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UNDERSTANDING COMPLEX MATH C PROBLEMS Written Specifically for the New TEKS


RALLY! EDUCATION 22 Railroad Avenue Glen Head, NY 11545 888-99-RALLY Fax: 1-516-671-7900 www.RALLYEDUCATION.com LESLIE@RALLYEDUCATION.com ………………..………………………………………………………………………………………………………………………

Understanding Math Series Understanding Math in the Real World

Understanding Complex Math Problems

+

This book is…

Understanding Complex Math Problems Written Specifically for the TEKS Grades 3 – 8 Multi-Step math problems for the New TEKS - Instruction helps students develop a process to solve complex math problems Buy the series separately… Includes 25 copies of Complex math plus the Teacher Guide Price: 25-pack: $179 Level C D E F G H

Grade 3 4 5 6 7 8

25-Pack Item # 8400-7 8404-5 8408-3 8412-0 8416-8 8420-5

Or Buy the Complete Program $229 or $349 15 or 25 Copies of Each Product Understanding Math in the Real World Understanding Complex Math Problems

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888-99-RALLY www.RALLYEDUCATION.com


UNDERSTANDING COMPLEX MATH C PROBLEMS Written Specifically for the New TEKS


ISBN 978-1-4204-8398-7 R 8398-7 Copyright ©2014 RALLY! EDUCATION. All rights reserved. No part of the material protected by this copyright may be reproduced in any form by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the copyright owner. Printed in the U.S.A. 0913.MAQ RALLY! EDUCATION • 22 Railroad Avenue, Glen Head, NY 11545 • (888) 99-RALLY

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4

Part A: Solving Complex Math Problems . . 5 Problem Page 1 (Modeled) TEKS 4A, 4K. . . . . . . . . . . . . . . 6 2 (Guided) TEKS 4A, 7C . . . . . . . . . . . . . . . 10 3 (Guided) TEKS 4H, 4K . . . . . . . . . . . . . . . 12 4 TEKS 3A, 3B . . . . . . . . . . . . . . . . . . . . . . 14 5 TEKS 3H, 4H . . . . . . . . . . . . . . . . . . . . . . 16 6 TEKS 6C, 6D . . . . . . . . . . . . . . . . . . . . . . 18 7 TEKS 4H, 6C . . . . . . . . . . . . . . . . . . . . . . 20 8 TEKS 3E, 4H . . . . . . . . . . . . . . . . . . . . . . 22 9 TEKS 4A, 6C . . . . . . . . . . . . . . . . . . . . . . 24 10 TEKS 5A, 5D . . . . . . . . . . . . . . . . . . . . . 26 11 TEKS 3A . . . . . . . . . . . . . . . . . . . . . . . . 28 12 TEKS 3D, 4A . . . . . . . . . . . . . . . . . . . . . 30 13 TEKS 5B, 5C . . . . . . . . . . . . . . . . . . . . . 32 14 TEKS 5E, 6C. . . . . . . . . . . . . . . . . . . . . . 34 15 TEKS 4A . . . . . . . . . . . . . . . . . . . . . . . . 36 16 TEKS 4A, 4K . . . . . . . . . . . . . . . . . . . . . 38 17 TEKS 2D, 5E . . . . . . . . . . . . . . . . . . . . . 40 18 TEKS 4H, 6B . . . . . . . . . . . . . . . . . . . . . 42 19 TEKS 4D, 4E . . . . . . . . . . . . . . . . . . . . . 44 20 TEKS 8A, 8B . . . . . . . . . . . . . . . . . . . . . 46

Part B: On Your Own . . . . . . . . . . . . . . . . . . 49 Problem Page 1 TEKS 3F, 3H . . . . . . . . . . . . . . . . . . . . . . 50 2 TEKS 3A . . . . . . . . . . . . . . . . . . . . . . . . . 51 3 TEKS 4K, 5B. . . . . . . . . . . . . . . . . . . . . . . 52 4 TEKS 5D, 6C . . . . . . . . . . . . . . . . . . . . . . 53 5 TEKS 5D, 6C . . . . . . . . . . . . . . . . . . . . . . 54 6 TEKS 4A, 5A . . . . . . . . . . . . . . . . . . . . . . 55 7 TEKS 4A, 4K . . . . . . . . . . . . . . . . . . . . . . 56

