Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 1
UNDERSTANDING COMPLEX MATH F PROBLEMS Written Specifically for the New TEKS N O TI A IC PY L B O U C P - OF E PR RO P
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 2
ISBN 978-1-4204-8410-6 R 8410-6 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. 1013.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 7D, 8D . . . . . . . . . . . . . . 6 2 (Guided) TEKS 6C, 10A . . . . . . . . . . . . . . 10 3 (Guided) TEKS 2D, 8D . . . . . . . . . . . . . . 12 4 TEKS 2D, 12C . . . . . . . . . . . . . . . . . . . . . 14 5 TEKS 4B, 5A . . . . . . . . . . . . . . . . . . . . . . 16 6 TEKS 12C, 13A. . . . . . . . . . . . . . . . . . . . . 18 7 TEKS 7D, 10A . . . . . . . . . . . . . . . . . . . . . 20 8 TEKS 5B, 7A . . . . . . . . . . . . . . . . . . . . . . 22 9 TEKS 4B, 5A . . . . . . . . . . . . . . . . . . . . . . 24 10 TEKS 4E, 4G . . . . . . . . . . . . . . . . . . . . . 26 11 TEKS 3D, 5A . . . . . . . . . . . . . . . . . . . . . 28 12 TEKS 3E, 7A. . . . . . . . . . . . . . . . . . . . . . 30 13 TEKS 2B, 2D . . . . . . . . . . . . . . . . . . . . . 32 14 TEKS 3D, 6C . . . . . . . . . . . . . . . . . . . . . 34 15 TEKS 2C, 10B . . . . . . . . . . . . . . . . . . . . . 36 16 TEKS 2B, 11 . . . . . . . . . . . . . . . . . . . . . . 38 17 TEKS 5A, 8D . . . . . . . . . . . . . . . . . . . . . 40 18 TEKS 3D, 11 . . . . . . . . . . . . . . . . . . . . . 42 19 TEKS 6C, 7A . . . . . . . . . . . . . . . . . . . . . 44 20 TEKS 2C, 6C . . . . . . . . . . . . . . . . . . . . . 46
Part B: On Your Own . . . . . . . . . . . . . . . . . . 49 Problem Page 1 TEKS 5B, 7A . . . . . . . . . . . . . . . . . . . . . . 50 2 TEKS 3E, 4H . . . . . . . . . . . . . . . . . . . . . . 51 3 TEKS 5A, 5B . . . . . . . . . . . . . . . . . . . . . . 52 4 TEKS 3D, 6C . . . . . . . . . . . . . . . . . . . . . . 53 5 TEKS 8C, 8D . . . . . . . . . . . . . . . . . . . . . . 54 6 TEKS 3A, 10A . . . . . . . . . . . . . . . . . . . . . 55 7 TEKS 4B, 5A . . . . . . . . . . . . . . . . . . . . . . 56 8 TEKS 6B, 7A . . . . . . . . . . . . . . . . . . . . . . 57
9 TEKS 6C, 10B . . . . . . . . . . . . . . . . . . . . . . 58 10 TEKS 2C, 3E. . . . . . . . . . . . . . . . . . . . . . 59 11 TEKS 5B, 7A . . . . . . . . . . . . . . . . . . . . . 60 12 TEKS 3E, 7A. . . . . . . . . . . . . . . . . . . . . . 61 13 TEKS 5B, 6C . . . . . . . . . . . . . . . . . . . . . 62 14 TEKS 2B, 2C . . . . . . . . . . . . . . . . . . . . . 63 15 TEKS 8C, 11. . . . . . . . . . . . . . . . . . . . . . 64 16 TEKS 4D, 6C . . . . . . . . . . . . . . . . . . . . . 65 17 TEKS 8C, 8D . . . . . . . . . . . . . . . . . . . . . 66 18 TEKS 5A, 5B . . . . . . . . . . . . . . . . . . . . . 67 19 TEKS 2D, 8C . . . . . . . . . . . . . . . . . . . . . 68 20 TEKS 6C, 10B . . . . . . . . . . . . . . . . . . . . . 69 21 TEKS 3D, 7A . . . . . . . . . . . . . . . . . . . . . 70 22 TEKS 3A, 3E. . . . . . . . . . . . . . . . . . . . . . 71 23 TEKS 4D, 4G . . . . . . . . . . . . . . . . . . . . . 72 24 TEKS 6C, 7C . . . . . . . . . . . . . . . . . . . . . 73 25 TEKS 3D, 8D . . . . . . . . . . . . . . . . . . . . . 