UNDERSTANDING COMPLEX MATH E PROBLEMS Written Specifically for the New 2012 TEKS
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Understanding Math Series Understanding Math in the Real World
Understanding Complex Math Problems
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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
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888-99-RALLY www.RALLYEDUCATION.com
UNDERSTANDING COMPLEX MATH E PROBLEMS Written Specifically for the New 2012 TEKS
ISBN 978-1-4204-8406-9 R 8406-9 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. 0813.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 3E, 3I . . . . . . . . . . . . . . . 6 2 (Guided) TEKS 7, 3E . . . . . . . . . . . . . . . . 10 3 (Guided) TEKS 3H, 3I, 3J . . . . . . . . . . . . . 12 4 TEKS 6B, 6A . . . . . . . . . . . . . . . . . . . . . . 14 5 TEKS 3G . . . . . . . . . . . . . . . . . . . . . . . . . 16 6 TEKS 2B, 3E . . . . . . . . . . . . . . . . . . . . . . . 18 7 TEKS 3H, 3L . . . . . . . . . . . . . . . . . . . . . . 20 8 TEKS 3F, 6B . . . . . . . . . . . . . . . . . . . . . . . 22 9 TEKS 3I, 4H . . . . . . . . . . . . . . . . . . . . . . . 24 10 TEKS 2C, 7 . . . . . . . . . . . . . . . . . . . . . . 26 11 TEKS 2A, 4H . . . . . . . . . . . . . . . . . . . . . 28 12 TEKS 3I, 4H . . . . . . . . . . . . . . . . . . . . . . 30 13 TEKS 3K, 7 . . . . . . . . . . . . . . . . . . . . . . 32 14 TEKS 7 . . . . . . . . . . . . . . . . . . . . . . . . . 34 15 TEKS 9A, 9C . . . . . . . . . . . . . . . . . . . . . 36 16 TEKS 3E, 3G . . . . . . . . . . . . . . . . . . . . . 38 17 TEKS 4F . . . . . . . . . . . . . . . . . . . . . . . . 40 18 TEKS 3E . . . . . . . . . . . . . . . . . . . . . . . . 42 19 TEKS 3G, 6B . . . . . . . . . . . . . . . . . . . . . 44 20 TEKS 3A, 3L . . . . . . . . . . . . . . . . . . . . . 46
Part B: On Your Own . . . . . . . . . . . . . . . . . . 49 Problem Page 1 TEKS 2B, 3K . . . . . . . . . . . . . . . . . . . . . . 50 2 TEKS 3L . . . . . . . . . . . . . . . . . . . . . . . . . 51 3 TEKS 3A, 3H . . . . . . . . . . . . . . . . . . . . . . 52 4 TEKS 3A, 3H . . . . . . . . . . . . . . . . . . . . . . 53 5 TEKS 4F, 4H . . . . . . . . . . . . . . . . . . . . . . . 54 6 TEKS 4H, 6B . . . . . . . . . . . . . . . . . . . . . . 55 7 TEKS 3B, 7 . . . . . . . . . . . . . . . . . . . . . . . 56 8 TEKS 3H, 3L . . . . . . . . . . . . . . . . . . . . . . 57
9 TEKS 2B . . . . . . . . . . . . . . . . . . . . . . . . . 58 10 TEKS 3I, 3L . . . . . . . . . . . . . . . . . . . . . . 59 11 TEKS 3I, 3L . . . . . . . . . . . . . . . . . . . . . . 60 12 TEKS 2A, 3E . . . . . . . . . . . . . . . . . . . . . 61 13 TEKS 3G, 4H, 6A . . . . . . . . . . . . . . . . . . 62 14 TEKS 3A, 3E, 3G . . . . . . . . . . . . . . . . . . 63 15 TEKS 3H, 3I . . . . . . . . . . . . . . . . . . . . . . 64 16 TEKS 3I, 4F . . . . . . . . . . . . . . . . . . . . . . 65 17 TEKS 3A, 3K . . . . . . . . . . . . . . . . . . . . . 66 18 TEKS 3K, 6B . . . . . . . . . . . . . . . . . . . . . 67 19 TEKS 3H, 3I . . . . . . . . . . . . . . . . . . . . . . 68 20 TEKS 7 . . . . . . . . . . . . . . . . . . . . . . . . . 69 21 TEKS 3E, 4F . . . . . . . . . . . . . . . . . . . . . . 70 22 TEKS 3H, 4F . . . . . . . . . . . . . . . . . . . . . 71 23 TEKS 3E, 3I . . . . . . . . . . . . . . . . . . . . . . 72 24 TEKS 3I, 4H . . . . . . . . . . . . . . . . . . . . . . 73 25 TEKS 3C, 7 . . . . . . . . . . . . . . . . . . . . . . 74 26 TEKS 3A, 3H . . . . . . . . . . . . . . . . . . . . . 75 27 TEKS 3I, 3K . . . . . . . . . . . . . . . . . . . . . . 76 28 TEKS 3H, 3I . . . . . . . . . . . . . . . . . . . . . . 77 29 TEKS 2A, 4F. . . . . . . . . . . . . . . . . . . . . . 78 30 TEKS 9A, 9C . . . . . . . . . . . . . . . . . . . . . 79 31 TEKS 3A, 3H . . . . . . . . . . . . . . . . . . . . . 80 32 TEKS 2B . . . . . . . . . . . . . . . . . . . . . . . . 81 33 TEKS 2C, 6B . . . . . . . . . . . . . . . . . . . . . 82 34 TEKS 3A, 3L . . . . . . . . . . . . . . . . . . . . . 83 35 TEKS 3A, 3C . . . . . . . . . . . . . . . . . . . . . 84 36 TEKS 3I . . . . . . . . . . . . . . . . . . . . . . . . . 85 37 TEKS 3A, 3E . . . . . . . . . . . . . . . . . . . . . 86 38 TEKS 2B . . . . . . . . . . . . . . . . . . . . . . . . 87 39 TEKS 4B, 7 . . . . . . . . . . . . . . . . . . . . . . 88 40 TEKS 3C, 4F . . . . . . . . . . . . . . . . . . . . . 89 41 TEKS 5 . . . . . . . . . . . . . . . . . . . . . . . . . 90 42 TEKS 3A, 4F. . . . . . . . . . . . . . . . . . . . . . 91 43 TEKS 5 . . . . . . . . . . . . . . . . . . . . . . . . . 92 44 TEKS 4C . . . . . . . . . . . . . . . . . . . . . . . . 93
