UNDERSTANDING COMPLEX MATH H PROBLEMS Written Specifically for the New TEKS
RALLY! EDUCATION 22 Railroad Avenue Glen Head, NY 11545 888-99-RALLY Fax: 1-516-671-7900 www.RALLYEDUCATION.com LESLIE@RALLYEDUCATION.com ………………..………………………………………………………………………………………………………………………
Understanding Math Series Understanding Math in the Real World
Understanding Complex Math Problems
+
This book is…
Understanding Complex Math Problems Written Specifically for the TEKS Grades 3 – 8 Multi-Step math problems for the New TEKS - Instruction helps students develop a process to solve complex math problems Buy the series separately… Includes 25 copies of Complex math plus the Teacher Guide Price: 25-pack: $179 Level C D E F G H
Grade 3 4 5 6 7 8
25-Pack Item # 8400-7 8404-5 8408-3 8412-0 8416-8 8420-5
Or Buy the Complete Program $229 or $349 15 or 25 Copies of Each Product Understanding Math in the Real World Understanding Complex Math Problems
-----------------------------------------------------
888-99-RALLY www.RALLYEDUCATION.com
UNDERSTANDING COMPLEX MATH H PROBLEMS Written Specifically for the New TEKS
ISBN 978-1-4204-8418-2 R 8418-2 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. 1213.MAQ RALLY! EDUCATION • 22 Railroad Avenue, Glen Head, NY 11545 • (888) 99-RALLY
Page 2 © 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 .
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4
Part A: Solving Complex Math Problems . . 5 Problem Page 1 (Modeled) TEKS 2B, 8C. . . . . . . . . . . . . . . 6 2 (Guided) TEKS 2B, 7C . . . . . . . . . . . . . . . 10 3 (Guided) TEKS 2B, 4C . . . . . . . . . . . . . . . 12 4 TEKS 10A, 10C . . . . . . . . . . . . . . . . . . . . . 14 5 TEKS 10C, 10D . . . . . . . . . . . . . . . . . . . . . 16 6 TEKS 8C, 8D . . . . . . . . . . . . . . . . . . . . . . 18 7 TEKS 2B, 7C . . . . . . . . . . . . . . . . . . . . . . . 20 8 TEKS 2B, 7D . . . . . . . . . . . . . . . . . . . . . . . 22 9 TEKS 5G, 7A. . . . . . . . . . . . . . . . . . . . . . . 24 10 TEKS 5I, 9A . . . . . . . . . . . . . . . . . . . . . . 26 11 TEKS 9A. . . . . . . . . . . . . . . . . . . . . . . . . 28 12 TEKS 4C, 5A . . . . . . . . . . . . . . . . . . . . . . 30 13 TEKS 2B, 8C . . . . . . . . . . . . . . . . . . . . . . 32 14 TEKS 2B, 2C . . . . . . . . . . . . . . . . . . . . . . 34 15 TEKS 2B, 2C . . . . . . . . . . . . . . . . . . . . . . 36 16 TEKS 4C, 5A . . . . . . . . . . . . . . . . . . . . . . 38 17 TEKS 4A, 5I, 7D . . . . . . . . . . . . . . . . . . . 40 18 TEKS 8C . . . . . . . . . . . . . . . . . . . . . . . . 42 19 TEKS 5D, 11A . . . . . . . . . . . . . . . . . . . . . 44 20 TEKS 2B . . . . . . . . . . . . . . . . . . . . . . . . . 46
Part B: On Your Own . . . . . . . . . . . . . . . . . . 49 Problem Page 1 TEKS 2B, 8C . . . . . . . . . . . . . . . . . . . . . . . 50 2 TEKS 5I, 9A . . . . . . . . . . . . . . . . . . . . . . . 51 3 TEKS 2B, 2C . . . . . . . . . . . . . . . . . . . . . . . 52 4 TEKS 2B, 2C . . . . . . . . . . . . . . . . . . . . . . . 53 5 TEKS 4C . . . . . . . . . . . . . . . . . . . . . . . . . 54 6 TEKS 4A, 5I, 7D . . . . . . . . . . . . . . . . . . . . 55 7 TEKS 2B, 7C . . . . . . . . . . . . . . . . . . . . . . . 56 8 TEKS 2B, 7D . . . . . . . . . . . . . . . . . . . . . . . 57 9 TEKS 5D, 11A . . . . . . . . . . . . . . . . . . . . . . 58
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
TEKS 2B, . . . . . . . . . . . . . . . . . . . . . . . . 59 TEKS 4C, 5A . . . . . . . . . . . . . . . . . . . . . . 60 TEKS 5G, 7A. . . . . . . . . . . . . . . . . . . . . . 61 TEKS 2B, 4C . . . . . . . . . . . . . . . . . . . . . . 62 TEKS 10A, 10C . . . . . . . . . . . . . . . . . . . . 63 TEKS 10C, 10D . . . . . . . . . . . . . . . . . . . . 64 TEKS 8C, 8D . . . . . . . . . . . . . . . . . . . . . 65 TEKS 8A, 8C . . . . . . . . . . . . . . . . . . . . . . 66 TEKS 8A, 8C . . . . . . . . . . . . . . . . . . . . . . 67 TEKS 5I, 9A . . . . . . . . . . . . . . . . . . . . . . 68 TEKS 2B, 7C . . . . . . . . . . . . . . . . . . . . . . 69 TEKS 4C, 5A . . . . . . . . . . . . . . . . . . . . . . 70 TEKS 2B, 7C . . . . . . . . . . . . . . . . . . . . . . 71 TEKS 2B, 2C . . . . . . . . . . . . . . . . . . . . . . 72 TEKS 2B, 2C . . . . . . . . . . . . . . . . . . . . . . 73 TEKS 4C, 8C . . . . . . . . . . . . . . . . . . . . . . 74 TEKS 4A, 5I, 7D . . . . . . . . . . . . . . . . . . . 