74 26 TEKS 2C, 9C . . . . . . . . . . . . . . . . . . . . . 75 27 TEKS 2B, 11 . . . . . . . . . . . . . . . . . . . . . . 76 28 TEKS 6C, 7A . . . . . . . . . . . . . . . . . . . . . 77 29 TEKS 5A, 8D . . . . . . . . . . . . . . . . . . . . . 78 30 TEKS 4C, 5A . . . . . . . . . . . . . . . . . . . . . 79 31 TEKS 6C, 7D . . . . . . . . . . . . . . . . . . . . . 80 32 TEKS 4G, 5A . . . . . . . . . . . . . . . . . . . . . 81 33 TEKS 3E, 4H . . . . . . . . . . . . . . . . . . . . . 82 34 TEKS 4E, 5B . . . . . . . . . . . . . . . . . . . . . . 83 35 TEKS 2B, 11 . . . . . . . . . . . . . . . . . . . . . . 84 36 TEKS 5B, 6C . . . . . . . . . . . . . . . . . . . . . 85 37 TEKS 4G, 5B . . . . . . . . . . . . . . . . . . . . . 86 38 TEKS 2B, 2C . . . . . . . . . . . . . . . . . . . . . 87 39 TEKS 6C, 10B . . . . . . . . . . . . . . . . . . . . . 88 40 TEKS 4C, 6C . . . . . . . . . . . . . . . . . . . . . 89 41 TEKS 7A, 8C . . . . . . . . . . . . . . . . . . . . . 90 42 TEKS 5B, 6C . . . . . . . . . . . . . . . . . . . . . 91 43 TEKS 2C, 3A . . . . . . . . . . . . . . . . . . . . . 92 44 TEKS 4E, 5A. . . . . . . . . . . . . . . . . . . . . . 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 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 Tia wrapped a gift in a 4-inch wide by 4-inch deep by 6-inch tall cardboard box. She measured the wrapping paper to cover the surface area of the box plus an additional 4 square inches. What was the total area of the wrapping paper?
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●1
STEP
IDENTIFY: What are you being asked to find? You are being asked to find the area of the wrapping paper that is used to cover the box.
●2 STEP
FIND: What do you need to solve the problem? • The box is 4 inches wide. • The box is 4 inches deep. • The box is 6 inches tall. The wrapping paper covers the surface area plus 4 square inches.
●3 STEP
CHOOSE: How will you solve the problem? The problem requires you to first find the area of each side of the box. Then, find the sum of the areas. Next, add 4 square inches to that value to find the area of the wrapping paper.
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●4 STEP
SOLVE: Solve the problem. First, find the area of the front of the box. 4 inches × 6 inches = 24 square inches Note that the back will have the same area. Next, find the area of the side of the box. 4 inches × 6 inches = 24 square inches Note that the other side will have the same area. Find the area of the bottom of the box. 4 inches × 4 inches = 16 square inches Note that the top will have the same area. Then find the sum of the areas. 24 in.2 + 24 in.2 + 24 in.2 + 24 in.2 + 16 in.2 + 16 in.2 = 128 in.2 The surface area of the cardboard box is 128 square inches. Find the area of the wrapping paper. Area of the wrapping paper = surface area of the box + 4 in.2 Area of the wrapping paper = 128 in.2 + 4 in.2 = 132 in.2
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●5 STEP
CHECK and JUSTIFY: Check and justify your answer. Use the formula for the surface area of a rectangular prism.
