Ebook Matematika: Imapact Mathematic 1 UNDUH IMPACT Math 1

Algebra and More

Course 1
impactmath.com

interactive student edition

Algebra and More

Course 1

Algebra and More

Course 1

Developed by
Education Development Center, Inc.
Peter Braunfeld, Ricky Carter, Haim Eshach, Sydney Foster, Susan Janssen,
Phil Lewis, Joan Lukas, Michelle Manes, Cynthia J. Orrell, Melanie Palma,
Faye Nisonoff Ruopp, Daniel Lynn Watt

Glencoe/McGraw-Hill
The algebra content for Impact Mathematics was adapted from the series, Access to Algebra, by Neville Grace,
Jayne Johnston, Barry Kissane, Ian Lowe, and Sue Willis. Permission to adapt this material was obtained from
the publisher, Curriculum Corporation of Level 5, 2 Lonsdale Street, Melbourne, Australia.
Copyright © 2004 by the McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of
America. Except as permitted under the United States Copyright Act, no part of this publication may be
reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without
prior written permission from the publisher.
Send all inquiries to:
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, OH 43240
ISBN: 0-07-860909-7

1 2 3 4 5 6 7 8 9 10 058/055 12 11 10 09 08 07 06 05 04

Impact Mathematics Project Reviewers
Education Development Center appreciates all the feedback from the curriculum specialists and teachers who
participated in review and testing.
Special thanks to:

Peter Braunfeld
Professor of Mathematics Emeritus
University of Illinois

Sherry L. Meier
Assistant Professor of Mathematics
Illinois State University

Judith Roitman
Professor of Mathematics
University of Kansas

Marcie Abramson
Thurston Middle School
Boston, Massachusetts

Alan Dallman
Amherst Middle School
Amherst, Massachusetts

Steven J. Fox
Bendle Middle School
Burton, Michigan

Denise Airola
Fayetteville Public Schools
Fayetteville, Arizona

Sharon DeCarlo
Sudbury Public Schools
Sudbury, Massachusetts

Kenneth L. Goodwin Jr.
Middletown Middle School
Middletown, Delaware

Chadley Anderson
Syracuse Junior High School
Syracuse, Utah

David P. DeLeon
Preston Area School
Lakewood, Pennsylvania

Fred E. Gross
Sudbury Public Schools
Sudbury, Massachusetts

Jeanne A. Arnold
Mead Junior High
Elk Grove Village, Illinois

Jacob J. Dick
Cedar Grove School
Cedar Grove, Wisconsin

Penny Hauben
Murray Avenue School
Huntingdon, Pennsylvania

Joanne J. Astin
Lincoln Middle School
Forrest City, Arkansas

Sharon Ann Dudek
Holabird Middle School
Baltimore, Maryland

Jean Hawkins
James River Day School
Lynchburg, Virginia

Jack Beard
Urbana Junior High
Urbana, Ohio

Cheryl Elisara
Centennial Middle School
Spokane, Washington

Robert Kalac
Butler Junior High
Frombell, Pennsylvania

Chad Cluver
Maroa-Forsyth Junior High
Maroa, Illinois

Patricia Elsroth
Wayne Highlands Middle School
Honesdale, Pennsylvania

Robin S. Kalder
Somers High School
Somers, New York

Robert C. Bieringer
Patchogue-Medford School Dist.
Center Moriches, New York

Dianne Fink
Bell Junior High
San Diego, California

Darrin Kamps
Lucille Umbarge Elementary
Burlington, Washington

Susan Coppleman
Nathaniel H. Wixon Middle School
South Dennis, Massachusetts

Terry Fleenore
E.B. Stanley Middle School
Abingdon, Virginia

Sandra Keller
Middletown Middle School
Middletown, Delaware

Sandi Curtiss
Gateway Middle School
Everett, Washington

Kathleen Forgac
Waring School
Massachusetts

Pat King
Holmes Junior High
Davis, California

v

Kim Lazarus
San Diego Jewish Academy
La Jolla, California

Karen Pizarek
Northern Hills Middle School

Grand Rapids, Michigan

Kathy L. Terwelp
Summit Public Schools
Summit, New Jersey

Ophria Levant
Webber Academy
Calgary, Alberta
Canada

Debbie Ryan
Overbrook Cluster
Philadelphia, Pennsylvania

Laura Sosnoski Tracey
Somerville, Massachusetts

Mary Lundquist
Farmington High School

Farmington, Connecticut

Sue Saunders
Abell Jr. High School
Midland, Texas

Ellen McDonald-Knight
San Diego Unified School District
San Diego, California

Ivy Schram
Massachusetts Department of Youth
Services
Massachusetts

Ann Miller
Castle Rock Middle School
Castle Rock, Colorado

Robert Segall

Windham Public Schools
Willimantic, Connecticut

Julie Mootz
Ecker Hill Middle School
Park City, Utah

Kassandra Segars
Hubert Middle School
Savannah, Georgia

Jeanne Nelson
New Lisbon Junior High
New Lisbon, Wisconsin

Laurie Shappee
Larson Middle School
Troy, Michigan

DeAnne Oakley-Wimbush
Pulaski Middle School
Chester, Pennsylvania

Sandra Silver
Windham Public Schools
Willimantic, Connecticut

Tom Patterson
Ponderosa Jr. High School
Klamath Falls, Oregon

Karen Smith
East Middle School
Braintree, Massachusetts

Maria Peterson
Chenery Middle School
Belmont, Massachusetts

Kim Spillane
Oxford Central School
Oxford, New Jersey

Lonnie Pilar
Tri-County Middle School
Howard City, Michigan

Carol Struchtemeyer
Lexington R-5 Schools
Lexington, Missouri

vi

Marcia Uhls
Truesdale Middle School
Wichita, Kansas
Vendula Vogel
Westridge School for Girls
Pasadena, California
Judith A. Webber
Grand Blanc Middle School
Grand Blanc, Michigan
Sandy Weishaar
Woodland Junior High
Fayetteville, Arkansas
Tamara L. Weiss
Forest Hills Middle School
Forest Hills, Michigan
Kerrin Wertz
Haverford Middle School
Havertown, Pennsylvania
Anthony Williams
Jackie Robinson Middle School
Brooklyn, New York
Deborah Winkler
The Baker School
Brookline, Massachusetts
Lucy Zizka
Best Middle School
Ferndale, Michigan

Chapter One

Chapter Two

All About Patterns . . . . . . . . . . . . .2

All About Numbers . . . . . . . . . . . .74

Lesson 1.1: Looking for Patterns . . . . . . . . . . . . . . .4
Investigation 1: Pascal’s Triangle and Sequences . 5
On Your Own Exercises . . . . . . . . . . . . . . . . . .10

