## Envision Math Common Core 3rd Grade Answers Key Topic 11 Use Operations with Whole Numbers to Solve Problems

Essential Question:
What are ways to solve 2-step problems? enVision STEM Project: Engineering Design
Do Research Use the Internet or other sources to find information about kites. Find two designs for building a kite. What materials do you need for each design? How much do those materials cost?
Journal: Write a Report Include what you found. Also in your report:

• Find the total cost for each design.
• Decide which design is less expensive.
• Write an equation to show how much less expensive that design is.

Review What You Know

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

• equation
• product
• quotient
• unknown

Question 1.
The equal sign shows that the left side of a(n) __________ has the same value as the right side.
The equal sign shows that the left side of a(n) equation has the same value as the right side.

Question 2.
A question mark can be a symbol for a(n) _________ value.
A question mark can be a symbol for a(n) unknown value.

Question 3.
The answer to a division problem is the _________.
The answer to a division problem is the Quotient .

Question 4.
739 – 104 Question 5.
512 + 216 Question 6.
710 – 569 Question 7.
104 + 67 Question 8.
664 + 78 Question 9.
825 – 477 Multiplication and Division

Question 10.
60 ÷ 6
60 ÷ 6 = 10
Explanation : Question 11.
40 × 4
40 × 4 = 160
Explanation : Question 12.
7 × 3
7 × 3 = 21 .

Question 13.
(3 × 10) × 6 =
A. (3 × 10) + (6 × 10)
B. (3 × 6) + (10 × 6)
C. 3 × (10 × 6)
D. (10 + 10 + 10) + 6
Option C.
(3 × 10) × 6 = 3 × 10 × 6 = 180 .
Explanation :
All signs are multiplication signs in the Equation so, just remove the brackets .

Question 14.
4 × (20 × 2) =
A. (4 × 20) × 2
B. (4 × 20) + (4 × 2)
C. 4 + (20 + 20)
D. (4 + 20) + (4 + 2)
Option A .
4 × (20 × 2) = 4 × 20 × 2 = 160
Explanation :
All signs are multiplication signs in the equation so, just remove the brackets .

Model with Math

Question 15.
Caleb has 8 toy cars. Each car has 4 wheels. He wants to know how many wheels are on all of his cars. Represent this problem using a bar diagram and an equation. Then solve.
Number of Car toys = 8
Number of wheels for each car toy = 4 wheels .
Total number of wheels for 8 car toys = 8 × 4 = 32 wheels .

Pick a Project

PROJECT 11A
Why do stores have sales and other promotions?
Project: Write a Skit About a Sale PROJECT 11B
How did grapefruit first come to Florida?
Project: Create a Poster About Citrus Groves PROJECT 11C
How would you make a budget for selling lemonade in a lemonade stand? 3-ACT MATH PREVIEW

Math Modeling
Cash Bucket I can … model with math to solve a problem that involves computing with whole numbers..

### Lesson 11.1 Solve 2-Step Word Problems: Addition and Subtraction

Solve & Share
Jen buys a backpack and a sleeping bag. She gets $10 off if the total is more than$200. What is the final cost of Jen’s order? Complete the bar diagram below. Then draw another diagram to solve the problem.
I can… draw diagrams and write equations to show how the quantities in a problem are related. Look Back! How can you use an estimate to show that the final cost of the backpack and sleeping bag you found makes sense? Essential Question
How Can You Use Diagrams to Solve 2-Step Problems?

Visual Learning Bridge
The results of a car survey are shown in the table. How many fewer cars in the survey have poor fuel efficiency than have either good or average fuel efficiency? Use estimation to check the answer. Find and answer the hidden question.
How many total cars in the survey have either good or average fuel efficiency? How many fewer cars in the survey have poor fuel efficiency? Convince Me! Critique Reasoning Jane used estimation to check the reasonableness of the work above. Explain whether Jane’s work makes sense.
165 and 98 is about 200.
224 minus 200 equals 24, which is close to 39.