Surface area = 2 (length × width) + 2 (length × height) + 2 (width × height) Surface area = 2 (4 × 4) + 2 (4 × 6) + 2 (4 × 6) = 2 (16) + 2 (24) + 2 (24) Evaluate. 32 in.2 + 48 in.2 + 48 in.2 = 128 in.2 The surface area of the cardboard box is 128 square inches. Then, find the area of the wrapping paper used to cover the box. Area of the wrapping paper = surface area of the box + 4 square inches Area of the wrapping paper = 128 in.2 + 4 in.2 = 132 in.2
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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 Mr. Paulson’s sixth-grade class is taking a field trip during the last week of the school year. The students will spend time at the local water park and recreation center. The distance from the school to the water park is 20 miles. It takes 25 minutes to get from the school to the water park by bus. After spending 4 hours at the park, the students are taken to the local recreation center to continue their day of activities. If the bus is driven at the same speed and the students arrive at the recreation center 30 minutes later, how far is the recreation center from the water park?
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●1 STEP
IDENTIFY: What are you being asked to find? You are being asked to find the distance from the water park to the recreation center.
●2 STEP
FIND: What do you need to solve the problem? • It is 20 miles from the school to the water park. It takes 25 minutes to get to the water park from the school. • The bus is driven at the same speed for both parts of the field trip. • It takes 30 minutes to get to the recreation center from the water park.
●3 STEP
CHOOSE: How will you solve the problem? The problem requires you to use the distance formula. Begin by finding the rate that the bus was traveling. Next, substitute that rate to find the distance between the water park and the recreation center.
●4 ●5 STEP
SOLVE: Solve the problem.
STEP
CHECK and JUSTIFY: Check and justify your answer.
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Problem 3 Brandy wants to draw a picture on her wall. Her mother tells her that she can’t take up an area that is larger than 8.5 inches by 11 inches, but she can use whatever dimensions she wants to use. Brandy draws a picture that is 9 inches by 10 inches. Does Brandy’s picture take up too much space?
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●1 STEP
IDENTIFY: What are you being asked to find? You are being asked to find the area of Brandy’s picture and to decide if it fits the area that her mother told her she could use.
●2 STEP
FIND: What do you need to solve the problem? • The area of the picture must be less than 8.5 inches by 11 inches. • Brandy’s picture is 9 inches by 10 inches.
●3 STEP
CHOOSE: How will you solve the problem? First, find the area of 8.5 inches by 11 inches. Next, find the area of 9 inches by 10 inches. Finally, compare the two areas to see if the area of 9 inches by 10 inches is smaller than the area of 8.5 inches by 11 inches.
●4 ●5 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 Mustafa has quiz scores of 35%, 75%, 60%, 90%, and 90%. His teacher gives him the option of either using the mean or median of his quiz scores as his final quiz grade. Which statistic should Mustafa request to use as his final quiz grade?
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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 and JUSTIFY: Check and justify your answer.
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Problem 5 Sylvia takes 6 hours to make 4 bracelets. She gives 3 of the bracelets away, and she keeps 1 for herself. Her friend’s mother asks Sylvia to make her 14 bracelets. She promises to pay Sylvia $7 for each bracelet. How many hours will it take Sylvia to complete the order?
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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 and JUSTIFY: Check and justify your answer.
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Problem 6 Heather used a dot plot to represent the number of stickers that her classmates earned in 1 week of science class (Monday through Friday). Use Heather’s dot plot to find the median number of stickers that were earned by her classmates.
0
1
2
3
4
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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 and JUSTIFY: Check and justify your answer.
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Problem 7 Mr. Carver shows his math students a jar of jellybeans that he has counted. He will give the jellybeans to the first student who can tell him the correct number of jellybeans in the jar. He tells his students that an equation for the number of jellybeans in the jar is 2(2x + 2y) = 5,000. He also tells them that x is equal to the number of jellybeans in the jar, and y is equal to 4x. Using the information given by Mr. Carver, how many jellybeans are in the jar?