Problem Page 8 TEKS 4K, 5D . . . . . . . . . . . . . . . . . . . . . . 57 9 TEKS 4A, 4K . . . . . . . . . . . . . . . . . . . . . . 58 10 TEKS 5A, 7E. . . . . . . . . . . . . . . . . . . . . . 59 11 TEKS 3H, 4K . . . . . . . . . . . . . . . . . . . . . 60 12 TEKS 3C, 4H . . . . . . . . . . . . . . . . . . . . . 61 13 TEKS 4G . . . . . . . . . . . . . . . . . . . . . . . . 62 14 TEKS 4H, 4K . . . . . . . . . . . . . . . . . . . . . 63 15 TEKS 4G, 4J . . . . . . . . . . . . . . . . . . . . . . 64 16 TEKS 4A, 4K . . . . . . . . . . . . . . . . . . . . . 65 17 TEKS 3C, 4H . . . . . . . . . . . . . . . . . . . . . 66 18 TEKS 7C . . . . . . . . . . . . . . . . . . . . . . . . 67 19 TEKS 3C, 4H . . . . . . . . . . . . . . . . . . . . . 68 20 TEKS 3B, 3G . . . . . . . . . . . . . . . . . . . . . 69 21 TEKS 3H, 7C . . . . . . . . . . . . . . . . . . . . . 70 22 TEKS 4F, 4G. . . . . . . . . . . . . . . . . . . . . . 71 23 TEKS 4A . . . . . . . . . . . . . . . . . . . . . . . . 72 24 TEKS 3H . . . . . . . . . . . . . . . . . . . . . . . . 73 25 TEKS 5B, 5E . . . . . . . . . . . . . . . . . . . . . . 74 26 TEKS 4H . . . . . . . . . . . . . . . . . . . . . . . . 75 27 TEKS 5A . . . . . . . . . . . . . . . . . . . . . . . . 76 28 TEKS 4D, 5D . . . . . . . . . . . . . . . . . . . . . 77 29 TEKS 3D, 4E . . . . . . . . . . . . . . . . . . . . . 78 30 TEKS 4A, 4K . . . . . . . . . . . . . . . . . . . . . 79 31 TEKS 2D, 4A . . . . . . . . . . . . . . . . . . . . . 80 32 TEKS 4K, 6D . . . . . . . . . . . . . . . . . . . . . 81 33 TEKS 4E, 4F . . . . . . . . . . . . . . . . . . . . . . 82 34 TEKS 4A, 4K . . . . . . . . . . . . . . . . . . . . . 83 35 TEKS 4H, 4K . . . . . . . . . . . . . . . . . . . . . 84 36 TEKS 8A, 8B . . . . . . . . . . . . . . . . . . . . . 85 37 TEKS 4D, 4K . . . . . . . . . . . . . . . . . . . . . 86 38 TEKS 4A, 4K . . . . . . . . . . . . . . . . . . . . . 87 39 TEKS 8A, 8B . . . . . . . . . . . . . . . . . . . . . 88 40 TEKS 6B, 6C . . . . . . . . . . . . . . . . . . . . . 89

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Introduction Welcome to Understanding Complex Math Problems. To be successful in mathematics, students must become good problem solvers. Using step-bystep instruction, this book teaches students problem-solving skills and shows them how to apply those skills to different types of math problems. Each item aligns to the new TEKS. Understanding Complex Math Problems is made up of two parts.

Part A: Solving Complex Math Problems The 5-Step Problem-Solving Plan Good problem solvers always follow a plan. The plan discussed in this book consists of five steps: Step Step Step Step Step

1: 2: 3: 4: 5:

IDENTIFY: What are you being asked to find? FIND: What do you need to solve the problem? CHOOSE: How will you solve the problem? SOLVE: Solve the problem. CHECK and JUSTIFY: Check and justify your answer.

This section introduces students to the plan that they will use to solve all problems in the book and uses modeled and guided instruction to demonstrate how to use the 5-Step Problem-Solving Plan.

Part B: On Your Own Solving Mathematical Problems In Part B, students solve each problem on their own using the same 5-Step Problem-Solving Plan. When students finish Understanding Complex Math Problems they will be better problem solvers and test takers.

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Part A Solving Complex Math Problems: The 5-Step Problem-Solving Plan Part A introduces students to the 5-Step Problem-Solving Plan that they will use to solve all the problems in this book.

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Modeled Instruction When solving problems, you need to follow a plan. This helps you to organize information and choose a strategy to solve the problem. The example below shows how to follow the 5-Step Problem-Solving Plan. Read the math problem and follow each step of the plan.

Problem 1 Olivia wants to get new carpet for her hallway. She has $100 to spend. Her hallway is 8 feet long and 6 feet wide. The carpet she likes costs $2 per square foot. Will she have enough money to buy the carpet she likes? If there is enough money, how much money will be left over?

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●1

STEP

IDENTIFY: What are you being asked to find?

●2

The problem tells you the length and width of the hall, the price of the carpet, and how much money Olivia can spend. You are asked to find if Olivia has enough money for the carpet she likes. If so, you are also asked to find out how much money will be left over.

STEP

FIND: What do you need to solve the problem?

●3

• Olivia’s hallway is 8 feet long and 6 feet wide.

• She has $100 to spend.

• The carpet she likes is $2 per square foot.

STEP

CHOOSE: How will you solve the problem? First, multiply the length by width to find the area of the hall. Then, multiply the area by the price of the carpet per square foot. If the cost of the carpet is less than the amount of money Olivia has then subtract the cost from $100 to find out how much money she will have left over.

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●4

STEP

SOLVE: Solve the problem. First, multiply the length by the width of the hallway to find the area of the hallway, in square feet. Length × Width = Area 8 feet × 6 feet = 48 square feet

The area of the hallway is 48 square feet. Multiply this area by the price per square foot of the carpet. Total square feet × Price per square feet = Total cost 48 × $2 = $96 Since Olivia has $100, she has enough money for the carpet, with money left over. To find how much money is leftover, subtract the total cost of the carpet from the amount of money she has. Original Amount of Money

Total Cost of Carpet

=

Amount of Money left over

$100 – $96 = $4

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●5

STEP

CHECK and JUSTIFY: Check and justify your answer. To check your answer, find out how many square feet of carpet Olivia can buy with $100. The cost of each square foot of carpet is $2, so divide her starting amount by $2 to find the number of square feet she can pay for. Original Amount of Money

÷

Cost per square foot

=

Number of square feet Olivia can pay for

$100 ÷ $2 = 50 square feet Then, compare the number of square feet she can pay for to the number of square feet in her hallway. She can pay for 50 square feet, and her hallway is 48 square feet. That means she has enough money and will have money left over.