75 TEKS 8A, 8C . . . . . . . . . . . . . . . . . . . . . . 76 TEKS 8A, 8C . . . . . . . . . . . . . . . . . . . . . . 77 TEKS 5I, 9A . . . . . . . . . . . . . . . . . . . . . . 78 TEKS 9A. . . . . . . . . . . . . . . . . . . . . . . . . 79 TEKS 2B, 8C . . . . . . . . . . . . . . . . . . . . . . 80 TEKS 5G, 7A. . . . . . . . . . . . . . . . . . . . . . 81 TEKS 2B, 4C . . . . . . . . . . . . . . . . . . . . . . 82 TEKS 7C, 10C . . . . . . . . . . . . . . . . . . . . . 83 TEKS 3C, 10C . . . . . . . . . . . . . . . . . . . . . 84 TEKS 8C, 8D . . . . . . . . . . . . . . . . . . . . . 85 TEKS 2B, 7C . . . . . . . . . . . . . . . . . . . . . . 86 TEKS 2B, 7D . . . . . . . . . . . . . . . . . . . . . . 87 TEKS 5D, 11A . . . . . . . . . . . . . . . . . . . . . 88 TEKS 2B . . . . . . . . . . . . . . . . . . . . . . . . . 89
Page 3 © 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 .
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.
Page 4 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 .
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.
Page 5 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 .
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 Jamie went to the farmers market last Saturday. She bought 1 small apricot and 5 large apricots that weighed a total of 26 ounces. If each large apricot weighed 3 ounces more than the small apricot, then how much did the small apricot weigh? Write your answer as a decimal.
Page 6 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? You are told that Jamie bought small and large apricots at a farmers market. You are told the total number of ounces of apricots she bought. You are asked to find the weight of the small apricot in decimal form.
●2 STEP
FIND: What do you need to solve the problem? • The number of small apricots Jamie bought is 1. • The number of large apricots Jamie bought is 5. • The total weight of the apricots Jamie bought: 26 ounces. • A large apricot weighs 3 ounces more than a small apricot.
●3 STEP
CHOOSE: How will you solve the problem? First, write an equation that describes the total weight of the apricots that Jamie bought in terms of the weight of a small apricot. Then, solve this equation. Finally, express your answer as a decimal.
Page 7 © 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 .
●4 STEP
SOLVE: Solve the problem. Let a represent the weight of a small apricot in ounces. Since each large apricot weighs 3 ounces more than a small apricot, a large apricot weighs a + 3 ounces. Because the total weight of 1 small apricot and 5 large apricots is 26 ounces, you can write the equation shown below. a + 5(a + 3) = 26 To solve this equation, first expand the expression on the left-hand side using the Distributive Property, and then combine like terms. a + 5a + 15 = 26 6a + 15 = 26 Next, subtract 15 from both sides of the equation, and then divide both sides of the equation by 6. 6a + 15 – 15 = 26 – 15 6a = 11 11 a = 6 11 To convert into a decimal, use long division until you find that 6 the decimal repeats. 1.83 6 1 1 0 .0 The small apricot that Jamie bought weighs 1.83 ounces.
Page 8 © 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 .
●5 STEP
CHECK and JUSTIFY: Check and justify your answer. To check your answer, plug the weight of a small apricot into the original equation for a. Make sure the left-hand side is equal to the right-hand side. 11 11 + 5( +3) = 26 6 6 11 55 + + 15 = 26 6 6 66 + 15 = 26 6 11 + 15 = 26 26 = 26
Page 9 © 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 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 Stephen is standing on the street across from his house and wants to know how far it is to his window. He knows that he is 5 meters away from his house and his window is 3 meters above the ground. Find the distance between Stephen and his window. Between what two whole numbers is this distance?
Page 10 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? You are told that Stephen wants to know the distance from where he is standing to his window. You are told the distance that he is from his house and the height of his window from the ground. You are asked to find the distance between Stephen and his window. You also need to identify the two whole numbers that are between your answer.
