Math Solutions Grade 7

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Open-Ended Math Problem Solving

………….. Math Solutions Grade Levels 3–8

• Provides instruction for answering open-ended math problems on tests.

• Students are taught to use a 5-Step Problem-Solving Plan when answering open-ended math questions.

Level Grade Level 25-Pack 100-Pack

C 3 4114-7 4115-4 D 4 4117-8 4118-5 E 5 4120-8 4121-5 F 6 4123-9 4124-6 G 7 4126-0 4127-7 H 8 4129-1 4130-7

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2 © RALLY! EDUCATION. NO PART OF THIS DOCUMENT MAY BE REPRODUCED WITHOUT WRITTEN PERMISSION OF THE PUBLISHER.

Math Solutions: Open-Ended Problem Solving • Level G

AcknowledgmentsExecutive Editor: Amy CollinsProject Manager: Edward NaselloEditors: Jeff Brannon

Linda LiDestri

ISBN 978-1-4204-4125-3R 4125-3

Copyright ©2008 RALLY! EDUCATION. All rights reserved. No part of the material protected by this copyright may bereproduced in any form by any means, electronic or mechanical, including photocopying, recording, or by any informationstorage and retrieval system, without permission in writing from the copyright owner. Printed in the U.S.A.

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Table of Contents

INTRODUCTION 4

How to Use This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4A Helpful List of Mathematical Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5How to Answer Short-Response and Extended-Response Questions . . . . . . . 6Mathematics Reference Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

INSTRUCTION 8

The 5-Step Problem-Solving Plan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

ON YOUR OWN 12

Number Sense and Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Number Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Equations and Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Units of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Statistics and Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Organization and Display of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Predictions from Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

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INTRODUCTIONTo the TeacherMath Solutions: Open-Ended Problem Solving provides instructional practice in answering open-endedmath problems for both short-response and extended-response questions. This book contains the sametypes of questions students are likely to encounter on tests. Math problems are formatted so thatstudents will learn to solve problems using a logical, organized plan.

How to Use This BookWe have arranged the problems in this book by math strands. The questions for each strand have beengrouped by mathematical-concept bands. This format allows teachers and students to emphasize thoseareas where the most practice is needed. In this book, the teacher has an opportunity to model problemsolving and guide students using a 5-Step Problem-Solving Plan. This book may also be used to providestudents with an opportunity to work on problems independently. The teacher can then review withthe students each problem and the strategies used by students to solve these problems. In addition,students do not need to solve these problems in the order presented here. The students (or teacher) canselect the specific strands they want to cover and proceed in any order.

Follow These Steps to Use This Book:Step 1: Model the 5-Step Problem-Solving Plan.Step 2: Choose a strand to study.Step 3: Choose a band to work on or review.Step 4: Have students solve a problem.Step 5: Review the problem and the solution.Step 6: Do another problem.

Math problems are formatted so that students can use a variety of problem-solving strategies inconjunction with the 5-Step Problem-Solving Plan. This teaches students to answer problems using alogical, organized, and thoughtful approach. This will help students answer problems even when a factscenario is presented in a new or unusual way. Finally, students will have had an opportunity to learn,perform, and practice the kinds of problems that are on actual math tests. All of this is designed so thatyour students will be successful test takers.

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A Helpful List of Mathematical Language (Grade 7)Listed below are some commonly used mathematical terms and expressions that 7th Grade studentsshould be familiar with. You may want to review this list with your students.

Number Sense and Operationscoefficientcounting numbersexponentsintegersintegralirrational numbersmultiplenatural numbersnon-perfect squaresperfect squarepi (≠)prime factorizationrational numbersreal numbersscientific notationsquare rootwhole numbers

Algebrabinomialequationevaluatefunctioninequalityinput valueslike termsmonomialspolynomialtable of valuestrinomial

Geometryadjacentcircumferencediameterexterior anglehypotenuseinterior anglespolygonPythagorean theoremradiusright angle

Measurement

central anglecustomary units of massexchange rate tablemap scalemetric units of massprotractorrelative error

Statistics and Probability

circle graphconductdouble bar graphdouble line graphsmeasures of central tendencyrangevalidity of sample methods

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To the Student

How to Answer Short-Response and Extended-Response Questions:Use the 5-Step Problem-Solving PlanWhen solving math problems, you need to follow a plan. This helps you to organize information andchoose a strategy to solve the problem. Follow these five steps to help you make your plan.