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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 and JUSTIFY: Check and justify your answer.
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Problem 8 Frank’s mother wants to teach him about loans and interest fees. He has $24.50 available to spend on a shopping trip with his mother. He puts 2 phone cases, a large bag of chips, a gallon of fruit punch, and a pack of batteries in the shopping cart. His total, including taxes, is $29.40. His mother loans him the difference and tells Frank that he owes her a 20% fee for the amount of money that she paid. Including the fee, how much money does Frank need to pay his mother?
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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 and JUSTIFY: Check and justify your answer.
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Problem 9 Sam starts a lemonade stand during the summer. His recipe uses 3 cups of sugar syrup for every 1 cup of lemon juice. His pitcher holds 8 cups, and he made 10 pitchers of lemonade on the first day. How many cups of lemon juice did he use?
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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 and JUSTIFY: Check and justify your answer.
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Problem 10 John has decided to wash cars for extra money over the summer. It takes him 3 hours to wash 4 cars. He charges $10 for each car. If he makes $60 on the first day, how long did he spend washing cars on the first day?
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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 and JUSTIFY: Check and justify your answer.
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Problem 11 Chrissy is selling cookies and makes the following table to see at a glance how much to charge people buying multiple boxes of cookies. Each box of cookies costs the same amount. How much does 1 box of cookies cost? How much does 9 boxes of cookies cost? Number of Boxes of Cookies
Cost
1 3
$12
5
$20
7
$28
9
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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 29
●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 and JUSTIFY: Check and justify your answer.
Page 29 © 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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 30
Problem 12 Tricia’s teacher asked her to create a mathematical expression that represented the amount of money that she would have if she started with 2 dimes and added 26 dimes. What expression should Tricia use, and how much money would she have?
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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 31
●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 and JUSTIFY: Check and justify 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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 32
Problem 13 Carl’s mother showed him a notebook that she uses to keep track of her finances. He noticed that most of the numbers on the page were negative. His mother said that the negative numbers represent money that she spent. She said that –30 represents less money spent than –45. In school, Carl learned that –30 is greater than –45. Use mathematical reasoning to explain how it is possible for Carl and his mother to be correct.
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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 33
●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 and JUSTIFY: Check and justify 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 .
Understanding Complex Math TX F_ACCS Understanding 10/29/13 2:01 PM Page 34
Problem 14 Rissa has 10 stickers next to her name on a name tag. Rissa is told that the amount of candy in a small bag is equal to 5 less than 2 times the number of stickers next to her name. Rissa’s teacher offers to give her the bag of candy if Rissa writes an expression for “5 less than 2 times the number, x, of stickers,” and then correctly solves the equation to find the number of pieces of candy in the bag. How many pieces of candy are in the bag? What is the coefficient of x in the expression?
Page 34 © 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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 35
●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 and JUSTIFY: Check and justify 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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 36
Problem 15 The temperature on Wednesday was –3˚F. The temperature on Thursday was 2˚F. Use mathematical reasoning to explain which day had the warmer temperature.
Page 36 © 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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 37
●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 and JUSTIFY: Check and justify 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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 38
Problem 16 For an art project in a woodworking class, Annie’s teacher plotted the points (–3, 4) and (–1, –3) on a coordinate plane. She does this to show the pattern a router bit will make in the boards when they are joined in a dovetail corner. The teacher told Annie to reflect the points across the y-axis, and connect the dots to see the pattern. What are the coordinates of the new points that were reflected across the y-axis?
Page 38 © 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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 39
●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 and JUSTIFY: Check and justify 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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 40
Problem 17 Cameron wants to make a model of his bookshelf for his art project. He uses the scale of 2 inches for every foot in the actual dimensions. The 2 actual bookshelf is 3 feet tall by 2ᎏᎏ feet wide by 1 foot deep. What is the 5 volume of his model?