●2 STEP
FIND: What do you need to solve the problem? • The distance from Stephen to his house is 5 meters. • The height of Stephen’s window from the ground is 3 meters.
●3 STEP
CHOOSE: How will you solve the problem? Make a diagram of the situation. Then, use the Pythagorean Theorem to find the distance between Stephen and his window. Finally, find the two whole numbers that this distance is between.
●4 ●5 STEP
SOLVE: Solve the problem.
STEP
CHECK and JUSTIFY: Check and justify your answer.
Page 11 © 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 Clevon goes to the store to buy a candy bar. The graph below shows his progress on the way to the store, in the store, and on the way home. The x-axis represents time, and the y-axis represents the distance he is from home. What was his rate, in miles per minute, on the way to the store? What was his rate, in miles per minute, on the way home? What was the percentage increase in his rate on the way home?
Distance (miles)
1
0.5
0
0
10
20
30
40
50
60
Time (minutes)
Page 12 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? You are told that Clevon went from his home to the store to buy a candy bar. You are given a graph of his progress and told to find his rate on the way to the store and on his way home.
●2 STEP
FIND: What do you need to solve the problem? You need to use the graph of Clevon’s trip to and from the store in order to answer the questions.
●3 STEP
CHOOSE: How will you solve the problem? Recall that rate equals the absolute value of the slope of a distancetime graph. You can find Clevon’s rate going to and from the store by using the graph to find the slopes for each segment of his trip.
●4 STEP
SOLVE: Solve the problem.
●5 STEP
CHECK and JUSTIFY: Check and justify your answer.
Page 13 © 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 Follow the 5-Step Problem-Solving Plan to answer questions 4–20.
Problem 4 Brian and David are neighbors. Their homes are shaped like the trapezoids pictured below. However their homes do not look the same. For each home, the front door is located at the top of the shape. David’s Home
Brian’s Home
a
b
112°
68°
112°
c
c 68°
68°
c
c 112°
68° b
112° a
David claims that the sides with the front doors are not congruent; therefore the homes are not congruent. Brian claims that the homes are congruent. Is David or Brian correct? For any congruence you identify, explain the sequence of rotations and/or reflections that lead to it.
Page 14 © 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 and JUSTIFY: Check and justify your answer.
Page 15 © 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 Alice was looking at a wallpaper sample and saw a pattern of 3 similar triangles that repeated. The repeated portion is pictured below. Find the geometric transformation that takes the black triangle to the gray triangle. Then, find the transformation that takes the gray triangle to the green triangle.
y
10 8 6 4 2 -8
-4
0
4
8
x
Page 16 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 and JUSTIFY: Check and justify your answer.
Page 17 © 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 Leslie is making a mosaic with tiles in the shape of identical triangles. She starts the mosaic by placing a few triangle tiles end to end in a row, all in the same direction. Then, she rotates a few additional triangles 180° from those in the first row, and fits them on top of the triangles in the first row. Her first 2 rows look like the diagram below.
(3x – 4)° x°
(4x – 8)°
The triangles always form straight rows. Let x represent the measure of the smallest angle. If the other 2 angles are 3x − 4 and 4x − 8 respectively, then what is the measure of the smallest angle of the triangular tile that is used in Leslie’s mosaic?
Page 18 © 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 and JUSTIFY: Check and justify your answer.
Page 19 © 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 Allegra is building a square picture frame as shown below. She wants to know the area of the open space in the frame.
Each of the 4 right triangles that make up the frame has a base of 5 inches and a height of 12 inches. Allegra decides to use the Pythagorean Theorem to find the length of the third side of each triangle, and then use her answer to find the area of the open space in the frame. Then, she moves the 4 triangular pieces around and notices that she can rearrange them within the same larger square as shown below. Allegra decides to use this arrangement to find the area of the open space in the frame.
Calculate the area of the open space in the frame using both of the methods just described. Do you get the same answer for both methods?
Page 20 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 and JUSTIFY: Check and justify your answer.
Page 21 © 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 Phil and Chris are trying to determine who lives closer to their school. They make a map of the location of their homes and their school where each square on the map corresponds to one square mile. Each location is labeled on the map. Using a straight line between each boy’s home and the school, who lives closer to their school?
14
Chris’ Home
12
Miles North
10
School
8 6 4 2 0
Phil’s Home 2
4
6
8
10
12
14
16
18
20
Miles East
Page 22 © 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 and JUSTIFY: Check and justify your answer.
Page 23 © 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 Niko is doing a science experiment and needs to use different sizes of cylindrical beakers. To measure the volume, V, of a cylinder, he learns the formula V = r 2h where r represents the radius of the cylinder and h represents the height of the cylinder. All of his cylinders are 6 inches tall. However, he has cylinders with 3 different radii: 1 inch, 2 inches, and 3 inches. Use the formula to find the volume of each of his 3 cylinders. Is the volume formula for a cylinder with a height of 6 inches a linear function? For this problem, use the approximation 3.14.
