 700 .

Question 5.
Select all the comparisons that are true.
04.15 > 4.051
1.054 > 1.45
5.14 < 5.041
5.104 < 5.41
5.014 < 5.41
Answer:  04.15 > 4.051, 5.104 < 5.41 and 5.014 < 5.41

Explanation:
Because of their place value ,
04.15 > 4.051
1 > 0
So, 04.15 > 4.051

5.104 < 5.41
1 < 4
So, 5.104 < 5.41

5.014 < 5.41
0 < 4
So, 5.014 < 5.41

Question 6.
Luke shaded 20 squares on his hundredths grid. Bekka shaded 30 squares on her hundredths grid.

A. Whose grid represents the larger decimal?
B. Write two decimals equivalent to Luke’s decimal.
Answer: 0.2 < 0.3, The two decimals equivalent to Luke’s decimal are 0.20 and 0.21

Explanation:
Given, Luke shaded 20 squares on his hundredths grid, \(\frac{20}{100}\) that is 0.2
Bekka shaded 30 squares on her hundredths grid. \(\frac{30}{100}\)that is 0.3
Bekka has largest decimal.

The two decimals equivalent to Luke’s decimal are 0.20 and 0.21

Question 7.
Which statements about the values of 2.044 and 20.44 are true? Select all that apply.
2.044 is \(\frac{1}{10}\) of 20.44.
2.044 is \(\frac{1}{100}\) of 20.44.
20.44 is 10 times 2.044.
20.44 is 100 times 2.044.
2.044 is 10 times 20.44.
Answer:  20.44 is 10 times 2.044.

Explanation:
Given, 20.44 is 10 times 2.044.
20.44 × 10 = 2.044.

Question 8.
The weight of Darrin’s phone is 3.405 ounces. What is 3.405 written in expanded form?
A. 3 × 1 + 4 × \(\frac{1}{10}\) + 5 × \(\frac{1}{1,000}\)
B. 3 × 10 + 4 × \(\frac{1}{10}\) + 5 × \(\frac{1}{1,000}\)
C. 3 × 10 + 4 × \(\frac{1}{10}\) + 5 × \(\frac{1}{100}\)
D. 3 × 1 + 4 × \(\frac{1}{100}\) + 5 × \(\frac{1}{1,000}\)
Answer:  D

Explanation:
Given, 3.405
The expanded form, 3 × 1 + 4 × \(\frac{1}{100}\) + 5 × \(\frac{1}{1,000}\)
= 3 +0.04 + 0.005
= 3.405

Question 9.
Elaine has a piece of wire that is 2.16 meters long. Dikembe has a piece of wire that is 2.061 meters long. Whose piece of wire is longer? How can you tell?
Answer: Dikembe has the longest wire.

Explanation:
Given, 2.16 and 2.061
2.16
2.061
Comparing the place values of the digits after decimal
1 > 0
So, 2.16 > 2.061
Dikembe has the longest wire.

Question 10.
In a basketball tournament, Dimitri averaged 12.375 rebounds per game. What is 12.375 written in expanded form? How is it written with number names?
Answer: The expanded form is (12 × 1) + {3 × \(\frac{1}{10}\) + 7 × \(\frac{1}{100}\) + 5 × \(\frac{1}{1,000}\)}

Explanation:
Given, 12.375
The expanded form is (12 × 1) + {3 × \(\frac{1}{10}\) + 7 × \(\frac{1}{100}\) + 5 × \(\frac{1}{1,000}\)}
12 + 0.3 + 0.07 + 0.005
= 12.375.
The number name is twelve thousand and three hundred and seventy five.

Question 11.
The numbers below follow a pattern.
0.006 0.06 0.6 6 ____ _____
A. What are the next two numbers in the pattern?
B. What is the relationship between the terms in the pattern?
Answer:  60 and 600 , The terms relate to each other in that the next term is 10 times the previous term.

Explanation:
Relationship between term: The next term is ten times the previous term.
0.006 = 6/1000
0.06 = 6/100
0.6 = 6/10
6 = 6/1
This is a geometric sequence, where the common ratio is 10. 0.6/0.06 = 6/0.6 = 0.06/0.006 = 10
To get next number we simply multiply the previous number by 10.
The terms relate to each other in that the next term is 10 times the previous term. 

Question 12.
Kendra and her horse completed the barrel racing course in 15.839 seconds. What is this number rounded to the nearest tenth? Explain how you decided.
Answer: The rounding number is 16.

Explanation:
Given, 15.839
5 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 16
The rounding number is 16.

Topic 1 Performance Task

Fruits and Vegetables
Henry recorded how many pounds of fruits and vegetables his family bought during the past two months.

Question 1.
Pick four fruits and list them in the table below.
Part A
Round each fruit’s weight to the nearest 0.1 pound. Write the rounded weight in the next column.

Answer:

Part B
Explain how you rounded the weights of the fruits.
Answer:  detailed explanation is given below.

Explanation:
For Apples, given 2.068
5 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 2
The rounding number is 2.

For Blueberries 1.07
0 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 1.1
The rounding number is 1.1.

For lemons 1.031
0 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 1
The rounding number is 1.

For oranges 3.502
0 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 3.5
The rounding number is 3.5.

For peaches 2.608
0 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 2.6
The rounding number is 2.6.

For pears 3.592
5 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 3.6
The rounding number is 3.6.

Question 2.
Pick four vegetables and list them in the table below.

Part A
Round each vegetable’s weight to the nearest 0.01 pound. Write the rounded weight in the next column.

Answer:

Part B
Explain how you rounded the weights of the vegetables.
Answer: Detailed explanation is given below

Explanation:

For Beets 1.862
6 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 1.86
The rounding number is 1.86.

For Celery 1.402
0 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 1.4
The rounding number is 1.4.

For Corn 2.556
5 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 2.56
The rounding number is 2.56.

For potatoes 3.441
4 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 3.44
The rounding number is 3.44.

Question 3.
Use <, >, or= to compare the weights of blueberries and lemons.
Answer:

Question 4.
When rounded to the nearest hundredth, two items will round to the same decimal. What two items are they?
Answer:

Question 5.
How does writing the weight for potatoes in expanded form show why the same digit can have different values?
Answer: Because of the place value of the numbers in the decimals

Explanation:
For potatoes 3.441
4 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 3.44
The rounding number is 3.44.

The first 4 has the place value of tenths and the last 4 has the place value of hundredths.

Question 6.
What is the relationship between the values of the two 4s in the weight of the potatoes?
Answer: The first 4 has the place value of tenths and the last 4 has the place value of hundredths. and the first 4 is ten times the value of second 4.

Explanation:

For potatoes 3.441
4 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 3.44
The rounding number is 3.44.

The first 4 has the place value of tenths and the last 4 has the place value of hundredths.
and the first 4 is ten times the value of second 4.

Question 7.
Write the number of pounds of celery Henry’s family bought using number names and in expanded form.
Answer: The number name is 1.4 pounds
The expanded form is 14 × \(\frac{1}{10}\)

Explanation:
For Celery 1.402
0 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 1.4
The rounding number is 1.4.

The number name is 1.4 pounds
The expanded form is 14 × \(\frac{1}{10}\)

Question 8.
The store where Henry’s family shops sold 103 times as many pounds of corn as Henry’s family bought.
Answer:263.68 pounds have sold.

Explanation:
For Corn 2.556
5 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 2.56
The rounding number is 2.56.

Given, Henry’s family shops sold 103 times as many pounds of corn as Henry’s family bought.
2.56 × 103 = 263.68

Part A
How many pounds of corn did the store sell? Write your answer in standard form and with number names.

Answer: The number form is twenty six thousand and three hundred and sixty eight.

Explanation:
Store has sold 263.68
The number form is twenty six thousand and three hundred and sixty eight.

Part B
Explain how you found your answer.
Answer:  detailed explanation is given below

Explanation:
For Corn 2.556
5 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 2.56
The rounding number is 2.56.

Given, Henry’s family shops sold 103 times as many pounds of corn as Henry’s family bought.
2.56 × 103 = 263.68

Go through the enVision Math Common Core Grade 5 Answer Key Topic 11 Understand Volume Concepts regularly and improve your accuracy in solving questions.

enVision Math Common Core 5th Grade Answers Key Topic 11 Understand Volume Concepts

Essential Questions: What is the meaning of volume of a solid? How can the volume of a rectangular prism be found?

enVision STEM Project: Everyday Energy
Do Research Use the Internet and other sources to learn more about these five types of energy: electrical, light, mechanical, sound, and thermal. Make a table of the various types of energy you use every day. Include at least one example of how you use each type of energy.

Journal: Write a Report Include what you found. Also in your report:
• Draw a diagram of your classroom and label where and how 3 types of energy are used.
• Estimate how far your desk is from a light energy source and add this dimension to your sketch.
• Use your diagram to make up and solve problems involving measurements such as the volume of your classroom.

Review What You Know

A-Z Vocabulary

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

• compensation
• partial products
• rectangle
• unit fraction

Question 1.
Adjusting a number to make a computation easier and balancing the adjustment by changing another number is called ____
Answer:

Adjusting a number to make a computation easier and balancing the adjustment by changing another number is called compensation

Question 2.
A fraction with a numerator of 1 is called a ____
Answer:

A fraction with a numerator of 1 is called a  unit fraction

Question 3.
A quadrilateral with 2 pairs of parallel sides that are the same length and 4 right angles is a ____
Answer:

A quadrilateral with 2 pairs of parallel sides that are the same length and 4 right angles is a rectangle

Area
Find the area of each figure.

Question 4.

Answer:

From the figure,

Length = 10h

breadth =6h

Area of rectangle = Length x Breadth = 10h x 6h = 60h^2

Question 5.

Answer:

From the figure,

Length = 12cm

breadth = 8cm

Area of rectangle = Length x Breadth = 12cm x 8cm = 96cm^2

Operations

Find each product or quotient.

Question 6.
16 × 6
Answer:

96

Question 7.
3 × 42
Answer:

126

Question 8.
216 ÷ 3
Answer:

Question 9.
128 ÷ 4
Answer:

Question 10.
(5 × 6) × 3
Answer:

(5 × 6) × 3

(30) x 3

90

Question 11.
(6 × 6) × 6
Answer:

(6 × 6) × 6

(36) x 6

=196

Question 12.
Joanie has two 12-inch-long wood pieces and two 16-inch-long wood pieces. What is the combined length of the wood pieces?
A. 28 inches
B. 32 inches
C. 56 inches
D. 192 inches
Answer:

Given,

Number of 12 inch wood pieces = 2 = 12 x 2 = 24

Number of 16 inch wood pieces = 2 = 16 x 2 = 32

combined length of the wood pieces = 24 + 32 = 56

C. 56 inches is correct

Finding Area

Question 13.
Niko used square tiles to make a rectangle with 2 rows and 7 tiles in each row. Explain how you can find the area of the rectangle.
Answer:

Pick a Project

PROJECT 11A
How big are skyscrapers?
Project: Build a Skyscraper with Unit Cubes

PROJECT 11B
Why do cats climb into boxes?
Project: Design a Cat Tree

PROJECT 11C
Why are trucks useful for transporting packages?
Project: Model a Truck’s Capacity

3-ACT MATH PREVIEW

Math Modeling

Video

Fill ‘er Up

Before watching the video, think:
Ice is frozen below 32°F (or 0°C), but most picnics and cookouts happen when it’s warm out. The insulated walls of a cooler help keep ice from melting, which keeps my juice nice and cold!

Lesson 11.1 Model Volume

Activity

Solve & Share
Gina is building a rectangular prism out of sugar cubes for her art class project. She started by drawing a diagram of the rectangular prism that is 4 cubes high, 4 cubes long and 2 cubes wide. How many cubes does she use to make the prism? Solve this problem any way you choose.

Use Appropriate Tools You can use cubes to build a rectangular prism and then draw a picture to show the different faces. Show your work!

Look Back! Gina decided to change her art project and build a rectangular prism that is 3 cubes long, 4 cubes wide, and 2 cubes high. Use the picture to determine the number of cubes she used.

Visual Learning Bridge

Essential Question How Can You Measure Space Inside a Solid Figure?

A.
Volume is the number of cubic units needed to pack a solid figure without gaps or overlaps. A cubic unit is the volume of a cube measuring 1 unit on each edge. What is the volume of this rectangular prism?

Each cube of a solid figure is 1 cubic unit.

B.
Use unit cubes to make a model.

Count the number of cubes.
There are 15 unit cubes in the bottom layer. The volume of the bottom layer is 15 cubic units.

C.
There are two layers.

Multiply the volume of the bottom layer by 2.
The volume of the prism is 2 × 15 or 30 cubic units.

Convince Me! Reasoning in the picture below, how many unit cubes does it take to make the rectangular prism below without gaps or overlaps? How many 2-cube towers does it take to make the rectangular prism?

Guided Practice

Do You Understand?

Question 1.
Make a model of a rectangular prism with a bottom layer that is 3 cubes long by 3 cubes wide. Make a top layer that is the same as the bottom layer. Then draw a picture of your model. What is the volume?
Answer:

Each cube of a solid figure is 1 cubic unit.

There are 9 unit cubes in the bottom layer. The volume of the bottom layer is 9 cubic units.

Multiply the volume of the bottom layer by 2.

The volume of the prism is 2 × 9 or 18 cubic units.

Question 2.
A-Z Vocabulary What is the difference between a unit cube and a cubic unit?
Answer:

A unit cube, more formally a cube of side 1, is a cube whose sides are 1 unit long. The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units.

Do You Know How?

In 3 and 4, use unit cubes to make a model of each rectangular prism. Find the volume.

Question 3.

Answer:

Each cube of a solid figure is 1 cubic unit.

There are 4 unit cubes in the bottom layer. The volume of the bottom layer is 4 cubic units.

Multiply the volume of the bottom layer by 2.

The volume of the prism is 2 × 4 or 8 cubic units.

Question 4.

Answer:

Each cube of a solid figure is 1 cubic unit.

There are 12 unit cubes in the bottom layer. The volume of the bottom layer is 12 cubic units.

The volume of the prism is 12 × 1 or 12 cubic units.

Independent Practice

In 5-13, find the volume of each solid. Use unit cubes to help.