Page 40 © 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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 41
●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 and JUSTIFY: Check and justify 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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 42
Problem 18 Mr. Collins draws 4 points on a coordinate plane. He tells his students that each point corresponds to a corner of the supply room for the math department. The points are at (3, 4), (3, –5), (–2, 4), and (–2, –5). Each unit length in the coordinate plane equals 1 foot. What is the shape of the supply room? What is its area?
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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 43
●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 and JUSTIFY: Check and justify 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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 44
Problem 19 Faith had $15.88 left on her music gift card. She downloaded 3 songs. One song cost $1.99. The other 2 songs were $0.99 each. Then, she downloaded 6 more songs that cost $0.99 each. How many additional $1.99 songs can Faith download?
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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 45
●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 and JUSTIFY: Check and justify 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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 46
Problem 20 Last year, Justin offered to keep a running total of the number of students who enter 5th grade at his school. There were 74 students last year and 81 students this year. Justin does not know how many 5th grade students there will be next year. Use a number line to represent the possible total number of students who enter 5th grade over the three-year period.
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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 47
●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 and JUSTIFY: Check and justify 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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 48
Page 48 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 .
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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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 50
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 Katie asked her mother to take her to the closest mall. Her mother promised to take her if she solved an equation that can be used to find the distance from their house to the mall. Her mother told her that she could find the distance from her house to the closest mall by adding 4 miles to the distance from their house to the school and then multiplying that answer by 2. The distance in miles from Katie’s house to the school is equal to one-half of Katie’s age. Katie is 12 years old. What is the distance from Katie’s house to the closest mall?
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 .
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Problem 2 Ben poured some orange juice from a container into a glass. When Ben 1 was done pouring his glass of orange juice, there was a ᎏᎏ gallon of juice 2 left in the container for other people to drink. Thirty minutes later, his 1 older brother drank ᎏᎏ of the remaining juice, and 1 hour later, his younger 4 2 brother drank ᎏᎏ of what was left. How much juice was left in the container 3 after each brother had orange juice to drink?
Page 51 © 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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 1:59 PM Page 52
Problem 3 Ms. Carlton’s class has a 3:4 ratio of boys to girls. There are 16 girls in the class. In a recent survey, 75% of the boys are 4 feet or taller and 80% of the girls are 4 feet or taller. Based on this information, determine the number of boys in the class who are 4 feet or taller.
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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 2:00 PM Page 53
Problem 4 Ms. Jenkins has 3 packages of red ribbons. Each package of red ribbons has the same number of ribbons. She also has 2 packages of blue ribbons. Each package of blue ribbons has the same number of ribbons. Write an equation to describe the total number of ribbons that Ms. Jenkins has. Then, solve the equation if there are 8 ribbons in each red ribbon package and 10 ribbons in each blue ribbon package.
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 .
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Problem 5 Dana is planting flowers around the base of a statue. To determine how many flowers to plant, she drew a simple picture of the space available. Both the space available and the base of the statue are square, so she draws a large square with a smaller square inside of it. One side of the small square is 2 feet long, and 1 side of the larger square is 5 feet long. Find the area of the space that is outside the small square but still inside the large square.
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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 2:00 PM Page 55
Problem 6 Alexis wants to make a mini cake that is one-half the size of the cake she usually makes. She plans to give the mini cake to her grandmother. Alexis divides all of the measurements for the original recipe in half. She puts 1 ᎏᎏ teaspoon of vanilla extract in the batter for the mini cake. How much 4 vanilla extract is used in the original recipe?
Page 55 © 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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 2:00 PM Page 56
Problem 7 Tanesha takes 2 hours to make 6 friendship bracelets. She has decided to make bracelets for her basketball team. She wants to make a bracelet for each of her 14 teammates and she wants to keep 1 for herself. How long will it take her to make the bracelets?