Page 24 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 and JUSTIFY: Check and justify your answer.
Page 25 © 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 Liz is saving money to buy a video game that costs $38. She has already saved $5. Each week, she saves $7.50 from her allowance to buy the game. She makes a graph, as shown below, to figure out how many weeks she will need to save money before she can afford to buy the game. Write equations for the two lines on the graph, and find the solution using the graph. 40 35
Money Saved
30 25 20 15 10 5 0
1
2
3
4
5
Weeks
Page 26 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 and JUSTIFY: Check and justify your answer.
Page 27 © 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 Ariel’s family had dinner at their favorite Mexican restaurant. Ariel and her sisters each ordered tacos, while her parents ordered burritos. The cost of a burrito is $1 less than the cost of 3 tacos. If Ariel’s family ordered 7 tacos and 2 burritos for a total cost of $30.50, then what is the individual cost of a taco and a burrito?
Page 28 Š R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .
●1 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 .
Problem 12 Santino and Greg both drive their cars very often and they are concerned about how much gasoline they each use. Santino kept a log of his gasoline use over the past week.
Gallons of Gasoline Used
Miles Driven by Santino
5
113
10
226
15
339
20
452
Greg describes the amount of gasoline that his car uses with the equation m = 19g, where m represents the number of miles that Greg drove and g represents the number of gallons of gasoline that Greg used. Who has a more fuel-efficient car?
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 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 .
Problem 13 Jack and Ellen went out for lunch and split the $18 bill. Let x represent 1 the amount that Jack paid. Ellen paid (x + 6). How much did Jack and 4 Ellen each pay? Express your answers as decimals.
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 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 .
Problem 14 The human body is made up of cells that account for most of our weight. An average cell weighs only 1 × 10−12 kilogram. However, there are about 6 × 1013 cells in the average human body. What is the total weight of all of the cells in an average human body? And if the average human weighs 80 kilograms, what percentage of the weight of an average human being is cells?
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 .
●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 .
Problem 15 In the 1920s and 1930s, many countries built large rigid airships that looked like blimps but were called zeppelins. One of these ships, called the Hindenburg, was filled with 3.34 × 105 kilograms of hydrogen. This allowed the Hindenburg to float in the air. If each hydrogen atom weighs about 1.67 × 10−27 kilogram, how many hydrogen atoms filled the ship? And if the ship carried 100 people, how many hydrogen atoms were there per person?
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 .
●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 .
Problem 16 Hannah went on a hike and made a graph of her progress during the first hour, as shown below. Her friend Sara went on a hike and gave the 1 equation d = t to describe her progress, where d represents her distance 18 in miles and t represents her time in minutes. How many minutes will it take each of the girls to walk 4 miles at their respective rates? Who is walking faster? 4
Distance (miles)
3
2
1
0
4
8
12
16
20
24
28
32
36
40
44
48
52
56
60
Time (minutes)
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 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 .
Problem 17 Tom likes playing pinball. His favorite place to play pinball, the Silverball Arcade, has a graph that displays the cost for a given number of games of pinball, as shown below. Use the points (0, 2) and (4, 5) to calculate the slope of the line which represents the cost per game at the Silverball Arcade. Then, use the points (0, 2) and (8, 8) to calculate the slope of the line. Show that these slopes are the same by using the length between each pair of points as the hypotenuse of a right triangle. Write an equation of the line displayed on the graph. 12 11 10 9
Cost (dollars)
8 7 6 5 4 3 2 1 0
1
2
3
4
5
6
7
8
9
10
11
12
13
Number of Games
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 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 .
Problem 18 Two friends, Brett and Ross, are arguing over a math problem. They were both given the equation shown below. 2 1 (2x + 15) = x – ( x – 6) 5 5 Brett claims that 10 is the solution, while Ross claims that 5 is the solution. Test both claims by substituting 10 for x and then 5 for x. Then, simplify the equation to explain your results. Which solution is correct?
Page 42 © R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .
●1 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 .
Problem 19 Frank collects data on 10 students in his gym class. He measures how far they can jump horizontally and how high they can jump vertically. He wants to see if there is a connection between a student’s ability to jump horizontally and his or her ability to jump vertically, so he makes the scatter plot shown below.
Vertical Jump (feet)
1.2
1.0
0.8
10
12
14
16
Horizontal Jump (feet)
Frank finds that the equation of the line of best fit for the data presented in 9 1 the scatter plot is y = x + . Is there a positive or negative correlation 125 20 between a student’s ability to jump horizontally and vertically? How high of a vertical jump does his model predict for a student who can jump 15 feet horizontally?
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 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 .