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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 2012 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: Check 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 Sixteen students have volunteered to stay after school to help set up for a pep rally. The coordinator of the pep rally plans to order pizza for the students. A pizza pie has 8 slices and costs $13.49. What is the total cost 1 of the pizza that is needed if each student has 1 slices? Write your answer 2 in decimal form.
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●1 STEP
IDENTIFY: What are you being asked to find? You are asked to find the total cost of the pizza. You are told that each pizza pie has 8 slices and costs $13.49. You are told that there 1 are 16 students and each student wants to have 1 slices. 2
●2 STEP
FIND: What do you need to solve the problem? • The number of students staying after school = 16 • The number of slices of pizza in each pizza pie = 8 • The cost of a pizza pie = $13.49 1 • The number of slices of pizza each student will get = 1 2
●3 STEP
CHOOSE: How will you solve the problem? First, you need to figure out how many slices of pizza are needed. Write a number sentence to multiply the number of students by the number of slices each student will eat. Then, write a number sentence to divide the number of slices of pizza needed by the number of slices in a pie to figure out the number of pies needed. Finally, write a number sentence to multiply the number of pies needed by the cost of a pie.
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●4 STEP
SOLVE: Solve the problem. First, write a number sentence to figure out how many slices are needed. Number of Students
×
Number of slices per student
16
×
1 1 2
16 + 8
=
Total number of slices needed
1 = (16 × 1) + (16 × ) 2 =
24
Next, divide the total number of slices needed by the number of slices in a pizza pie. Number of slices needed
÷
Number of slices in a pie
=
Number of pies needed
24 ÷ 8 = 3 pies Finally, multiply the cost of a pizza pie by the number of pizza pies needed. Cost of a pizza pie
×
Number of pies needed
=
Total cost of the pizza
$13.49
×
3
=
$40.47
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●5 STEP
CHECK: Check your answer. To check your answer, work backwards. First, figure out how many students you can feed with $40.47 worth of pizza. Divide the total cost by the cost of a single pie. $40.47 ÷ $13.49 = 3 Multiply your answer by the number of slices in each pie to figure out how many slices you can buy with $40.47. 3 × 8 = 24 Finally, check that your answer makes sense. Can all 16 students be 1 fed from 24 slices of pizza if each student eats 1 slices? 2 Each pie is divided into 8 pieces (3 × 8 = 24), and in 1 pie the pieces 1 are cut in half. You can count 16 servings of 1 pieces. 2 16 half slices
16 whole slices
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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 Sally is making a model of the White House for her American history class. She decides to use red ribbon for wallpaper borders. The first room needs 170 centimeters of ribbon. The second room needs 130 centimeters of ribbon. Each meter of ribbon costs $5. How much money does Sally need to buy ribbon for her model? Use the measurement conversion below to help answer this question. 1 meter = 100 centimeters
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●1 STEP
IDENTIFY: What are you being asked to find? You are asked to find the total cost of the ribbon Sally needs to make her model. You are told how much ribbon is needed for each room, how much 1 meter of ribbon costs, and how many centimeters are in a meter.