Question 5.

Answer:

Each cube of a solid figure is 1 cubic unit.

There are 9 unit cubes in the bottom layer. The volume of the bottom layer is 9 cubic units.

Multiply the volume of the bottom layer by 3.

The volume of the prism is 9 × 3 or 27 cubic units.

Question 6.

Answer:

Each cube of a solid figure is 1 cubic unit.

There are 8 unit cubes in the bottom layer. The volume of the bottom layer is 8 cubic units.

Multiply the volume of the bottom layer by 3.

The volume of the prism is 8 × 3 or 24 cubic units.

Question 7.

Answer:

Each cube of a solid figure is 1 cubic unit.

There are 20 unit cubes in the bottom layer. The volume of the bottom layer is 20 cubic units.

Multiply the volume of the bottom layer by 2.

The volume of the prism is 20 × 2 or 40 cubic units.

Question 8.

Answer:

Each cube of a solid figure is 1 cubic unit.

There are 8 unit cubes in the bottom layer. The volume of the bottom layer is 8 cubic units.

Multiply the volume of the bottom layer by 2.

The volume of the prism is 8 × 2 or 16 cubic units.

Question 9.

Answer:

Each cube of a solid figure is 1 cubic unit.

There are 2 unit cubes in the bottom layer. The volume of the bottom layer is 2 cubic units.

Multiply the volume of the bottom layer by 5.

The volume of the prism is 2 × 5 or 10 cubic units.

Question 10.

Answer:

Each cube of a solid figure is 1 cubic unit.

There are 6 unit cubes in the bottom layer. The volume of the bottom layer is 6 cubic units.

Multiply the volume of the bottom layer by 3.

The volume of the prism is 6 × 3 or 18 cubic units.

Question 11.

Answer:

Each cube of a solid figure is 1 cubic unit.

There are 12 unit cubes in the bottom layer. The volume of the bottom layer is 12 cubic units.

Multiply the volume of the bottom layer by 3.

The volume of the prism is 12 × 3 or 36 cubic units.

Question 12.

Answer:

Each cube of a solid figure is 1 cubic unit.

There are 12 unit cubes in the bottom layer. The volume of the bottom layer is 12 cubic units.

Multiply the volume of the bottom layer by 2.

The volume of the prism is 12 × 2 or 24 cubic units.

Question 13.

Answer:

Each cube of a solid figure is 1 cubic unit.

There are 10 unit cubes in the bottom layer. The volume of the bottom layer is 10 cubic units.

The remaining cubes are 4 x 3 or 12 and the volume is 12 cubic units

The volume of the prism is 10 + 12 or 22 cubic units.

Problem Solving

In 14-18, use the table.

Compare the volumes of the prisms. Write >, <, or = for each

Question 14.
Prism A Prism B
Answer:

Prism A > Prism B

Question 15.
Prism B Prism C
Answer:

Prism B = Prism C

Question 16.
Prism C Prism A
Answer:

Prism C < Prism A

Question 17.
If you added another layer of unit cubes on top of Prism A, what would the volume of the new solid be in cubic units?
Answer:

Each cube of a solid figure is 1 cubic unit.

There are 8 unit cubes in the bottom layer. The volume of the bottom layer is 8 cubic units.

Multiply the volume of the bottom layer by 2.

The volume of the prism is 8 × 2 or 16 cubic units.

Question 18.
If you put Prism C on top of Prism A, what would the volume of the new solid be in cubic units?
Answer:

The volume of A is 16 cubic units

The volume of C is 6 cubic units

If you put Prism C on top of Prism A, Then the volume is 16 + 6 or 22 cubic units

Question 19.
Reasoning In an election, 471 people voted. Candidate B received \(\frac{2}{3}\) of the votes. How many votes did Candidate B receive?
Answer:

Number of votes voted = 471

Candidate B received 2/3 of the votes

Number of votes B received = 2/3 x 471 =314

Question 20.
Higher Order Thinking Ms. Kellson’s storage closet is 3 feet long, 3 feet wide, and 7 feet high. Can she fit 67 boxes that each have a volume of 1 cubic foot in her closet? Explain your answer.
Answer:

No.

Each cube of a solid figure is 1 cubic unit.

There are 9 unit cubes in the bottom layer. The volume of the bottom layer is 9 cubic units.

Multiply the volume of the bottom layer by 7.

The volume of the prism is 9 × 7 or 63 cubic units.

She can not fit 67 boxes that each have a volume of 1 cubic foot in her closet.

Assessment Practice

Question 21.
Natalie made the solid figures shown using unit cubes. Which statement about these models is true?

A. Model X and Model Y have the same volume.
B. The volume of Model X is 9 cubic units greater than the volume of Model Y.
C. The volume of Model X is 19 cubic units greater than the volume of Model Y.
D. The volume of Model X and Model Y combined is 45 cubic units.
Answer:

C. The volume of Model X is 19 cubic units greater than the volume of Model Y.

Visual Learning Bridge

Essential Question How Can You Use a Formula to Find Stor the Volume of a Rectangular Prism?

A.
Remember that volume is the number of cubic units (units”) needed to pack a solid figure without gaps or overlaps.
Find the volume of the rectangular prism if each cubic unit represents 1 cubic foot.

A formula is a rule that uses symbols to relate two or more quantities.

You can find the volume of a rectangular prism by counting cubes or using a formula.

B.
If the dimensions of a rectangular prism are given as length l, width w, and height h, then use this formula to find the volume V:

Volume = length × width × height
V = l × w × h
V = (6 × 4) × 3
V = 24 × 3
V = 72
The volume of the rectangular prism is 72 cubic feet or 72 ft³.

C.
Another formula for the volume of a rectangular prism is V = b × h, where b is the area of the base.
V = b × h.
V = 24 × 3
V = 72 ft³

Convince Me! Reasoning Give the dimensions of a different rectangular prism that also has a volume of 72 ft³. Explain how you decided.

Guided Practice

Do You Understand?

Question 1.
In the Example on page 462, could you first multiply the width by the height? Explain.
Answer:

Question 2.
A wooden block measures 5 centimeters tall, 3 centimeters wide, and 2 centimeters long. The area of the base is 6 centimeters. Draw a rectangular prism to show the block and label it. What is the volume of the block?
Answer:

Do You Know How?

In 3 and 4, find the volume of each rectangular prism.

Question 3.

Answer:

Volume = length × width × height
V = l × w × h
V = (9 × 2) × 4
V = 18 × 4
V = 72
The volume of the rectangular prism is 72 cubic inch or 72 in³.

Question 4.

Answer:

The volume of a rectangular prism is V = b × h, where b is the area of the base.
V = b × h.
V = 24 × 9
V = 216 yd³

Independent Practice

In 5-10, find the volume of each rectangular prism.

Question 5.

Answer:

Volume = length × width × height
V = l × w × h
V = (7 × 4) × 3
V = 28 × 3
V = 84

The volume of the rectangular prism is 84 cubic cm or 72 cm³.

Question 6.

Answer:

Volume = length × width × height
V = l × w × h
V = (4 × 4) × 5
V = 16 × 5
V = 80
The volume of the rectangular prism is 80 cubic inch or 80 in³.

Question 7.

Answer:

The volume of a rectangular prism is V = b × h, where b is the area of the base.
V = b × h.
V = 52 × 3
V = 156 m³

Question 8.

Answer:

The volume of a rectangular prism is V = b × h, where b is the area of the base.
V = b × h.
V = 64 × 4
V = 256 cm³

Question 9.

Answer:

Volume = length × width × height
V = l × w × h
V = (7 × 7) × 7
V = 49 × 7
V = 343
The volume of the rectangular prism is 343 cubic meter or 343 m³.

Question 10.

Answer:

The volume of a rectangular prism is V = b × h, where b is the area of the base.
V = b × h.
V = 153 × 7
V = 1071 cm³

Problem Solving

Question 11.
The dictionary is 3 inches thick. What is the volume of the dictionary?

Answer:

Volume = length × width × height
V = l × w × h
V = (7 × 3) × 9
V = 21 × 9
V = 189
The volume of the rectangular prism is 189 cubic inch or 189 in³.

Question 7.

Question 12.
Higher Order Thinking Two ovens have measurements as shown. Which oven has a greater volume? How much greater is its volume? Show your work.

Answer:

The volume of a First oven is V = b × h, where b is the area of the base.
V = b × h.
V = 576 × 15
V = 8640 in³

The volume of a second oven is V = b × h, where b is the area of the base.
V = b × h.
V = 672 × 14
V = 9408 in³

The second oven has greater volume and 768 in³ greater than the first one.

Question 13.
The perimeter of an equilateral triangle is 51 feet. What is the length of one of its sides? Explain your work.
Answer:

The formula of the perimeter of an equilateral triangle is 3a, where a is the length of one side.

Given, The perimeter of an equilateral triangle is 51 feet

3a = 51

a = 51/3 = 17

Therefore, the length of one of its sides is 17 and all sides are equal.

Question 14.
Reasoning Harry is in line at the store. He has three items that cost $5.95, $4.25, and $1.05. Explain how Harry can add the cost of the items mentally before he pays for them.
Answer:

Given,

Harry has three items that cost $5.95, $4.25, and $1.05.

$5.95 approximately equals $6

$4.25  approximately equals $4

$1.05 approximately equals $1

Therefore, by approximation, Harry can add the cost of the items mentally before he pays for them.

$6 + $4 + $1 = $11. (approximately)

Assessment Practice

Question 15.
Choose all the Expressions that can be used to find the volume of this wooden box.

(6 × 4) < 3
(6 × 4) + 3
6 × 4
6 × (4 × 3)
24 × 3
Answer:

6 × 4

24 × 3

Lesson 11.3 Combine Volumes of Prisms

Activity

Solve & Share
Ariel is thinking of a three-dimensional figure that is made by combining two rectangular prisms. What is the volume of this three-dimensional figure? Solve this problem any way you choose.

Use Structure You can find the volumes of the rectangular prisms that make up the solid figure. Show your work!

Look Back! How did you separate the solid into simpler rectangular prisms? Write the dimensions of each of the prisms.

Visual Learning Bridge

Essential Question How Can You Find the Volume of a Solid Figure question Composed of Two Rectangular Prisms?

A.
The shape and size of a storage building are shown in the figure. The building supervisor wants to find the volume to determine how much storage space is available. What is the volume of the building?

You can find the volume of this figure by finding the volume of two rectangular prisms that make up the figure.

B.
The building can be separated into two rectangular prisms as shown. Identify the measurements for the length, width, and height of each prism.

C.
Use the formula V = l × w × h to find the volume of each rectangular prism.

Add to find the total volume.
180 + 630 = 810
The volume of the storage building is 810 cubic meters.

Convince Me! Reasoning What is another way to divide the solid above into two rectangular prisms? What are the dimensions of each prism?

Guided Practice

Do You Understand?

In 1 and 2, use the solid below. The dashed line separates it into two rectangular prisms, A and B.

Question 1.
What are the length, width, and height of Prism A? What are the length, width, and height of Prism B?
Answer:

length, width, and height of Prism A are 4 ft, 3 ft, 12 ft respectively.

length, width, and height of Prism B are 3 ft, 2 ft, 5 ft respectively.

Question 2.
What is another way you could separate the shape into two rectangular prisms? What are each prism’s dimensions?
Answer:

Bisecting the figure horizontally, another way I could separate the shape into two rectangular prisms.

length, width, and height of Prism A are 4 ft, 3 ft, 7 ft respectively.

length, width, and height of Prism B are 6 ft, 3 ft, 5 ft respectively.

Do You Know How?

In 3 and 4, find the volume of each solid figure.

Question 3.

Answer:

Volume = length × width × height
Volume of A = l × w × h
V = (5 × 7) × 8
V = 35 × 8
V = 280

The volume of the A is 280 cubic in or 280 in³.

Volume = length × width × height
Volume of B  = l × w × h
V = (10 × 7) × 2
V = 70 × 2
V = 140

The volume of the B is 140 cubic in or 140 in³.

Combined Volume  of Solid Figure is 280 in³ + 140 in³  = 420 in³ 

Question 4.

Answer:

Volume = length × width × height
Volume of A = l × w × h
V = (5 × 8) × 5
V = 40 × 5
V = 200

The volume of the A is 200 cubic cm or 200 cm³.

Volume = length × width × height
Volume of B  = l × w × h
V = (7 × 8) × 2
V = 56 × 2
V = 112

The volume of the B is 112 cubic cm or 112 cm³.

Combined Volume  of Solid Figure is 200 cm³ + 112 cm³  = 312 in³

Independent Practice

In 5-7, find the volume of each solid figure.

Question 5.

Answer:

Volume = length × width × height
Volume of A = l × w × h
V = (8 × 4) × 7
V = 32 × 7
V = 224

The volume of the A is 224 cubic cm or 224 cm³.

Volume = length × width × height
Volume of B  = l × w × h
V = (6 × 4) × 3
V = 24 × 3
V = 72

The volume of the B is 72 cubic cm or 72 cm³.

Combined Volume  of Solid Figure is 224 cm³ + 72 cm³  = 296 cm³

Question 6.

Answer:

Volume = length × width × height
Volume of A = l × w × h
V = (6 × 10) × 5
V = 60 × 5
V = 300

The volume of the A is 300 cubic ft or 300 ft³.

Volume = length × width × height
Volume of B = l × w × h
V = (6 × 2) × 4
V = 12 × 4
V = 48

The volume of the B is 48 cubic ft or 48 ft³.

Combined Volume  of Solid Figure is 300 ft³ + 48 ft³  = 348 ft³

Question 7.

Answer:

Volume = length × width × height
Volume of A = l × w × h
V = (4 × 6) × 12
V = 24 × 12
V = 288

The volume of the A is 288 cubic m or 288 m³.

Volume = length × width × height
Volume of B  = l × w × h
V = (9 × 5) × 12
V = 45 × 12
V = 540

The volume of the B is 540 cubic m or 540 m³.

Combined Volume  of Solid Figure is 288 m³ + 540 m³  = 828 m³

Problem Solving

For 8-10, use the drawing of the solid figure.