Problem 20 Alex is curious as to whether or not there is a connection between students who walk to school and students who bring their own lunches. Alex surveys 120 students at his school and records his findings in the following table. Walk to School
Take Bus/Get a Ride
Total
Bring Lunch
15
25
40
Buy School Lunch
15
65
80
Column Total
30
90
120
What percentage of the 120 students surveyed bring their own lunch to school? What percentage of the 30 students who walk to school bring their own lunches? What percentage of the 90 students who take the bus or get a ride bring their lunch to school? Are the students who walk to school more likely to bring their own lunch than the students who take the bus or get a ride to school?
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 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 .
Page 48 © R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .
Part B On Your Own: Solving Mathematical Problems In Part B, students solve each problem on their own using the 5-Step Problem-Solving Plan.
Page 49 Š R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .
Directions: Answer questions 1-40. Use the 5-Step Problem-Solving Plan. Be sure to show all of your work. If you need more room, use a separate sheet of paper. Remember to check your answers.
Problem 1 Marilyn goes to the grocery store with 2 coupons. One coupon allows her to get 2 free boxes of cereal and the other coupon allows her to take 50¢ off her cereal purchase. Since cereal costs $3 a box, Marilyn uses the equation below to describe the price, p, of x boxes of cereal. p = 3(x − 2) − 0.50 However, when Marilyn is checking out, the cashier uses the formula below to calculate the price of her cereal purchase. 24 p = 3x – 11 Can you use both of these formulas to calculate the price of Marilyn’s cereal purchase, or did the cashier make a mistake?
Page 50 © R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .
Problem 2 Chloe and her friends go shopping at the mall. Together, they buy 3 bracelets and 4 tubes of lip gloss for a total cost of $36. The cost of a bracelet is $3 less than twice the cost of a tube of lip gloss. What is the cost of a bracelet? What is the cost of a tube of lip gloss?
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 The human body is home to many good bacteria that help the body function normally. There are approximately 1 Ă— 1014 bacteria cells in the human body. In fact, there are so many bacteria in the human body that they account for 2% of the weight of the average human. If the average weight of a human is 180 pounds, about how much does an average bacteria cell weigh? Express your answer in scientific notation.
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 Chemists often use a unit called a mole. A mole is the number of atoms in 12 grams of carbon. Since atoms are very small, there are 6 Ă— 1023 atoms in 12 grams of carbon. How many atoms are in 1 gram of carbon? How many atoms are in 1 kilogram of carbon?
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 Kirk and Mia both have summer jobs. Kirk scoops ice cream at an ice cream parlor. He makes a graph of the hours he works and the money that he makes, as shown below.
Money Earned (dollars)
150
125
100
75
50
25
0
2.5
5
7.5
10
12.5
15
17.5
20
Hours Worked
Mia works at a shoe store where she gets paid a salary and a commission. She describes her wages for last week with the equation y = 6.50x + 30, where x represents the hours she worked and y represents the money she earned. If Kirk and Mia each worked 20 hours last week, then who made more money? Who earns more money per hour?
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 Kimberly decides to rent a bike for the day. She goes to the bike rental stand and sees the following graph that describes the cost per minute to rent a bike.
48 42
Cost (dollars)
36 30 24 18 12 6 0
15
30
45
60
75
90
105 120 135 150 165 180
Minutes
Use the points (0, 12) and (30, 18) to calculate the slope of the line. Then, use the points (0, 12) and (60, 24) to calculate the slope of the line. Show that these slopes are the same by using the length between each pair of points as the hypotenuse of a right triangle. Use the Pythagorean Theorem to find the length of each hypotenuse. Write an equation of the line displayed on the graph.
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 Andrea has a square amulet made out of 4 right triangles, as shown in the sketch below. She wants to know the total area of the amulet including the hole in the center.
3 cm
4 cm
Andrea knows that the 4 right triangles that make up the frame of the amulet all have a base of 3 centimeters and a height of 4 centimeters. She also knows that the center square of the amulet measures 1 centimeter by 1 centimeter, because its length is the difference of the 2 legs of the right triangles. Find the total area of the amulet by adding the area of the center square and the area of the 4 right triangles. What is the length of a side of the square amulet?
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 As a science project on Monarch butterflies, Henry’s class did a survey to see where the Monarch’s main food supply, the milkweed plant, is located, and how far the butterflies would need to fly in order to find food. The students find out that 2 houses have the milkweed plant in their backyards, and that the school also has a small butterfly garden with the milkweed plant. Henry makes the map shown below of the locations of the milkweed plants. Each square on the map corresponds to 1 square mile. What is the distance from the school to each of the houses? Which house is located closer to the school?
14 1st Milkweed plant location
12
Miles North
10
2nd Milkweed plant location
8 6 4 Henry’s School
2 0
2
4
6
8
10
12
14
16
18
20
22
Miles East
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 Dylan collects information on the study habits of 54 8th graders. He asks each of them how many hours per week they spend studying and how many hours per week they spend watching television. He then plots the data in the graph below.