Lesson 2.1: Factors and Multiples . . . . . . . . . . . . . .76
Investigation 1: Factors . . . . . . . . . . . . . . . . . .77
Investigation 2: Prime Numbers . . . . . . . . . . . .79
Investigation 3: Common Factors . . . . . . . . . . . .82
Investigation 4: Multiples . . . . . . . . . . . . . . . . .85
Lab Investigation: A Locker Problem . . . . . . . . .88
On Your Own Exercises . . . . . . . . . . . . . . . . . .90

Lesson 1.2: Following Rules . . . . . . . . . . . . . . . . . .14
Investigation 1: Sequences and Rules . . . . . . . . .15
Investigation 2: Order of Operations . . . . . . . . .19
On Your Own Exercises . . . . . . . . . . . . . . . . . .23
Lesson 1.3: Writing Rules for Patterns . . . . . . . . . .28
Investigation 1: Finding Rules . . . . . . . . . . . . . .29
Investigation 2: Connecting Numbers . . . . . . . . .32
On Your Own Exercises . . . . . . . . . . . . . . . . . .36
Lesson 1.4: Patterns in Geometry . . . . . . . . . . . . .42
Investigation 1: Polygons . . . . . . . . . . . . . . . . .42
Investigation 2: Angles . . . . . . . . . . . . . . . . . . .46
Investigation 3: Classifying Polygons . . . . . . . . .50
Investigation 4: Triangles . . . . . . . . . . . . . . . . .54
Lab Investigation: Polygons to Polyhedra . . . . . .58
On Your Own Exercises . . . . . . . . . . . . . . . . . .61
Review and Self-Assessment . . . . . . . . . . . . . . . . .69

Lesson 2.2: Patterns in Fractions . . . . . . . . . . . . . .96
Investigation 1: Visualizing Fractions . . . . . . . . .97
Investigation 2: Equivalent Fractions . . . . . . . . .99
Investigation 3: Comparing Fractions . . . . . . . .102
Investigation 4: Estimating with Fractions . . . .104
On Your Own Exercises . . . . . . . . . . . . . . . . .106
Lesson 2.3: Patterns in Decimals . . . . . . . . . . . . .112
Investigation 1: Understanding Decimals . . . . .113
Investigation 2: Measuring with Decimals . . . . .117
Investigation 3: Comparing and Ordering
Decimals . . . . . . . . . . . . . . . . . . . . . . . . . .120
On Your Own Exercises . . . . . . . . . . . . . . . . .123
Lesson 2.4: Fractions and Decimals . . . . . . . . . . . .128
Investigation 1: Estimating Fraction and
Decimal Equivalents . . . . . . . . . . . . . . . . . .128
Investigation 2: Changing Fractions to Decimals 131
Investigation 3: Patterns in Fractions and
Decimals . . . . . . . . . . . . . . . . . . . . . . . . . .134
On Your Own Exercises . . . . . . . . . . . . . . . . .137
Lesson 2.5: Negative Numbers . . . . . . . . . . . . . . .142
Investigation 1: Understanding Negative
Numbers . . . . . . . . . . . . . . . . . . . . . . . . . .143
On Your Own Exercises . . . . . . . . . . . . . . . . .146
Review and Self-Assessment . . . . . . . . . . . . . . . .148

vii

Chapter Three

Chapter Four

Working with Fractions and
Decimals . . . . . . . . . . . . . . . .152

Making Sense of Percents . . . . . .224

Lesson 3.1: Adding and Subtracting Fractions . . . . .154
Investigation 1: Adding and Subtracting with
Fraction Pieces . . . . . . . . . . . . . . . . . . . . .154
Investigation 2: Adding and Subtracting Using
Common Denominators . . . . . . . . . . . . . . . .157
Investigation 3: Adding and Subtracting
Mixed Numbers . . . . . . . . . . . . . . . . . . . . .161
Lab Investigation: Using a Fraction Calculator . .164
On Your Own Exercises . . . . . . . . . . . . . . . . .166
Lesson 3.2: Multiplying and Dividing
with Fractions . . . . . . . . . . . . . . . . . . . . . . . .172
Investigation 1: Multiplying Fractions and
Whole Numbers . . . . . . . . . . . . . . . . . . . . .172
Investigation 2: A Model for Multiplying
Fractions . . . . . . . . . . . . . . . . . . . . . . . . .175
Investigation 3: More Multiplying with
Fractions . . . . . . . . . . . . . . . . . . . . . . . . .178
Investigation 4: Dividing Whole Numbers
by Fractions . . . . . . . . . . . . . . . . . . . . . . .182
Investigation 5: Dividing Fractions
by Fractions . . . . . . . . . . . . . . . . . . . . . . .185
On Your Own Exercises . . . . . . . . . . . . . . . . .189
Lesson 3.3: Multiplying and Dividing with Decimals 198
Investigation 1: Multiplying Whole Numbers
and Decimals . . . . . . . . . . . . . . . . . . . . . . .198
Investigation 2: Multiplying Decimals as
Fractions . . . . . . . . . . . . . . . . . . . . . . . . .201
Investigation 3: Multiplying Decimals in
Real Life . . . . . . . . . . . . . . . . . . . . . . . . . .204
Investigation 4: Dividing Decimals . . . . . . . . . .207
Investigation 5: Multiplying or Dividing? . . . . .210
On Your Own Exercises . . . . . . . . . . . . . . . . .213
Review and Self-Assessment . . . . . . . . . . . . . . . .221

viii

Lesson 4.1: Using Percents . . . . . . . . . . . . . . . . .226
Investigation 1: Understanding Percents . . . . . .227
Investigation 2: Making a Circle Graph . . . . . . .230
Investigation 3: Parts of Different Wholes . . . .234
Investigation 4: Linking Percents, Fractions,
and Decimals . . . . . . . . . . . . . . . . . . . . . . .236
On Your Own Exercises . . . . . . . . . . . . . . . . .240
Lesson 4.2: Finding a Percent of a Quantity . . . . . .248
Investigation 1: Modeling with a Grid . . . . . . .249
Investigation 2: Using a Shortcut . . . . . . . . . . .252
On Your Own Exercises . . . . . . . . . . . . . . . . .256
Lesson 4.3: Percents and Wholes . . . . . . . . . . . . .260
Investigation 1: Finding the Percent . . . . . . . . .261
Investigation 2: Finding the Whole . . . . . . . . . .264
Lab Investigation: Playing Percent Ball . . . . . . .268
On Your Own Exercises . . . . . . . . . . . . . . . . .270
Review and Self-Assessment . . . . . . . . . . . . . . . . .27