●2 STEP
FIND: What do you need to solve the problem? • The amount of ribbon needed for the first room = 170 cm • The amount of ribbon needed for the second room = 130 cm • The cost of ribbon = $5 per meter • The number of centimeters in a meter = 100
●3 STEP
CHOOSE: How will you solve the problem? First, find the total amount of ribbon needed for both rooms. Next, convert the total amount of ribbon needed to meters. Finally, multiply the amount of ribbon needed by the cost of a meter of ribbon.
●4 ●5 STEP
SOLVE: Solve the problem.
STEP
CHECK: Check your answer.
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Problem 3 1 Fred is baking a cake for his grandmother. The recipe calls for cup of 6 1 1 brown sugar, cup of white sugar, and 2 cups of flour. Fred wants to 3 4 cut the recipe in half, so he only needs to use half the amount of ingredients specified in the recipe. How many cups of sugar will Fred need for the cake?
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●1 STEP
IDENTIFY: What are you being asked to find? The problem tells you how much brown sugar and how much white sugar the recipe calls for. You are told that Fred will only use half of the sugar the recipe calls for. You are asked for the total amount of sugar that Fred will use for the cake.
●2 STEP
FIND: What do you need to solve the problem? 1 • The amount of brown sugar in the recipe = cups 6 1 • The amount of white sugar in the recipe = cups 3 • Fred has to cut the amount of sugar in the recipe by half. • You do NOT need to know the amount of flour in the recipe. You are only asked to calculate the amount of sugar Fred will use.
●3 STEP
CHOOSE: How will you solve the problem? First, figure out the total amount of sugar the recipe calls for. Add the amount of brown sugar and the amount of white sugar. Then, cut the amount of sugar in the recipe in half by dividing the total amount of sugar by 2.
●4 ●5 STEP
SOLVE: Solve the problem.
STEP
CHECK: Check your answer.
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Directions Follow the 5-Step Problem-Solving Plan to answer questions 4–20.
Problem 4 Jane is volunteering at a food pantry. She has to pack 2 wooden crates with cube-shaped cartons of soup. Each carton contains 1 liter of soup. She is able to fit 4 layers of 12 cartons in the first crate. She is able to pack 32 cartons in the second crate. Both crates are packed without any space between the cartons. What is the total volume, in liters, of the 2 crates?
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●1 STEP
IDENTIFY: What are you being asked to find?
●2 STEP
FIND: What do you need to solve the problem?
●3 STEP
CHOOSE: How will you solve the problem?
●4 STEP
SOLVE: Solve the problem.
●5 STEP
CHECK: Check your answer.
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Problem 5 4 On Halloween, Jimmy and his 4 friends each collected of a pound of 5 candy. When Jimmy and his friends returned from trick-or-treating, Jimmy’s mom made them share their candy with his younger sister Jennifer. If the children shared the candy evenly, how much candy would each child get? Write your answer as a simplified fraction.
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●1 STEP
IDENTIFY: What are you being asked to find?
●2 STEP
FIND: What do you need to solve the problem?
●3 STEP
CHOOSE: How will you solve the problem?
●4 STEP
SOLVE: Solve the problem.
●5 STEP
CHECK: Check your answer.
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Problem 6 James is shopping for licorice at the super market. He has enough money for 1 large package of licorice that weighs 1.443 pounds or 2 smaller packages that weigh 0.72 pounds each. If James wants as much licorice as possible, should he buy the 1 large package or the 2 smaller packages?
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●1 STEP
IDENTIFY: What are you being asked to find?