Question 8.
How would you find the volume of the figure shown?
Answer:

Question 9.
Algebra Write two Expressions that can be added to find the volume of the solid figure.
Answer:

Volume = length × width × height
Volume of A = l × w × h
V = (5 × 4) × 10
V = 20 × 10
V = 200

The volume of the A is 200 cubic m or 200 m³.

Volume = length × width × height
Volume of B  = l × w × h
V = (5 × 2) × 7
V = 10 × 7
V = 70

The volume of the B is 70 cubic m or 70 m³.

Question 10.
What is the volume of the solid figure?
Answer:

Combined Volume  of Solid Figure is 200 m³ + 70 m³  = 270 m³

Question 11.
Higher Order Thinking A solid figure is separated into two rectangular prisms. The volume of Rectangular Prism A is 80 cubic feet. Rectangular Prism B has a length of 6 feet and a width of 5 feet. The total volume of the solid figure is 200 cubic feet. What is the height of Rectangular Prism B? Show your work.
Answer:

Question 12.
Model with Math The Peters family will drive 615 miles to reach their vacation destination. If they drive 389 miles the first day, how many miles will they drive the second day? Complete the bar diagram to help.

Answer:

Assessment Practice

Question 13.
A horizontal line separates the solid figure at the right into two rectangular prisms. Write an Expression for the volume of the solid figure.

Answer:

Volume of A  = length × width × height
V = l × w × h
V = (6 × 10) × 5
V = 60 × 5
V = 300

The volume of the rectangular prism is 300 cubic in or 300 in³.

Volume of B  = length × width × height
V = l × w × h
V = (10 × 10) × 2
V = 100 × 2
V = 200

The volume of the rectangular prism is 200 cubic in or 200 in³.

Total volume =  200 in³ + 300 in³  = 500 in³

Lesson 11.4 Solve Word Problems Using Volume

Activity

Solve & Share
A school has two wings, each of which is a rectangular prism. The school district is planning to install air conditioning in the school and needs to know its volume. What is the volume of the school? Solve this problem any way you choose.

Model with Math Write a multiplication Expression for the volume of each wing of the building.

Look Back! Write a mathematical Expression that can be used to find the total volume of the school.

Visual Learning Bridge

Essential Question How Can You Use Volume Formulas to Solve Real-World Problems?

A.
The nature center has a large bird cage called an aviary. It consists of two sections, each shaped like a rectangular prism. There needs to be 10 cubic feet of space for each bird. How many birds can the nature center have in the aviary?

You can make sense of the problem by breaking it apart into simpler problems.

B.
Find the volume of each section. Use the formula V = l × w × h.
Small section:
V = 4 × 3 × 8 = 96
Large section:
V = 10 × 6 × 8 = 480
Add to find the total volume:
96 + 480 = 576
The combined volume is 576 cubic feet.

C.
Divide to find the number of birds that will fit.

576 ÷ 10 = 57.6
The nature center can put 57 birds in the aviary.

Convince Me! Critique Reasoning Tom solved the problem in a different way. First, he found the total area of the floor, and then he multiplied by the height. Does Tom’s method work? Explain.

Guided Practice

Do You Understand?

Question 1.
How can you find the volume of the china cabinet?

Question 2.
What is the height of the top section of the china cabinet? Explain.

Height of the top section of the china cabinet is 4 ft

Because, 7 ft – 3 ft = 4 ft

Question 3.
Find the volume of the china cabinet.
Answer:

Volume of china cabinet  = length × width × height
V = l × w × h
V = (4 × 1) × 4
V = 4 × 4
V = 16

The volume of the rectangular prism is 16 cubic ft or 16 ft³.

Do You Know How?

Question 4.
Find the volume of the building below.

Answer:

Volume of A = length × width × height
V = l × w × h
V = (35 × 54) × 82
V = 1890 × 82
V = 154,980

The volume of the rectangular prism is 154,980 cubic ft or 154,980 ft³.

Volume of B = length × width × height
V = l × w × h
V = (40 × 30) × 50
V = 1200 × 50
V = 60000

The volume of the rectangular prism is 60000 cubic ft or 60000 ft³

Therefore, Combined volume is  154,980 ft³ + 60000 ft³  = 214,980 ft³

Question 5.
The nature center has a fish tank shaped like a rectangular prism that measures 6 feet long by 4 feet wide by 4 feet high. It can be stocked safely with 3 small fish in each cubic foot of water. How many small fish can safely fit in the tank?
Answer:

Volume of rectangular prism  = length × width × height
V = l × w × h
V = (6 × 4) × 4
V = 24 × 4
V = 96

The volume of the rectangular prism is 96 cubic ft or 96 ft³.

If it can be stocked safely with 3 small fish in each cubic foot of water. Then 96/3 = 12

12 fish can be stocked safely with 3 small fish in each cubic foot of water.

Independent Practice

Question 6.
Sophie built a house out of building blocks. Find the volume of the house Sophie built.

Answer:

Volume of A = length × width × height
V = l × w × h
V = (3 × 5) × 10
V = 15 × 10
V = 150

The volume of the rectangular prism is 150 cubic cm or 150 cm³.

Volume of B = length × width × height
V = l × w × h
V = (20 × 24) × 20
V = 480 × 20
V = 9600

The volume of the rectangular prism is 9600 cubic cm or 9600 cm³

Therefore, Combined volume is  150 cm³ + 9600 cm³  = 9750 cm³

Question 7.
How many cubic inches of concrete would it take to make these stairs?

Answer:

Volume of first stair = length × width × height
V = l × w × h
V = (40 × 10) × 7
V = 400 × 7
V = 2800

The volume of the first stair is 2800 cubic in or 2800 in³.

Volume of Second stair = length × width × height
V = l × w × h
V = (40 × 10) × 14
V = 400 × 14
V = 5600

The volume of the second stair is 5600 cubic in or 5600 in³

Volume of third stair = length × width × height
V = l × w × h
V = (40 × 10) × 21
V = 400 × 21
V = 8400

The volume of the rectangular prism is 8400 cubic in or 8400 in³

Therefore, Combined volume is  2800 in³ + 5600 in³ + 8400 in³  = 16800 in³

16800 cubic inches of concrete it takes to make these stairs

Problem Solving

Question 8.
A floor plan of Angelica’s bedroom and closet is shown at the right. The height of the bedroom is 9 feet. The height of the closet is 7 feet. What is the total volume of the bedroom and the closet?

Answer:

Volume of bedroom = length × width × height
V = l × w × h
V = (14 × 12) × 9
V = 168 × 9
V = 1512

The volume of the second stair is 1512 cubic ft or 1512 ft³

Volume of  closet = length × width × height
V = l × w × h
V = (3 × 4) × 7
V = 12 × 7
V = 84

The volume of the rectangular prism is 84 cubic ft or 84 ft³

Therefore, Combined volume is  1512 ft³ + 84 ft³  = 11596 ft³

Question 9.
Critique Reasoning Does it make sense for Angelica to find the combined area of the bedroom floor and closet floor before finding the total volume? Explain your thinking.
Answer:

Yes. the total volume is sum of the bedroom and closet of Angelica room

Question 10.
Higher Order Thinking An office building surrounds a rectangular open air courtyard. What is the volume of the building? How did you find the answer?

Answer:

Volume of building = length × width × height
V = l × w × h
V = (64 × 60) × 28
V = 3840 × 28
V = 107,520

The volume of the second stair is 107,520 cubic m or 107,520 m³

Volume of open air courtyard = length × width × height
V = l × w × h
V = (18 × 30) × 28
V = 540 × 28
V = 15120

The volume of the rectangular prism is 15120 cubic m or 15120 m³

Therefore, Total volume of building is  107,520 m³ – 15120 m³  = 92400 m³

Assessment Practice

Question 11.
Mrs. Bhatia’s closet consists of two sections, each shaped like a rectangular prism. She plans to buy mothballs to keep the moths away. She needs one box for every 32 cubic feet of space. How many boxes should she buy?
A. 6 boxes
B. 7 boxes
C. 8 boxes
D. 10 boxes

Answer:

Volume of first section = length × width × height
V = l × w × h
V = (5 × 4) × 8
V = 20 × 8
V = 160

The volume of the first section is 160 cubic ft or 160 ft³

Volume of open second section = length × width × height
V = l × w × h
V = (3 × 2) × 8
V = 6 × 8
V = 48

The volume of the second section is 48 cubic ft or 48 ft³

Therefore, the total volume of the building is  160 ft³ + 48 ft³  = 208 ft³

If she needs one box for every 32 cubic feet of space. Then Number of boxes = 208 / 32 = 6.5

She needs 6 or 7 boxes.

Lesson 11.5 Use Appropriate Tools

Problem Solving

Solve & Share
A space station is being built from 24 cubic modules. The space station can be any shape but the modules must be placed together so that entire faces match up with each other. Choose a tool to create two different plans for the space station. Explain why you chose the tool you selected.

Thinking Habits
Be a good thinker! These questions can help you.
• Which tools can I use?
• Why should I use this tool to help me solve the problem?
• Is there a different tool I could use?
• Am I using the tool appropriately?

Look Back! Use Appropriate Tools How did you decide which tool to use?

Visual Learning Bridge

Glossory

Essential Question How Can You Use Appropriate Tools question to Solve Volume Problems?

A.
Jeremiah needs to build a display of boxes that is 4 feet tall.
The boxes he uses are cubes that measure 1 foot on each edge. His display needs to look like a pyramid, with just one box in the top layer.
How many boxes will Jeremiah need to make his display?

What do I need to do?
I need to choose an appropriate tool to solve this problem.

Here’s my thinking…

B.
How can I use appropriate tools strategically to help me solve this problem?
I can
• decide which tool is appropriate.
• use cubes to solve this problem.
• use the tool correctly.

C.
I could use grid paper, but I will use cubes because building a display will make it easier to count the cubes.
Each cube represents 1 box in the display. My display will have 4 layers because it needs to be 4 feet tall, and each box is 1 foot high.
The display has 1 + 4 + 9 + 16 = 30 cubes.

So, Jeremiah needs 30 boxes in all to make his display.

Convince Me! Use Appropriate Tools What tools other than cubes could you use to solve this problem? Explain.

Guided Practice

Use Appropriate Tools
A paint store manager is going to build a display with same-sized cubes. The display will look like a staircase with 5 steps. Each step in the display will be 6 cubes long. The store manager will build the staircase display with 1-foot plastic cubes.

Question 1.
What tool might the manager use to be sure that there is enough space for the display? Explain.
Answer:

Question 2.
What is the volume of the display? Explain how you used tools to decide.
Answer:

Independent Practice

Use Appropriate Tools

Cindy plans to make a jewelry box shaped like a rectangular prism. She wants it to have a volume of 96 cubic inches.

Question 3.
How can you find possible dimensions of the box?
Answer:

Think about a tool you can use to help represent and solve the problem.

Question 4.
What could the dimensions of that jewelry box be?
Answer:

Question 5.
Can Cindy build the box so that it is twice as wide as it is tall?
Answer:

Question 6.
Cindy has some ribbon to decorate the jewelry box. What tool might help her decide how much of the jewelry box she can decorate?
Answer:

Problem Solving
Performance Task

Flower Planters
An architect is designing flower planters for a park. Each planter consists of a border of 1-foot concrete cubes surrounding a square opening. Each concrete cube weighs 120 pounds. The diagram below shows the top view of some of the planters.

Remember to think about which tools make sense for these problems.

Question 7.
Use Structure What is the total volume of a planter that has a 6 × 6 opening?
Answer:

Question 8.
Use Structure What is the total volume of a planter that has an 8 × 8 opening?
Answer:

Question 9.
Use Appropriate Tools What will be the total volume of a planter that has a 12 × 12 opening? Can you determine this by just using paper and pencil? Explain.
Answer:

Question 10.
Model with Math Each concrete cube used to make the planters costs $3.00. What is the total cost of the cubes needed for two planters with 6 × 6 openings, two with 8 × 8 openings, and two with 12 × 12 openings? Write an Expression that represents the total cost.
Answer:

Topic 11 Fluency Practice

Activity

Find a Match

Work with a partner. Point to a clue. Read the clue.
Look below the clues to find a match. Write the clue letter in the box above the match.
Find a match for every clue.

Answer:

Topic 11 Vocabulary Review

Word List

Glossory

Understand Vocabulary

In 1-3, choose the best term from the Word List. Write it on the blank.
• area
• cubic unit
• formula
• rectangular prism
• unit cube
• volume

Question 1.
The number of same-size unit cubes that fill a solid figure without overlaps or gaps in the figure’s ____
Answer:

The number of same-size unit cubes that fill a solid figure without overlaps or gaps in the figure’s volume

Question 2.
A solid figure with 6 rectangular faces that are not all squares is a(n) ___
Answer:

A solid figure with 6 rectangular faces that are not all squares is a rectangular prism

Question 3.
A rule that uses symbols to relate two or more quantities is a(n) ___.
Answer:

A rule that uses symbols to relate two or more quantities is a formula

Question 4.
Cross out the Expression(s) below that do NOT represent the volume of the prism.

Answer:

Draw a line from each three-dimensional figure in Column A to its volume in Column B.

Answer:

Question 8.
One box is 3 inches by 4 inches by 5 inches. A second box is 4 inches by 4 inches by 4 inches. Explain how to decide which box holds more.
Answer:

Volume of the first box is, V = 3 x 4 x 5 = 60 cubic inches.

Volume of the second box is, V = 4 x 4 x 4 = 64 cubic inches.

The second box holds more because it has more volume.

Topic 11 Reteaching

Set A
pages 457-460

Find the number of cubes needed to make this rectangular prism.

Multiply to find the total number of cubes.
3 × 5 × 3 = 45
The volume is 45 cubic units.

Remember, you can multiply the numbers in any order!

Remember that you can find the number of cubes in each layer and then multiply by the number of layers.

Find each volume. You may use cubes to help.

Question 1.

Answer:

Each cube of a solid figure is 1 cubic unit.

There are 14 unit cubes in the bottom layer. The volume of the bottom layer is 14 cubic units.

The volume of the prism is 14 × 2 or 28 cubic units.

Question 2.

Answer:

Each cube of a solid figure is 1 cubic unit.

There are 10 unit cubes in the bottom layer. The volume of the bottom layer is 10 cubic units.