Chapter Five

Chapter SIx

Exploring Graphs . . . . . . . . . . . .276

Analyzing Data . . . . . . . . . . . . .340

Lesson 5.1: Interpreting Graphs . . . . . . . . . . . . . .278
Investigation 1: Using Points to Display
Information . . . . . . . . . . . . . . . . . . . . . . . .278
Investigation 2: Interpreting Points . . . . . . . . .282
Investigation 3: Interpreting Lines and
Curves . . . . . . . . . . . . . . . . . . . . . . . . . . .286
On Your Own Exercises . . . . . . . . . . . . . . . . .292

Lesson 6.1: Using Graphs to Understand Data . . . .342
Investigation 1: Using Line Graphs to Solve
a Mystery . . . . . . . . . . . . . . . . . . . . . . . .343
Investigation 2: Using Bar Graphs to
Analyze Data . . . . . . . . . . . . . . . . . . . . . .346
Investigation 3: Making Histograms . . . . . . . .350
On Your Own Exercises . . . . . . . . . . . . . . . . .353

Lesson 5.2: Drawing and Labeling Graphs . . . . . . .300
Investigation 1: Drawing Your Own Graphs . . .301
Investigation 2: Plotting Points . . . . . . . . . . . .302
Investigation 3: Choosing Scales . . . . . . . . . . .306
On Your Own Exercises . . . . . . . . . . . . . . . . .311

Lesson 6.2: What Is Typical? . . . . . . . . . . . . . . . .362
Investigation 1: Mode and Median . . . . . . . . . .362
Investigation 2: Stem-and-Leaf Plots . . . . . . . .366
Investigation 3: The Meaning of Mean . . . . . . .370
Investigation 4: Mean or Median? . . . . . . . . . .373
Investigation 5: Analyzing Data . . . . . . . . . . . .377
On Your Own Exercises . . . . . . . . . . . . . . . . .380

Lesson 5.3: Using Graphs to Find Relationships . . .316
Investigation 1: Looking for a Connection . . . . .317
Investigation 2: How Some Things Grow . . . . .319
Investigation 3: Making Predictions
From Graphs . . . . . . . . . . . . . . . . . . . . . . .320
Lab Investigation: Graphing with
Spreadsheets . . . . . . . . . . . . . . . . . . . . . .324
On Your Own Exercises . . . . . . . . . . . . . . . . .327

Lesson 6.3: Collecting and Analyzing Data . . . . . . .390
Investigation 1: Planning Your Analysis . . . . . .391
Investigation 2: Carrying Out Your Analysis . . .394
Lab Investigation: Statistics and Spreadsheets . .396
On Your Own Exercises . . . . . . . . . . . . . . . . .399
Review and Self-Assessment . . . . . . . . . . . . . . . .402

Review and Self-Assessment . . . . . . . . . . . . . . . .336

ix

Chapter Seven

Chapter Eight

Variables and Rules . . . . . . . . . .408

Geometry and Measurement . . . .464

Lesson 7.1: Patterns and Variables . . . . . . . . . . . .410
Investigation 1: Sequences, Rules, and
Variables . . . . . . . . . . . . . . . . . . . . . . . . .411
Investigation 2: Are These Rules the Same? . . .414
Investigation 3: What’s My Rule? . . . . . . . . . .419
On Your Own Exercises . . . . . . . . . . . . . . . . .422

Lesson 8.1: Angles . . . . . . . . . . . . . . . . . . . . . . .466
Investigation 1: Measuring Angles . . . . . . . . . .467
Investigation 2: Investigating Angle
Relationships . . . . . . . . . . . . . . . . . . . . . .472
On Your Own Exercises . . . . . . . . . . . . . . . . .477

Lesson 7.2: Rules in Real Life . . . . . . . . . . . . . . .430
Investigation 1: Rules in Context . . . . . . . . . . .431
Lab Investigation: Crossing a Bridge . . . . . . . . .433
Investigation 2: Translating Words into
Symbols . . . . . . . . . . . . . . . . . . . . . . . . . .436
Investigation 3: Equivalent Rules . . . . . . . . . . .439
On Your Own Exercises . . . . . . . . . . . . . . . . .443
Lesson 7.3: Explaining Number Relationships . . . . .450
Investigation 1: Think of a Number . . . . . . . . .451
Investigation 2: Consecutive Numbers . . . . . . .455
On Your Own Exercises . . . . . . . . . . . . . . . . .457
Review and Self-Assessment . . . . . . . . . . . . . . . .460

Lesson 8.2: Measuring Around . . . . . . . . . . . . . . .482
Investigation 1: Finding Perimeter . . . . . . . . . .482
Investigation 2: Approximating π . . . . . . . . . .486
On Your Own Exercises . . . . . . . . . . . . . . . . .490
Lesson 8.3: Areas and Squares . . . . . . . . . . . . . .494
Investigation 1: Counting Square Units . . . . . . .495
Investigation 2: Squaring . . . . . . . . . . . . . . . .498
Investigation 3: More about Squaring . . . . . . .501
Investigation 4: Taking Square Roots . . . . . . . .504
On Your Own Exercises . . . . . . . . . . . . . . . . .508
Lesson 8.4: Calculating Areas . . . . . . . . . . . . . . . .514
Investigation 1: Areas of Parallelograms . . . . .515
Investigation 2: Areas of Triangles . . . . . . . . . .518
Investigation 3: Areas of Circles . . . . . . . . . . .522
Lab Investigation: Using a Spreadsheet to
Maximize Area . . . . . . . . . . . . . . . . . . . . .525
On Your Own Exercises . . . . . . . . . . . . . . . . .528
Lesson 8.5: The Pythagorean Theorem . . . . . . . . .536
Investigation 1: Right Triangles and Squares . . .536
Investigation 2: Using the Pythagorean
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . .540
On Your Own Exercises . . . . . . . . . . . . . . . . .544
Review and Self-Assessment . . . . . . . . . . . . . . . .551

x

Chapter Nine

Chapter Ten

Solving Equations . . . . . . . . . . . .556

Understanding Probability . . . . . .602

Lesson 9.1: Understanding Equations . . . . . . . . . .558
Investigation 1: Equations and Inequalities . . . .559
Investigation 2: Equations with Variables . . . . .560
Lab Investigation: Just Undo It! . . . . . . . . . . . .563
On Your Own Exercises . . . . . . . . . . . . . . . . .565

Lesson 10.1: The Language of Chance . . . . . . . . . .604
Investigation 1: Probability in Everyday Life . . .605
Investigation 2: Theoretical Probability . . . . . .608
Lab Investigation: The Spinning Top Game . . . . .613
On Your Own Exercises . . . . . . . . . . . . . . . . .615

Lesson 9.2: Backtracking . . . . . . . . . . . . . . . . . . .570
Investigation 1: Learning to Backtrack . . . . . . .571
Investigation 2: Practicing Backtracking . . . . . .574
Investigation 3: Using Backtracking
to Solve Problems . . . . . . . . . . . . . . . . . . .576
On Your Own Exercises . . . . . . . . . . . . . . . . .579