●2 STEP
FIND: What do you need to solve the problem?
●3 STEP
CHOOSE: How will you solve the problem?
●4 STEP
SOLVE: Solve the problem.
●5 STEP
CHECK: Check your answer.
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Problem 7 Alex and his brother Bob have brought home a small blueberry pie to share 5 1 with their 2 sisters. Alex eats of the pie and Bob eats of the pie. If the 12 4 remaining pie is divided evenly between the 2 sisters, how much pie does each sister get?
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●1 STEP
IDENTIFY: What are you being asked to find?
●2 STEP
FIND: What do you need to solve the problem?
●3 STEP
CHOOSE: How will you solve the problem?
●4 STEP
SOLVE: Solve the problem.
●5 STEP
CHECK: Check your answer.
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Problem 8 Sandra is sending her family and friends gift baskets for the holidays. The gift baskets will include blocks of gourmet chocolate. There are 21 people who will get gift baskets. Sandra has purchased a large block of chocolate that has a base of 24 square inches and a height of 7 inches. If Sandra divides the chocolate evenly among the baskets, how many cubic inches of chocolate will each basket contain?
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●1 STEP
IDENTIFY: What are you being asked to find?
●2 STEP
FIND: What do you need to solve the problem?
●3 STEP
CHOOSE: How will you solve the problem?
●4 STEP
SOLVE: Solve the problem.
●5 STEP
CHECK: Check your answer.
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Problem 9 Eva is using cloth for a school project. She has 6 rectangular pieces of 1 cloth. Each piece has a length of 3 feet and a width of 1 feet. 2 How many square feet of cloth does Eva have?
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●1 STEP
IDENTIFY: What are you being asked to find?
●2 STEP
FIND: What do you need to solve the problem?
●3 STEP
CHOOSE: How will you solve the problem?
●4 STEP
SOLVE: Solve the problem.
●5 STEP
CHECK: Check your answer.
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Problem 10 John and Melanie are designing a science experiment for the school science fair. Their experiment requires a large amount of copper wire. They need 22 spools of copper wire that each contains 31 meters of wire. How many kilometers of wire do John and Melanie need for their experiment? Round your answer to the nearest tenth of a kilometer. Use the measurement conversion below to help answer this question. 1 kilometer = 1,000 meters
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●1 STEP
IDENTIFY: What are you being asked to find?
●2 STEP
FIND: What do you need to solve the problem?
●3 STEP
CHOOSE: How will you solve the problem?
●4 STEP
SOLVE: Solve the problem.
●5 STEP
CHECK: Check your answer.
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Problem 11 Sue is making a small-scale copy of a famous sculpture. She estimates that she will need 11.4 cubic inches of clay. The art store sells clay in three different size containers. The largest container holds 10 cubic inches of clay, the medium size container holds 1 cubic inch of clay, and the smallest 1 container holds cubic inches of clay. If Sue does not want to purchase 10 more clay than is necessary, what is the least amount of containers she can purchase?
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●1 STEP
IDENTIFY: What are you being asked to find?
●2 STEP
FIND: What do you need to solve the problem?
●3 STEP
CHOOSE: How will you solve the problem?
●4 STEP
SOLVE: Solve the problem.
●5 STEP
CHECK: Check your answer.
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Problem 12 Diana is buying wooden building blocks for her younger brother. She wants to buy enough blocks to fill half of her brother’s plastic toy bin. The volume of the plastic bin is 162 cubic inches. Each wooden block is a cube with edges that measure 1-inch in length. How many blocks does Diana need to buy?
Page 30 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 STEP
IDENTIFY: What are you being asked to find?
●2 STEP
FIND: What do you need to solve the problem?
●3 STEP
CHOOSE: How will you solve the problem?
●4 STEP
SOLVE: Solve the problem.
●5 STEP
CHECK: Check your answer.
Page 31 © 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 Brittany is making soup for her brother’s birthday dinner. In her pantry, there are several cube-shaped containers of packaged broth. Each container holds 1 liter of broth. There are 4 containers of chicken broth, 7 containers of beef broth, and 3 containers of vegetable broth. If there are approximately 61 cubic inches in 1 liter, how many cubic inches of broth does Brittany have?
Page 32 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 STEP
IDENTIFY: What are you being asked to find?
●2 STEP
FIND: What do you need to solve the problem?