The volume of the prism is 10 × 2 or 20 cubic units.

Question 3.

Answer:

Each cube of a solid figure is 1 cubic unit.

There are 12 unit cubes in the bottom layer. The volume of the bottom layer is 12 cubic units.

The volume of the prism is 12 × 6 or 72 cubic units.

Set B
pages 461-464
Find the volume of this rectangular prism.

Volume = length × width × height V=l×w×h
= 9 cm × 4 cm × 2 cm
V = 72 cubic centimeters or 72 cm³
The volume of the prism is 72 cm³.

Remember if you know the area of the base of a rectangular prism, use the formula V = b × h, where b is the area of the base.

Find each volume. You may use cubes to help.

Question 1.
Area of the base, b = 42 square meters and height = 3 meters
Answer:

The volume of a rectangular prism is V = b × h, where b is the area of the base.
V = b × h.
V = 42 × 3
V = 126 m³

Question 2.
Area of the base = 75 square inches and height = 15 inches
Answer:

The volume of a rectangular prism is V = b × h, where b is the area of the base.
V = b × h.
V = 75 × 15
V = 1125 in³

Question 3.

Answer:

Volume = length × width × height
Volume = l × w × h
V = (8 × 4) × 3
V = 32 × 3
V = 96

The volume of the B is 96 cubic ft or 70 ft³.

Set C
pages 465-468, 469-472

Some solid figures can be separated into two rectangular prisms.

Add the volume of each prism to find the total volume of the solid figure.
V = (4 × 4 × 5) + (10 × 4 × 5)
= 80 + 200
= 280
The volume of the solid figure is 280 cubic inches.

Remember to identify the length, width, and height of each prism, so that you can calculate the volume of each part.

Question 1.
Find the volume.

Answer:

Volume = length × width × height
Volume of A = l × w × h
V = (10 × 6) × 2
V = 60 × 2
V = 120

The volume of the A is 120 cubic cm or 120 cm³.

Volume = length × width × height
Volume of B  = l × w × h
V = (2 × 6) × 3
V = 12 × 3
V = 36

The volume of the B is 36 cubic cm or 36 cm³.

Combined Volume  of Solid Figure is 120 cm³ + 36 cm³  = 156 cm³

Question 2.
An office building has the dimensions shown. What is the volume of the building?

Answer:

Volume = length × width × height
Volume of A = l × w × h
V = (70 × 50) × 90
V = 3500 × 90
V = 315,000

The volume of the A is 315,000 cubic m or 315,000 m³.

Volume = length × width × height
Volume of B  = l × w × h
V = (60 × 50) × 20
V = 3000 × 20
V = 60000

The volume of the B is 60000 cubic m or 60000 m³.

Combined Volume  of Solid Figure is 315,000 m³ + 60000 m³  = 375,000 m³

Set D
pages 473-476
Think about these questions to help you use appropriate tools strategically.

Thinking Habits
Molly used 1-inch cubes to build the structure shown. She left a 3-inch by 1-inch opening in both layers of the structure.
• Which tools can I use?
• Why should I use this tool to help me solve the problem?
• Is there a different tool I could use?
• Am I using the tool appropriately?

Remember that tools such as place-value blocks, cubes, and grid paper can help you solve problems involving volume.

Question 1.
What tools could you use to model the problem?
Answer:

cubes

Question 2.
What is the total volume of the structure?
Answer:

Total blocks in bottom layer = 5 x 3 =15

Height =2 ; total volume is 15 x 2 = 30.

blocks left = 3 x 1 x 2 = 6

Therefore, total volume = 30 – 6 = 24 cubic units

Topic 11 Assessment Practice

Question 1.
Julio used unit cubes to make a rectangular prism. What is the volume of the prism?

A. 18 cubic units
B. 72 cubic units
C. 54 cubic units
D. 108 cubic units
Answer:

B. 72 cubic units

Question 2.
Select the possible dimensions for a prism with each given volume.

Answer:

Question 3.
A. A swimming pool is 50 meters long, 15 meters wide, and 3 meters deep. What is the volume of the pool?
A. 4,500 cubic meters
B. 2,250 cubic meters
C. 900 cubic meters
D. 750 cubic meters
B. After filling the pool for several minutes, the water is 1 meter deep. What is the volume of water in the pool?
Answer:

A.

Volume = length × width × depth
Volume of Swimming pool = l × w × h
V = (50 × 15) × 3
V = 750 × 3
V = 2250

The volume of the Swimming pool is 2250 cubic m or 2250 m³.

B.

Volume = length × width × depth
Volume of Swimming pool after filling the pool for several minutes, the water is 1 meter deep =
V = (50 × 15) × 4
V = 750 × 4
V = 3000

The volume of the Swimming pool is 3000 cubic m or 3000 m³.

Question 4.
A small building has the dimensions shown.
A. Write an Expression for the total volume of the building.

B. What is the volume of the building?
Answer:

A.

Volume = length × width × height
Volume of A = l × w × h
V = (8 × 12) × 24
V = 96 × 24
V = 2304

The volume of the A is 2304 cubic ft or 2304 ft³.

Volume = length × width × height
Volume of B  = l × w × h
V = (32 × 12) × 16
V = 384 × 16
V = 6114

The volume of the B is 6114 cubic ft or 6114 ft³

B.

Combined Volume  of Solid Figure is 2304 ft³ + 6114 ft³  = 8418 ft³

Question 5.
A. Choose all the Expressions that could NOT be used to find the volume of the bale of hay.

100 × 40
(100 × 40) × 20
4,000 × 20
(100 × 40) + 20
(100 + 40) + 20
B. Another 10 cm of hay is added to the top of the bale. What is the volume of the bale of hay now?
Answer:

A.

(100 × 40) × 20
4,000 × 20

B.

Volume = length × width × height
Volume of bale of hay  = l × w × h
V = (100 × 40) × 30
V = 4000 × 30
V = 120000

The volume of the bale of hay is 120000 cubic cm or 120000 cm³

Question 6.
Madeline made the wooden steps shown. What is the volume of the steps?

A. 72 cubic inches
B. 540 cubic inches
C. 840 cubic inches
D. 1,080 cubic inches
Answer:

Volume = length × width × height
Volume of A = l × w × h
V = (10 × 9) × 6
V = 90 × 6
V = 540

The volume of the A is 540 cubic in or 540 in³.

Volume = length × width × height
Volume of B  = l × w × h
V = (10 × 5) × 6
V = 50 × 6
V = 300

The volume of the B is 300 cubic in or 300 in³

Combined Volume  of Solid Figure is 540 in³ + 300 in³  = 840 in³

Question 7.
A. What is the volume of the trunk shown?

B. Which equation was used to find the volume of the trunk?
A. V = b × h
B. V = l × w × h
C. V = l × w
D. V = b × b × h
Answer:

A.

The volume of a rectangular prism is V = b × h, where b is the area of the base.
V = b × h.
V = 750 × 25
V = 18750 in³

B.

Option A. V = b × h

Question 8.
For her science project, Jada wants to build a rectangular prism out of foam blocks. The prism should have a volume of 350 cubic inches and a height of 5 inches.
What does the area of the base of the prism need to be for the given volume and height? Give one pair of possible whole-number dimensions for the base.
Answer:

The volume of a rectangular prism is V = b × h, where b is the area of the base.
V = b × h.
350 = b × 5
b = 350/5 = 70

The area of the base of the prism needs to be for the given volume and height is 70 in²

Question 9.
Martin’s suitcase has a volume of 1,080 cubic inches. Lily’s suitcase measures 9 inches wide, 13 inches long, and 21 inches high. What is the combined volume of the two suitcases?
Answer:

The volume of the first suitcase is 1080 cubic in or 1080 in³.

Volume = length × width × height
Volume of Second suitcase is  = l × w × h
V = (9 × 13) × 21
V = 117 × 21
V = 2457

The volume of the second suitcase is 2457 cubic in or 2457 in³

The combined volume of both suitcases is 1080 in³ + 2457 in³  = 3537 in³

Question 10.
Select all the Expressions that can be used to find the volume of the box in cubic centimeters.

8 × 6
(4 × 8) × 6
32 × 6
46 × 8
(4 × 8) + 6
Answer:

(4 × 8) × 6
32 × 6

Topic 11 Performance Task

Sporting Goods
Hiroto works in a sporting goods store. 1. Hiroto stacks identical boxes of golf balls to form a rectangular prism. Each box is a cube.

Part A
How many boxes are in the Golf Ball Display?

Answer:

There are 84 boxes in the Golf Ball Display
Part B
Explain how the number of boxes you found in Part A is the same as what you would find by using the formula V= l × w ×h.

Answer:

Each block has volume of 1 cubic unit.

Length = 7 boxes

Breadth = 4 boxes

Height = 3 boxes

Volume is length x breadth x height

V= l × w ×h = 7 x 4 x 3

v = 28 x 3 = 84 cubic units.

Part C
Hiroto needs to restack the boxes so the display is 2 layers high, less than 14 inches wide, and less than 30 inches long. The size of each box is shown in at the right. What is one way Hiroto can stack the boxes? Justify your answer.

Part D
What is the volume of the golf ball display in cubic inches? Explain how you solved.
Answer:

Question 2.
Hiroto builds two displays using rectangular foam blocks.
Part A
What is the volume of the foam block used for the Baseball Hats and Helmets Display? Explain how to solve using the formula V = b × h.

Part B
Hiroto used two blocks to build the Baseball Uniforms Display. What is the combined volume of the blocks? Explain how you solved.

Part C
Explain how you knew which units to use for your answer to Part B.
Answer:

Go through the enVision Math Common Core Grade 3 Answer Key Topic 6 Connect Area to Multiplication and Addition regularly and improve your accuracy in solving questions.

enVision Math Common Core 3rd Grade Answers Key Topic 6 Connect Area to Multiplication and Addition

Essential Question:
How does area connect to multiplication and addition?

enVision STEM Project: Design Solutions
Do Research There are different designs that help protect against weather, such as as lightning rods, flood-defense barriers, and wind-resistant roofs. Use the Internet or another source to gather information about these kinds of designs and how they work.

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

• Tell how some window or door designs can help protect against weather.
• Use a grid to draw one of the window or door designs. Count the number of unit squares your design measures. Label your drawing to show how the design works to protect against weather.

Review What You Know

Vocabulary
Choose the best term from the box. Write it on the blank.
• addend
• equal groups
• multiply
• array

Question 1.
When you skip count to get the total number, you ________.
Answer:
When you skip count to get the total number, you __add______.

Question 2.
Dividing apples so everyone gets the same number of apples is an example of making __________.
Answer:
Dividing apples so everyone gets the same number of apples is an example of making _equal groups____.

Question 3.
When you display objects in rows and columns, you make a(n). ___________.
Answer:
When you display objects in rows and columns, you make a(n). multiply

Division as Sharing

Question 4.
Chen has 16 model cars. He puts them in 4 rows. Each row has an equal number of cars. How many columns are there?
Answer:

Question 5.
Julie has 24 glass beads to give to 4 friends. Each friend gets an equal share. How many glass beads does each friend get?
Answer:

Arrays

Question 6.
Write an addition equation and a multiplication equation for the array shown at the right.

Answer:
Number of rows = 6
Number of columns = 5
We add columns value as many times as rows.
We add 6 times 5.
Addition equation = 5 + 5 + 5 + 5 + 5 + 5 = 30
Multiplication equation= Rows × Columns = 6 × 5 = 30.

Relating Multiplication and Division

Question 7.
There are 12 team members. They line up in 3 equal rows. Which multiplication equation helps you find how many are in each row?
A. 2 × 6 = 12
B. 1 × 12 = 12
C. 3 × 4 = 12
D. 3 × 12 = 36
Answer:
Total number of team members = 12
Number of rows = 3
Number of members in each row = 12 ÷ 3
12 ÷ 3 = 4
There are 4 members in each row.
3 × 4 = 12.
Option C. 3 × 4 = 12.

Question 8.
There are 20 bottles of juice lined up in 4 equal rows. Explain how you can use a multiplication equation to find out how many bottles of juice are in each row.
Answer:
Number of juice bottles = 20
Number of row the juice bottles lined up = 4
Number of bottles in each row = 20 ÷ 4
20 ÷ 4 = 5
4 × 5 = 20.
There are 5 juice bottles in each row.

Pick a Project

PROJECT 6A
How are cities built?
Project: Plan a Dog Park

PROJECT 6B
What are community gardens?
Project: Design a Community Garden

PROJECT 6C
What are carpenters?
Project: Draw a School Floor Plan

PROJECT 6D
How do you play the game?
Project: Make an Area Game

+

Lesson 6.1 Cover Regions

Solve & Share
Look at Shapes A-C on Area of Shapes Teaching Tool. How many square tiles do you need to cover each shape? Show your answers below. Explain how you decided.
I can … count unit squares to find the area of a shape.

Look Back! Can you be sure you have an accurate answer if there are gaps between the tiles you used? Explain.

Essential Question
How Do You Measure Area?

Visual Learning Bridge
Emily made a collage in art class. She cut shapes to make her design. What is the area of this shape?

Count the unit squares that cover Emily’s shape. The count is the area of the shape.

36 unit squares cover the shape. The area of the shape is 36 square units.

Sometimes you can estimate the area. You can combine partially filled squares to estimate full squares.
Count the unit squares that cover this shape.
About 27 unit squares cover the shape.
The area of the shape is about 27 square units.

Convince Me! Construct Arguments Karen says these shapes each have an area of 12 square units. Do you agree with Karen? Explain.

Answer:
I don’t agree with Karen statement that areas of these shapes is 12 square units each. Because the area of the shape 1
That is circle is approximately 10 square units and the area of the Second shape is 12 square units.
Both the shape have different areas.

Another Example!
Emily wants to cover this octagon.

If she tries to cover it using unit squares, there will be gaps or overlaps.
Emily can break the square into two same-size triangles. She can cover the shape completely using this triangle:

Fourteen triangles cover the octagon. The area of the octagon is 7 square units.