Lesson 10.2: Analyzing Games . . . . . . . . . . . . . . .620
Investigation 1: Who’s Greater? . . . . . . . . . . .621
Investigation 2: Dice Sums . . . . . . . . . . . . . . .623
Investigation 3: Rolling Differences . . . . . . . . .624
Investigation 4: Geometric Probability . . . . . . .626
On Your Own Exercises . . . . . . . . . . . . . . . . .631

Lesson 9.3: Guess-Check-and-Improve . . . . . . . . .586
Investigation 1: Using Guess-Check-andImprove . . . . . . . . . . . . . . . . . . . . . . . . . .586
Investigation 2: Solving Problems Using
Guess-Check-and-Improve . . . . . . . . . . . . .589
Investigation 3: Choosing a Method . . . . . . . . .592
On Your Own Exercises . . . . . . . . . . . . . . . . .594

Lesson 10.3: Making Matches . . . . . . . . . . . . . . .638
Investigation 1: Matching Colors . . . . . . . . . . .639
Investigation 2: Matching Cards . . . . . . . . . . .643
On Your Own Exercises . . . . . . . . . . . . . . . . .646

Review and Self-Assessment . . . . . . . . . . . . . . . .599

Review and Self-Assessment . . . . . . . . . . . . . . . .652

Glossary . . . . . . . . . . . . . . . . . 656
Index . . . . . . . . . . . . . . . . . . . 666
Photo Credits . . . . . . . . . . . . . . 675

1

All about Patterns
Real-Life Math
A Bee Tree Although a female honeybee has two parents, a male honeybee has
only a mother. The family tree of a male honeybee’s ancestors reveals an interesting pattern of numbers.
M

1 male bee

F

1 parent
F

M
F
M
F M

2 grandparents

M
F

F

F
F

M

M
F

3 great-grandparents
F

F M

5 great-great-grandparents
F

8 great-great-great-grandparents

The numbers of bees in the generations—1, 1, 2, 3, 5,
8, and so on—form a famous list of numbers known
as the Fibonacci sequence.

Think About It Can you discover a pattern in
the family tree or the list of numbers that
will help you find the next two or three
numbers in the Fibonacci sequence?

Family Letter
Dear Student and Family Members,
Our class is about to begin an exciting year of mathematics. Don’t
worry—mathematics is more than just adding and subtracting numbers.
Mathematics has been called the “science of patterns.” Recognizing and
describing patterns and using patterns to make predictions are important
mathematical skills.
We’ll begin by looking for patterns in Pascal’s triangle, a number triangle
containing many patterns.
1
1
1
1
2
1
1
3
3
1
1
4
6
4
1
1
5
10
10
5
1
Can you describe any patterns in the triangle? Try to predict the numbers
in the next row of the table. Don’t worry if you can’t find any patterns yet.
We’ll be learning lots about this triangle in the next few days.
We’ll also be looking for shape patterns. For example, the surface of a
honeycomb, like the one shown here in the background, is made up of a
pattern of hexagons that fit together with no overlaps. Can you make a
similar pattern with squares? How about with triangles?

Vocabulary Along the way, you’ll be learning about these new terms:
angle
concave polygon
line symmetry
order of operations
polygon

regular polygon
sequence
term
triangle inequality
vertex

What can you do at home?
During the next few weeks, your student may show interest in patterns
and rules. Ask him or her to think about common occurrences of patterns
and rules, such as this rule to estimate how many miles you are from a
lightning strike: Count the number of seconds between seeing the lightning
and hearing the thunder, and then divide by 5.

impactmath.com/family_letter

3

Looking for
Patterns
Patterns are everywhere! You can see patterns in wallpaper, fabric, buildings,
flowers, and insects. You can hear patterns in music and song lyrics and
even in the sound of a person’s voice. You can follow patterns to catch a
bus or a train or to locate a store with a particular address.
Patterns are an important part of mathematics. You use them every time
you read a number, perform a mathematical operation, interpret a graph,
or identify a shape. In this lesson, you will search for, describe, and
extend many types of patterns.

Explore
In this diagram, you can begin
at “Start” and trace a path,
following the arrows, to any
of the letters.

Start
A
C

How many different paths are
there from Start to A? Describe
each path.
How many paths are there
from Start to D? Describe each
path.

F
J

B
D

G
K

E
H

L

I
M

N

How many paths are there from Start to G? Describe each path.
There are four paths from Start to K. Describe all four.
Add another row of circles to a copy of the diagram, following the
pattern of arrows and letters. How many paths are there from Start
to S? Describe them.
On a new copy of the diagram, replace each letter with the number
of paths from Start to that letter. For example, replace A with 1 and
K with 4.
The triangle of numbers you just created is quite famous. You will learn
more about the triangle and the patterns it contains in Investigation 1.

4 CHAPTER 1

All about Patterns

Investigation 1

Pascal’s Triangle
and Sequences

The number triangle you created in the Explore has fascinated mathematicians for centuries because of the many patterns it contains. Chinese and
Islamic mathematicians worked with the triangle as early as A.D. 1100.
Blaise Pascal, a French mathematician who studied it in 1653, called it
the arithmetic triangle. It is now known as Pascal’s triangle in his honor.

A

Problem Set

Below is a copy of the diagram you made in the Explore. The word
“Start” has been replaced by the numeral 1, and the rows have been
labeled.
1
The number triangle as it appears
in Precious Mirror of the Four
Elements, written by Chinese
mathematician Chu Shih-Chieh
in 1303

1
1
1
1
1

2
3

4
5

1
1
3
6

10

1
4

10

1
5

1

Row 0
Row 1
Row 2
Row 3
Row 4
Row 5

There are many patterns in this triangle. For example, each row reads the
same forward as it does backward.
1.

Describe as many patterns in the triangle as you can.

2.

To add more rows to the triangle, you could count paths as you did
in the Explore—but that might take a lot of time. Instead, use some
of the patterns you found in Problem 1 to extend the triangle to
Row 7. You may not be able to figure out all the numbers, but fill
in as many as you can.

3.

One way to add new rows to the triangle is to consider how each
number is related to the two numbers just above it to the left and right.
Look at the numbers in Rows 3 and 4. Describe a rule for finding the
numbers in Row 4 from those in Row 3. Does your rule work for
other rows of the triangle as well?

4.

Use your rule from Problem 3 to complete the triangle to Row 9.