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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 2:00 PM Page 57
Problem 8 A phone tree refers to a method of using a telephone to call many people starting with just one person. The first person calls a few people. Then, each of these people calls a few more people, and so on. Isaiah’s teacher gave him the following table to show how one possible phone tree would work. She explained that the values on the left represent the level on the tree, and the values on the right are the total number of people called at that level of the tree. Each person in this phone tree calls the same number of people. How many people is each person supposed to call? How many people will be called at the 5th level of the phone tree? Level of the
Number of people called at
phone tree
each level of the phone tree
1
3
2
9
3
27
4
81
5
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 .
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Problem 9 Min had $15.90 left on a gift card for electronic games and music. She downloaded 2 electronic games. The electronic games cost $1.99 each. Then she downloaded 4 songs that cost $1.99 each. How many $0.99 songs can Min download with the money that is left on her gift card?
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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 2:00 PM Page 59
Problem 10 Rachel is using twine to make a wind chime. She’s using a pattern she printed from the Internet that uses a number line with 0, 1 and 2 marked 1 on it. She needs to leave ᎏᎏ of each length of twine at the top in order to 4 fasten each chime to the hanger. How can she use the number line to 1 approximate the position of ᎏᎏ on the twine? 4
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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 2:00 PM Page 60
Problem 11 Rico had $20. He went to the store with his older brother. The total cost of Rico’s purchases was $25, so his brother paid for the amount that Rico did not have. His brother told Rico that he had to pay him back this difference plus an additional 10% when he received his allowance. Including the fee, how much money does Rico need to pay his brother?
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 .
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Problem 12 Bentley left an unopened candy bar on the counter while he went to the 1 store with his mother. While Bentley was gone, his older sister ate ᎏᎏ of 5 1 the candy bar. His younger brother ate ᎏᎏ of the remaining amount of 4 the candy bar. How much of his candy bar was left when Bentley returned home?
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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 2:00 PM Page 62
Problem 13 Karen wants to buy a bike that costs $109 plus tax. The sales tax is 8%. Karen earned $15 babysitting her cousin. She put the $15 in her piggy bank to start saving for the bike. She receives $12 a week for her allowance. If her allowance is the only money that she adds to the $15, how many weeks later will she be able to buy the bike? Round your answer to the nearest week.
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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 2:00 PM Page 63
Problem 14 Steve’s father shows him a notebook that he uses to keep track of the money that he spends each week. His father says that the negative numbers represent money that he spent. He says that –50 on one line in the notebook represents less money spent than –65 on another line. In school, Steve learned that –50 is greater than –65. Use mathematical reasoning to explain how it is possible for both Steve and his father to be correct.
Page 63 © 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 .
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Problem 15 Ms. Webb draws 3 points on a coordinate plane. She tells her students that each point corresponds to a corner of her closet. The points are at (–3, 4), (–3, –3), and (2, –3). Each unit length in the coordinate plane equals 1 foot. What is the shape of the closet? What is its area?
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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 2:00 PM Page 65
Problem 16 It takes Caroline’s mother 10 minutes to travel from the mall to the restaurant driving at a speed of 0.5 miles per minute. After eating at the restaurant, Caroline’s mother drives at a speed of 0.5 miles per minute to get home. The distance from the restaurant to their house is twice the distance between the mall and the restaurant. How long did it take Caroline’s mother to travel from the restaurant to her home?
Page 65 © 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 .
Understanding Complex Math TX F_ACCS Understanding 10/24/13 2:00 PM Page 66
Problem 17 Brian is working on a mural project where he will be transferring a small drawing to a large wall. On the small drawing, he draws a rectangle that has a length of x 2 inches and a width of x inches. He writes an expression to represent the area of the rectangle, then he evaluates the expression for x = 4, to see if he has enough wall space for the large mural. Find the expression that Brian used, and then solve for the area of the rectangle on his small drawing.
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 .