Guided Practice

Do You Understand?
Question 1.
How do you know the area of the octagon is 7 square units?
Answer:
The squares are divided into two equal triangles.
The unit square is 1 unit long

There are 14 triangles filling the Octagon
14 triangles
2 triangles makes 1 square unit
14 ÷ 2 = 7.
The area of the octagon = 7 square unit

Question 2.
Explain how finding the area of a shape is different from finding the length of a shape.
Answer:
Area of a shape
The area of a shape is the number of unit squares that cover the surface of a closed figure. Area is measured in square units.
Length of a shape
Length is a measurement of the distance around something.

Do You Know How?
In 3 and 4, count to find the area. Tell if the area is an estimate.
Question 3.

Answer:

In the above image each square represents 1 unit long
A rectangular shape is represented.
The shape covers 4 rows and 5 columns
Number of square = 5 + 5 + 5 + 5 = 20 units.
Area of the shape  = 5 × 4 = 20 square unit.

Question 4.

Answer:
In the above image each square represents 1  unit long
A Triangle shape is represented in the square table.
The shape covers approximately 5 squares.
Area of the shape  = 5 square units.

Independent Practice

In 5-7, count to find the area. Tell if the area is an estimate.
Question 5.

Answer:

The shape showed in the image  is a T shaped.
Here each square is 1 square unit
Number of squares covered in the horizontal line of T = 4 + 4 = 8 unit
Number of squares covered in the vertical line of T = 2 + 2 = 4
Area of the shape = 8 + 4 = 12 square units.

Question 6.

Answer:

Here each square represents 1 unit long.
An oval shape is represented in the table.
Number of squares covered by the shape = 15 squares.
Area of the shape = 15 square units.

Question 7.

Answer:

Here each square is 1 unit long.
A square is divided into 2 equal triangles.
In the shape above
Number of squares covered = 8 units
Number of triangle covered = 2
2 equal triangles is equal to 1 square.
Area of the shape = 8 units + 1 unit = 9 square units.

Problem Solving

Question 8.
Maggie buys 4 sketch pads. She pays with a 20-dollar bill. How much change does Maggie get back?

Answer:
Number of sketch pads Maggie bought = 4
Cost of each sketch pads = $3
Cost of all the sketch pads = 4 × $3 = $12
$3 + $3 + $3 + $3 = $12
Money she paid = 20 dollar bill.
Amount of change she got back = $20 – $12 = $8.
Maggie got $8 cash back.

Question 9.
Critique Reasoning Janet covers the red square with square tiles. She says, “I covered this shape with 12 unit squares, so I know it has an area of 12 square units.” Do you agree with Janet? Explain.

Answer:
As we can see the tiles covers more area than the red square. I don’t agree with Janet’s saying “I covered this shape with 12 unit squares, so I know it has an area of 12 square units.”

Question 10.
Higher Order Thinking Chester drew this picture of a circle inside a square. What would be a good estimate of the green-shaded area of the square? How did you calculate your answer?

Answer:

Question 11.
Number Sense Arthur puts 18 erasers into equal groups. He says there are more erasers in each group when he puts the erasers in 2 equal groups than when he puts the erasers in 3 equal groups. Is Arthur correct? Explain.
Answer:
Total number of erasers Arthur has = 18 erasers.
Dividing the 18 erasers into 2 equal groups = 18 ÷ 2 = 9 erasers.
Dividing the 18 erasers into 3 equal groups = 18 ÷ 3 = 6 erasers.
Arthur says there are more erasers in each group when he puts the erasers in 2 equal groups than when he puts the erasers in 3 equal groups.
I agree with the Arthur’s sayings that there will be more erasers in 2 equal groups than in 3 equal groups.
There are 9 erasers in 2 groups and 6 erasers in 3 groups
The difference between the 2 groups and 3 groups = 9 – 6 = 3.
There are 3 more erasers in group 2.

Assessment Practice

Question 12.
Daryl draws this shape on grid paper. Estimate the area of the shape Daryl draws.

A. About 4 square units
B. About 5 square units
C. About 6 square units
D. About 7 square units
Answer:
Total number of squares = 6 × 5 = 30 squares.
Number of uncovered squares = 22 squares.
Number of covered squares = total squares – uncovered squares
30 – 22 = 8 squares.
There are 8 covered squares.
But there is white area in the covered shape squares.
So, The area of the Daryl shape is about 7 square units.

Lesson 6.2 Area: Nonstandard Units

Solve & Share
Find the area of the postcard on each grid. What do you notice about the size of the postcard on each grid? What do you notice about the area of the postcard on each grid? Explain.
I can … count unit squares to find the area of a shape.

Look Back! Are the measurements of the areas of the postcard shown above the same? Explain.
Answer:

Explanation:
Image 1
Number of squares = 6 × 10 = 60 squares.
Number of uncovered squares = 36 squares
Number of postcard covered squares = 60 – 36 = 24 squares.
Area of the postcard = 24 square units.

Image 2
Number of squares = 3 × 5 = 15 squares.
Number of uncovered squares = 9 squares
Number of postcard covered squares = 15 – 9 = 6 squares.
Area of the postcard = 6 square units.

1 Square in image 2 is equal to 4 squares in image 1
As shown in the image above.
So, both the postcards have the same area in both the images.

Essential Question
How Can You Measure Area Using Non-Standard Units?

Visual Learning Bridge
Tran designs a bookmark for a book. How can he use unit squares to find the area of the bookmark?

You can count the number of unit squares.

There are 32 unit squares.
Area = 32 square units

You can use a different unit square.

Convince Me! Reasoning How are the areas of these two squares alike and how are they different?

Guided Practice

Do You Understand?
Question 1.
Which of these shapes has an area of 5 square units? How do you know?

Answer:

Do You Know How?
Question 2.
Draw unit squares to cover the figures and then find the area. Use the unit squares shown.

Answer:

Independent Practice

In 3-5, draw unit squares to cover the figures and find the area. Use the unit squares shown.
Question 3.

Answer:

Question 4.

Answer:

Question 5.

Answer:

Problem Solving

Question 6.
Ben finds that the area of this figure is 14 square units. Draw unit squares to cover this figure.

Answer:

Question 7.
Luke eats 6 grapes from the bowl. Then Juan and Luke equally share the grapes that are left. How many grapes does Juan eat? Show how you used reasoning to solve the problem.

Answer:
Total number of grapes = 24 grapes.
Number of grapes Luke ate = 6 grapes.
Number of remaining grapes in the bowl = 24 grapes – 6 grapes = 18 grapes.
Luke and Juan eat the remaining grapes equally.
Number of grapes Juan and Luke eat equally are = 18 ÷ 2 = 9 grapes.
Juan and Luke eat 9 grapes each.

Question 8.
Construct Arguments Riaz estimates that the area of this figure is 45 square units. Martin estimates the area is 48 square units. Whose estimate is closer to the actual measure? Explain.

Answer:

Question 9.
Higher Order Thinking Theo wants to cover the top of a small table with square tiles. The table is 12 square tiles long and 8 square tiles wide. How many tiles will Theo need to cover the table?
Answer:

Assessment Practice

Question 10.
Rick used the smaller unit square and found that the area of this shape is 18 square units. If he used the larger unit square, what would the area of the shape be?

A. 1 square unit
B. 2 square units
C. 3 square units
D. 4 square units
Answer:

Lesson 6.3 Area: Standard Units

Solve & Share
Draw a square to represent 1 unit square. Use your unit square to draw a rectangle that has an area of 8 square units. Compare your shape with a partner’s shape. What is the same? What is different?
I can … measure the area of a shape using standard units.

Look Back! Are the sizes of your and your partner’s shapes something that is the same or something that is different? Explain.

Essential Question
How Can You Measure Area Using Question Standard Units of Length?

Visual Learning Bridge
Meg bought this sticker. What is the area of the sticker in square centimeters?

Count the unit squares.

6 unit squares cover the sticker. The sticker is measured in square centimeters.
So, the area of the sticker is 6 square centimeters.

Convince Me! Be Precise If square inches rather than square centimeters were used for the problem above, would more unit squares or fewer unit squares be needed to cover the shape? Explain.

Guided Practice

Do You Understand?
Question 1.
If Meg’s sticker on the previous page measured 2 inches by 3 inches, what would its area be?
Answer:
The measurement of Meg’s sticker are = 2 inches by 3 inches.
Area = 2 inches × 3 inches = 6 inches.
Area of Meg’s sticker is 6 inches.

Question 2.
Zoey paints a wall that measures 8 feet by 10 feet. What units should Zoey use for the area of the wall? Explain.
Answer:
Measurements of Zoey paint on the wall = 8 feet by 10 feet
Area of the paint on the wall = 8 feet × 10 feet  = 80 square feet.

Do You Know How?
In 3 and 4, each unit square represents a standard unit. Count the shaded unit squares. Then write the area.
Question 3.

Answer:
Number of shaded unit squares = 4 × 3 = 12 squares.
Area of the shaded squares = 12 square ft.
square ft because 1 square is measure is ft.

Question 4.

Answer:
Number of shaded squares = 6 × 2 = 12 squares.
Area of the shaped squares = 12 square m.
square m because 1 square is measure is m.

Independent Practice

In 5-10, each unit square represents a standard unit. Count the shaded unit squares. Then write the area.
Question 5.

Answer:
Number of shaded unit squares = 4 × 4 = 16 unit squares.
Area of shaded unit squares = 16 squares in.
square in because 1 square is measure is in.

Question 6.

Answer:
Number of shaped unit squares = 3 × 3 = 9 unit squares.
Area of the shaded unit squares = 9 square ft.
square ft because 1 square is measure is ft.

Question 7.

Answer:
Number of shaded unit squares = 6 × 3 = 18 unit squares.
Area of shaded unit squares = 18 square in.
square in because 1 square is measure is in.

Question 8.

Answer:
Number of shaded unit squares = 9 × 7 = 63 unit squares.
Area of the shaded unit squares = 63 square m.
square m because 1 square is measure is m.

Question 9.

Answer:
Number of shaded unit squares = 7 × 5 = 35 unit squares.
Area of the shaded unit squares = 35 square cm.
square cm because 1 square is measure is cm.

Question 10.

Answer:
Number of shaded unit squares = 4 ×5 = 20 unit squares.
Area of the shaded unit squares = 20 square ft.
square ft because 1 square is measure is ft.

Problem Solving

Question 11.
Reasoning Mr. Sanchez grows three types of vegetables in his garden. What is the area of the garden that Mr. Sanchez uses to grow lettuce and cucumbers? Explain how to use the units in this problem.

Answer:
Number of squares covered Cucumbers = 4 × 2 = 8 squares
Area of garden used to grow Cucumber= 8 square ft.
Number of square covered to grow Lettuce = 3 × 6 = 18 squares
Area of garden used to grow Lettuce = 18 square ft.
Total area used to grow Lettuce and cucumber in Mr. Sanchez garden = 8 square ft + 18 square ft = 26 square ft.
Area used to grow Lettuce and cucumber is 26 square ft.

Question 12.
Lisa received 34 text messages on Monday and 43 text messages on Tuesday. She received 98 text messages on Wednesday. How many more text messages did Lisa receive on Wednesday than on Monday and Tuesday combined?
Answer:
Number of text messages Lisa received on Monday = 34
Number of text messages Lisa received on Tuesday = 43
Number of text messages Lisa received on Wednesday = 98
Total number of text messages Lisa received on Monday and Tuesday combined = 34 + 43 = 77 text messages.
Number of more text messages did Lisa receive on Wednesday than on Monday and Tuesday combined are = 98 – 77 = 21
Lisa receive 21 more text messages  on Wednesday than on Monday and Tuesday combined.

Question 13.
Monica buys a postage stamp. Is the area of the stamp more likely to be 1 square inch or 1 square meter? Explain.
Answer:
Generally the dimensions of the postage stamp are in inches.
So , the area of the postage stamp Monica bought is 1 square inch.

Question 14.
Algebra Which operation can you use to complete the equation below? 8 = 56 ☐ 7
Answer:
To complete the given equation we use division.
56 ÷ 7 = 8
8 × 7 = 56
So, we use Division.

Question 15.
Higher Order Thinking Brad says a square that has a length of 9 feet has an area of 18 square feet. Is Brad correct? Why or why not?
Answer:
No, Brad is not  correct.
The area of a square is side × side
Here the length of the square = 9 feet
Area of the square = 9 ft  × 9 ft = 81 square ft.
So, what Brad said is wrong the area of the square is 81 square ft not 18 square ft.

Assessment Practice

Question 16.
Each of the unit squares in Shapes A-C represent 1 square foot. Select numbers to tell the area of each shape.

Answer:

Lesson 6.4 Area of Squares and Rectangles

Solve & Share
Jorge is carpeting two rooms. One room is a square with a side that measures 6 meters. The other room is a rectangle with sides that measure 3 meters and 12 meters. How many square meters of carpet does Jorge need?
I can … find the area of squares and rectangles by multiplying.

Look Back! What do you notice about the lengths of the sides and the areas of the two rooms Jorge is carpeting?

Essential Question
How Can You Find the Question Area of a Figure?

Visual Learning Bridge
Mike paints a rectangular wall in his room green. The picture shows the length and width of Mike’s wall. A small can of paint covers 40 square feet. Does Mike need more than one small can to paint the wall of his room?

One Way
Count the unit squares to find area.

There are 48 unit squares. The area of Mike’s wall is 48 square feet.

Another Way
Count the number of rows and multiply by the number of squares in each row. There are 8 rows and 6 squares in each row.

The area of Mike’s wall is 48 square feet. He will need more than one small can of paint.

Convince Me! Model with Math Mike plans to paint a wall in his living room blue. That wall measures 10 feet tall and 8 feet wide. What is the area of the wall Mike plans to paint blue? How many cans of paint will he need?

Another Example!
The area of another wall in Mike’s room is 56 square feet. The wall is 8 feet high. How wide is the wall?

56 = 8 × ?
You can use division: 56 ÷ 8 = ?
56 ÷ 8 = 7
The wall is 7 feet wide.

Guided Practice

Do You Understand?
Question 1.
Suji’s garden is 4 yards long and 4 yards wide. What is the area of Suji’s garden?
Answer:
Length of Suji’s garden = 4 yards
Width of Suji’s garden = 4 yards
Area of Suji’s garden = length × width = 4 yards × 4 yards = 16 square yards.