Step 1. Decide what you are being asked to do.Step 2. Find what you need to know.Step 3. Choose a strategy for solving the problem.Step 4. Solve the problem.Step 5. Check your solution and answer.

Use a Problem-Solving StrategyWhen solving problems throughout this book, you will need to use different problem-solving strategies.Sometimes more than one strategy can be used to solve a problem. Knowing how to choose the beststrategy for solving a problem will make you successful in math. The following is a list of somestrategies you may want to use.

• Draw a Picture• Organize—Make a Chart or List, Graph, Sort Data• Work Backwards• Use Manipulatives or Act Out• Compute—Write a Number Sentence, Use Ratio or Proportion• Use a Formula or Rule• Guess and Check• Make It Simpler• Make Estimates• Look for a Pattern

Helpful Tips for Answering Questions in This Book:• You can use your calculator to help solve any of the questions.

• You can use the Mathematics Reference Sheet to help solve some questions.

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Mathematics Reference SheetFORMULAS

rd

s

r

h

hl

w

h

l

s

w

CONVERSIONS

1 centimeter � 10 millimeters

1 meter � 100 centimeters � 1,000 millimeters

1 kilometer � 1,000 meters

1 gram � 1,000 milligrams

1 kilogram � 1,000 grams

1 pound � 16 ounces

1 ton � 2,000 pounds

1 cup � 8 fluid ounces

1 pint � 2 cups

1 quart � 2 pints

1 gallon � 4 quarts

1 liter � 1,000 milliliters

1 kiloliter � 1,000 liters

Circle Area � � r2

Circumference � 2�r

Cube Total Surface Area � 6s2

Volume � s3

Right Circular Cylinder Total Surface Area � 2�rh � 2�r2

Volume � �r2h

Right Rectangular Prism Total Surface Area � 2wl � 2 lh � 2wh

Volume � lwh

Right Triangular Prism Total Surface Area � wh � lw � lh � ls

Volume � �12

�wh � l

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INSTRUCTION

The 5-Step Problem-Solving PlanDirections: Use the 5-Step Problem-Solving Plan to help you solve Question A.

QUESTION A

Charles must pack a lunch to bring on the school picnic. He must choose one sandwich, one snack,and one drink. The chart below shows all of his choices.

Lunch Choices

How many lunch combinations does Charles have to choose from?

Step 1: Decide what you are being asked to do.

This problem asks you to determine the number of possible lunch combinations Charles hasto choose from.

Step 2: Find what you need to know.

You need the following information to solve this problem:

There are 2 sandwiches to choose from (ham & cheese, peanut butter & jelly).There are 2 snacks to choose from (fruit, crackers).There are 2 drinks to choose from (milk, juice).

Step 3: Choose a strategy for solving the problem.

In order to solve problems that ask “how many combinations,” it is helpful to use the problem-solvingstrategy called Organize: Make a Chart or List, Graph, Sort Data.

Drink

milk

juice

Snack

fruit

crackers

Sandwich

ham & cheese

peanut butter & jelly

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Step 4: Solve the problem.

In the space below, make a tree diagram or a list of all the possible lunch combinations.Count how many lunch combinations are possible.

Step 5: Check your solution and answer.

Check all of the lunch combinations in your tree diagram or list. Make sure all of the possiblecombinations appear once and that no combination appears more than once.


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Directions: Use the 5-Step Problem-Solving Plan to solve Question B on your own.

QUESTION B

Tashi needs to choose what to wear to her friend’s house. She must choose one pair of shoes, one pairof pants, and one shirt. The chart below shows the different color choices she has to choose from foreach article of clothing.

Clothes Tashi Can Choose From

How many combinations of shoes, pants, and shirts does Tashi have to choose from?




Shirts

yellow

green

Pants

black

gray

brown

Shoes

blue

white

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ON YOUR OWN

Strand: Number Sense and Operations

Band: Number Systems . . . . . . . . . . . . . . . . . . 13

Band: Operations . . . . . . . . . . . . . . . . . . . . . . . 17

The 5-Step Problem-Solving Plan






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Band: Number Systems

The 5-Step Plan: 1. Decide 2. Find 3. Choose 4. Solve 5. Check


1 The table below shows geographic information about Africa.

Geography of Africa

Write the numbers, in standard form, for the lowest elevation and the highest elevationof Africa.