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Directions Use the 5-Step Problem-Solving Plan to answer the following questions. The first three steps have already been completed to guide you through the process. Complete Steps 4 and 5 on your own.

Problem 2 Kamea’s dance recital was supposed to start at 7:00 P.M. It began 15 minutes late. During the recital there were 5 dances that lasted 7 minutes each. After the recital, it took Kamea and her family 10 minutes to drive to a diner. They spent 38 minutes eating dinner and then 14 minutes driving back home. What time did Kamea and her family arrive back at their house?

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●1

STEP

IDENTIFY: What are you being asked to find? The problem tells you that Kamea’s recital started 15 minutes after 7:00. It tells you how long each activity of the night lasted. You are asked to find out what time Kamea and her family got home that night.

●2

Use the measurement conversion below to help answer this question.

STEP

1 hour = 60 minutes

FIND: What do you need to solve the problem? • Kamea’s dance recital was supposed to start at 7:00 P.M.

●3

• It started 15 minutes late. • There were 5 dances that lasted 7 minutes each. • The drive to the diner took 10 minutes.

• Eating dinner took 38 minutes. • The drive home took 14 minutes.

STEP

CHOOSE: How will you solve the problem?

●45 ●

You need to find out how much time Kamea’s family spent during each activity. Add the minutes from the first activity to the start time to see what time it ended. Then, add the minutes from the next activity to the new start time to see what time it ended, and so on. Do this until you get the end time after the final activity.

STEP

SOLVE: Solve the problem.

STEP

CHECK and JUSTIFY: Check and justify your answer. Page 11

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Problem 3 Mrs. Clark is preparing tables in her classroom for the first day of school. She will have 20 students in her class. Each table will have 5 students. Mrs. Clark wants to put some supplies in the center of each table. She has 48 pencils. She has 20 packs of crayons and 16 packs of markers. She has 1 ᎏᎏ as many packs of colored pencils as she does packs of markers. She 2 wants to put an equal amount of supplies at each table. How many of each type of supply can Mrs. Clark put at each table?

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●1

STEP

IDENTIFY: What are you being asked to find?

●2

You are asked to find how many of each supply Mrs. Clark can put on each table. You are told how many of each supply she has, except for colored pencils. You will need to find how many colored pencils there are. Then, you will need to find how many tables there are.

STEP

FIND: What do you need to solve the problem? • There are 20 students in the class. • Each table will have 5 students.

●3

• Mrs. Clark wants to put an equal amount of supplies at each table. • She has 48 pencils, 20 packs of crayons, and 16 packs of markers.

1 • She has ᎏᎏ as many colored pencils as she does markers. 2

●4 ●5

STEP

CHOOSE: How will you solve the problem? Use division to find out how many tables there are and how many supplies are at each table.

STEP

SOLVE: Solve the problem.

STEP

CHECK and JUSTIFY: Check and justify your answer.

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Directions Follow the 5-Step Problem-Solving Plan to answer questions 4–20.

Problem 4 Malik is measuring string to make a mobile. He wants the center pieces of string to be different lengths. His mom draws a number line and asks Malik 4 6 to mark and label the fractions ᎏᎏ and ᎏᎏ on the number line. Then, she 8 8 asks him to mark and label the fraction that is exactly halfway between 4 6 ᎏᎏ and ᎏᎏ. 8 8 0

1

Malik decides he wants to add 4 more pieces to the mobile. He wants them all to be the same length. He has 1 piece of string left. To measure it, Malik draws a second number line of the interval from 0 to 1, divided into 4 equal sections. He must label the marks dividing the number line with the correct fractions, using fourths. What do Malik’s 2 number lines look like? What fraction on his second number line is in the same spot as the 4 fraction ᎏᎏ on the first number line? 8

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●1 ●2

STEP

IDENTIFY: What are you being asked to find?

●3

STEP

FIND: What do you need to solve the problem?

●4

STEP

CHOOSE: How will you solve the problem?

●5

STEP

SOLVE: Solve the problem.

STEP

CHECK and JUSTIFY: Check and justify your answer.

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Problem 5 1 Zoe and her sister Taylor are baking cookies. They use ᎏᎏ cup of sugar 4 3 2 and ᎏᎏ cup of flour. Do they use more sugar or flour? They also use ᎏᎏ 4 3 2 tablespoon of baking soda and ᎏᎏ tablespoon of salt. Do they use more 4 baking soda or salt? Taylor tells her sister that they will make 24 cookies. Zoe wants to know how many people they can share with if each person gets 3 cookies. How many people can share 24 cookies if each person gets 3?

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●1 ●2

STEP

IDENTIFY: What are you being asked to find?

●3

STEP

FIND: What do you need to solve the problem?

●4

STEP

CHOOSE: How will you solve the problem?

●5

STEP

SOLVE: Solve the problem.

STEP

CHECK and JUSTIFY: Check and justify your answer.

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Problem 6 The town of Fairview is planning to build a park. The shaded boxes in the grid below show the area that the park will cover. Each square unit is equal to a square mile. Draw a line to divide the shape into two separate rectangles. Use these rectangles to help you find the area that the park will cover.

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●1 ●2

STEP

IDENTIFY: What are you being asked to find?

●3

STEP

FIND: What do you need to solve the problem?

●4

STEP

CHOOSE: How will you solve the problem?