●3 STEP
CHOOSE: How will you solve the problem?
●4 STEP
SOLVE: Solve the problem.
●5 STEP
CHECK: Check your answer.
Page 33 © 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 Michelle is conducting a science experiment for school. She is recording the daily rain amounts in her county. Today, she recorded 21.3 millimeters of rain. Michelle’s teacher has asked her to convert her observations to centimeters and to round to the nearest centimeter. What should Michelle record for today’s rain? Use the measurement conversion below to help answer this question. 1 centimeter = 10 millimeters
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 STEP
IDENTIFY: What are you being asked to find?
●2 STEP
FIND: What do you need to solve the problem?
●3 STEP
CHOOSE: How will you solve the problem?
●4 STEP
SOLVE: Solve the problem.
●5 STEP
CHECK: Check your answer.
Page 35 © 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 Carla is conducting an experiment on pond water for science class. She has 3 beakers filled with different amounts of pond water. The first beaker has 1 3 liter, the second beaker has 1 liter, and the third beaker has liter. 4 4 Make a line plot that shows the amount of water in each of the three beakers. If the water were equally redistributed to the 3 beakers, how much water would each beaker contain?
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 STEP
IDENTIFY: What are you being asked to find?
●2 STEP
FIND: What do you need to solve the problem?
●3 STEP
CHOOSE: How will you solve the problem?
●4 STEP
SOLVE: Solve the problem.
●5 STEP
CHECK: Check your answer.
Page 37 © 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 Saul plans to sell wool socks that he knitted at a local craft fair. Saul 4 must use of a ball of yarn to knit 1 pair of socks. A ball of yarn costs $3, 5 and he sells each pair of socks for $12.75. He has 8 balls of yarn that he plans to use to make socks. How much money will Saul receive for the socks he is able to knit from the 8 balls of yarn?
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 STEP
IDENTIFY: What are you being asked to find?
●2 STEP
FIND: What do you need to solve the problem?
●3 STEP
CHOOSE: How will you solve the problem?
●4 STEP
SOLVE: Solve the problem.
●5 STEP
CHECK: Check your answer.
Page 39 © 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 Serge is at a grocery store shopping for hand soap. The store sells 1-ounce 4 bars of Super Suds soap, -ounce bars of Sudsy Goodness soap, and 5 8 -ounce bars of Bubblelicious soap. Each type of soap is sold in packs 7 of 5. List the different brands of soap in order, from greatest to least, to show which pack weighs the most and which pack weighs the least.
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 STEP
IDENTIFY: What are you being asked to find?
●2 STEP
FIND: What do you need to solve the problem?
●3 STEP
CHOOSE: How will you solve the problem?
●4 STEP
SOLVE: Solve the problem.
●5 STEP
CHECK: Check your answer.
Page 41 © 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 Jim and Sandy are buying ground meat so they can make meatloaf. Their recipe calls for 1.3 pounds of ground lamb, 0.7 pounds of ground pork, and 0.4 pounds of ground beef. Ground pork and ground lamb both cost $4 per pound and ground beef costs $2 per pound. Write a numerical expression that represents the cost of the meat Jim and Sandy need. Evaluate this expression and round it to the nearest dollar.
Page 42 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 STEP
IDENTIFY: What are you being asked to find?
●2 STEP
FIND: What do you need to solve the problem?
●3 STEP
CHOOSE: How will you solve the problem?
●4 STEP
SOLVE: Solve the problem.
●5 STEP
CHECK: Check your answer.
Page 43 © 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 Alice is in charge of buying mulch for her neighborhood’s community garden. The garden store sells mulch in packages that are shaped like rectangular prisms. Each package has a base of 30 square inches and a height of 8 inches. Alice plans on buying 100 packages of mulch. How many cubic inches of mulch is Alice planning to buy?
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 STEP
IDENTIFY: What are you being asked to find?
●2 STEP
FIND: What do you need to solve the problem?
●3 STEP
CHOOSE: How will you solve the problem?
●4 STEP
SOLVE: Solve the problem.
●5 STEP
CHECK: Check your answer.
Page 45 © 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 20 David needs to read a book for his English literature class. It takes him 1 hour to read each chapter and each chapter has an average of 27 pages. 2 At this rate, how many pages can David read in 12 hours?
Page 46 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 STEP
IDENTIFY: What are you being asked to find?
●2 STEP
FIND: What do you need to solve the problem?