Question 2.
The area of Michi’s garden is 32 square feet. The garden is 8 feet long. How wide is Michi’s garden?

Answer:
Area of Michi’s garden = 32 square feet
Length of the garden = 8 feet
width of the garden = ? feet
Area = length × width
32 = 8 × width
width = 32 ÷ 8 = 4 feet.
8 ft × 4 ft = 32 square ft
Michi’s garden is 4 ft wide.

Do You Know How?
In 3 and 4, find the area of each figure. Use grid paper to help.
Question 3.

Answer:

Question 4.

Answer:

Independent Practice

In 5 and 6, find the area. In 7, find the missing length. Use grid paper to help.
Question 5.

Answer:
Length of the figure = 3 cm
Width of the figure = 1 cm
Area of the figure = 3 cm × 1 cm = 3 square cm.

Question 6.

Answer:
Length of the figure = 4 ft
Width of the figure = 9 ft
Area of the figure = 4 ft × 9 ft = 36  square ft.

Question 7.

Answer:

Problem Solving

Question 8.
Jen’s garden is 4 feet wide and has an area of 28 square feet. What is the length of Jen’s garden? How do you know?
Answer:
Area of Jen’s garden = 28 square ft
width of Jen’s garden = 4 ft
Length of Jen’s garden = area ÷ width
28 sq ft ÷ 4 ft = 7 ft.
Length of Jen’s garden = 7 ft.

Question 9.
Make Sense and Persevere Briana has 2 grandmothers. She mailed 2 cards to each of them. In each card she put 6 photographs. How many photographs did Briana mail in all?
Answer:
Number of grandmothers Briana has = 2 grandmothers
Number of cards she mailed to them = 2 each
total number of cards she mailed = 2 + 2 = 4
Number of photographs in each card = 6 photographs
Total number of photographs Briana mailed in all = 4 × 6 = 24 photographs.

Question 10.
Kevin thinks he found a shortcut. He says he can find the area of a square by multiplying the length of one side by itself. Is Kevin correct? Why or why not?
Answer:
Area of a square is side × side = side square.
Kevin says he can find the area of a square by multiplying the length of one side by itself.
I agree with Kevin’s saying.
Multiplying the length of one side by itself is same as multiplying side × side.
Because the length of all the side of a square are same.

Question 11.
Higher Order Thinking Ryan measures a rectangle that is 9 feet long and 5 feet wide. Teo measures a rectangle that has an area of 36 square feet. Which rectangle has the greater area? Explain how you found the answer.
Answer:
Length of Ryan rectangle = 9 ft
Width of Ryan rectangle = 5 ft
Area of rectangle = length × width = 9 ft  × 5 ft = 45 ft.
Area of the rectangle Teo measured = 36 square ft.
Ryan rectangle has the greater area.
On comparing the areas of both the rectangles. Ryan rectangle area is 45 square ft and Teo rectangle area is 36 square ft
45 – 36 = 9 square ft greater.
Ryan rectangle area is 9 square ft greater than Teo’s rectangle area.

Assessment Practice

Question 12.
Marla makes maps of state preserves. Two of her maps of the same preserve are shown. Select all the true statements about Marla’s maps.

☐ You can find the area of Map A by counting the unit squares.
☐ You can find the area of Map B by multiplying the side lengths.
☐ The area of Map A is 18 square feet.
☐ The area of Map B is 18 square feet.
☐ The areas of Maps A and B are NOT equivalent.
Answer:
☐ You can find the area of Map A by counting the unit squares.
☐ You can find the area of Map B by multiplying the side lengths.
☐ The area of Map A is 18 square feet.
☐ The area of Map B is 18 square feet.
All the above mentioned statements are true.

Lesson 6.5 Apply Properties: Area and the Distributive Property

Solve & Share
The new reading room floor is a rectangle that is 8 feet wide by 9 feet long. Mrs. Wallace has a rectangular rug that is 8 feet wide by 5 feet long. What area of the reading room floor will not be covered by the rug?
I can … use properties when multiplying to find the area of squares and rectangles.

Look Back! Does your strategy change if the rug is in the corner of the room or in the center? Explain why or why not.

Essential Question
How Can the Area of Rectangles Represent Question the Distributive Property?

Visual Learning Bridge
Gina wants to separate this rectangle into two smaller rectangles. Will the area of the large rectangle equal the sum of the areas of the two small rectangles?

Separate the 8-unit side into two parts.

7 × 8 = 7 × (5 + 3) = (7 × 5) + (7 × 3)

So, the area of the large rectangle is equal to the sum of the areas of the two small rectangles.

Convince Me! Generalize Find another way to separate this rectangle into two smaller parts. Write an equation you can use to find the areas of the two smaller rectangles. Is the area of the large rectangle still the same? What can you generalize?

Guided Practice

Do You Understand?
Question 1.
Describe a way to separate a 6×6 square into two smaller rectangles.
Answer:

6 × 6 = 36
6 × 6 = 6 × (2 + 4 ) = 6 × 2 + 6 × 4 = 12 + 24 = 36.

Question 2.
What multiplication facts describe the areas of the two smaller rectangles you identified in Exercise 1?
Answer:
6 × 6 = 36
6 × 6 = 6 × (2 + 4 ) = 6 × 2 + 6 × 4 = 12 + 24 = 36.

Do You Know How?
Complete the equation that represents the picture.
Question 3.

Answer:

Explanation:
Given 6 × 5
Using distributive property to find the product of 6 × 5.
By breaking one of the facts to get the product.
Here we break 5 in to 2 and 3
Then the equation will be
6 × (2 + 3 )= (6 × 2 ) + (6 × 3)

Independent Practice

In 4 and 5, complete the equation that represents the picture.
Question 4.

Answer:

Explanation:
Given 5 × 7
Using distributive property to find the product of 5 × 7
By breaking one of the facts to get the product.
Here 7 is broken into 4 and 3
Then the equation will be
5 × 7 = 5 × (4 + 3) = (5 ×4) + (5 × 3 ) = 20 + 15 = 35.

Question 5.

Answer:

Explanation:
Given 3 × 8
Using distributive property to find the product of 3 × 8
By breaking one of the facts to get the product.
Here 8 is broken into 4 and 4
Then the equation will be
3 × 8 = 3 × (4 + 4) = (3 ×4) + (3 × 4 ) = 12 + 12 = 24..

In 6, write the equation that represents the picture.
Question 6.

Answer:

Explanation:
Given 4 × 6
Using distributive property to find the product of 4× 6
By breaking one of the facts to get the product.
Here 6 is broken into 2 and 4
Then the equation will be
4 × 6 = 4 × (2 + 4) = (4 ×2) + (4 × 4 ) = 8 + 16 = 24

Problem Solving

Question 7.
Amit sold 3 shells last week for $5 each and 2 more shells this week for $5 each. Show two ways to determine how much money Amit made in the two weeks.
Answer:
Number of shells Amit sold last week = 3 shells
Cost of each shell he sold = $5 each
Total Money Amit made in last week = 3 × $5 = $15
Number of shells he sold this week = 2 shells
Cost of each shell he sold = $5 each.
Total money Amit made this week = 2 × $5 = $10
Total money Amit made in two weeks = $15 + $10 =$25.
Other way
Total number of shells he sold in two weeks = 3 + 2
Cost of each shell = $5
Total money Amit earned in two weeks =
=$5 × (3 + 2)
= ($5 × 3) + ($5 × 2 )
= $15 + $10
= $25.

Question 8.
enVision® STEM Claudia wants to replace the roof of a dog house with a new wind-resistant material. The roof has two rectangular sides that are 6 feet by 4 feet. What is the total area of the roof?
Answer:
Number of rectangular sides = 2
Sizes of the rectangular sides = 6 feet by 4 feet
Area of 1 rectangular sides = 6 ft × 4 ft = 24 square ft
Total area of the roof = 2 × 24 square ft = 48 square ft.

Question 9.
Use Structure Chiya has an 8 x 6 sheet of tiles. Can she separate the sheet into two smaller sheets that are 8 x 4 and 8 x 2? Do the two smaller sheets have the same total area as her original sheet? Explain.

Answer:
The dimensions of Chiya sheet of tiles = 8 × 6
Area of the sheet = 8× 6 = 48.
The sheet is divided into two smaller sheet
Using distributive property and dividing into two smaller parts
8 × 6 = 8 ×(4 + 2) = (8 × 4) + (8 × 2) = 32 + 16 = 48.
In both the cases the area is same.

Question 10.
Higher Order Thinking List all possible ways to divide the rectangle at the right into 2 smaller rectangles.

Answer:
The dimensions of the picture = 4 rows with 5 squares
Total number of squares = 4 × 5 = 20 squares.
Using distributive property and finding all the possible ways.
4 × 5 = 4 × (1 + 4 )= (4 × 1) + (4 × 4) = 4 + 16 = 20
4 × 5 = 4 × (2 + 3 )= (4 × 2) + (4 ×3) = 8  + 12 = 20
4 × 5 = 4 × (3 + 2 )= (4 × 3) + (4 × 2) =12 + 8 = 20
4 × 5 = 4 × (4 + 1 )= (4 × 4) + (4 × 1) = 16 + 4 = 20.

Assessment Practice

Question 11.
Which equation represents the total area of the green shapes?

A. 4 × 8= 4 × (6 + 2) = (4 × 6) + (4 × 2)
B. 4 × 7 = 4 × (3 + 4) = (4 × 3) + (4 × 4)
C. 4 × 7 = 4 × (4 + 3) = (4 × 4) + (4 × 3)
D. 4 × 7 = 4 × (5 + 2) = (4 × 5) + (4 × 2)
Answer:
Number of rows = 4 with 7 square each
Area of the green shapes = 4 × 7 = 28 square units.
Breaking 7 into smaller part
7 = 1 + 6
7 = 2 + 5
7 = 3 + 4
7 = 4 + 3
7 = 5 + 2
7 = 6 + 1
the equations are
B. 4 × 7 = 4 × (3 + 4) = (4 × 3) + (4 × 4)
C. 4 × 7 = 4 × (4 + 3) = (4 × 4) + (4 × 3)
D. 4 × 7 = 4 × (5 + 2) = (4 × 5) + (4 × 2)
which represents the green shapes.

Lesson 6.6 Apply Properties: Area of Irregular Shapes

Solve & Share
Mrs. Marcum’s desk is shaped like the picture below. The length of each side is shown in feet. Find the area of Mrs. Marcum’s desk.
I can … use properties to find the area of irregular shapes by breaking the shape into smaller parts.

Look Back! How can you check your answer? Is there more than one way to solve this problem? Explain.

Essential Question
How Can You Find the Area of an Irregular Shape?

Visual Learning Bridge
Mr. Fox is covering a miniature golf course putting green with artificial grass. Each artificial grass square is 1 square foot. What is the area of the putting green that Mr. Fox needs to cover?

One Way
You can draw the figure on grid paper. Then count the unit squares to find the area.

The area of the putting green is 56 square feet.

Another Way
Divide the putting green into rectangles. Find the area of each rectangle. Then add the areas.

Rectangle A 4 × 3= 12
Rectangle B 4 × 3 = 12
Rectangle C 4 × 8 = 32
12 + 12 + 32 = 56. The area of the putting green is 56 square feet.

Convince Me! Use Structure Find another way to divide the putting green into smaller rectangles. Explain how you can find the area of the putting green using your smaller rectangles.

Answer:

Here dividing the image as per the above image. into 3 rectangles.
Rectangle A = 8 ft × 3 ft = 24 square ft
Rectangle B = 8 ft × 3 ft = 24 square ft
Rectangle C = 4 ft × 2 ft = 8 square ft
Total area of the image = 24 +24 + 8 = 56 square ft.

Guided Practice

Do You Understand?
Question 1.
Explain why you can find the area of the putting green on the previous page using different rectangles.
Answer:

Question 2.
Explain what operation you use to find the total area of the smaller rectangles.
Answer:
I used distributive property.
I divide the shape into smaller shapes and then added all the shapes to find the total shape.
To find the area of the irregular shape find the areas of the smaller area and then add them together.

Do You Know How?
In 3 and 4, find the area of each figure. Use grid paper to help.
Question 3.

Answer:

Question 4.

Answer:

Independent Practice

In 5-8, find the area of each figure. Use grid paper to help.
Question 5.

Answer:

Question 6.

Answer:

Question 7.

Answer:

Question 8.

Answer:

Problem Solving

Question 9.
Reasoning Mrs. Kendel is making a model house. The footprint for the house is shown at the right. What is the total area? Explain your reasoning.

Answer:

Explanation:
Here to find the area of Mrs. Kendel Model House i divided the model into two sections.
Living Section and Sleeping Section.
Finding the area of Living Section and Sleeping Section and then adding them gives the total area of the model house.

Question 10.
Vocabulary Fill in the blanks. Mandy finds the _________ of this shape by dividing it into rectangles. Phil gets the same answer by counting ________.

Answer:
Mandy finds the _Area__ of this shape by dividing it into rectangles. Phil gets the same answer by counting _squares__.

Question 11.
Algebra Write an equation. Use a question mark to represent the unknown quantity for the phrase “six times a number is 24.” Solve your equation.
Answer:
“six times a number is 24.”
4 + 4 + 4 + 4 + 4 + 4 = 24
six times 4 is 24.

Question 12.
Higher Order Thinking Mr. Delancy used 3-inch square tiles to make the design at the right. What is the area of the design he made? Explain how you found it.

Answer:
Dimensions of 1 square = 3 in
Area of 1 square = 3 in × 3 in = 9 square in.
Number of green squares = 4 squares
Number of blue squares = 4 squares.
Total number of squares =4 + 4 = 8 squares.
Total Area of all squares = 8 × 9 square in = 72 square in.

Assessment Practice

Question 13.
Jared drew the figure at the right. Draw lines to show how you can divide the shape to find the area. Then select the correct area for the figure at the right.

A. 6 square inches
B. 24 square inches
C. 30 square inches
D. 33 square inches
Answer:

Lesson 6.7 Problem Solving

Look For and Use Structure
Solve & Share
Mr. Anderson is tiling his kitchen floor. He will not need tiles for the areas covered by the kitchen island or the counter. How many square meters of tiles does Mr. Anderson need?
I can … use the relationships between quantities to break a problem into simpler parts.