●5

STEP

SOLVE: Solve the problem.

STEP

CHECK and JUSTIFY: Check and justify your answer.

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Problem 7 Lily has 12 board games in her room. She wants to put them on the 3 shelves in her closet. She wants an equal number of games on each shelf. How many board games will Lily put on each shelf? Lily also needs to find the area of each shelf. The drawing below shows the area of 1 shelf in square units. Each shelf is the same size. What is the area of each of Lily’s shelves?

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●1 ●2

STEP

IDENTIFY: What are you being asked to find?

●3

STEP

FIND: What do you need to solve the problem?

●4

STEP

CHOOSE: How will you solve the problem?

●5

STEP

SOLVE: Solve the problem.

STEP

CHECK and JUSTIFY: Check and justify your answer.

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Problem 8 Diego and his 3 friends are collecting stones from the river so they can paint them. They collected 24 stones and put them into piles. Each pile has an equal number of stones. If they made 1 pile for each person, how many stones would be in each pile? The boys painted 12 of the stones that day. Each person painted the same number of stones from his pile. How many stones did each person paint? What fraction of the stones in each pile did each person paint?

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●1 ●2

STEP

IDENTIFY: What are you being asked to find?

●3

STEP

FIND: What do you need to solve the problem?

●4

STEP

CHOOSE: How will you solve the problem?

●5

STEP

SOLVE: Solve the problem.

STEP

CHECK and JUSTIFY: Check and justify your answer.

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Problem 9 Luke’s dad bought 3 rugs for their porch. The rugs are shown in the diagrams below. What is the area of each rug? Use the measurement conversion below to help answer this question. 1 square = 1 square foot

Rug A

Rug B

Rug C

The length and width of the porch are shown in the diagram below. What area of the porch will NOT be covered with the new rugs? Porch

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●1 ●2

STEP

IDENTIFY: What are you being asked to find?

●3

STEP

FIND: What do you need to solve the problem?

●4

STEP

CHOOSE: How will you solve the problem?

●5

STEP

SOLVE: Solve the problem.

STEP

CHECK and JUSTIFY: Check and justify your answer.

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Problem 10 Mia was getting ready for her birthday party. She made punch with soda and fruit juice. She added 2 liters of juice and 4 times that many liters of soda. There were 3 liters of punch left after the party. How much punch did people drink? What fraction of the punch was soda? Plot the fraction on a number line.

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●1 ●2

STEP

IDENTIFY: What are you being asked to find?

●3

STEP

FIND: What do you need to solve the problem?

●4

STEP

CHOOSE: How will you solve the problem?

●5

STEP

SOLVE: Solve the problem.

STEP

CHECK and JUSTIFY: Check and justify your answer.

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Problem 11 Keisha has a candy bar that she wants to share with a friend. She split the candy bar into halves.

Her friend says that they can split the candy bar other ways and still each have an equal amount of the candy bar. What is another way to split the candy bar equally? What fraction describes the way that her friend may have split the candy bar?

Page 28 Š R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


●1 ●2

STEP

IDENTIFY: What are you being asked to find?

●3

STEP

FIND: What do you need to solve the problem?

●4

STEP

CHOOSE: How will you solve the problem?

●5

STEP

SOLVE: Solve the problem.

STEP

CHECK and JUSTIFY: Check and justify your answer.

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Problem 12 1 The track at the school is ᎏᎏ mile long. On Wednesday, Sara ran around the 4 track 12 times. How many miles did Sara run on Wednesday? Sara ran around the track 16 times on Saturday. How many more miles did Sara run on Saturday than on Wednesday?

Page 30 © R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


●1 ●2

STEP

IDENTIFY: What are you being asked to find?

●3

STEP

FIND: What do you need to solve the problem?

●4

STEP

CHOOSE: How will you solve the problem?

●5

STEP

SOLVE: Solve the problem.

STEP

CHECK and JUSTIFY: Check and justify your answer.

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Problem 13 Max’s math classroom is set up with desks in 3 rows as shown below. His teacher asks Max to write an equation to describe the number of desks.

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

Then, she separates 2 of the rows from the third row. She asks Max to write a different equation to describe the number of desks. What equations should Max write?

Page 32 © R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


●1 ●2

STEP

IDENTIFY: What are you being asked to find?

●3

STEP

FIND: What do you need to solve the problem?

●4

STEP

CHOOSE: How will you solve the problem?

●5

STEP

SOLVE: Solve the problem.

STEP

CHECK and JUSTIFY: Check and justify your answer.

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Problem 14 Cassie and her mom are picking out new tile for the kitchen floor. The kitchen is 9 feet long and 10 feet wide. How many square feet of tile will they need for the kitchen? They forget to bring the measurements to the tile store, but they remember the total area of the floor. What are other possible width and length measurements that the kitchen floor could have?

Page 34 Š R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


●1 ●2

STEP

IDENTIFY: What are you being asked to find?

●3

STEP

FIND: What do you need to solve the problem?

●4

STEP

CHOOSE: How will you solve the problem?

●5

STEP

SOLVE: Solve the problem.

STEP

CHECK and JUSTIFY: Check and justify your answer.

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Problem 15 Zach’s dog had puppies. There are 3 yellow puppies, 4 brown puppies, and 3 black puppies. What fraction of the puppies are brown? What fraction of the puppies are yellow? Zach found homes for 1 black puppy and 1 brown puppy. What fraction of the puppies still need to find homes?