Guided Practice

Do You Understand?
Question 1.
How do the bar diagrams help you write equations for the problem on the previous page?
A bar is used to represent the whole. So we label it with the amount of the whole. .Finally, we indicate the required number of fractional parts with a question mark and determine the value represented.
In the above problem two quantities are given and total is calculated .

Do You Know How?
Question 2.
Josie has $145. She buys a bike for$127. The next week she saves $15. How much money does Josie have now? Complete the bar diagrams and write equations to solve. Answer: Independent Practice In 3, use the map. Draw diagrams and write equations to solve. Use letters to represent unknown quantities. Question 3. Manuel’s family drove from Louisville to Indianapolis to Detroit and then directly back to Louisville. How much farther did they drive going to Detroit than returning from Detroit? Answer: Distance from Louisville to Indianapolis = 114 miles . Distance from Indianapolis to Detroit = 283 miles . Distance from Louisville to Indianapolis to Detroit = t = 114 + 283 = 397 miles . Question 4. How can you estimate to check if your answer above is reasonable? Explain. Answer: Problem Solving Question 5. Write equations to find how many more tickets were sold for the roller coaster on Saturday than for the swings on both days combined. Use letters to represent the unknown quantities. You can draw diagrams to help. Answer: Question 6. Higher Order Thinking Write a two-step problem about buying or selling books that can be solved using addition or subtraction. Solve your problem. Answer: Question 7. enVision® STEM Lindsay has 6 boxes of toothpicks. There are 80 toothpicks in each box. Lindsay uses all of the toothpicks to build a model bridge. How many toothpicks does Lindsay use in all? Answer: Question 8. Model with Math Matt had 327 plastic bottles for recycling. He recycled 118 bottles on Monday. He recycled 123 bottles on Tuesday. How many bottles does Matt have left to recycle? Use representations such as bar diagrams or equations to model with math. Use letters to represent the unknown quantities. Estimate to check your work. Answer: Assessment Practice Question 9. Ukie has 142 leaves in her collection. She gives 25 to her brother and then collects 19 more. Create and solve equations to find the number of leaves Ukie has now. Answer: Question 10. Richard had$236 in his savings account. He got $45 for his birthday and saved all but$16 of it. Create and solve equations to find the amount in Richard’s savings account now.

### Lesson 11.2 Solve 2-Step Word Problems: Multiplication and Division

Solve & Share
Two friends decide to share equally all of the apples they picked. They filled the bags shown with 4 apples in each bag. How many apples will each friend get?
I can … draw diagrams and write equations to show how the quantities in a problem are related. Look Back! Tell why multiplication can be used to find the total number of apples.

Essential Question
How Can You Use Diagrams to Solve 2-Step Problems?

Visual Learning Bridge
The teams for the City Baseball Tournament are divided equally into 3 leagues. Each league is divided into 2 regions with the same number of teams in each region. How many teams are in each region? Step 1
Find and answer the hidden question.
How many teams are in each league? t = 24 ÷ 3
t = 8
There are 8 teams in each league.

Step 2
Use the answer to the hidden question to answer the original question.
How many teams are in each region? r = 8 ÷ 2
r = 4
There are 4 teams in each region.

Convince Me! Make Sense and Persevere Another tournament has 2 leagues with 9 teams in each league. An equal number of teams will play on each of the 3 days of the tournament. Each team will play once. How many teams will play on each day?

Guided Practice

Do You Understand?
Question 1.
Why do you need to solve the hidden question first for the problem in Box A on the previous page?

Question 2.
What multiplication equation could be written for the bar diagram in Box C on the previous page? Do You Know How?
In 3, complete the bar diagrams and write equations to solve.
Question 3.
There are 8 students in 2 school vans with the same number of students in each van. The admission fee for each student is $5. What is the total admission fee for all of the students in one van? Answer: Independent Practice In 4, draw diagrams and write equations to solve. Use letters to represent the unknown quantities. Question 4. Arif saves$4 each week. After 6 weeks, he spends all the money he saved on 3 items. Each item costs the same amount. How much does each item cost?