Thinking Habits
Be a good thinker! These questions can help you.

• What patterns can I see and describe?
• How can I use the patterns to solve the problem?
• Can I see expressions and objects in different ways?
  

Look Back! Use Structure is the tiled area greater than or less than the total area of the kitchen? Explain.

Essential Question
How Can You Use Structure to Solve Problems?

Visual Learning Bridge
Janet is painting a door. She needs to paint the entire door except for the window.
What is the area of the part of the door that needs paint?

What do I need to do to solve this problem?
I need to find the area of the door without the window.

How can I make use of structure to solve this problem?
I can

• break the problem into simpler parts.
• find equivalent expressions.

I will subtract the area of the window from the total area.
Here’s my thinking…)
Find the area of the whole door.
4 feet × 9 feet = 36 square feet

Find the area of the window.
2 feet × 2 feet = 4 square feet

Subtract to find the area that needs paint.
36 – 4 = 32 square feet

The area of the part of the door that needs paint is 32 square feet.

Convince Me! Use Structure Janet thinks of a different way to solve the problem. She says, “I can divide the area I need to paint into 4 smaller rectangles. Then I will find the area of each smaller rectangle and add the 4 areas.” Does Janet’s strategy make sense? Explain.

Guided Practice

Use Structure Lil glued beads on the border of the frame. What is the area of the part she decorated with beads?

Question 1.
How can you think about the total area of the frame?
Answer:

Question 2.
Use what you know to solve the problem.

Answer:
Given a frame and a image
To find the  decorated area we subtract the image area from  total area.

Independent Practice

Use Structure A keypad has 10 rubber buttons. Each button is 1 centimeter by 2 centimeters. The rest is made out of plastic. Is the area of the plastic greater than the area of the rubber buttons?

Answer:

Question 3.
How can you break the problem into simpler parts? What is the hidden question?
Answer:
Area of the buttons is calculated. Then
Total area of the keypad is calculated.
Finally the area of the plastic is calculated by subtracting buttons area from the total area.

Question 4.
How can you find the area of all the rubber buttons?
Answer:
The area of the rubber = Area of the keypad – Area of the 10 buttons
= 70 square cm – 20 square cm
= 50 square cm.

Question 5.
Use what you know to solve the problem.
Answer:
We know the
Dimensions of the button
Dimensions of the keypad.
Using these dimensions area of buttons and keypad are calculated.
Subtracting the button area from keypad area. Plastic area is calculated.

Problem Solving

Performance Task
Place Mat Genevieve is designing a placemat. The center measures 8 inches by 10 inches. A 2-inch border goes around the center. Genevieve cuts the corners to make the placemat an octagon. She wants to find the area of the placemat.

Question 6.
Use Structure What are the lengths and widths of each rectangular border piece?
Answer:

Question 7.
Use Appropriate Tools How can Genevieve find the area of the 4 corner pieces using grid paper?

Answer:

Area of the corners is 8 square in.

Question 8.
Model with Math What equation can Genevieve use to find the area of the center? Find the area of the center using your equation.

Answer:
Given
The center measures 8 inches by 10 inches.
Area of the center = 8 in × 10 in = 80 square in.

Question 9.
Reasoning How are the quantities in this problem related?
Answer:
They share one side of the other rectangles.

Question 10.
Be Precise Solve the problem. Explain what unit you used for your answer.
Answer:
We used inches. to measure the answer.

Topic 6 Fluency Practice Activity

Follow the Path
Shade a path from START to FINISH. Follow the quotients that are odd numbers. You can only move up, down, right, or left. Once you complete the path, write the fact families for each of the squares you shaded.
I can … multiply and divide within 100.

Topic 6 Vocabulary Review

Understand Vocabulary
Choose the best term from the Word List. Write it on the blank.
Word List

• area
• column
• Distributive Property
• estimate
• multiplication
• product
• row
• square unit
• unit square

Question 1.
A(n) __________ has sides that are each 1 unit long.
Answer:
A(n) _unit square__ has sides that are each 1 unit long.

Question 2.
__________ is the number of unit squares that cover a region or shape.
Answer:
__Area__ is the number of unit squares that cover a region or shape.

Question 3.
You can use the ___________ to break apart facts and find the __________.
Answer:
You can use the __Distributive Property__ to break apart facts and find the ___Area_______.

Question 4.
A unit square has an area of 1 _________.
Answer:
A unit square has an area of 1 __Square unit_______.

Question 5.
When you __________, you give an approximate answer.
Answer:
When you _multiplicate_________, you give an approximate answer.

Write always, sometimes, or never.

Question 6.
Area is __________ measured in square meters.
Answer:
Area is __sometimes_______ measured in square meters.

Question 7.
Multiplication __________ involves joining equal groups.
Answer:
Multiplication __sometimes________ involves joining equal groups.

Question 8.
The area of a shape can __________ be represented as the sum of the areas of smaller rectangles.
Answer:
The area of a shape can __always________ be represented as the sum of the areas of smaller rectangles.

Use Vocabulary in Writing
Question 9.
What is the area of this rectangle? Explain how you solved the problem. Use at least 3 terms from the Word List in your answer.

Answer:
Number of rows = 4
Number of squares in each row = 6 squares.
Area of the rectangle = 4 × 6 = 24 square units.

Topic 6 Reteaching

Set A pages 209-212

A unit square has sides that are 1 unit long.
Count the unit squares that cover the shape. The count is the area of the shape.

Seventeen unit squares cover the shape. The area of the shape is 17 square units.
Sometimes you need to estimate to find the area. First count the full squares. Then estimate the number of partially filled squares.

Remember that area is the number of unit squares needed to cover a region with no gaps or overlaps.

In 1 and 2, count to find the area. Tell if the area is an estimate.
Question 1.

Answer:

Question 2.

Answer:

Set B pages 213-216

Unit squares can be different sizes. The size of a unit square determines the area.

The measurements are different because different sizes of unit squares were used.

Remember that you can use unit squares to measure area.

Draw unit squares to cover the figures and find the area. Use the unit squares shown.
Question 1.

Answer:

Question 2.

Answer:

Set C pages 217-220

The unit squares below represent square inches.
What is the area of the figure below?

Twenty-four unit squares cover the figure. The area of the figure is measured in square inches.
So, the area of the figure is 24 square inches.

Remember that you can measure using standard or metric units of area for unit squares.

In 1 and 2, each unit square represents a standard unit. Count the unit squares. Then write the area.
Question 1.

Answer:

Question 2.

Answer:

Set D pages 221-224

You can find area by counting the number of rows and multiplying by the number of squares in each row.

There are 5 rows.
There are 4 squares in each row.
5 × 4 = 20
The area of the figure is 20 square inches.

Remember that you can multiply the number of rows by the number of squares in each row to find the area.

In 1-3, find the area of each figure. Use grid paper to help.
Question 1.

Answer:

Question 2.

Answer:

Question 3.

Answer:

Set E pages 225-228

Reteaching Continued

You can use the Distributive Property to break apart facts to find the product.
Separate the 5 unit side into two parts.

Area of the large rectangle: 6 × 5 = 30
Areas of the small rectangles:
6 × 2 = 12
6 × 3 = 18
Add the two areas: 12 + 18 = 30
You can write an equation to show that the area of the large rectangle is equal to the sum of the areas of the two small rectangles.
6 × 5 = 6 × (2 + 3) = (6 × 2) + (6 × 3)

Remember that you can separate a rectangle into two smaller rectangles with the same total area.

In 1-3, write the equations that represent the total area of the red shapes. Find the area.
Question 1.

Answer:

Question 2.

Answer:

Question 3.

Answer:

Set F pages 229-232

Find the area of this irregular shape.

You can place the shape on grid paper and count the unit squares. The area of the shape is 21 square inches.

You also can divide the shape into rectangles. Find the area of each rectangle and add.
5 × 3 = 15 square inches
3 × 2 = 6 square inches
15 + 6 = 21 square inches

Remember that you can add smaller areas to find a total area.

In 1 and 2, find the area of each shape.
Question 1.

Answer:

Question 2.

Answer:

Set G pages 233-236

Think about these questions to help you use structure in solving problems.
Thinking Habits

• What patterns can I see and describe?
• How can I use the patterns to solve the problem?
• Can I see expressions and objects in different ways?
  

Remember to look for simpler ways of representing an area.

Debra made this design from 1-inch square tiles. What is the area of the blue tiles?

Question 1.
How can you express the area of the blue tiles?
Answer:
Area of the blue tile = area of the outer rectangle – area of the inner rectangle
= 7 in × 7 in – 5 in × 3 in
= 49 square in – 15 square in
= 34 square in.
Area of the blue tile is 34 square in.

Question 2.
Solve the problem. Explain how you solved.
Answer:
Area of the outer rectangle = 7 in × 7 in = 49 square in
Area of the inner rectangle = 5 in × 3 in = 15 square in
Area of the blue tile = area of the outer rectangle – area of the inner rectangle
= 49 square in – 15 square in
= 34 square in.
Area of the blue ti,e is 34 square in.

Topic 6 Assessment Practice

Question 1.
Count to find the area of the shape. Tell if the area is exact or an estimate.

Answer:
There are 10 squares covered by the shape. out of ten 4 are half covered with the shape.
So, area of the shape is around 8 squares unit.

Question 2.
Use the Distributive Property to write the equation that represents the picture. Then give the area of each smaller rectangle and the large rectangle.

Answer:
Area of the larger rectangle = 3 × 5 = 15
Area of the smaller rectangles |
3 × 3 = 9
3 × 2 = 6
adding them
9 + 6 = 15 .
The equation
3 × 5 = 3 × (2 + 3) = (3 × 2) + (3 × 3 ) = 6 + 9 = 15.

Question 3.
Lewis says that the figure below has an area of 4 square meters. Is he correct? Explain.

Answer:
No, Lewis is wrong. As he says the area of the figure is 4 square m. But the  square represent cm.
The area of the figure is = 7 square cm.
There are 4 complete squares and 6 half divided squares.
adding them
4 + (6 ÷ 2 ) = 4 + 3 = 7 square cm.
Area of the figure is 7 square cm.

Question 4.
Select all of the ways to break apart the area of the large rectangle into the sum of the areas of two smaller rectangles. Then give the area of the large rectangle.

☐ 5 × 7 = 5 × (1 + 5) = (5 × 1) + (5 × 5)
☐ 5 × 7 = 5 × (3 + 4) = (5 × 3) + (5 × 4)
☐ 5 × 7 = 5 × (2 + 3) = (5 × 2) + (5 × 3)
☐ 5 × 7 = 5 × (1 + 6) = (5 × 1) + (5 × 6)
☐ 5 × 7 = 5 × (2 + 5) = (5 × 2) + (5 × 5)
Area = ☐ square inches
Answer:
7 is divided into two parts
7 = (1 + 6)
7 = (2 + 5)
7 = (3 + 4)
7 = (4 + 3)
7 = (5 + 2)
7 = (6 + 1)
Area of the larger rectangle = 5 in × 7 in = 35 square in
5 × 7 = 5 × (3 + 4) = (5 × 3) + (5 × 4) = 15 + 20 = 35
5 × 7 = 5 × (1 + 6) = (5 × 1) + (5 × 6) = 5 + 30 = 35
5 × 7 = 5 × (2 + 5) = (5 × 2) + (5 × 5) = 10 + 25 = 35
all these equations are correct.

Question 5.
What is the total area of the design below?

A. 4 × 4 = 16 square inches
B. (4 × 4) + (2 × 2) = 20 square inches
C. (4 × 4) + (2 × 2) + (2 × 2) = 24 square inches
D. 8 × 4 = 32 square inches
Answer:
The design is divided into 3 rectangles
(4 in × 4 in) + (2 in × 2 in) + (2 in × 2 in)
16 square in + 4 square in + 4 square in
24 square in.
So the answer is C. (4 × 4) + (2 × 2) + (2 × 2) = 24 square inches.

Question 6.
Jared draws a rectangle. Explain how to find the area using the Distributive Property.

Answer:
Area of the rectangle using distributive property
Dividing the rectangle into smaller parts
9 ft is divided into 5 ft and 4 ft
Area of the smaller rectangles
5 × 9 = 5 × (5 + 4)
= (5 × 5) + (5 × 4)
= 25 + 20
= 45 square ft.
Area of the rectangle Jared drew is 45 square ft.

Question 7.
Fran has a square flower bed. One side of the flower bed is 3 feet long. How can you find the area of the flower bed?
Answer:
Shape of Fran flower bed is Square.
Square have 4 equal sides.
One side of the flower bed = 3 ft.
Area of a square = side × side
Area of the flower bed = 3 ft × 3 ft = 9 square ft.
Fran square flower bed is 9 square ft.

Question 8.
Find the missing side length. Then find the area and explain how to find it.

Answer:
The opposite side of the missing side has a length of 6 ft.
Here the missing length having 1 ft length above and below it.
6 ft id distributed as
6 = 1 + 1 + ?
? = 6 – 1 – 1
? = 6 – 2
? = 4 ft.
The missing side length is 4 ft.
On adding the lengths 1 ft + 4 ft + 1 ft = 6 ft .
same as the opposite side.

Question 9.
This rectangle has an area of 56 square centimeters. What is the missing length? Use an equation to explain.

Answer:
Given
Area of the rectangle is 56 square cm.
Area of a rectangle = length × width
56 square cm = 7 cm × ? cm
? cm = 56 square cm ÷ 7 cm
? cm = 8 cm.
The missing length is 8 cm.
7 cm × 8 cm = 56 square cm.

Question 10.
What is the area of Ron’s figure? Explain.

Answer:

Question 11.
Maddie makes a mosaic with 1-inch glass squares, as shown below. Which color of glass has the greatest area in Maddie’s mosaic?

Answer:
Number of violet squares = 1 + 3 +5 + 5 + 3 + 1 = 9 + 9 = 18 squares
Area of violet squares = 18 square in
Number of yellow squares = 2 + 6 + 2 = 10 squares.
Area of yellow squares = 10 square in
Number of white squares = 4 + 4 = 8 squares.
Area of white squares = 8 square in

Violet color squares have greater area.