Page 36 Š R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


●1 ●2

STEP

IDENTIFY: What are you being asked to find?

●3

STEP

FIND: What do you need to solve the problem?

●4

STEP

CHOOSE: How will you solve the problem?

●5

STEP

SOLVE: Solve the problem.

STEP

CHECK and JUSTIFY: Check and justify your answer.

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Problem 16 David is at the book fair with his friends. He plans to buy a book about sea turtles that costs $9. He wants to buy 4 other books, and he has $25 total to spend. If David spends the same amount on each of the other 4 books, how much does each book cost? A friend tells David that if he puts back the $9 book, he can get 6 books that cost $5 each. Is the friend correct?

Page 38 Š R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


●1 ●2

STEP

IDENTIFY: What are you being asked to find?

●3

STEP

FIND: What do you need to solve the problem?

●4

STEP

CHOOSE: How will you solve the problem?

●5

STEP

SOLVE: Solve the problem.

STEP

CHECK and JUSTIFY: Check and justify your answer.

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Problem 17 Principal Smith is looking at different rooms to use as the school’s computer lab. He needs a room that is at least 75 square feet. Room A is 8 feet long and 8 feet wide. Room B is 8 feet long and 10 feet wide. Room C is 8 feet long and 12 feet wide. How does the area of each room change as the width gets larger? Which room(s) can Principal Smith use for the computer lab?

Page 40 Š R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


●1 ●2

STEP

IDENTIFY: What are you being asked to find?

●3

STEP

FIND: What do you need to solve the problem?

●4

STEP

CHOOSE: How will you solve the problem?

●5

STEP

SOLVE: Solve the problem.

STEP

CHECK and JUSTIFY: Check and justify your answer.

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© R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Problem 18 Mrs. Long was describing a mystery shape to her class. These are the clues she gave: • It is a quadrilateral. • It has exactly 1 pair of parallel sides. Margo says that the shape is a rhombus. Peter says the shape is a trapezoid. Which student is correct? Why? 3 There are 20 students in the class and ᎏᎏ of them guessed that the mystery 4 shape was a trapezoid. How many students said the shape is a trapezoid?

Page 42 © R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


●1 ●2

STEP

IDENTIFY: What are you being asked to find?

●3

STEP

FIND: What do you need to solve the problem?

●4

STEP

CHOOSE: How will you solve the problem?

●5

STEP

SOLVE: Solve the problem.

STEP

CHECK and JUSTIFY: Check and justify your answer.

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© R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Problem 19 The students at Douglas Elementary School are collecting canned food for a food drive. The pictograph below shows the number of cans collected by each grade level during week 1 of the food drive. Cans Collected During Week 1 of Food Drive Grade

Cans Collected

3 4 5

Key: Each

= 5 cans collected

How many cans did all of the grades collect during week 1? Write a multiplication equation showing how many cans Grade 3 brought in during Week 1. How does the equation change if each can symbol in the pictograph stands for 4 cans?

Page 44 Š R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


●1 ●2

STEP

IDENTIFY: What are you being asked to find?

●3

STEP

FIND: What do you need to solve the problem?

●4

STEP

CHOOSE: How will you solve the problem?

●5

STEP

SOLVE: Solve the problem.

STEP

CHECK and JUSTIFY: Check and justify your answer.

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Problem 20 Jamie was working on a project for art class. He measured the lengths of all the pieces of scrap paper in his art classroom. Then, he made a line plot to show the lengths he measured. Here is what his line plot looks like: Lengths of Pieces of Paper X

3

X X

X X

1 3— 2

4

X X X 1 4— 2

5

1 5— 2

X

X

6

1 6— 2

X X X 7

1 7— 2

X X X

X X 8

1 8— 2

9

X

X

1 10 10 — 1 11 9— 2 2

How many pieces of paper did Jamie measure? How many pieces of paper were longer than 6 inches? How many were 6 inches or shorter? 1 Jamie’s art teacher brought him 2 more pieces of paper that were each 4ᎏᎏ 2 inches long. Are there now more pieces of paper that measure 6 inches or shorter, or more pieces of paper that measure longer than 6 inches?

Page 46 © R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


●1 ●2

STEP

IDENTIFY: What are you being asked to find?

●3

STEP

FIND: What do you need to solve the problem?

●4

STEP

CHOOSE: How will you solve the problem?

●5

STEP

SOLVE: Solve the problem.

STEP

CHECK and JUSTIFY: Check and justify your answer.

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© R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Page 48 © R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Part B On Your Own: Solving Mathematical Problems In Part B, students solve each problem on their own using the 5-Step Problem-Solving Plan.

Page 49 Š R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Directions Answer questions 1–40. Use the 5-Step Problem-Solving Plan. Be sure to show all of your work. If you need more room, use a separate sheet of paper. Remember to check your answers.

Problem 1 Yuna is making a poster for her class project. Her teacher told the class that at least half of the poster should have writing on it, not pictures. Yuna has a lot of writing and doesn’t want to run out of space. She divided her poster board into 8 equal sections.

She used 2 of the sections for drawings, 1 for a photo, and 5 for writing. What fraction of the poster board did Yuna use for writing? Did she follow her teacher’s instructions? Write a number sentence using < or > or = to compare the fraction of the poster board that should have writing to the fraction of the poster board that Yuna actually wrote on.