Question 5.
Georgia says that because she knows 5 × 4 = 20, she knows Arif will have saved more than $20. Is Georgia correct? Explain. Answer: Problem Solving Question 6. Be Precise One Grade 3 class collected 86 pounds of newspaper. The other Grade 3 class collected 65 pounds of newspaper. Which grade collected the greatest number of pounds of newspaper? Answer: Question 7. Six pounds of pecans are shared equally between 2 friends. Each pound has 60 pecans. How many pecans does each friend get? Write equations to solve. Use letters to represent the unknown quantities. Answer: Question 8. Higher Order Thinking A store can buy boxes of 8 calculators each for$32 or boxes of 5 calculators each for $30. How much less I CICIC is each calculator if the store buys boxes of 8 calculators each instead of boxes of 5 calculators each? Complete the bar diagrams and solve. Answer: Assessment Practice Question 9. In a basketball game, Morgan made 9 shots, each worth 2 points. Jim scored the same number of points in the game. Jim’s shots were worth 3 points each. How many shots did Jim make? A. 4 shots B. 6 shots C. 8 shots D. 9 shots Answer: Question 10. Mia saved an equal amount of money each week for 6 weeks. Heather saved$3 a week for 8 weeks. Mia and Heather saved the same total amount. Which equation could you use to help find how much Mia saved each week?
A. s = 8 × 6
B. s = 6 × 3
C. s = 6 ÷ 3
D. s = 24 ÷ 6

### Lesson 11.3 Solve 2-Step Word Problems: All Operations

Solve & Share

An aquarium had 75 clownfish in a large water tank. The clownfish represented in the graph were added to this tank. How many clownfish are in the tank now? Write and explain how you found the answer.
I can … solve two-step word problems involving different operations. Look Back! What operations did you use to solve this problem? Tell why you needed those operations.

Essential Question
How Can You Solve 2-Step Problems?

Visual Learning Bridge
Jill can rent a car and GPS device for $325 for 7 days. What is the cost to rent the car for a week without the GPS device? Use estimation to check the answer. a = cost to rent the GPS for 7 days b = cost to rent the car without the GPS for 7 days To solve a two-step problem, you first have to find and answer the hidden question. The hidden question is: How much does it cost to rent the GPS for 7 days? Write and solve an equation for the hidden question. Use a letter for the unknown quantity. 7 ×$9 = a
$63 = a It costs$63 to rent the GPS for 7 days.

Write and solve an equation for the problem. Use a letter for the unknown quantity
$325 –$63 = b
$262 = b It costs$262 to rent the car without the GPS for 7 days.
You can use compatible numbers and mental math to estimate. 7 × 9 = 63. 63 is close to 75 and 325 – 75 is 250. 250 is close to 262, so the

Convince Me! Use Structure Jill can rent a different car and a DVD player for $384 for 7 days. She wants to know the cost to rent the car for a week without the DVD player. Explain how this problem is different from the problem above. Then solve. Guided Practice Do You Understand? Question 1. Dotty says you also can use$63 + b = $325 instead of$325 – $63 = b for the problem on the previous page. Can this equation be used to get the correct answer? Explain. Answer: Question 2. When solving a two-step problem, why should you answer the hidden question first? Answer: Do You Know How? In 3, write equations to solve. Use letters to represent unknown quantities. Question 3. Look at Box A on the previous page. How much would it cost Jill to rent the car for a week with the GPS and DVD player? Answer: Independent Practice In 4 and 5, write equations to solve. Use letters to represent the unknown quantities. Question 4. Trish bought 4 yards of rope to make a swing. Judy spent$18 on rope. How much did the two girls spend in all? Question 5.
Martha has 12 stamps. Toni has 21 stamps. Toni divides her stamps into 3 equal groups. She gives one group to Martha. How many stamps does Martha have now? Problem Solving

In 6 and 7, use the fruit shown at the right. Question 6.
Maurice needs 36 apples for his party. How much will the apples cost?

Question 7.
Higher Order Thinking Delia bought 24 lemons and 63 apples. How much did she spend on fruit?

Question 8.
Vocabulary Fill in the blank. When you ________ 72 to the nearest ten, you get 70.