Hours Watching Television
40
30
20
10
10
20 Hours Studying
30
Dylan finds that the equation of the line of best fit for the data that is displayed on scatter plot is y = −x + 42.5. Based on the data shown, is there a positive or negative correlation between watching television and studying? How many hours of television per week does the model predict for a student who studies 20 hours per week?
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 Fred is collecting data on who prefers MP3s and who prefers CDs. He stands outside of a coffee shop on a Saturday and asks 80 people for their preference and their age. He records his findings in the table shown below. Prefers MP3s
Prefers CDs
Total
Under 25
25
5
30
25 or Older
35
15
50
Column Total
60
20
80
What percentage of the 80 people prefers MP3s? Of the 50 people who are age 25 or older, what percentage prefers MP3s? Of the 30 people who are under age 25, what percentage prefers MP3s? Based on Fred’s data, do younger people prefer MP3s more than older people do?
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 Raphael wants to put his money in a savings account and is comparing two different banks. The Bank of Yonkers gives him a chart that shows him how much interest he will earn on various account balances (the interest rate is constant).
Bank of Yonkers Account Balance (Dollars)
Interest Earned (Dollars)
10
0.20
15
0.30
20
0.40
25
0.50
Instead of a chart, the Bank of Brooklyn gives him an equation to describe the amount of interest he will earn. The equation is y = 0.03x, where y represents the amount of interest and x represents the account balance. Which bank pays more in interest?
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 Michael is making snowballs of various sizes and is curious about their volume. To measure the volume, V, of a sphere, he uses the formula 4 V = ( ) r 3, where r represents the radius of the sphere. He made 3 3 snowballs with different sized radii: 2 inches, 3 inches, and 4 inches. Use the formula to find the volume of his 3 snowballs. Is the volume formula for a sphere a linear function? For this problem use the approximation 3.14, and round your answers to 2 decimal places.
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 Amelia is walking to the pharmacy. On the way to the pharmacy, she runs into her friend Gretchen. They stop and talk for a few minutes, and then walk to the pharmacy together. Below is a graph of their progress. The x-axis represents the time in minutes that Amelia spends on her trip to the pharmacy, and the y-axis represents her distance in miles from home. 1.25
Distance (miles)
1
0.75
0.5
0.25
0
5
10
15
20
25
30
Time (minutes)
What was Amelia’s rate in miles per minute before she ran into Gretchen? What was her rate in miles per minute when she was walking to the store with Gretchen? What was the percentage decrease in her rate after meeting up with Gretchen?
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 Kacey goes to a museum and sees 2 paintings next to each other that are both shaped like parallelograms. A sketch of what she saw is shown below. 3 feet
3 feet 120°
60°
60°
120°
1.5 feet 60°
1.5 feet 120°
120°
60°
Are the paintings congruent? If so, what rotations and/or reflections will take one painting to the other?
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 .
Problem 15 Armin had to design a flag for an art project. He drew the sketch of the flag that is made from three similar triangles on graph paper as shown below. Find the geometric transformation that takes the green triangle to the gray triangle. Then, find the transformation that takes the gray triangle to the black triangle. y 16
12
8
4
0
5
10
15
20
25
30
x
–4
–8
–12
–16
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 Mariel is staring at the rows of tiles on the wall in her math classroom. She notices that the tiles are all parallelograms. She sketches the pattern below in her notebook.
4(x – 50)° x°
Let x represent the measure of the smaller angle of the parallelograms. If the larger angle of the parallelograms is 4(x − 50), then what is the measure of the smaller angle of the parallelograms?
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 .
Problem 17 Kristina goes to the greenhouse with her mother. They buy 4 bags of garden soil and 2 bags of fertilizer. When her mom is checking out, the cashier asks if they would like help carrying the 132 pounds of bagged garden products to their car. If a bag of garden soil weighs 1 pound less than 8 times the weight of a bag of fertilizer, how much does a bag of fertilizer and a bag of garden soil each weigh?
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 Gail and her brother, Andy, combined their money to buy a birthday cake for their mother. The cake cost $21. Let x represent the amount that Andy pays. Let 4(x − 5.95) represent the amount Gail pays. How much did each of them pay? Express your answers as decimals.
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 Akako is at the arcade and wants to buy a life-sized stuffed panda that costs 210 tickets. So far she has received 20 tickets from playing a skeeball game. If she receives an average of 9.5 tickets for each new game of skeeball that she plays, then how many games does she need to play before she has enough tickets to buy the stuffed panda? Write equations for the two lines in Akako’s graph, and find the solution using the graph.
200
Tickets Earned
150
100
50
0
5
10
15
20
Games Played
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 Malik is building a ladder to go up to the tree house in his backyard. He knows that the tree house is 13 feet above the ground and wants to place his ladder on the ground 4 feet away from the base of the tree. Use the Pythagorean Theorem to find the total length of the ladder. Between what two whole numbers is the length of the ladder?