●3 STEP
CHOOSE: How will you solve the problem?
●4 STEP
SOLVE: Solve the problem.
●5 STEP
CHECK: Check your answer.
Page 47 © 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 1–44. 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 Robin needs to purchase plastic containers to store her clothes. Storage Maniacs, a storage specialty store, sells 3 different size containers. The large container has a volume of 6.41 cubic meters, the medium size container has a volume of 4.02 cubic meters, and the small container has a volume of 2.38 cubic meters. If Robin wants to maximize the volume, should she buy 1 large container or a medium container and a small container?
Page 50 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 2 Phil shares a lawn-mowing business in the summer with his 4 best friends. 1 Each day they mow 4 lawns and use gallon of gas. If they share the cost 2 of gas evenly and 1 gallon of gas costs $5, how much does each friend spend on gas per day?
Page 51 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 3 Cory works for a bakery that sells cake by weight. One customer wants 1 3 11 of a chocolate cake, 2 yellow cakes, and of a white cake. A whole 3 4 12 cake in any of these flavors costs $15 and weighs 450 grams. How many grams of cake does the customer want to buy?
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 Cindy and her 2 brothers want to divide the leftover pie from their party. 1 2 There is of a blueberry pie left over and of an apple pie left over. 2 5 Both pies were the same size originally. If they divide the leftover pie evenly, what fraction of a whole pie will each person get?
Page 53 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 5 George and 4 of his friends are planting radishes in a garden patch. The 1 garden patch is 6 feet wide and 2 feet long. Each square foot of garden 3 can hold 10 radish plants and they want to use the entire garden patch for radishes. They decide to divide up the number of radishes each person plants evenly among the group. How many radishes does each person plant?
Page 54 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 6 John is selling handmade dice at a board game convention. Each die is a wooden cube whose sides are all 1 centimeter. John is storing the dice in a rectangular prism plastic container. The container is 2 centimeters high, so he is able to fit 2 layers of dice in the container. Each layer contains 36 dice. The rectangular prism is 12 centimeters long and 3 centimeters wide. Count the dice that John can fit into the plastic container to determine the volume of the container.
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 Elizabeth collects model trains. She recently bought a large box of model train tracks. The box contains 133 segments of track that are each 22 centimeters long. One meter equals 100 centimeters. How many meters of train tracks does the box contain?
Page 56 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 8 Terrence has just bought 7 tubes of hobby glue to use on his model 1 airplanes. For each airplane, he uses of a tube on the airplane’s body 3 1 and of a tube on the airplane’s wings. How many airplanes can Terrence 6 make from his 7 tubes?
Page 57 © 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 Jason’s school band is hosting a fish fry fundraiser to buy new uniforms. Jason is in charge of buying the frying oil to make the fish and chips. The supermarket sells two sizes of oil bottles for the same price. The first bottle 11 holds 1 liter of oil and the second bottle holds liters of oil. A liter of 10 oil is enough to fry 25 batches of fish and chips. If Jason needs to buy 7 bottles of oil, which kind of bottle should he buy to get the most oil?
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 1 Jack and 4 of his friends have purchased pound of cashews. They want 2 to share the cashews and the cost of the cashews equally among themselves. A pound of cashews costs $20. How many pounds of cashews will Jack get and how much will he pay for it?
Page 59 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 11 1 Justin has gallon of skim milk. He wants to distribute the amount of milk 2 he drinks evenly over the next 5 days. If milk costs $3 a gallon, what is the value of the milk that Justin drinks in 1 day?
Page 60 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 12 Andrew has a large amount of $1 bills, dimes, and pennies. His friend Josh asks to borrow $5.72. What is the fewest number of pennies, dimes, and bills that Andrew can give to Josh to total this amount?
Page 61 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 13 Christina has invented a new measure of volume that she calls blurgs. One blurg is equal to 17 cubic centimeters. In Christina’s kitchen, there is a box packed completely with sugar cubes. Each cube is 1 cubic centimeter, and the box contains 272 sugar cubes. How many blurgs of sugar cubes are in the box?
Page 62 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 14 Melissa needs to buy 100 bolts for an art project. The hardware store sells packs of 20 bolts for $1.95 and packs of 50 bolts for $4.95. Should Melissa purchase 4 packs of 20 bolts or 2 packs of 50 bolts if she wants to spend the least amount of money?