Answer lowest elevation meters

highest elevation meters

�1.56 � 102 meters

5.895 � 103 meters

Lowest Elevation

Highest Elevation

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2 Look at the table below. It shows approximate measurements for the highest and lowestpoints on Earth.

Earth Elevations

Write the elevations in standard form.

Answer highest point feet

lowest point feet

Highest Point

Lowest Point

Location

Mt. Everest, Nepal

Dead Sea, Israel/Jordan

Elevation (compared to sea level)

2.9 � 104 feet

�1.4 � 103 feet

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3 The table below shows the average distance between Earth and its moon and the averagedistance between Earth and the sun.

Distance from Earth

How much greater, in kilometers, is the distance from Earth to the sun than the distancefrom Earth to the moon? Write your answer in standard form.

Show your work.

Answer kilometers

Name

Moon

Sun

Average Distance

3.844 � 105 kilometers

1.49 � 108 kilometers


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4 The table shows the distance that each of the first 5 planets is from the sun.

Distance of Planets from the Sun

In standard form, write approximately how much farther Mars is from the sun than Earth isfrom the sun.

Show your work.

Answer miles

Planet

Mercury

Venus

Earth

Mars

Jupiter

Distance from the Sun (miles)

3.59 � 107

6.72 � 107

9.29 � 107

1.42 � 108

4.84 � 108

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5 The Broadway Middle School PTA conducted a book sale to raise money for the musicprogram. They sold 115 fiction books, 67 biographies, 91 travel books, and 36 hobbybooks. Fiction books were sold for $2 each, biographies were $4 each, and travel bookswere $5 each. The PTA raised $1,169 at the book sale.

Part AEach hobby book was sold for the same price. What was the price of each hobby book?

Show your work.

Answer $

Part BWhat is the difference between the amount of money raised from selling the type of bookthat generated the most money in sales and the type of book that generated the leastamount of money in sales?

Show your work.

Answer $


Band: Operations



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6 Jacob’s scout troop sold boxes of greetings cards to raise money for their camping trips.Jacob sold 75 boxes, Frank sold 86 boxes, Dashawn sold 51 boxes, and Harjinder sold62 boxes. In all, these 4 campers raised $3,562 by selling greetings cards.

Part AIf each box of greetings cards was sold for the same price, what was the cost of one box?

Show your work.

Answer $

Part BWhat was the amount raised by the 2 top selling scouts?

Show your work.

Answer $

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7 The lowest temperature ever recorded in Buffalo, New York, was �20°F (Fahrenheit) inFebruary, 1961. The average low temperature in Buffalo, New York, for the month ofFebruary is 19°F.

Part AOn the number line below, plot the lowest recorded temperature and the averagelow temperature in Buffalo, New York, for the month of February. Be sure to label bothpoints appropriately.

Part BHow many degrees colder was the coldest day in February than the average lowtemperature for February?

Show your work.

Answer °F

220222224 218 216 214 212 210 28 26 24 22 0 2 4 6 8 10 12 14 16 18 20 22 24


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8 Over a period of 26 years, the average high temperature in Antarctica during the month ofMarch was 6°F, and the average low temperature was �4°F.

Part AOn the number line below, plot and label the average high and low temperatures inAntarctica for the month of March during this 26-year period.

Part BHow many degrees colder was the average low temperature than the averagehigh temperature?

Show your work.

Answer °F

210 29 28 27 26 25 24 23 22 21 0 1 2 3 4 5 6 7 8 9 10

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Strand: Algebra

Band: Equations and Inequalities. . . . . . . . . . . 22








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9 Isabel and her friend, Hannah, go to the movies. A movie ticket costs $9. Snacks at themovies cost $3 each. Write an equation Isabel can use to determine the number of snacks, s,she can buy if she has $27 before she buys tickets for herself and her friend.

Equation

Using the equation above, find the number of snacks Isabel can buy.

Show your work.

Answer snacks

Strand: Algebra

Band: Equations and Inequalities


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10 Inez does landscaping in her community. She charges $140 for materials and $15 per hourfor labor. Write an equation that can be used to determine the number of hours, h, sheplans to work for a job she estimated to cost $230.

Equation

Use your equation, to find the number of hours, h, she plans to work on this job.

Show your work.

Answer hours


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Strand: Geometry

Band: Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . 25







Strand: Geometry

Band: Shapes


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11 Brian’s tool chest is shaped like a rectangular prism, as shown below.

[not drawn to scale]

Determine the volume of the tool chest. Round your answer to the nearest tenth.

Show your work.