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 For her middle school graduation, Rashida receives some gift money. She wants to invest in some stocks instead of putting the money in a savings account. She is considering two companies and wants to invest in the one that has the higher rate of return on its expenses. The first company, Pizza Inc., provides her with the following information:
Pizza Inc. Expenses (Thousands of Dollars)
Earnings (Thousands of Dollars)
1,000
1,200
2,000
2,400
3,000
3,600
4,000
4,800
The second company, Veggieburger Corp., describes its rate of return with the equation y = 1.15x, where y represents the earnings in thousands of dollars and x represents the expenses in thousands of dollars. Which company has a higher rate of return?
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 For Earth Day, Jordan plants a sapling maple tree in his family’s front yard. To help support the sapling as it grows, he ties twine to the trunk of the tree and stakes it to the ground a few inches away from the base of the tree. The twine is 3 feet long and is tied to the ground 1 foot from the base of the tree. Use the Pythagorean Theorem to find the height of the point on the trunk where the twine is fastened to the tree. Approximate your answer to the nearest tenth.
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 Beach volleyball is a version of indoor volleyball played on sand. There are 3 × 107 grains of sand in each cubic foot of sand, and there are 2 × 103 cubic feet of sand in a beach volleyball court. How many grains of sand are there in a beach volleyball court? If there are 3 × 108 people in the United States, what is the ratio of the number of grains of sand in a beach volleyball court to the U.S. population?
Page 72 © 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 The sun is the closest star to Earth. However, the sun is 9.3 Ă— 107 miles away from our planet. If you wanted to travel from Earth to the sun in a year, then what would your average speed have to be in miles per hour? Express your answer in scientific notation rounded to two decimal places.
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 Jeff went for a long walk on a Saturday. His brother, Daniel, started walking after Jeff had walked 1.13 miles and tried to catch up with him, following the same path. Daniel knows that he walks 4 miles per hour. He graphed Jeff’s progress relative to his own walk, as shown below. What was Jeff’s walking rate? When will Daniel catch up to Jeff?
12 10
Miles
8 6 4 2 0
0.25 0.5 0.75
1
1.25 1.5 1.75
2
2.25 2.5 2.75
3
Hours
Page 74 © 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 Olivia is trying to decide if she wants to join a monthly movie discount program. The club’s membership pamphlet has the following graph that describes its costs.
80 72 64
Cost (dollars)
56 48 40 32 24 16 8 0
1
2
3
4
5
6
7
8
9
10
Movies Viewed Each Month
Use the points (0, 22) and (1, 28) to calculate the slope of the line to find the cost per movie for someone who is a member of the discount program. Then, use the points (0, 22) and (3, 40) to calculate the slope of the line. Show that these slopes are the same by using the length between each pair of points as the hypotenuse of a right triangle. Then, write an equation of the line displayed on the movie discount program’s graph.
Page 75 © 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 Clara goes to the supermarket and buys 3 eggplants and 5 acorn squash. When she is checking out the cashier tells her that she bought a total of 1 1 7 pounds of vegetables. If an acorn squash weighs of a pound less 2 2 than 3 times the weight of an eggplant, then how much does a single eggplant and a single acorn squash weigh?
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 Two friends, Ron and Peyton, go on a road trip and are splitting the $57 cost of gasoline. Since Ron did most of the driving, he feels that he should pay less than Peyton. Let x represent the amount that Ron pays. An expression for the amount that Peyton pays is 7(x − 11). How much did Ron and Peyton each pay?
Page 77 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 29 Julie is saving money to buy a bike that costs $220. So far she has saved $22. Each week, she saves $8.25 from her allowance to buy the bike. She makes a graph to figure out how many weeks she will need to save before she can afford to buy the bike. Write equations for the two lines on the graph, and then find the solution using the graph.
Money Saved (dollars)
200
150
100
50
0
5
10
15 Weeks
20
25
Page 78 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 30 Carter’s family goes out for ice cream. His mother, father, and sister order ice cream cones. Carter and his brother order sundaes. The cost of a sundae is $2 less than 3 times the cost of an ice cream cone. If the total bill is $29.75, how much does a single cone cost and how much does a single sundae cost?
Page 79 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 31 3 Stephanie bought fabric to make drapes for her room. She bought 7 feet 4 of fabric. Some of the fabric is white and some of it is checkered. The 1 amount of white fabric she bought is of a foot less than 5 times the 4 amount of checkered fabric she bought. How many feet of checkered fabric did she buy? Express your answer as a decimal.
Page 80 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 32 Tim is at the doctor’s office and notices that there are different sizes of paper cones to use as disposable water cups. He is curious about the volume of each of these cups. He knows that the volume, V, of a cone 1 is given by the formula V = ( ) r 2h, where r represents the radius of 3 the cone, and h represents the height of the cone. While all of the paper cones are 12 centimeters tall, they have three different possible radii: 3 centimeters, 4 centimeters, and 5 centimeters. Use the formula for the volume of a cone to find the volume of the three cones. Is the volume of a cone with a height of 12 centimeters a linear function? For this problem, use the approximation 3.14.