LESSON 1.1

Looking for Patterns 5

Pascal’s triangle has many interesting patterns in it. You have probably
worked with other patterns in the form of puzzles like these:
Fill in the blanks.
Puzzle A: 2, 5, 8, 11, __, __ , __
Puzzle B: 16, 8, 4, 2, __, __, __
Puzzle C: 3, 2, 3, 2, __, __, __
Puzzle D: ★, ✻, ★, ✻, __, __, __

V O C A B U L A R Y
sequence
term

To solve these puzzles, you need to find a pattern in the part of the list
given and use it to figure out the next few items. Ordered lists like these
are called sequences. Each item in a sequence is called a term. Terms
may also be referred to as stages.

&

Think Discuss
Here is Puzzle A. Describe a rule you can follow to get from one
term to the next.
2, 5, 8, 11, __, __ , __
According to your rule, what are the next three terms?
Now look at Puzzle B. Describe the pattern you see.
16, 8, 4, 2, __, __, __
According to the pattern you described, what are the next three
terms?
What pattern do you see in Puzzle C: 3, 2, 3, 2, __, __, __?
According to the pattern, what are the next three terms?
Sequences don’t always involve numbers. Look at Puzzle D, for
example.
★, ✻, ★, ✻, __, __, __
Describe the pattern, and give the next three terms.
In Puzzles A and B, each term is found by applying a rule to the term
before it. In Puzzles C and D, the terms follow a repeating pattern. In the
next problem set, you will explore more sequences of both types.

6 CHAPTER 1

All about Patterns

M A T E R I A L S
• toothpicks
(optional)
• counters (optional)

Problem Set
1.

B

The sequences in Parts a–e follow a repeating pattern. Give the next
three terms or stages of each sequence.
a.

b.

3, 6, 9, 3, 6, 9, 3, 6, . . .

c.
d.

7, 1, 1, 7, 1, 1, 7, 1, 1, . . .

1 2 1 2 1 2
e. ᎏ2ᎏ, ᎏ3ᎏ, ᎏ2ᎏ, ᎏ3ᎏ, ᎏ2ᎏ, ᎏ3ᎏ,

2.

...

In Parts a–e, each term in the sequence is found by applying a rule
to the term before it (the preceding term). Give the next three terms
of each sequence.
a.

3, 6, 9, 12, . . .

b.

c.

100, 98.5, 97, . . .

d.

3, 5, 8, 12, . . .

1 1 1 1
e. ᎏ2ᎏ, ᎏ3ᎏ, ᎏ4ᎏ, ᎏ5ᎏ,

...

LESSON 1.1

Looking for Patterns 7

3.

Below are two sequences, one made with toothpicks and the other
with counters. You and your partner should each choose a different
sequence. Do Parts a–c on your own using your sequence.
Sequence A

Sequence B

4.

5.

a.

Make or draw the next three terms of your sequence.

b.

How many toothpicks or counters will be in the tenth term?
Check by making or drawing the tenth term.

c.

Give a number sequence that describes the number of toothpicks
or counters in each term of your pattern.

d.

Compare your answers to Parts a–c with your partner’s. What is
the same about your answers? What is different?

Describe the pattern in each number sequence, and use the pattern to
fill in the missing terms.
a.

5, 12, 19, 26, __ , __ , __

b.

0, 9, 18, 27, __ , __ , __

c.

125, 250, __ , 1,000, __ , __ , 8,000

d.

1, 0.1, __, 0.001, __, 0.00001, __

e.

4, 6, 9, 11, 14, 16, 19, __, __, __

Consider this sequence of symbols:
⌬, ⌬, ⌬, ⍀, ⍀, ⌬, ⌬, ⌬, ⍀, ⍀, ⌬, ⌬, ⌬, ⍀, ⍀, . . .

Just
the

facts

The symbols in Problem
5 are letters of the
Greek alphabet. ⌬ is
the letter delta, and ⍀
is the letter omega.
Greek letters are
used frequently in
physics and advanced
mathematics.
8 CHAPTER 1

a.

If this repeating pattern continues, what will the next six
terms be?

b.

What will the 30th term be?

c.

How could you find the 100th term without drawing 100 symbols?
What will the 100th term be?

All about Patterns

6.

The sequence below is known as the Fibonacci sequence after the
mathematician who studied it. The Fibonacci sequence is interesting
because it appears often in both natural and manufactured things.
1, 1, 2, 3, 5, 8, 13, . . .

Just
the

facts

The Fibonacci
numbers—the numbers
in the sequence—can
be found in the arrangements of leaves and
flowers on plants and
of scales on pine cones
and pineapples.

a.

Study the sequence carefully to see whether you can discover the
pattern. Give the next three terms of the sequence.

b.

Write instructions for continuing the Fibonacci sequence.

&

Share

Summarize
1. The

diagram from the Explore on page 4 is repeated below. How
is Pascal’s triangle related to the number of paths from Start to
each letter in this diagram?
Start
A
D

C
F
J

B

G
K

E
H

L

I
M

N

2. You

discovered that each number in Pascal’s triangle is the sum of
the two numbers just above it. Explain what this means in terms of
the number of paths to a particular letter in the diagram above.

3. Describe

some strategies you use when searching for a pattern in
a sequence.

LESSON 1.1

Looking for Patterns 9

On Your Own Exercises

&
Apply

Practice

1.

Here are the first few rows of Pascal’s triangle:
1
1
1
1

2.

Row 0
Row 1
Row 2
Row 3

1
2

3

1
3

1

a.

How many numbers are in each row shown?

b.

How many numbers are in Row 4? In Row 5? In Row 6?

c.

If you are given a row number, how can you determine how many
numbers are in that row?

d.

In some rows, every number appears twice. Other rows have a
middle number that appears only once. Will Row 10 have a middle number? Will Row 9? How do you know?

A certain row of Pascal’s triangle has 252 as the middle number and
210 just to the right of the middle number.
...

?

?

?

?

252

210

?

?

?

...

a.

What is the number just to the left of the middle number? How do
you know?

b.

What is the middle number two rows later? How do you know?

Describe the pattern in each sequence, and use the pattern to find the next
three terms.
3.

3, 12, 48, 192, __ , __ , __

4.

0.1, 0.4, 0.7, 1.0, __ , __ , __

5.

2, 5, 4, 7, 6, 9, __, __, __

6.

⌬, ⬁, ⌬, ⌬, ⬁, ⌬, ⌬, ⌬, ⬁, __, __, __

7.

⫺5, ⫺4, ⫺3, ⫺2

8.

a, c, e, g, __, __, __

, __ , __, __

Just
the

facts

In mathematics, the
symbol ⌬ represents
the amount of change
in a quantity, and the
symbol ⬁ represents
infinity.
10 C H A P T E R 1

All about Patterns

impactmath.com/self_check_quiz

&
Extend

Connect

9.

Some patterns in Pascal’s triangle appear in unexpected ways. For
example, look at the pattern in the sums of the rows.