Answer cubic feet

6 ft.

3.6 ft.

2.4 ft.


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12 Lydia took the measurements of a can of soup and drew a picture of it.


Find the volume of the can of soup to the nearest tenth of a cubic centimeter.

Show your work.

Answer cubic centimeters

7 cm.

11 cm.

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13 Steven uses rust-proof paint to cover the cylindrical tank shown below.


Part AWhat is the surface area of the entire tank to the nearest square foot?

Show your work.

Answer square feet

Part BOne can of rust-proof paint will cover 20 square feet. How many cans of paint must Stevenpurchase to cover the entire surface area of the tank?

Show your work.

Answer cans of paint

h = 10 ft.

r = 4 ft.


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14 Julie ships a present to her grandmother in a box. The dimensions of the box areshown below.


What is the total surface area of the box?

Show your work.

Answer square inches

16 in.

8 in.

10 in.

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Strand: Measurement

Band: Units of Measurement . . . . . . . . . . . . . . 30

Band: Estimation. . . . . . . . . . . . . . . . . . . . . . . . 34









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15 England’s Paradise Amusement Park requires each person to weigh at least 25 kilograms inorder to ride its roller coaster. What is the minimum amount of weight, in grams, a personmust weigh to be allowed to ride the coaster?

Show your work.

Answer grams

Strand: Measurement

Band: Units of Measurement


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16 Brett weighed some pebbles he had found at the beach. He found a brown pebble thatweighed 1,824 milligrams and a pearl-colored pebble that weighed 2.75 grams. How muchmore did the pearl-colored pebble weigh, in milligrams, than the brown pebble?

Show your work.

Answer milligrams



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17 Use your protractor to help you solve this problem.

Maninder was asked to keep track of the amount of time he spent on different activitiesduring a typical day. The chart below shows the amount of time he spent doing eachactivity.

Maninder’s Typical Day

Use your protractor to create a circle graph that properly displays the information shown inthe chart. Be sure to label each section and indicate a title for the graph.

Show your work.

1

2

90

80

70

60

50

40

30

20

10

8 07 0 6 0 5 0 4 0 3 0 2 0

1 0100

110 120 130 140 150 160170

100

110

120

130

140

150

160

170

Activity

Sleeping

School

Homework and Chores

Other Activities

Percent of the Day

30%

25%

15%

30%

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18 Use your protractor to help you solve this problem.

Jason surveyed 120 students in the 7th grade to find out their favorite kinds of books.The table below shows the results of his poll.

7th Graders’ Favorite Books

Create a circle graph to display the data Jason collected. Be sure to label each section on thegraph and to give it a title.

Show your work.

Type of Book

Biography

Mystery

Science Fiction

Other

Number of Students

48

42

18

12

1

2

90

80

70

60

50

40

30

20

10

8 07 0 6 0 5 0 4 0 3 0 2 0

1 0100

110 120 130 140 150 160170

100

110

120

130

140

150

160

170

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19 The fish tank below contains 4 liters of water.

Estimate the capacity of the fish tank when the fish tank is completely filled with water.

Estimation liters

On the lines below, describe how you can estimate the capacity of the fish tank withoutusing a ruler.

Strand: Measurement

Band: Estimation


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20 The Casey family has an above ground pool that is in the process of being filled with water.The diagram shows how much water is in the pool so far. The weight of the water shown is7,200 pounds.

Estimate what the weight of the water would be if the pool was completely filled.

Estimation pounds

On the lines below, describe how you arrived at your estimate without using a ruler orknowing the measurements.


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Strand: Statistics and Probability

Band: Organization and Display of Data. . . . . 37

Band: Predictions from Data . . . . . . . . . . . . . . 43

Band: Probability . . . . . . . . . . . . . . . . . . . . . . . 45








Band: Organization and Display of Data


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21 At the Mather Middle School, there are 46 students in the Computer Club and 39 students inthe Science Club. Of these students, 10 are in both the Computer Club and the Science Club.

Part AUsing the information provided above, complete the Venn diagram. Be sure to indicate atitle, label each circle, and place the appropriate number in each section of the diagram.

Part BWhat is the total number of students who participate in the clubs at Mather Middle School?

Answer students


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22 The list below shows the number of students in Mr. Kelly’s class who participate in bandand chorus.

• A total of 65 students participate in the band.

• A total of 78 students participate in the chorus.

• There are 18 students who participate in both activities.