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 Jasmine had a carrot cake with cream cheese frosting for her birthday. She 1 1 and her friend each ate of the cake and Jasmine’s brother ate of the 4 3 cake. Jasmine’s mom and dad shared the remaining cake evenly. What fraction of the cake did Jasmine’s parents each eat?
Page 64 © 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 Carlos is building a clubhouse that will have a carpeted floor. The carpet he wants to buy costs $3 a square meter. The clubhouse will be 4 meters 1 long, 2 meters wide, and 5 meters high. How much will it cost 2 Carlos to buy enough carpet for the clubhouse?
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 Tara is measuring 20 soil samples for her science class. The mass of the first 3 soil samples is 57 milligrams each, and the mass of the rest of the 17 samples is 63 milligrams each. What is the total mass of all of the samples? Round your answer to the nearest 10 milligrams.
Page 66 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 18 Grace bought 2 cartons of tomato soup. The first carton was 8 inches tall, 2 inches wide, and 4 inches long. The second carton was 12 inches tall, 3 inches wide, and 4 inches long. How many cubic inches of soup did Grace just buy?
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 Todd has to write an essay for his history class and an essay for his English 2 class. Both essays are due in 12 days. Todd plans to write of a page of 5 1 the European history essay and of a page of the English essay every day. 3 How many pages will Todd have written in total on the day that the essays are due?
Page 68 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 20 Matt has 3 jars, each containing 145 tablespoons of honey. There are 16 tablespoons in a cup. How many cups of honey does Matt have?
Page 69 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 21 Jeffrey is sending out a total of 33 holiday cards to friends and family members. Each card costs $0.42, and it costs $0.33 to mail each card. How much will it cost Jeffrey to buy and mail all of the cards?
Page 70 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 22 Alexa invites 3 friends over to have ice cream. In Alexa’s freezer, there are 5 1 pint of strawberry ice cream and pint of chocolate ice cream. If Alexa 6 2 and her friends share the ice cream equally, how much ice cream does each person get?
Page 71 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 23 Edwin is selling magazine subscriptions to raise money for a school camping trip. Each magazine subscription costs $35. Half of the money collected goes towards the magazines and the other half goes towards the camping trip. If Edwin sells 44 magazine subscriptions, how much money has he raised for the camping trip?
Page 72 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 24 Simon is making gift boxes filled with hard candies to give to his teachers 1 at the end of the school year. Each gift box holds cubic foot of hard 3 candies. Simon has 1 large container of hard candies. The container’s base 1 is 2 square feet and its height is 1 feet. How many gift boxes can Simon 2 fill with the hard candies in the large container?
Page 73 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 25 Nancy is throwing a dinner party for 8 guests plus herself. She will be serving shrimp as an appetizer and wants to be sure that there is enough for everyone. She has purchased 0.911 kilograms of shrimp from the fish market. If everyone gets the same amount of shrimp how many grams of shrimp will each person get? There are 1,000 grams in a kilogram. Leave your answer as a mixed number.
Page 74 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 26 In preparation for a junior triathlon, Caleb is planning his training routine 1 1 for the next 5 days. He can either do hour of weight training and 2 3 hour of cardiovascular training every day or he can attend a fitness class that lasts an hour each day. Caleb wants to maximize the amount of time he spends training. Which routine involves spending more time training over the next 5 days?
Page 75 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 27 Richard is in charge of arranging vases of flowers on tables for his sister’s wedding reception. Each table must receive 1 vase. There are 13 people sitting at each table, and there are 429 people attending the reception. 4 Each vase needs to be filled with cup of water. How much water does 5 Richard need to ensure that every table has a vase?
Page 76 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 28 Greg and Meagan are running a bake sale to raise money for their school. They need to determine if they have enough table space to hold all of the items being sold. They have 2 tables. The first table has a surface area of 7 3 square meters. The second table is 3 meters long and meters wide. 8 2 How much table top space do they have between the 2 tables?
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Problem 29 Lucy is working on an experiment for science class. She has 2 bottles of saline solution. There is 0.74 liter of solution in the first bottle and 1.42 liters of solution in the second bottle. Find the total volume of saline solution that Lucy has. Rewrite the volume in expanded form and use this to determine how many 1-liter beakers are necessary to hold all of the solution.