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Problem 18 Ms. Murphy’s track team has a 3:2 ratio of children with brown eyes to children with blue eyes. There are 12 children with brown eyes on the team. At a recent track meet, 80% of the children with brown eyes ran in events and 75% of the children with blue eyes ran in events. The children who didn’t run in events at that meet participated in other events, like the shot put and the high and long jumps. Based on this information, determine the number of children with blue eyes who ran in an event.
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Problem 19 Cory wants to paint a mural on a wall in his room. His mother tells him that he can’t take up an area that is larger than 4 feet by 8 feet, but he can use whatever dimensions he wants to use. Cory paints a mural that is 5 feet by 7 feet. Does Cory’s mural take up too much space?
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Problem 20 Robin’s piano teacher puts a star sticker next to her name whenever Robin plays a piece without making a mistake. She also gives Robin a bag of candy at the end of each month of lessons. The number of pieces of candy in the bag is equal to 6 more than 3 times the number of stickers by her name. Robin has 7 stickers by her name. How many pieces of candy are in the bag?
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Problem 21 Ms. Sony has 4 packages of green paper clips. Each package of green paper clips has the same number of paper clips. She has 3 packages of white paper clips. Each package of white paper clips has the same number of paper clips. Write an expression to describe the total number of paper clips that Ms. Sony has. Then, solve the expression if there are 10 paper clips in each green paper clip package, and 15 paper clips in each white paper clip package.
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Problem 22 Alanna has only enough sugar to bake half a batch of cookies. She divides 3 all the measurements for the original cookie recipe in half. She puts ᎏᎏ cup 4 of flour in the mix. How much flour is used in the original recipe?
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Problem 23 Malik has decided to mow lawns for extra money over the summer. It takes him 2 hours to mow 5 lawns. He charges $15 for each lawn. If he makes $90 on the first day, how long did he spend mowing lawns?
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Problem 24 Mrs. Culkin shows James a jar of lemon drops. She tells James that she has counted the lemon drops and will give him the jar if he tells her how many lemon drops are in the jar. She tells James that an equation for the number of lemon drops in the jar is 2(2x + y) = 2,300. She also tells him that x is equal to the number of lemon drops in the jar and y is equal to 3x. How many lemon drops are in the jar? What is the coefficient of y in the sum (2x + y)?
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Problem 25 Flora is helping her dad build a doghouse. To plan where it will go in their small yard, she draws a picture of the yard as a large rectangle with a small square inside of it to represent the doghouse. The doghouse is square and one side has a length of 3 feet. The yard is 7 feet by 5 feet. How much remaining space will there be in the yard once the doghouse has been put in place? Find the area of the space that will be outside of the doghouse.
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Problem 26 Josh is keeping a running total of the number of free throws he makes each day. He began on Monday and scored 11 points. Tuesday he scored 15 points. His total for Wednesday and Thursday combined was 18 points. The number of points Josh will score on Friday is unknown. Use a number line to show the total number of points Josh may score over the 5-day period.
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Problem 27 Jesse’s parents are planning to plant trees on their property in a way that mirrors the trees they already have planted. His parents use a coordinate plane to plot the points where their existing trees are: (–3, 2) and (4, 2). Now they ask him to find out where they should plant the new trees if the points are reflected over the x-axis. What are the coordinates of the points where they will plant the new trees?
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Problem 28 In his science class, Anthony’s teacher asks him to create an expression that represents a colony of bacteria that starts with 2 cells and triples in number each hour. His teacher asks him to use the expression to figure out how many cells the colony will have after 4 hours. What expression should Anthony use, and how many cells will the colony have after 4 hours?
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Problem 29 Carlos wants to make a miniature model of the fish tank in his art classroom. He uses the scale of 3 inches for every foot in the actual 1 dimensions. The actual fish tank is 2ᎏᎏ feet tall by 6 feet wide by 1 foot 3 deep. What is the volume of his model?