1
1 ⴙ
1 ⴙ 4
1 ⴙ 5 ⴙ

10.

1
ⴙ
2
ⴙ
6
ⴙ

1
ⴙ
3
ⴙ
10

1
ⴙ 1
4 ⴙ 1
ⴙ 5 ⴙ 1

Row 0 Sum ⴝ 1
Row 1 Sum ⴝ 2
Row 2 Sum ⴝ 4

a.

Find the sum of each row shown above.

b.

Describe the pattern in the row sums.

The pattern below involves two rows of numbers. If the pattern
were continued, what number would be directly to the right of 98?
Explain how you know.
1

11.

1
ⴙ
3
ⴙ
10

3
2

4

6
5

7

9
8

12
11

10

13

15
14

16

18
17

Look at this pattern of numbers. If it were continued, what number
would be directly below 100?

10

5
11

2
6
12

1
3
7
13

4
8
14

LESSON 1.1

9
15

16

Looking for Patterns 11

12.

For this problem, you may want to draw the shapes on graph paper.
a.

Find the next term in this sequence:

Term 1
b.

Squares in
Bottom Row
1
3

c.

Look at your table carefully. Describe the pattern of numbers in
the second column. Use your pattern to extend the table to show
the number of squares in the bottom rows of Terms 5 and 6.

d.

Predict the number of squares in the bottom row of Term 30.

e.

Now make a table to show the total number of squares in each of
the first five terms.
Term
1
2
3
4
5

In y o u r

own

words

12 C H A P T E R 1

Term 3

This table shows the number of squares in the bottom rows of
Terms 1 and 2. Copy and complete the table to show the number
of squares in the bottom rows of the next two terms.

Term
1
2
3
4

What is a pattern?
Is every sequence
of numbers a
pattern? Is every
sequence of
shapes a pattern?
Explain your
answers.

Term 2

f.

Total Number of Squares
1
4

Look for a pattern in your table from Part e. Use the pattern to
predict the total number of squares in Term 10.

All about Patterns

13.

Imagine that an ant is standing in the square labeled A on the grid
below. The ant can move horizontally or vertically, with each step
taking him one square from where he started.

A

Mixed

Review

Remember
Writing a number in
standard form means
writing it using digits.
For example, standard
form for seventeen is 17.

Remember

a.

On a copy of the grid, color each square (except the center
square) according to the least number of steps it takes the ant to
get there. Use one color for all squares that are one step away,
another color for all squares that are two steps away, and so on.

b.

What shapes are formed by squares of the same color? How
many squares of each color are there? What other patterns do
you notice?

Find each sum or difference without using a calculator.
14.

5,853 ⫺ 788

17.

Write thirty-two thousand, five hundred sixty-three in standard form.

18.

Write fourteen million, three hundred two thousand, two in standard
form.

19.

Write 324 in words.

15.

1,054 ⫹ 1,492

20.

16.

47,745 ⫺ 2,943

Write 12,640 in words.

21. Geometry

Imagine that you
have 12 square tiles, each measuring
1 inch on a side.

a.

In how many different ways can
you put all 12 tiles together to
make a rectangle? Sketch each
possible rectangle.

1 in.

1 in.

b.

Which of your rectangles has the greatest perimeter? What
is its perimeter?

c.

Which of your rectangles has the least perimeter? What is
its perimeter?

The perimeter of a
figure is the distance
around the figure.

LESSON 1.1

Looking for Patterns 13

Following Rules
Jing drew a rectangle. She then wrote a rule for generating a sequence of
shapes starting with her rectangle.
Starting rectangle:
Rule: Draw a rectangle twice the size of the preceding rectangle.
By following Jing’s rule, Caroline drew this sequence:

Jahmal followed Jing’s rule and drew this sequence:

&

Think Discuss
Could both sequences above be correct? Explain your answer.
Rosita also followed Jing’s rule. The sequence she drew was different from both Caroline’s and Jahmal’s. What might Rosita’s
sequence look like?
Rewrite the rule so that Caroline’s sequence is correct but Jahmal’s
is not. Try to make your rule clear enough that anyone following it
would get the sequence Caroline did.

14 C H A P T E R 1

All about Patterns

Investigation 1

Sequences and Rules

You have just seen how three students could follow the same rule yet
draw three different sequences. This is because Jing’s rule is ambiguous—
it can be interpreted in more than one way. In both mathematics and
everyday life, it is often important to state rules in such a way that
everyone will get the same result.

Problem Set

A

1.

Create a sequence of shapes in which each shape can be made by
applying a rule to the preceding shape. On a blank sheet of paper,
draw the first shape of your sequence and write the rule. Try to make
your rule clear enough that anyone following it will get the sequence
you have in mind.

2.

Exchange starting shapes and rules with your partner. Follow your
partner’s rule to draw at least the next three shapes in the sequence.

3.

Compare the sequence you drew in Problem 2 with your partner’s
original sequence. Are they the same? If not, describe how they
are different and why. If either your rule or your partner’s rule is
ambiguous, work together to rewrite it to make it clear.

Rules are often used to describe how two quantities are related. For example,
a rule might tell you how to calculate one quantity from another.
E X A M P L E

An adult dose of SniffleLess cold medicine is 2 ounces. The dose for
a child under 12 can be calculated by using this rule:
Divide the child’s age by 12, and multiply the result by 2 ounces.
This rule tells how a child’s dose is related to the age of the child. If
you know the child’s age, you can use the rule to calculate the dose.
You can apply the rule to calculate the dose for a 3-year-old child:
dose ⫽ 3 ⫼ 12 ⫻ 2 ounces
⫽ 0.25 ⫻ 2 ounces
⫽ 0.5 ounce

LESSON 1.2

Following Rules 15

In Problem Set B, you will look at some common rules for finding one
quantity from another.

Problem Set
1.

B

You can use this rule to estimate how many miles away a bolt of
lightning struck:
Count the seconds between seeing the lightning flash and hearing
the thunder, and divide by 5.
Use the rule to estimate how far away a bolt of lightning struck if
you counted 15 seconds between the flash and the thunder.

2.

Hannah’s grandmother uses this rule to figure out how many spoonfuls of tea to put in her teapot:
Use one spoonful for each person, and then add one extra spoonful.

Just

a.

How much tea would Hannah’s grandmother use for four people?

the

b.

Hannah’s grandfather thinks this rule makes the tea too strong.
Make up a rule he might like better.

c.

Using your rule from Part b,
how many spoonfuls of tea are
needed for four people?

d.

Hannah’s cousin Amy likes her
tea much stronger than Hannah’s
grandmother does. Make up a
rule Amy might like, and use it
to figure out how many spoonfuls of tea would be needed for
four people.

facts

It is believed that people
first began drinking hot
tea more than 5,000
years ago in China. Iced
tea wasn’t introduced
until 1904 at the St.
Louis World’s Fair, when
an Englishman named
Richard Blechynden
added ice to the drink
because no one was
buying his hot tea.
3.