Part AUse the list above to complete a Venn diagram that shows how many of Mr. Kelly’s studentsparticipate in these activities. Be sure to indicate a title, label each circle, and write theappropriate numbers in each section of the diagram.

Part BWhat is the total number of students who participate in these activities?

Answer students

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23 Vijay was looking at his bank statement to see how his savings has grown. The chart belowshows his savings account balance over the last several months.

Vijay’s Savings Account Balance

Based on the chart, which kind of graph would be most appropriate to display the data?

Answer

Use the space provided below to explain why the graph you chose is best to displaythis data.

Month

January

February

March

April

May

June

Amount

$100

$115

$125

$145

$170

$165


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24 High and low temperatures were recorded for five cities during the month of March.The table shows the average high and low temperatures during the month.

Average High and Low Temperatures

What type of graph would be best to display this data?

Answer

In the space provided, indicate why the graph you chose would be the best graph todisplay this data.

City, State

New York, NY

Miami, FL

Phoenix, AZ

Chicago, IL

Anchorage, AK

Average High Temperature

51° F

76° F

82° F

50° F

41° F

Average Low Temperature

38° F

61° F

46° F

36° F

30° F


25 Larry sells several different newspapers at his newsstand on weekdays. During one week hecompared sales for the Gazette News and the Sun Times. He compiled the sales data andrecorded it in the chart below.

Newspaper Sales

On the grid provided, draw a double-bar graph that properly reflects the data in the table.Remember to provide a title, a key, an appropriate scale, and to label the axes.

Day

Monday

Tuesday

Wednesday

Thursday

Friday

Gazette News

65

58

72

70

65

Sun Times

62

54

65

70

60


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26 A new company recorded the number of visitors to its website. The table below showsthe number of visits each day during the first and second weeks the website was availableto customers.

Visits to Website

On the grid below, create a double-bar graph that shows the number of visits to the websiteeach day during the first week and the second week.

Be sure to

• title the graph• label the axes• provide an appropriate key for the graph• provide a scale for the graph• graph all the data

Day

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

Week 1 (number of visits)

15

16

18

20

25

33

43

Week 2 (number of visits)

55

69

84

99

115

125

134


Band: Predictions from Data


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27 Lloyd recorded the number of sesame bagels sold each day at his store during one week.He graphed the data in two different ways.

Which one of the two graphs is misleading?

Answer

On the lines below, explain why the graph you chose is misleading.

Day of the Week

Day of the Week

Graph A

Graph B

Num

ber

ofSe

sam

eBa

gels

Sold

Num

ber

ofSe

sam

eBa

gels

Sold

0

25

50

100

Sat.Fri.Thur.Wed.Tues.Mon.Sun.

0

10

20

30

40

50

60

70

80

90

100

Sat.Fri.Thur.Wed.Tues.Mon.Sun.


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28 Mrs. Jones and Mr. Smith are managers at the Big A Amusement Park. Each of themrecorded and graphed the park attendance for a period of one week. Their graphs areshown below.

Whose graph is misleading?

Answer

In the space provided below, explain why the graph you chose is misleading.

Day of the Week

1000

1100

1200

1300

900

800

700

15001000

5000

Day of the Week

Mrs. Jones’ Graph Mr. Smith’s Graph

Att

enda

nce

Att

enda

nce

Sat.Fri.Thur.Wed.Tues.Mon.Sun. Sat.Fri.Thur.Wed.Tues.Mon.Sun.


Band: Probability


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29 Ms. Wilson opened a pet store in September. She advertised and worked hard to build asuccessful business. The table below shows the number of pets she sold in her store duringthe first several months she was in business.

Pets Sold in Ms. Wilson’s Store

Between which 2 months did pet sales increase by the greatest amount?

Answer between and

Based on the data in the table, predict pet sales for May and June and explain yourreasoning on the lines provided below.

Month

October

November

December

January

February

March

April

Number of Pets Sold

12

15

18

22

26

33

39


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30 Marisol opened a small restaurant. The table below shows the number of dinners she servedduring the first 5 weeks her restaurant was open.

Dinners Served at Marisol’s Restaurant

Between which 2 weeks did the number of dinners served increase the most?

Answer between and

Based on the information provided in the table, how many dinners should Marisol planto be able to serve during the 7th week? On the lines below, explain how you arrived atyour prediction.

Number of Dinners

210

255

300

352

407

Week

1

2

3

4

5

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Math Solutions Grade 7