Question 9.
Generalize Carla collected 328 shells. Dan collected 176 shells. How can you use compatible numbers to estimate how many shells they collected?

Question 10.
enVision® STEM Sasha is using the engineering design process to plan a scratching post for her cat. There will be 3 levels. Sasha will spend $10 on the pole. She will spend$7 on each level. Sasha’s plan to find the total cost is shown at the right. Is she correct? Explain.
3 × $10 =$30
$30 +$7 = $37 Answer: Assessment Practice Question 11. Use the fruit from Exercises 6 and 7. Kaylie bought 4 bags of oranges and 1 bag of apples. How many pieces of fruit did she buy? Write equations to solve. Use letters to represent any unknown quantities. Answer: ### Lesson 11.4 Problem Solving Critique Reasoning Solve & Share Concert tickets for adults cost$12. Concert tickets for students cost $9. Marie has$190. She wants to buy 1 adult and 20 student tickets.

Skip says, “$190 is enough for all the tickets because$9 × 20 = $180 and$180 is less than $190.” Does Skip’s reasoning make sense? Explain. I can … critique the reasoning of others using what I know about estimating. Thinking Habits Be a good thinker! These questions can help you. • What questions can I ask to understand other people’s thinking? • Are there mistakes in other people’s thinking? • Can I improve other people’s thinking? Look Back! Critique Reasoning What did Skip get correct in his reasoning and what did he get wrong? Essential Question How Can You Critique the Reasoning of Others? Visual Learning Bridge Gina has$68. She earns $9 an hour babysitting. She wants to buy a computer program for$130.
Will Gina have enough money to buy the program if she babysits for 6 hours?
Danielle solved this problem.
Her work is shown at the right.

6 × $9 =$54, which is about $60. Gina has$68, which is about $70.$60 + $70 =$130

What is Danielle’s reasoning to support her conclusion?

How can I critique the reasoning of others?
I can

• decide if the strategy used makes sense.
• look for flaws in estimates or calculations.

Here’s my thinking…
Danielle’s reasoning has flaws. Because Danielle rounded up each time, her total estimate adds up to more than the actual amount Gina will have.
6 × 9 = $54$54 + $68 =$122
Danielle’s conclusion is not correct because the actual amount Gina will have is less than $130. Convince Me! Critique Reasoning Tony says that if Gina babysits for 8 hours, she will have enough money. He reasons that 8 ×$9 = $72, which rounds to$70, and $70 +$60 = $130. Does his reasoning make sense? Explain. Guided Practice Critique Reasoning Miguel’s and Nita’s goal is to collect 600 box tops. When you critique Miguel collects 253 box tops in January and 158 box reasoning, you can look for tops in February. Nita collects 209 box tops in all. Teri estimates they have already collected more than clarify or improve their goal. She estimates that 250 + 150 = 400 and 400 + 200 = 600. the reasoning. Question 1. What is Teri’s argument? How does she support it? Answer: Question 2. Does Teri’s conclusion make sense? Explain. Answer: Independent Practice Critique Reasoning Gill gets 24 stickers on Monday. She gets the same number of stickers on Tuesday. Gill then shares all of her stickers equally among 8 friends. Liam concluded that each friend gets fewer than 5 stickers. His work is shown at the right. Liam’s work 8 × 3 = 24 So, 24 ÷ 8 = 3. Each friend gets 3 stickers. 3 is less than 5. Question 3. What is Liam’s argument? How does he support it? Answer: Question 4. Does Liam’s reasoning make sense? Explain. Answer: Question 5. Explain the strategy you would use to improve Liam’s work. Answer: Problem Solving Performance Task Selling Buttons A Grade 3 class is going to buy buttons like the ones shown. Each package costs$8. Each package is 40 cm long. They need to know if $50 is enough money to buy 200 buttons. Jim’s work 6 × 30 = 180 buttons, which is not enough. 7 × 30 = 210 buttons, so the class needs to buy 7 packages. 7 ×$8 = $48 48 < 50$50 is enough.

Question 6.
Make Sense and Persevere Which given information do you NOT need to solve this problem?