A cookbook gives this rule for roasting beef:
Cook for 20 minutes at 475°F. Then lower the heat to 375°F and
cook for 15 minutes per pound. If you like rare beef, remove the
roast from the oven. If you like it medium, cook it an additional
7 minutes. If you like it well done, cook it an additional 14 minutes.
What is the total cooking time for a 4-pound beef roast if you like it
medium?

16 C H A P T E R 1

All about Patterns

The rules in Problem Set B are fairly simple. Many rules involve more
complicated calculations. If you don’t need to find an exact value, you
can sometimes use a simpler rule to find an approximation.

Problem Set
1.

C

You can convert temperatures from degrees Celsius (°C) to degrees
Fahrenheit (°F) using this rule:
Multiply the degrees Celsius by 1.8 and add 32.

212°F

100°C

194

90

176

80

158

70

140

60

122

50

104

40

86

30

68

20

50

10

32°F

0°C

14

⫺10

Copy and complete
the table to show
some Celsius
temperatures and
their Fahrenheit
equivalents.

2.

Degrees
Celsius
0
10
20
30
40
50

Degrees
Fahrenheit
32
50

The Lopez family spent their summer vacation in Canada.
Ms. Lopez used this rule to convert Celsius temperatures to
Fahrenheit temperatures:
Multiply the degrees Celsius by 2 and add 30.
This rule makes it easy to do mental calculations, but it gives only
an approximation of the actual Fahrenheit temperature.
a.

Complete this
table to show the
approximate
Fahrenheit
temperatures this
rule gives for the
listed Celsius
temperatures.

Degrees
Celsius
0
10
20
30
40
50

Approximate
Degrees
Fahrenheit
30
50

b.

For which Celsius temperature do the two rules above give the
same result?

c.

For which Celsius temperatures in the table does Ms. Lopez’s rule
give a Fahrenheit temperature that is too high?

LESSON 1.2

Following Rules 17

3.

4.

One day the Lopez family flew from Toronto, where the temperature
was 37°C, to Winnipeg, where the temperature was 23°C.
a.

Use the rule from Problem 1 to find the exact Fahrenheit temperatures for the two cities.

b.

Use Ms. Lopez’s rule from Problem 2 to find the approximate
Fahrenheit temperatures for the two cities.

c.

For which city did Ms. Lopez’s rule give the more accurate
temperature?

Look back at your answers to Problems 2 and 3. What happens to
the Fahrenheit approximation as the Celsius temperature increases?

Toronto, the capital of the
province of Ontario

&

Share

Summarize
1. Below

are the first term and a rule for a sequence.

First term: 20
Rule: Write the number that is 2 units from the preceding number
on the number line.
a.

Give the first few terms of two sequences that fit the rule.

b.

Rewrite the rule so that only one of your sequences is correct.

2. At

the corner market, bananas cost 49¢ a pound. Write a rule for
calculating the cost of a bunch of bananas.

18 C H A P T E R 1

All about Patterns

Investigation 2

Order of Operations

A convention is a rule people have agreed to follow because it is helpful
or convenient for everyone to do the same thing. The rules “When you
drive, keep to the right” and “In the grocery store, wait in line to pay for
your selections” are two conventions.

Just
the

facts

Conventions are not
unchangeable like the
physical law “When you
drop an object, it falls
to the ground.” People
can agree to change
a convention and do
something different.

Reading across the page from left to right is a convention that Englishspeaking people have adopted. When you see the words “dog bites child,”
you know to read “dog” then “bites” then “child” and not “child bites
dog.” Not all languages follow this convention. For example, Hebrew is
read across the page from right to left, and Japanese is read down the
page from left to right.
To do mathematics, you need to know how to read mathematical expressions. For example, how would you read this expression?
5⫹3⫻7
There are several possibilities:
• Left to right: Add 5 and 3 to get 8, and then multiply by 7. The
result is 56.
• Right to left: Multiply 7 and 3 to get 21, and then add 5. The result
is 26.
• Multiply and then add: Multiply 3 and 7 to get 21, and then add 5.
The result is 26.

V O C A B U L A R Y
order of
operations

To communicate in the language of mathematics, people follow a convention for reading and evaluating expressions. The convention, called
the order of operations, says that expressions should be evaluated in
this order:
• Evaluate any expressions inside parentheses.
• Do multiplications and divisions from left to right.
• Do additions and subtractions from left to right.
To evaluate 5 ⫹ 3 ⫻ 7, you multiply first and then add:

Remember
Evaluating a mathematical expression means
finding its value.

5 ⫹ 3 ⫻ 7 ⫽ 5 ⫹ 21 ⫽ 26
If you want to indicate that the addition should be done first, you would
use parentheses:
(5 ⫹ 3) ⫻ 7 ⫽ 8 ⫻ 7 ⫽ 56
LESSON 1.2

Following Rules 19

E X A M P L E

These calculations follow the order of operations:
15 ⫺ 3 ⫻ 4 ⫽ 15 ⫺ 12 ⫽ 3
1 ⫹ 4 ⫻ (2 ⫹ 3) ⫽ 1 ⫹ 4 ⫻ 5 ⫽ 1 ⫹ 20 ⫽ 21
3⫹6⫼2⫺1⫽3⫹3⫺1⫽6⫺1⫽5

Another convention in mathematics involves the symbols used to represent multiplication. You are familiar with the ⫻ symbol. An asterisk or a
small dot between two numbers also means to multiply. So, each of these
expressions means “three times four”:
3⫻4

Problem Set

3ⴢ4

3*4

D

In Problems 1–4, use the order of operations to decide which of the
expressions are equal.

20 C H A P T E R 1

1.

8ⴢ4⫹6

(8 ⴢ 4) ⫹ 6

8 ⫻ (4 ⫹ 6)

2.

2⫹8ⴢ4⫹6

(2 ⫹ 8) ⫻ (4 ⫹ 6)

2 ⫹ (8 ⴢ 4) ⫹ 6

3.

(10 ⫺ 4) ⫻ 2

10 ⫺ (4 * 2)

10 ⫺ 4 * 2

4.

24 ⫼ 6 * 2

(24 ⫼ 6) ⫻ 2

24 ⫼ (6 ⴢ 2)

5.

Make up a mathematical expression with at least three operations,
and calculate the result. Then write your expression on a separate
sheet of paper, and swap expressions with your partner. Evaluate
your partner’s expression, and have your partner check your result.

6.

Most modern calculators follow the order of operations.
a.