Page 50 © R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Problem 2 1 Mr. Garza orders 8 pizzas for a class party. He asks that ᎏᎏ of the pizzas 4 3 1 be cheese only. He asks for ᎏᎏ to have pepperoni, and for ᎏᎏ to have 4 2 veggies. How many of the pizzas will have both pepperoni and veggies? How many will have only pepperoni?

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Problem 3 Jack has a 30-liter fish tank. He uses a 3-liter jug to fill the fish tank with water. How many times does he have to fill the jug in order to fill the tank? Jack wants to put 15 fish in his tank. The clerk at the pet store said that for every 6 liters of water Jack can have 1 fish in the tank. How many fish can Jack have in his current tank? Using this rule, how big of a tank would Jack need to have 15 fish?

Page 52 Š R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Problem 4 Josie wants to paint her room. Her mom tells her to find the area of the walls in her room so they can buy enough paint. One of the walls has a door that will be painted the same color as the walls. The diagram below shows one of Josie’s walls, divided into square units. Each square represents 1 square foot. All 4 of her walls are the same size and shape.

= 1 square foot

Using the diagram, what is the area of the walls in Josie’s room? A gallon of paint covers 90 square feet, and Josie buys 5 gallons of paint. Will she have any leftover paint?

Page 53 © R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Problem 5 Carson builds a rectangular skateboard ramp that is 4 feet wide and has an area of 20 square feet. His friend Ryan builds a skateboard ramp that is also 4 feet wide. Ryan’s ramp has double the area of Carson’s ramp. What are the lengths of the sides of each boy’s ramp?

Page 54 © R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Problem 6 Brooke went to the zoo with her aunt. She wanted to see all the animals. There were 998 animals at the zoo. In the morning, Brooke and her aunt saw 637 animals. Between lunch and dinner, she went to 8 different areas of the zoo and saw 32 animals in each area. How many animals at the zoo did Brooke not see?

Page 55 Š R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Problem 7 Ty’s mom took him and his friend Mark to a movie at 5:30 P.M. The ticket prices and food prices are shown in the tables below. Ticket Prices Child

$6

Adult

$9

Senior

$8

Food Prices Popcorn (small / large)

$3 / $5

Soda (small / large)

$2 / $3

Candy

$2

Each boy had a small popcorn and small soda. Ty’s mom had a large soda. How much did she pay for their movie tickets, popcorn, and sodas?

Page 56 © R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Problem 8 Ben buys an 84-ounce bag of cat food. He gives his cat 3 ounces of food, twice a day. How many days will the bag of cat food last? Ben is going out of town for 28 days. How many bags of cat food does he need to leave with the pet sitter while he is gone?

Page 57 Š R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Problem 9 To earn money, Ruby is doing chores for her neighbors. The table below shows how much money she earns for each type of chore. This week, Ruby washed 3 cars, mowed 2 lawns, and walked 5 dogs. How much money did Ruby earn this week? Chore

Amount earned

Dog Walking

$2

Car Washing

$4

Lawn Mowing

$5

Other Yard work

$3

Page 58 Š R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Problem 10 Camden’s backpack filled with books has a mass of 12 kilograms in the morning. After each class, he puts a book into his locker. Each book has a mass of 2 kilograms. Using Camden’s class schedule below, what is the mass of his backpack after Math? Period

Class

1

Language Arts

2

Science

3

Math

4

History

5

Spanish

6

Reading

Page 59 © R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Problem 11 Maria is making a bow for each girl in her dance class. She has 3 different types of ribbon that come in specific lengths. She has yellow ribbon that is 2 2 ᎏᎏ yard long. She has red ribbon that is ᎏᎏ yard long. Finally, she has blue 4 3 2 ribbon that is ᎏᎏ yard long. 6 Maria will use all 3 ribbons for each bow. She wants to start with the longest piece of ribbon for the bottom layer. Which piece of ribbon will she start with? There are 15 girls in her dance class. Each bow takes 3 pieces of ribbon. How many pieces of ribbon will Maria use?

Page 60 © R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Problem 12 Ava took a poll to find what kinds of sports her friends play. Here is what 1 1 1 she found: ᎏᎏ play soccer, ᎏᎏ play baseball, and ᎏᎏ play volleyball. Which 3 8 5 sport does the greatest number of her friends play? Which sport does the least number of her friends play? If 15 friends are in Ava’s poll, how many of them play volleyball?

Page 61 © R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Problem 13 Patrick’s brother gave him all of his old baseball cards. He gave Patrick 3 boxes with baseball cards inside. Each box had 2 different bags inside. Each bag had 50 baseball cards in it. How many baseball cards did Patrick’s brother give him?

Page 62 © R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Problem 14 Emma asks her 16 family members what they want to drink with 3 Thanksgiving dinner. á&#x17D;?á&#x17D;? of her family has given her their drink order. 4 How many people have ordered their drinks? How many drink orders does Emma still need to collect? Half of the people that have ordered would like to have water. How many people would like to have water to drink?

Page 63 Š R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Problem 15 Ines goes to swim practice 3 days a week. Swim practice lasts for 30 minutes. She has piano lessons 2 days a week. Piano lessons are 20 minutes long. She also has to practice piano at home for 10 minutes, 4 times a week. How much more time does Ines spend at swim practice than at piano lessons each week? How much more time does she spend at swim practice than at piano lessons and piano practice combined?

Page 64 Š R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Problem 16 Sydney is picking out fabric to make a blanket. Her blanket pattern is 6 feet long and 4 feet wide. The blue fabric she likes costs $2 per square foot. How much will it cost her to buy the fabric for her blanket? She also likes a red fabric that costs $3 per square foot. How much more will that fabric cost for her blanket than the blue fabric?