Page 81 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 33 Manuel went on a hike last week. He hiked for 90 minutes before taking a 30-minute break and then resumed hiking at a new pace. He made the following graph of his hiking progress.
Distance (miles)
7 6 5 4 3 2 1 0
10
20
30
40
50
60
70
80
90 100 110 120 130 140 150 160 170 180
Time (minutes)
What was Manuel’s hiking rate, in miles per minute, before his break? What was his hiking rate, in miles per minute, after his break? What was the percentage decrease in his hiking rate after his break?
Page 82 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 34 Heather goes to a skate park and notices two ramps in the shape of right triangles. A sketch of what she saw is shown below.
5 ft
5 ft
12 ft
13 ft
Are the ramps that Heather saw congruent? Explain your answer.
Page 83 Š R A L LY ! E D U C AT I O N . N O PA R T O F T H I S D O C U M E N T M AY B E R E P R O D U C E D W I T H O U T W R I T T E N P E R M I S S I O N O F T H E P U B L I S H E R .
Problem 35 Paul made a sketch of some buildings near his school on graph paper, as shown below. When he finished, he noticed that they were all similar rectangles. y 20
15
10
5
–15
–10
–5
0
5
10
15
x
Find the geometric transformation that takes the green rectangle to the gray rectangle. Then, find the transformation that takes the gray rectangle to the black rectangle. Write an equation for the transformation that takes each (x, y) coordinate in the green rectangle to the gray rectangle. Then, write an equation for the transformation that takes the gray rectangle to the black rectangle.
Page 84 © 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 36 Rebecca’s apartment building is made out of straight rows of bricks that are identical isosceles trapezoids, as shown below. Let x represent the measure of the larger angle in the trapezoids. If the measure of the smaller 1 angle is (x + 12), then what is the measure of the larger angle of the 3 trapezoids?
x°
1 (x + 12)° 3
Page 85 © 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 37 Sofia’s school is laid out like the diagram below. The largest square represents the main building which contains the classrooms. The courtyard is in the center and is in the shape of a right triangle. The smallest square represents the auditorium and the medium-sized square represents the gym. Is the area of the main school building larger or smaller than the area of the auditorium and the gym combined?
170 yards
150 yards
80 yards
Page 86 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 38 Jamal and Brittany are both going to visit their friend Erin. They make a map of the locations of their three houses, as shown below. On their map, 1 square corresponds to one 1 square mile. Using a straight line, how far is each of their houses from Erin’s house? Whose house is closer to Erin’s house?
14 Brittany’s House
12
Miles North
10
Jamal’s House
8 6 4 Erin’s House
2 0
2
4
6
8
10
12
14
16
18
20
22
Miles East
Page 87 © 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 39 Drew collects data about Chinese food deliveries from 36 people to investigate the connection between the delivery time and price of the food being delivered. He uses the x-axis for the price of the food that is being delivered and he uses the y-axis for the time that it takes for the food to be delivered.
Delivery Time (minutes)
50
40
30
20
25.0
37.5
50
62.5
Price of food ($)
Drew finds that the equation of the line of best fit for the data presented in the scatter plot is y = 0.8x + 1.2. Is there a positive or negative correlation between the delivery time and the price of the food ordered? How long will it take to receive a $50 order according to this model?
Page 88 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 40 Kwan is curious as to whether or not there is a connection between students who are vegetarians and those who ride bikes. To investigate this, he asks 144 students at his school about their eating habits and if they ride bikes. The results of his survey are shown in the table below. Rides a Bike
Doesn’t Ride A Bike
Total
Vegetarian
18
6
24
Not Vegetarian
78
42
120
Column Total
96
48
144
What percentage of the 144 students surveyed are vegetarians? What percentage of the 96 bicyclists are vegetarians? What percentage of the 48 students who don’t ride a bike are vegetarians? Based on Kwan’s survey, are bicyclists or non-bicyclists more likely to be vegetarians?
Page 89 © 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 90 © 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 91 © 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 92 © 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 .
To be successful in mathematics, students must become good problem solvers. Using step-bystep instruction, Understanding Complex Math Problems teaches students problem-solving skills and shows them how to apply those skills to different types of math problems. Each item in this book aligns to the new TEKS. In Part A of this book, students will learn to follow a 5-Step Problem Solving Plan. Step 1: IDENTIFY: What are you being asked to find? Step 2: FIND: What do you need to solve the problem? Step 3: CHOOSE: How will you solve the problem? Step 4: SOLVE: Solve the problem. Step 5: CHECK and JUSTIFY: Check and justify your answer. In Part B of this book, students will solve complex math problems on their own using the same 5-Step Problem-Solving Plan they have learned in Part A. When students finish Understanding Complex Math Problems they will be better problem solvers and test takers.
www.rallyeducation.com 22 Railroad Avenue, Glen Head, NY 11545