Question 7.
Critique Reasoning Jim solved the problem as shown above. Does Jim’s strategy make sense? Explain. Question 8.
Be Precise Are Jim’s calculations correct? Explain how you decided.

Question 9.
Use Appropriate Tools Can place-value blocks be used to check whether Jim’s math is correct? Explain.

### 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 next to the match.

Find a match for every clue.
I can… multiply and divide within 100.

Clues
A. The product is between 55 and 60.
B. The product is equal to 10 × 2.
C. The quotient has two digits.
D. The product is between 50 and 55.
E. The quotient is less than 5.
F. The product is between 30 and 40.
G. The quotient is a multiple of 3.
H. The quotient is equal to the divisor. ### Topic 11 Vocabulary Review

Understand Vocabulary
Choose the right term from the Word List. Write it in the blank.
Word List

• difference
• dividend
• divisor
• equation
• factor
• product
• quotient
• sum
• unknown

Question 1.
The missing number in an equation is a(n) ___________.

Question 2.
A multiplication problem has more than one __________.

Question 3.

Question 4.
In a division problem, you divide the _______ by the ________.

Draw a line to match the term to the result of a relationship between the numbers 80 and 4. Use Vocabulary in Writing
Question 9.
There are 52 cards in a deck. Al turns 4 cards face up. Then he deals the rest of the cards to 6 people. How many cards does each person get? Explain how to solve this problem. Use at least 2 terms from the word list.

### Topic 11 Reteaching

Set A pages 409-412

You can use more than one step to solve problems.
Use bar diagrams to help write equations.
A ship has 439 passengers. One hundred seventy-nine new passengers get on at a port. Then 250 passengers get off. How many passengers are on the ship now? Remember to check for reasonableness after each step.