Page 65 Š R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Problem 17 1 There are 80 students in the school band. ᎏᎏ of students in the band 4 3 are 4th graders and ᎏᎏ of students in the band are 5th graders. How 4 many band members are 4th graders? How many band members are 5th graders? 1 Only ᎏᎏ of the students who auditioned to be in the band became band 2 members. How many students auditioned for the school band?

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Problem 18 Joshâ&#x20AC;&#x2122;s band concert started at the time shown on the clock below. He had to do 3 things to get ready before the concert. First, he spent 10 minutes changing his clothes. Then, he ate dinner with his family, which took 45 minutes. Finally, he practiced with the band for 20 minutes. What time did Josh begin getting ready for his concert?

11 12

1

10

2

9

3

8

4 7

6

5

Use the measurement conversion below to help answer this question. 1 hour = 60 minutes

Page 67 Š R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Problem 19 Neil is setting up tables for his birthday party. There will be 9 children and 9 adults at the party. Each table can seat 6 people. How many tables will Neil need? Neil wants to put snacks in the center of each table. He has 30 water 1 bottles. He also has 27 packs of nuts and 18 cupcakes. He has ᎏᎏ as many 3 packs of chips as he does packs of nuts. How many water bottles and each type of snack can Neil put at each table?

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Problem 20 Lynn is measuring a desk for her room. Her ruler is very old and there is a fraction that is hard to read on it. This fraction is located exactly halfway 2 4 between ᎏᎏ and ᎏᎏ. What is the value of the missing fraction? Draw a 6 6 number line to show this fraction. On your number line, label 0 and 1. 2 4 Label the fractions ᎏᎏ and ᎏᎏ, and label the fraction exactly halfway 6 6 2 4 between ᎏᎏ and ᎏᎏ. 6 6 Lynn also needs to measure the desk to the nearest half inch. Draw a second number line divided into 2 equal sections between 0 and 1. Label 1 0, 1, and the ᎏᎏ mark. What fraction on your first number line is in the 2 1 same location as ᎏᎏ on your second number line? 2

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Problem 21 Ella is going to a friend’s house. For the first part of the trip, she goes west 1 6 for ᎏᎏ mile. Then, she turns right and goes north for ᎏᎏ mile. Does she 8 8 travel farther west or north on the first part of the trip? 3 On the second part of the trip, Ella goes east for ᎏᎏ mile. Then, she turns 5 3 left and goes north again for ᎏᎏ mile. Does she travel farther east or north 10 on the second part of the trip? It will take Ella 12 minutes to ride her bike from her house to her friend’s house. If she leaves her house at 12:34 P.M., what time will she arrive?

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Problem 22 Kate brought fun erasers to school to give to her friends. She brought 28 erasers. She is giving the erasers to 4 friends in her class and 3 friends in Mrs. Ford’s class. How many erasers will each friend get if Kate gives each friend the same number of erasers? 1 ᎏᎏ of the erasers Kate brought are scented. How many of the erasers are 2 scented? How many scented erasers will each friend get if Kate gives her friends each the same number of scented erasers?

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Problem 23 Sam’s uncle is building a new frame for Sam’s bed. The grid below shows the outline of Sam’s room. The shaded squares show the outline of his bed. What is the area of Sam’s room? What is the area of the frame for Sam’s bed? What is the area of the part of the room that is not covered by the frame?

= 1 square foot

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Problem 24 Summer break is 10 weeks long. Alex will spend 2 weeks of summer break at camp. What fraction of the summer will he not be at camp? 3 Alex will be at his aunt’s house for ᎏᎏ of the summer break. Will he spend 10 more time at his aunt’s house or at camp during summer break?

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Problem 25 Elaina made a chart to help her sister learn how to multiply numbers. The chart shows the total number of wheels for different numbers of cars. The chart shows that as the number of cars increases, the number of wheels also increases. Elaina told her sister that the numbers in the chart follow a pattern. Number of Cars

Number of Wheels

2

8

4

16

6

24

8

32

10

40

12

48

?

?

?

?

What pattern does the chart show? What will be the next 2 numbers in each row?

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Problem 26 Briana picked 24 apples. She picked 6 green apples, 6 red apples, and 12 yellow apples. She wants to put the same number of apples in each of her 3 baskets. How many apples will go in each basket? What fraction of the apples is red?

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Problem 27 1 The pool at the park is ᎏᎏ mile long. On Monday, Kevin swam the length 10 of the pool 15 times. How many miles did Kevin swim on Monday? Kevin swam the length of the pool 25 times on Sunday. How many more miles did Kevin swim on Sunday than on Monday?

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Problem 28 Mrs. Clark bought star stickers to put on her students’ papers. The sticker sheets have the stickers arranged as shown in the picture below. What number sentence does the picture show?

Draw a box around some of the stars to show a 4 × 4 array. Draw another box around the rest of the stars to show a 4 × 1 array. Based on the stars and the boxes you drew, complete the number sentence: (4 ×

)+(

× 1) = 4 ×

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Problem 29 Eli is helping his father plant a row of trees in their yard. His father gave him a drawing with a number line showing where each tree should go. The drawing got dirty and Eli can’t read all the numbers. What are the fractions for the points that are not labeled? 0 0 — 10

1 1 — 10

?

3 — 10

4 — 10

?

6 — 10

7 — 10

?