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Problem 30 Maria is making punch for the school dance. Her recipe uses 2 cups of a soda for every cup of orange juice. A pitcher holds 9 cups of punch, and she makes 10 pitchers of punch. How many cups of orange juice did she use?
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Problem 31 Kristin asked her mother to take her to the costume store for a Halloween costume. Her mother promised to take her if she solved an equation for the distance from their house to the costume store. Her mother told Kristin that she could find the distance from her house to the costume store by subtracting 10 miles from the distance from their house to the water park, then multiplying that answer by 3. The distance from Kristin’s house to the water park is equal to twice Kristin’s age. Kristin is 11 years-old. What is the distance from Kristin’s house to the costume store?
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Problem 32 Ariel scored 65% on her first test. Her teacher offered to let her retake the test. She scored a 75% on the second test. Each test had the same number of questions, and each question was worth 1 point. If Ariel got 26 questions correct on the first test, how many questions did she get correct on the second test?
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Problem 33 1 Aaron has ᎏᎏ pound of chocolate. He shares it with Tom and Brenda. He 2 3 1 gives himself ᎏᎏ of the chocolate and splits the other ᎏᎏ evenly between 4 4 Tom and Brenda. How many ounces of chocolate did Aaron give Brenda?
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Problem 34 Ashley listened to 45 songs in 3 hours. The songs were about the same length. 20% of the songs she listened to each hour were hip-hop. How many hip-hop songs did she listen to during the first hour?
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Problem 35 Javier played a game involving battleships using a coordinate plane. He has one ship at (2, –3), (2, –1), and (2, 1). He has a matching ship reflected over the y-axis. Find the coordinates of the matching ship.
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Problem 36 Travis has $60 to spend at the mall. He spends 25% of it at the food court. After buying the food, he splits his remaining money in a 1 to 2 ratio. He puts the smaller amount in his savings account and he spends the rest of the money on a gift card. How much money did Travis spend on the gift card?
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Problem 37 Hannah spent $10 to buy supplies to make lemonade. She opened a lemonade stand and used all of her supplies. If she earned $15, what percent is her profit?
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Problem 38 Jason says that negative temperatures can be compared the same as positive temperatures. He argues that 15˚C is warmer than 10˚C and –15˚C is warmer than –10˚C. Use mathematical reasoning to explain if Jason is correct that –15˚C is warmer than –10˚C.
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Problem 39 Arianna is 3 years younger than twice her sister’s current age. If Arianna is 11 years old, what is her sister’s current age?
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Problem 40 Vo broke her shoelace, so her sister gave her a piece of string. In order to make it the correct length for her shoe, Vo cut off 20% of the string. She threw away the 20% and kept the rest. If Vo’s string now measures 8 inches, how long was her original piece of string?
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Problem 41 For a birthday party, Valencia and her mother will be covering boxes full of gifts and candy to give away as party favors for the guests. To be sure they have enough construction paper for each box, they need to measure the surface area of a box. Valencia measured each side of the box and found that it was 8 inches wide by 2 inches deep by 10 inches high. What is the total surface area of the box?
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Problem 42 Billy wants to buy a video game that costs $79 plus tax. The sales tax is 6%. Billy has saved $25 and he receives $10 a week for his allowance. If his allowance is the only money that he adds to his savings, how many weeks later will he be able to buy the video game? Round your answer to the nearest whole week.
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Problem 43 Taylor is building a birdhouse from plans she got from her local wildlife association. On the plans, it shows a number line that has 0, 0.5, and 1 marked on it so that she can measure the width and height of the walls of the birdhouse. She will need to mark the wood along all sides 1 at ᎏᎏ, which is where the nails will go. How can she approximate the 8 1 position of ᎏᎏ on the number line without using a ruler? 8
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Problem 44 Juan scored 60% on his first test. His teacher offered to let him retake the test. He scored an 80% on the second test. Each test had the same number of questions, and each question was worth 1 point. If Juan got 21 questions correct on the first test, how many questions did he get correct on the second test?
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