In 1, draw bar diagrams and write equations to solve the problem.
Question 1.
Mr. Sato has $800. He spends$600 on rent. Then he spends $85 on groceries. How much money does he have left? Answer: Set B pages 413-416 You can use bar diagrams and equations to show how numbers are related. Roger reads a poem with 16 lines. Each line has 7 words. The poem is on 2 pages with the same number of lines on each page. How many words are on each page? 16 ÷ 2 = a lines on each page a = 8 lines 8 × 7 = w words on each page w = 56 words Remember to use unknowns to stand for the numbers you need to find. In 1, draw bar diagrams and write equations to solve the problem. Question 1. A rancher has 24 cows. He puts an equal number of cows in 4 fields. Each cow produces 5 gallons of milk. How much milk do the cows in one field produce? Answer: Set C pages 417-420 Ryan reads a book with 420 pages. He also reads 7 magazine articles. Each article has 6 pages. How many pages does Ryan read? First, write and solve an equation for the hidden question. Use a letter for the unknown quantity. n = number of article pages 7 × 6 = n 42 = n Then, write and solve an equation for the problem. Use a letter for the unknown quantity. t = total pages read 420 + 42 = t 462 = t Ryan reads 462 pages. Remember to perform the operations in the correct order for the problem. In 1 and 2, write equations to solve. Use letters to stand for unknown quantities. Question 1. Destiny gets$168 for selling mint cookies and chocolate cookies. She sells 8 boxes of chocolate cookies at $9 per box. How much does Destiny get for selling mint cookies? Answer: Question 2. Mary buys three books that cost$7 and one book that costs $12. How much did Mary spend on books? Answer: Set D pages 421-424 Think about these questions to help you critique the reasoning of others. Thinking Habits • What questions can I ask to understand other people’s thinking? • Are there mistakes in other people’s thinking? • Can I improve other people’s thinking? Remember to consider all parts of an explanation. Pat needs to practice guitar for at least 40 hours this month. He has practiced 9 hours a week for the last 3 weeks. He practiced for 15 hours this week. Pat says, “3 × 9 is less than 40. So I have not practiced enough.” Question 1. Does Pat’s reasoning make sense? Explain. Answer: Question 2. How can you clarify or improve Pat’s reasoning? Answer: ### Topic 11 Assessment Practice Question 1. Emma works at a diner. On Monday, she served 7 tables with 6 people at each table. On Tuesday, she served 72 people. She wants to know how many more people she served on Tuesday than on Monday. Choose the correct operations to represent this problem using equations. Write each operation on the blanks. 7 ____ 6 = m 72 _____ 42 = d. + – × ÷ Answer: Question 2. Madison has a jar of 160 jelly beans. She saves 88 for herself. She divides the rest equally among 8 friends. She wants to find how many jelly beans each friend gets. Which equations should she use? Select all that apply. ☐ 80 ÷ 8 = a ☐ 88 – 8 = b ☐ 160 – 88 = c ☐ 160 ÷ 80 = d ☐ 72 ÷ 8 = e Answer: Question 3. Alberto sells magazine subscriptions to his neighbors. Three neighbors each buy 3 subscriptions. There are 8 issues in each subscription. Write and solve equations to find the total number of magazine issues all of his neighbors get. Answer: Question 4. Ro’s diner has a jar of 585 toothpicks. At lunch, 315 toothpicks are used. At dinner, 107 toothpicks are used. Ro says that 315 – 107 = t toothpicks are used in all, and that 585 – 208 = r toothpicks are left. Is her answer reasonable? Explain why or why not. Answer: Question 5. Tyler has 127 trading cards in his collection. Chloe has 63 cards in her collection. Then Chloe gives her collection away. She divides it equally among Tyler and 8 other friends. How many cards does Tyler have now? Answer: Question 6. Ten people each bring 4 platters of food to a family reunion. The 120 guests all share the platters equally. A. How many guests share 1 platter? Answer: B. Landon thinks that the answer is 12 guests. He says, “10 + 4 = 14, and 14 rounds to 10. Then 10 × 12 = 120.” Do you agree with his reasoning? Explain. Answer: Question 7. Jeri collects flags of the United Nations. She wants to know how many cases she needs to display her collection. She can fit 9 flags in one case. There are 193 countries in the United Nations. Jeri does not have flags for 130 countries. Which equation could she use first to solve this problem? A. c = 193 + 130 B. c = 193 – 130 C. f = 72 ÷ 9 D. f = 9 × 7 Answer: Question 8. Mrs. Lazio lives 8 miles from her office. She drives to her office and back 5 days each week. On Saturday she also drives 173 miles to visit her sister. How many miles does Mrs. Lazio drive each week? Answer: Question 9. José counted beetles for a project. He saw 5 beetles at the park each month and 3 beetles in his backyard each month. How many beetles did he count in 4 months? A. 7 B. 32 C. 30 D. 8 Answer: Question 10. Gail wants to save$68 for a jacket. She saves money for 3 weeks. Each week she saves $5. What is a good estimate for how much more money Gail needs to save? Select all that apply. ☐$55
☐ $50 ☐$60
☐ $40 ☐$35

Filmmaking Camp
Mrs. Radner and Mr. Yu teach filmmaking at a summer camp. The students work in crews to make movies. The summer ends with the crew and actors watching all the movies.

Class Details

• Mrs. Radner helps the students who make the action and drama films.
• Mr. Yu helps the students who make the comedy films.
• There are 246 actors in all.

Use the Class Details list and Film Types table to answer Questions 1-3. Question 1.
There are 20 actors working on each drama film. How many actors are not working on drama films?

Question 2.
Mrs. Radner has read 139 pages of the action-film scripts. How many more pages does she need to read to finish reading all of the action-script pages?

Question 3.
Mr. Yu says, “I have read 169 pages of the scripts for each comedy film. I need to read 27 more pages to finish reading all of the pages.” Do you agree with his reasoning? Explain why or why not.

Question 4.
Mrs. Radner wants to find t, the total time to watch all the student films. Use the Film Lengths table to answer the following questions. Part A
Mrs. Radner estimates t = 810 minutes. She reasons, “There are 9 films. Most of the films are 90 minutes long. Nine times 90 equals 810.” Do you agree with her reasoning? Explain why or why not.

Part B
Use bar diagrams or equations to represent t. Then find t.