Use your calculator to compute 2 ⫹ 3 ⫻ 4. What is the result?
Did your calculator follow the order of operations?

b.

Use your calculator to compute 1 ⫹ 4 ⫻ 2 ⫹ 3. What is the
result? Did your calculator follow the order of operations?

All about Patterns

Problem Set

E

Mr. Conte gets electricity and gas from the Smallville Power Company.
The company uses this rule to calculate a customer’s bill:
Charge 12.05 cents per kilowatt-hour (kwh) of electricity used and
65.7 cents per therm of gas used.
1.

This month, Mr. Conte’s household used 726 units of electricity and
51.7 units of gas. How much should his bill be? Give your answer in
dollars and cents.

2.

The computer system at Smallville Power crashed, so the clerks
have to use calculators to determine the bills. The calculators do not
use the order of operations. Instead, they evaluate the operations in
the order they are entered. To figure out Mr. Conte’s bill, the clerk
enters the expression below. Will the result be correct, too little, or
too much? Explain.
726 ⫻ 12.05 ⫹ 51.7 ⫻ 65.7

3.

Suppose the clerk enters the calculation below instead. Will the
result be correct, too much, or too little? Explain.
12.05 ⫻ 726 ⫹ 65.7 ⫻ 51.7

LESSON 1.2

Following Rules 21

A fraction bar is often used to indicate division. For example, these
expressions both mean “divide 10 by 2” or “10 divided by 2”:
ᎏ120ᎏ

10 ⫼ 2

Sometimes a fraction bar is used in more complicated expressions:
2 ⫹ᎏ
3
ᎏ
4⫹4
In expressions such as this, the bar not only means “divide,” it also acts as
a grouping symbol—grouping the numbers and operations above the bar
and grouping the numbers and operations below the bar. It is as if the
expressions above and below the bar are inside parentheses.
2 ⫹ᎏ
3
The expression ᎏ
4 ⫹ 4 means “Add 2 ⫹ 3, then add 4 ⫹ 4, and divide the
results.” So this expression means ᎏ58ᎏ, or 0.625.

This more complete order of operations includes the fraction bar:
• Evaluate expressions inside parentheses and above and below
fraction bars.
• Do multiplications and divisions from left to right.
• Do additions and subtractions from left to right.

Problem Set

F

Find the value of each expression.
2 ⫹ᎏ
2
1. ᎏ
1⫹1

3.

2.

2
ᎏ
2⫹ᎏ
1⫹1

Your calculator does not have a fraction bar as a grouping symbol,
2 ⫹ᎏ
2
so you have to be careful when entering expressions like ᎏ
1 ⫹ 1.
a.

What result does your calculator give if you enter 2 ⫹ 2/1 ⫹ 1 (or
2 ⫹ 2 ⫼ 1 ⫹ 1)? Can you explain why you get that result?

b.

2 ⫹ᎏ
2
What should you enter to evaluate ᎏ
1 ⫹ 1?

&

Share

Summarize
Why is it important to learn mathematical conventions such as the
order of operations?

22 C H A P T E R 1

All about Patterns

On Your Own Exercises

&
Apply

Practice

Use the first term and rule given to create a sequence. Tell whether your
sequence is the only one possible. If it isn’t, give another sequence that
fits the rule.
1.

First term: 40
Rule: Divide the preceding term by 2.

2.

First term:
Rule: Draw a shape with one more side than the preceding shape.

3.

Starting with a closed geometric figure with straight sides, you can
use the rule below to create a design.
Find the midpoint (middle point) of each side of the figure. Connect
the midpoints, in order, to make a new shape. (It will be the same
shape as the original, but smaller.)
a.

Copy this square. Follow the rule three times, each time starting
with the figure you drew the previous time.

b.

Copy this triangle. Follow the rule three times, each time starting
with the figure you drew the previous time.

c.

Draw your own shape, and follow the rule three times to make
a design.

4. Measurement

Luis is making a dessert that requires three eggs
for each cup of flour.

a.

How many eggs does he need for three cups of flour?

b.

For a party, Luis made a large batch of his dessert using a dozen
eggs. How much flour did he use?

impactmath.com/self_check_quiz

LESSON 1.2

Following Rules 23

5. Economics

Althea uses this rule to figure out how much to charge
for baby-sitting:
Charge $5 per hour for one child, plus $2 per hour for each
additional child.

a.

Last Saturday she watched the Newsome twins for 3 hours. How
much money did she earn? Explain how you found your answer.

b.

Mr. Foster hires Althea to watch his three children for 2 hours.
How much will she charge?

c.

Does Althea earn more for watching two children for 3 hours or
three children for 2 hours?

d.

Althea hopes to earn $25 next weekend to buy her sister a birthday
present. Describe two ways she could earn at least $25 baby-sitting.

6. Measurement

You can convert speeds from kilometers per hour
to miles per hour by using this rule:
Multiply the number of kilometers per hour by 0.62.

a.

Convert each kilometers-per-hour value in the table below to
miles per hour.
Kilometers
per Hour
50
60
70
80
90
100
110
120

b.

Miles per
Hour

As part of his job, Mr. Lopez does a lot of driving in Canada. He
uses this rule to approximate the speed in miles per hour from a
given speed in kilometers per hour:
Divide the number of kilometers per hour by 2 and add 10.
Use Mr. Lopez’s rule to convert each kilometers-per-hour value in
the table to an approximate miles-per-hour value.

24 C H A P T E R 1

c.

For which kilometers-per-hour values from the tables are the
results for the two rules closest?

d.

For which kilometers-per-hour values in the table does
Mr. Lopez’s rule give a value that is too high?

All about Patterns

Evaluate each expression.
(3 ⫹ 3) ⫻ (2 ⫹ 2)
7⫹6⫺2ⴢ6
9. (3 ⫹ 3) ⫹ 2 ⫼ 2
10. ᎏᎏ
11 ⫺ 5 ⴢ 2
Tell whether each expression was evaluated correctly using the order of
operations. If not, give the correct result.
7.

&
Extend

Connect

3⫹3ⴢ2⫹2

8.

11.

10 ⫻ (1 ⫹ 5) ⫺ 7 ⫽ 8

13.

(16 ⫺ 4 ⴢ 2) ⫺ (14 ⫼ 2) ⫽ 5 14. 100 ⫺ 33 ⴢ 2 ⫺ (4 ⫹ 8) ⫽ 22

15.

You can produce a sequence of numbers by applying this rule to
each term:

12.

54 ⫺ 27 ⫼ 3 ⫽ 45

If the number is even, get the next number by dividing by 2. If the
number is odd, get the next number by multiplying by 3 and adding 1.
a.

Use this rule to produce a sequence with 1 as the first term.
Describe the pattern in the sequence.

b.

Now use the rule to produce a sequence with 8 as the f