Question 12.
Select the correct side length for each square given its area.

Answer:

Question 13.
Explain how to find the area of each rectangle and the total area of the rectangles.

Answer:
Number of squares Ryan rectangle covers = 6 × 4 = 24 squares.
Area of Ryan rectangle = 24 square cm
Number of squares Jodie rectangle covers = 3 × 2 = 6 squares.
Area of Jodie rectangle = 6 square cm.
Total area of the rectangles = 24 square cm + 6 square c,m = 30 square cm.

Question 14.
Some students in Springfield make a parade float with the letter S on it. Draw lines to divide the shape into rectangles. Then find how many square feet the letter is.

A. 28 square ft
B. 56 square ft
C. 54 square ft
D. 90 square ft
Answer:

Question 15.
Max draws 2 rectangles, each with an area of 24 square centimeters. What could be the side lengths of Max’s rectangles? Show how he could use the Distributive Property to represent the area in each case.
Answer:
Given
Number of rectangles Max draws = 2 rectangles
Area of each rectangles = 24 square cm
The lengths of the rectangle 1 = 4 cm × 6 cm
Using distributive property
to find the area of rectangle 1 =
=  4 cm × 6 cm
= 4 cm × (2 cm + 4 cm )
= (4 cm ×2 cm) + (4 cm × 4 cm)
= 8 square cm + 16 square cm
=24 cm
to find the area of rectangle 2 =
The lengths of the rectangle 2 = 3 cm × 8 cm
= 3 cm × 8 cm
= 3 cm × (3 cm + 5 cm)
= (3 cm × 3 cm ) + (3 cm × 5 cm)
= 9 square cm +15 square cm
= 24 square cm

Question 16.
A community center builds a new activity room in the shape shown below. Explain how to find the area of the room, and solve.

Answer:

Question 17.
Show 2 different unit squares that you can use to measure the area of these rectangles. Find the area with your unit squares.

Answer:

Question 18.
Ethan wants to know the area of the yellow part of this design.

A. Explain how you can break this problem into simpler problems.
Answer:
Finding the area of the outer rectangle.
Finding the area of the inner rectangle .
Then subtracting the area of inner rectangle from outer rectangle gives the area of the yellow region.

B. Find the yellow area. Show your work.
Answer:
Area of the outer rectangle = 6 ft × 4 ft = 24 square ft
Area of the inner rectangle = 3 ft × 2 ft = 6 square ft
Area of the yellow area = Area of outer rectangle – Area of inner rectangle
= 24 square ft – 6 square ft
= 18 square ft.
The yellow area is 18 square ft.

Topic 6 Performance Task

Banner Design
Jessie is designing a banner that has red, blue, and white sections.
The Banner Details list shows the rules for each color.
The Jessie’s Banner diagram shows the different sections of the banner.
Banner Details

• Red sections must have a total area greater than 5 in. 4in. 3 in. 40 square inches.
• Blue sections must have a total area greater than 30 square inches.
• The white section must have an area less than 40 square inches.

Use the Jessie’s Banner diagram to answer Question 1.

Question 1.
To check if his banner fits the rules, Jessie started this table. Complete the table. Use multiplication and addition as needed.

Answer:

Use the table above and the Banner Details list to answer Question 2.

Question 2.
Is Jessie’s banner within the totals in the Banner Details list? Explain.
Answer:

Banner Details

• Red sections must have a total area greater than 5 in. 4in. 3 in. 40 square inches.
• Blue sections must have a total area greater than 30 square inches.
• The white section must have an area less than 40 square inches.

Yes, Jessie banner is not within the totals in the Banner detail list.
Total red color area = 35 square inches + 21 square inches = 56 square inches.
56 square inches is greater than 40 square inches
Total White color area = 30 square inches.
30 square inches less than 40 square inches.
Total blue color area = 9 square inches + 25 square inches = 34 square inches.
34 square inches greater than 30 square inches.

Question 3.
Jessie makes a square patch to go on top of the banner.
Part A
Draw unit squares to cover the patch. How many unit squares cover the patch?

Answer:

Part B
Jessie says if he checks the area by multiplying, the area will be the same as if he counts each unit square. Is he correct? Explain.
Answer:

Yes, Jessie is correct.
Area of the patch by multiplying is same as area by counting the squares.

Question 4.
Jessie uses two colors to make the patch. The colors have different areas.
Part A
Explain how to separate the square into two smaller rectangles with different areas. Use multiplication to find the areas of each of the smaller rectangles.
Answer:

Part B
Is the area of the square equal to the total area of the two smaller rectangles? Use an equation to explain.
Answer:

Go through the enVision Math Common Core Grade 3 Answer Key Topic 7 Represent and Interpret Data regularly and improve your accuracy in solving questions.

enVision Math Common Core 3rd Grade Answers Key Topic 7 Represent and Interpret Data

Essential Question:
How can data be represented, analyzed, and interpreted?

enVision STEM Project: Seasons
Do Research Use the Internet or other sources to find information about patterns of temperature in the different seasons where you live. Include information about the average monthly temperatures and the record low and high temperatures.

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

• For one week, record the daily high and low temperatures in the area where you live. Make a graph displaying this information.
• Find the difference between the highest and lowest daily temperatures from your graph.

Review What You Know

Vocabulary
Choose the best term from the box. Write it on the blank.
• equal groups
• multiples
• multiplication
• number line

Question 1.
_______ have the same number of items.
Answer:
_Equal Groups__ have the same number of items.

Question 2.
A ________ can be used to help compare numbers.
Answer:
A _multiples_ can be used to help compare numbers.

Question 3.
________ is used to find a total when joining equal groups.
Answer:
_Multiplication__ is used to find a total when joining equal groups.

Multiplication

In 4 and 5, complete the equation.
Question 4.
5 × 3 = _______
Answer:
5 × 3 = 15.
Explanation:
When we multiply 5 with 3 times we get 15.
When you add 5 three times we get 15.
5 + 5 + 5 = 15.

Question 5.
3 × _______ = 21
Answer:
3 × 7 = 21
Explanation:
Given
one multiple and the product.
one multiple = 3
Product = 21
To find the other multiple we divide product by multiple
21 ÷ 3 = 7
3 × 7 = 21.

Question 6.
Make a bar diagram to represent 4 × 6.
Answer:

Multiplication on the Number Line

Question 7.
Ed bought 2 bags of grapefruit. There are 6 grapefruit in each bag. How many grapefruit did he buy? Draw jumps on the number line to find the answer.

Answer:

Question 8.
Show the multiplication fact 3 x 4 on the number line. Write the product.

Answer:

Finding Area
Question 9.
Find the area of the rectangle.

☐ = 1 square inch
Answer:
Number of squares = 4 × 5 = 20 squares.
Area = 20 square inches.

Pick a Project

PROJECT 7A
How is a book printed?
Project: Collect Data and Create Picture Graphs

PROJECT 7B
Would you like to live in a city?
Project: Make a Bar Graph About Cities and Towns

PROJECT 7C
What is your favorite animal?
Project: Develop a Picture Graph About Animals

3-ACT MATH PREVIEW

Math Modeling
Swings and Slides

+ `1

Solve & Share
Students in Jorge’s class took a survey of their favorite cereals and made this graph to show the results. Name at least three facts about the information in the graph.
I can … use picture graphs and bar graphs to answer questions about data sets.

Look Back! What do the two different symbols on the graph stand for?

Essential Question
How Can You Read Picture Graphs?

Visual Learning Bridge

Use the key
Look at the data for East Falls League.
There are 3 and 1 .
The 3 represent 3 × 2 = 6 teams.
The 1 represents 1 × 1 = 1 team.
6 + 1 = 7
There are 7 teams in the East Falls League.

How many more teams does the East Falls League have than the South Falls League?
Use the picture graph to write equations and compare the two rows.

The East Falls League has 3 more teams than the South Falls League.

Convince Me! Be Precise Tell something about each league you can find out from the picture graph.

Another Example!
A scaled bar graph uses bars to represent and compare information. This bar graph shows the number of goals scored by different players on a hockey team. The scale shows the units used.

On this bar graph, each horizontal grid line represents two units. Every other grid line is labeled: 0, 4, 8, and so on. For example, the line halfway between 4 and 8 represents 6 goals.

Guided Practice

Do You Understand?
In 1 and 2, use the bar graph above.
Question 1.
How many goals in all did Alex and Reggie score?
Answer:

Number of goals Alex scored = 18 goals.
Number of goals Reggie scored = 16 goals.
Total number of goals both scored in all = 18 + 16 = 34 goals.

Do You Know How?
Question 2.
Explain how to find how many fewer goals Cindi scored than Alex.
Answer:
Number of goals Alex scored = 18 goals
Number of goals Cindi scored = 14 goals
Number of fewer goals Cindi scored than Alex = 18 – 14 = 4 goals.
Cindi scored 4 fewer goals than Alex.

Independent Practice

In 3-5, use the picture graph.

Question 3.
Which area has lights on for exactly 50 hours each week?
Answer:
full bulb = 10 hours    Half bulb = 5 hours
Number of hours of light on each week in Exercise room = 7 full bulb and 1 half bulb
(7 × 10 hours) + (1 × 5 hours) = 70 + 5 = 75 hours.
Number of hours of light on each week in Locker room = 9 full bulbs
9 × 10 hours = 90 hours
Number of hours of light on each week in Swimming Pool = 5 full and 1 half
(5 × 10 hours) + (1 × 5 hours) = 50 + 5 = 55 hours.
Number of hours of light on each week in Tennis Court = 5 full
5 × 10 hours = 50 hours.
Tennis Court area has lights on for exactly 50 hours each week

Question 4.
What does the half bulb in the data for the exercise room represent?
Answer:
Half bulb in the data for exercise room represents 5 hours.

Question 5.
In one week, how many more hours are lights on in the exercise room than at the swimming pool?
Answer:
Number of hours of light on each week in Exercise room = 7 full bulb and 1 half bulb
(7 × 10 hours) + (1 × 5 hours) = 70 + 5 = 75 hours.
Number of hours of light on each week in Swimming Pool = 5 full and 1 half
(5 × 10 hours) + (1 × 5 hours) = 50 + 5 = 55 hours.
Number of more hours of light on in the exercise room than the swimming Pool = 75 hours – 55 hours = 20 hours.
20 hours of more light on in the exercise room than the swimming Pool

Problem Solving

In 6-8, use the picture graph.

Question 6.
Reasoning For which days can you use the expression 9 x 7 to find how many points were scored?
Answer:

From the data above
Number of points scored by a football team on October 3 = 9 × 7 = 63 points
Number of points scored by a football team on October 10 = 9 × 7 = 63 points
Number of points scored by a football team on October 17 = 6 × 7 = 42 points
Number of points scored by a football team on October 24  = 7 × 7 = 49 points
October 3  and October 10 days can  use the expression 9 x 7 to find number of  points  scored.
On both the days the team scored 9 × 7 points = 63 points.

Question 7.
On which days did the football team score fewer than 50 points?
Answer:
Number of points scored by a football team on October 17 = 6 × 7 = 42 points
Number of points scored by a football team on October 24  = 7 × 7 = 49 points
On October 17 and October 24 the football team scored points fewer than 50.

Question 8.
Higher Order Thinking How many more points were scored on October 10 and 24 combined than on October 3 and 17?
Answer:
Number of points scored by a football team on October 3 = 9 × 7 = 63 points
Number of points scored by a football team on October 17 = 6 × 7 = 42 points
Number of points scored on October 3 and October 17 combined = 63 + 42 = 105 points.
Number of points scored by a football team on October 10 = 9 × 7 = 63 points
Number of points scored by a football team on October 24  = 7 × 7 = 49 points
Number of points scored on October 10 and October 24 combined =63 + 49 = 112 points.
Number of more points scored on October 10 and 24 combined than on October 3 and 17 combined = 112 – 105 = 7 points.
7 more points scored on October 10 and 24 combined than on October 3 and 17 combined.

Assessment Practice

In 9 and 10, use the bar graph at the right.

Question 9.
How many more miles per hour is the top running speed of the cheetah than of the Cape hunting dog?
A. 25 miles per hour
B. 30 miles per hour
C. 35 miles per hour
D. 40 miles per hour
Answer:

Given the Top running speeds of animals
mile per hour – mph
The running speed of Cape Hunting Dog = 35 mph
The running speed of Cheetah = 70 mph
The running speed of Grizzly Bear = 30 mph
The running speed of Coyote = 40 mph
The top running speed of the Cheetah  more than of the Cape hunting dog = 70 mph – 35 mph
= 70 mph – 35 mph  = 35 mph.
Answer B. 35 mph more miles per hour is the top running speed of the Cheetah than of the Cape hunting dog.

Question 10.
How many fewer miles per hour is the top running speed of the grizzly bear than of the coyote and the Cape hunting dog combined?

A. 40 miles per hour
B. 45 miles per hour
C. 50 miles per hour
D. 55 miles per hour
Answer:
The top running speed of Grizzly Bear = 30 mph
The top running speed of Coyote = 40 mph
The top running speed of Cape Hunting Dog = 35 mph
The top running speed of Coyote and Cape Hunting Dog combined = 40 mph + 35 mph = 75 mph
The top running speed of the grizzly bear fewer than of the coyote and the Cape hunting dog combined = 75 mph – 30 mph
= 75 mph – 30 mph
= 45 mph
Answer B. 45 mph is fewer miles per hour is the top running speed of the grizzly bear than of the coyote and the Cape hunting dog combined.

Lesson 7.2 Make Picture Graphs

Solve & Share
Mary is helping her teacher count school playground equipment. She records the data in a frequency table. Use the data in the table to complete the picture graph. Write two statements about your completed graph.
I can … make a picture graph to record information and answer questions about a data set.

Look Back! How did you know the number of symbols to draw for jump ropes?

Essential Question
How Do You Make a Picture Graph?

Visual Learning Bridge
Sam recorded the number of each kind of bicycle a store sold during one month. He made a frequency table. Use the table to make a picture graph.
You can also collect data with a survey by asking people questions.

Write a title for the picture graph.
The title is “Kinds of Bicycles Sold.”
Choose a symbol for the key. Decide what each whole symbol and half-symbol will represent.