?

10 — 10

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Problem 30 Tim has 25 amusement park tickets. He uses some of his tickets on the games shown in the table below. He plays Skee Ball twice and the rest of the games one time each. Game

Number of tickets to play

Skee Ball

3

Water Races

2

Lucky Ducks

1

Balloon Pop

3

Ring Toss

3

After playing the games, he wants to spend the rest of his tickets on 2 rides. Which 2 rides can Tim go on with the number of tickets he has left? Rides

Number of tickets to ride

Ferris Wheel

8

Roller Coaster

7

Bumper Cars

5

Swings

5

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Problem 31 Sophie has 2 containers of water in front of her. Each container can hold 50 liters of water. How much liquid is in each container? How much liquid is in both of the containers combined? Is there enough space in 1 container to hold all the liquid from both containers?

50 L

50 L

45

45

40

40

35

35

30

30

25

25

20

20

15

15

10

10

5

5

Container A

Container B

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Problem 32 The picture below shows the surface of a swimming pool. The pool is 10 feet wide. The shallow end is 4 feet long and the deep end is 10 feet long. Find the area of the pool.

10 ft

4 ft

10 ft

Shallow

Deep

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Problem 33 Stephen tells his younger sister that if she can count by tens then she can multiply by 10. His sister does not believe him. Stephen says counting by any number other than 1, called skip counting, is a way to multiply. He says he can show this using a multiplication chart. What does Stephen mean? Is he correct?

Ă&#x2014;

1

2

3

4

5

6

7

8

9 10 11 12

1

1

2

3

4

5

6

7

8

9 10 11 12

2

2

4

6

8 10 12 14 16 18 20 22 24

3

3

6

9 12 15 18 21 24 27 30 33 36

4

4

8 12 16 20 24 28 32 36 40 44 48

5

5 10 15 20 25 30 35 40 45 50 55 60

6

6 12 18 24 30 36 42 48 54 60 66 72

7

7 14 21 28 35 42 49 56 63 70 77 84

8

8 16 24 32 40 48 56 64 72 80 88 96

9

9 18 27 36 45 54 63 72 81 90 99 108

10 10 20 30 40 50 60 70 80 90 100 110 120 11 11 22 33 44 55 66 77 88 99 110 121 132 12 12 24 36 48 60 72 84 96 108 120 132 144

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Problem 34 Grace went shopping with her friends. She had $50 to spend. She bought 2 pairs of shorts for $7 each. She also bought 5 pairs of earrings for $4 each. How much money did Grace have left?

Page 83 Š R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .


Problem 35 Evan and his cousin John collect stamps. They put equal numbers of stamps on pages in a binder. Evan has 24 stamps divided among 6 pages. John has 28 stamps divided among 7 pages. How many stamps does each 1 boy have on each page? How many stamps do they have in all? If ᎏᎏ of 4 their stamps are postcard stamps, how many are postcard stamps?

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Problem 36 Laura is measuring lengths of ribbons. Make a line plot using the data Laura recorded in the table below. How many pieces of ribbon did Laura measure? What is the most common length of ribbon? What is the least common length of ribbon? Ribbon length (inches) Number of ribbons

1 1 1 2ᎏᎏ 2ᎏᎏ 3ᎏᎏ 4 2 2 3

2

2

4

1 4ᎏᎏ 2

5

4

5

1

1 1 3 5ᎏᎏ 6ᎏᎏ 6ᎏᎏ 2 4 4 1

3

4

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Problem 37 Anna’s mother bought Anna 4 packs of new socks. Each pack had 3 different colors of socks in them. Each pack had 2 pairs of each of the colors. How many pairs of socks did Anna’s mother buy for Anna?

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Problem 38 Clara rides her bike to school 3 days a week. It takes her 15 minutes each way. On the 2 days that she doesnâ&#x20AC;&#x2122;t ride her bike, she walks to school. It takes her 25 minutes to walk to school each way. Does Clara spend more time walking or riding her bike to and from school each week? How much more?

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Problem 39 Mark took a poll. He asked some students to name their favorite school subject. He created a bar graph to show the data he collected. Favorite Subject at School

Number of Classmates

10 8 6 4 2 0

Math

Reading

Science

Writing

Subject How many students did Mark poll? How many more students chose reading than writing as their favorite subject?

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Problem 40 Lindsay has a new quilt. It is made up of squares that her grandmother cut from her favorite old T-shirts. Each of the squares that make up the quilt is the same size. The quilt is 5 squares long and 4 squares wide. What fraction describes each square of the quilt as a part of the whole quilt? Draw the quilt and shade in the fraction you named. Lindsay says that the quilt is a rectangle, but also a square. Is she correct? Why or why not?

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To be successful in mathematics, students must become good problem solvers. Using step-bystep instruction, Understanding Complex Math Problems teaches students problem-solving skills and shows them how to apply those skills to different types of math problems. Each item in this book aligns to the new TEKS. In Part A of this book, students will learn to follow a 5-Step Problem Solving Plan. Step 1: IDENTIFY: What are you being asked to find? Step 2: FIND: What do you need to solve the problem? Step 3: CHOOSE: How will you solve the problem? Step 4: SOLVE: Solve the problem. Step 5: CHECK and JUSTIFY: Check and justify your answer. In Part B of this book, students will solve complex math problems on their own using the same 5-Step Problem-Solving Plan they have learned in Part A. When students finish Understanding Complex Math Problems they will be better problem solvers and test takers.

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