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Problem 30 Lenny has 3 different partially empty bottles of soda. The first bottle has 1 1 liter of soda in it, the second bottle has liter of soda in it, and the 2 4 3 third bottle has liter of soda in it. Make a line plot describing the 4 amount of soda in each bottle. Use this line plot to determine how much soda would be in each bottle if each bottle had the same amount of soda.
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Problem 31 Tara and Herman have just returned from a farmers’ market where they 1 both bought dried fruit. Tara bought pound of dried cranberries and 2 3 Herman bought pound of dried apricots. They decide to mix their fruit 5 together and then share the fruit mix evenly. How much dried fruit does each person get?
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Problem 32 Simon has been keeping track of his spending. Over the past month he has spent $10.30 on candy. By accident, Simon wrote down that he spent $10.03 on candy. Is the amount Simon spent on candy greater or less than the amount he wrote?
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Problem 33 Arthur is supplying cold water to the players in a high school soccer game. He has a large rectangular cooler full of water. The area of the base of the cooler is 92 square inches and the height of the cooler is 45 inches. How much water is in the cooler? Round your answer to the nearest hundreds place.
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Problem 34 1 Becky has set up a candle-making studio in her basement. It takes her of 5 a workday to make 1 candle and she sells each candle for $35. How much money does Becky make from her candles over 6 workdays?
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Problem 35 Kelly is throwing a going away party for a friend. She wants to serve iced tea at the party. There are 42 people attending the party, and she wants to make sure that each person can have 4 cups of iced tea. How many gallons of iced tea should Kelly buy for the party? There are 16 cups in 1 gallon.
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Problem 36 Jordan is shopping for a cabinet to store his schoolbooks. He wants to buy the cabinet that has the greatest volume. The first cabinet he looks at is 2 feet wide, 3 feet long, and 1 foot tall. The second cabinet he looks at is 4 6 feet wide, 1 foot long, and feet tall. Determine which cabinet has the 5 greater volume without finding the actual volume of each cabinet.
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Problem 37 George has written and self-published an electronic comic book. His total publishing expenses came out to $200. The comic book sold 59 copies during the first week for a price of $7.50 per copy. How much profit has George made after the first week? Round your answer to the nearest dollar.
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Problem 38 Victor is shopping for apple juice at the grocery store. He can purchase 1 large carton of juice containing 1.500 liters or 6 smaller cartons containing 0.251 liters each. Which option should Victor choose if he wants to purchase the most juice?
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Problem 39 Jake used 100 blocks to build a tower that was 1.24 meters tall. Later, while Jake was taking a nap, his sister Amanda took away the top 68 centimeters of blocks to build her own tower. When Jake woke up, he rebuilt his tower, adding 85 centimeters of height to what remained after Amanda took some of the blocks away. How many meters tall was Jake’s tower now? Use the measurement conversion below to help answer this question. 1 meter = 100 centimeters
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Problem 40 In an art class, 5 students are going to make figurines out of clay. The teacher has a block of clay that is a right rectangular prism 9 inches long, 4 inches wide, and 3 inches high. If she divides the block so that each student has the same amount of clay, how many cubic inches of clay does each student have? Write your answer as a mixed number.
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Problem 41 Louis is cutting out shapes from construction paper. He has cut out several rectangles so far. He knows that all of the angles in the rectangles are right angles. Next, Louis cuts out a square and a parallelogram. How many right angles must there be in the square and in the parallelogram?
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Problem 42 John is buying lunch for himself and 4 friends. John is having a salad that costs $6 and nothing to drink. One of his friends is having a chicken sandwich that costs $7 and a lemonade that costs $1. The remaining 3 friends are each having a hamburger that costs $5 and a soda that costs $2. Write down and evaluate a numerical expression that describes the total cost of the lunch.
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Problem 43 Lauren is in charge of scenery for her school play. She is making wooden frames for 3 murals that she has not seen yet. She only knows that 1 mural will be painted on a square canvas, 1 mural will be painted on a rectangular canvas, and 1 mural will be painted on a trapezoidal canvas. As she designs the frames, which canvases can Lauren be sure have opposite sides of equal length?
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Problem 44 Justin is saving the money he gets for an allowance to buy concert tickets. Each week he puts $5.37 in his piggy bank and $10.74 in his bank account. He does not spend any of this money. Make a table showing how much money will be in his piggy bank and his bank account at the end of each week for 4 weeks. Describe the relationship between the amount of money in the piggy bank and the bank account at the end of each week.
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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 2012 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: Check 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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