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TENTH EDITION
MATHEMATICS
for elementary teachers
A Conceptual Approach
BENNETT BURTON NELSON EDIGER

TENTH EDITION
Mathematics
for Elementary
Teachers
A Conceptual Approach
Albert B. Bennett, Jr.

University of New Hampshire
Laurie J. Burton
Western Oregon University
L. Ted Nelson
Portland State University
Joseph R. Ediger
Portland State University
MATHEMATICS FOR ELEMENTARY TEACHERS: A CONCEPTUAL APPROACH, TENTH EDITION
Published by McGraw-Hill Education, 2 Penn Plaza, New York, NY 10121. Copyright © 2016 by McGraw-
Hill Education. All rights reserved. Printed in the United States of America. Previous editions © 2012, 2010,
2007, and 2004. 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 the prior written consent of McGraw-Hill Education,
including, but not limited to, in any network or other electronic storage or transmission, or broadcast for
distance learning.
Some ancillaries, including electronic and print components, may not be available to customers outside the
United States.
This book is printed on acid-free paper.
1 2 3 4 5 6 7 8 9 0 DOW/DOW 1 0 9 8 7 6 5
ISBN 978-0-07-803565-4
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Library of Congress Cataloging-in-Publication Data
Mathematics for elementary teachers : a conceptual approach / Albert B. Bennett, Jr., University of New
Hampshire, Laurie J. Burton, Western Oregon University, L. Ted Nelson, Portland State University, Joseph
R. Ediger, Portland State University. — Tenth edition.
pages cm
Includes bibliographical references and index.
ISBN 978-0-07-803565-4 (alk. paper) — ISBN 0-07-803565-1 (alk. paper) 1. Mathematics—Study and
teaching (Elementary) 2. Elementary school teachers—Training of. 3. Mathematics teachers—Training
of. I. Bennett, Albert B. II. Burton, Laurie J. III. Nelson, Leonard T. IV. Ediger, Joseph R.
QA39.3.B457 2015
510—dc23
2014027053
The Internet addresses listed in the text were accurate at the time of publication. The inclusion of a website
does not indicate an endorsement by the authors or McGraw-Hill Education, and McGraw-Hill Education
does not guarantee the accuracy of the information presented at these sites.
www.mhhe.com
Question for Cover Design
The cover of this book shows the five Platonic
Solids. Each one of these solids has a certain number
of faces, edges, and vertices. Because the cover has
“see-through” models of these solids, you can count
the numbers of faces, edges, and vertices (corners)
for each solid. For example, the cube has 6 faces,
12 edges (4 on top, 4 on the bottom, and 4 vertical
edges), and 8 vertices (4 corners on top and 4 corners
below). There is an interesting relationship between
the numbers of faces, edges, and vertices for the
cube that also holds for the numbers of faces, ver-
tices, and edges for each of the other Platonic Solids.
Count the numbers of faces, edges, and vertices for
the other Platonic Solids to experience the thrill of
discovering this famous relationship.
INTRODUCING
PROFESSOR JOSEPH EDIGER
We would like to welcome Joseph Ediger, Senior Instructor in the Fariborz Maseeh
Department of Mathematics and Statistics at Portland State University, as coauthor of
the tenth editions of Mathematics for Elementary Teachers: A Conceptual Approach and
Mathematics for Elementary Teachers: An Activity Approach. Joe has twelve years of
experience teaching middle school and high school mathematics in Chicago, in rural
Montana, and in Portland, Oregon. After completion of an M.S. in Mathematics from
Portland State University in 1994, Joe joined the PSU Mathematics Department faculty.
He teaches a variety of math courses for preservice and inservice elementary and middle
school teachers, and has a special interest in the history of mathematics and in recre-
ational mathematics. He also supervises graduate teaching assistants in the Math
Department and enjoys serving as a mentor for beginning teachers. During summers, Joe
and his wife Kathryn keep busy tending fruit trees, berries, and vegetable gardens in
their half-acre backyard.
iii
One-Page Math Activities
with Manipulatives
1.1 Peg-Jumping Puzzle 2
1.2 Pattern Block Sequences 20
2.1 Sorting and Classifying Attribute Pieces 40

2.2 Deductive Reasoning Game 60
3.1 Numeration and Place Value with Base-Five Pieces 82

3.2 Addition and Subtraction with Base-Five Pieces 100
3.3 Multiplication with Base-Five Pieces 121
3.4 Division with Base-Five and Base-Ten Pieces 144
4.1 Divisibility with Base-Ten Pieces 172

4.2 Factors and Multiples from Tile Patterns 192
5.1 Addition and Subtraction with Black and Red Tiles 216
5.2 Equality and Inequality with Fraction Bars 240
5.3 Operations with Fraction Bars 269
6.1 Decimal Place Value with Base-Ten Pieces and Decimal Squares 302
6.2 Decimal Operations with Decimal Squares 327
6.3 Percents with Percent Grids 352
6.4 Irrational Numbers on Geoboards 379
7.1 Forming Bar Graphs with Color Tiles 404

7.2 Averages with Columns of Tiles 435
7.3 Simulations in Statistics 460
8.1 Experimental Probabilities from Simulations 486

8.2 Determining the Fairness of Games 509
8.3 Counting Arrangements 529
9.1 Extending Tile Patterns 552

9.2 Slopes of Geoboard Line Segments 571
9.3 Sorting by Rules 592
10.1 Angles in Pattern Block Figures 618

10.2 Tessellations with Polygons 641
10.3 Views of Cube Figures 658
10.4 Symmetries of Pattern Block Figures 681
11.1 Perimeters of Pattern Block Figures 706

11.2 Areas of Pattern Blocks Using Different Units 729
11.3 Surface Area and Volume for Three-Dimensional Figures 753
12.1 Tracing Figures from Motions with Tiles 786

12.2 Rotating, Reflecting, and Translating Figures on Grids 812
12.3 Enlargements with Pattern Blocks 839
iv
Contents
Preface ix
CHAPTER 1 PROBLEM SOLVING 1

Spotlight on Teaching and Learning 1
1.1 Introduction to Problem Solving 3
Math Activity 1.1: Peg-Jumping Puzzle * Problem Solving and Polya’s Four Steps *
Making a Drawing * Guessing and Checking * Making a Table * Using a Model *
Working Backward * Tower Puzzle Applet
1.2 Patterns and Problem Solving 21
Math Activity 1.2: Pattern Block Sequences * Finding a Pattern * Patterns in Nature *
Number Patterns * Inductive Reasoning * Solving a Simpler Problem
Chapter 1 Test 37
CHAPTER 2 SETS AND REASONING 39

2.1 Sets and Venn Diagrams 41
Math Activity 2.1 Sorting and Classifying Attribute Pieces * Sets and Their
Elements * Venn Diagrams * Relationships between Sets * Operations on Sets
2.2 Introduction to Deductive Reasoning 61
Math Activity 2.2 Deductive Reasoning Game * Deductive Reasoning * Venn
Diagrams and Deductive Reasoning * Conditional Statements * Reasoning with
Conditional Statements
Chapter 2 Test 78
CHAPTER 3 WHOLE NUMBERS 81

3.1 Numeration Systems 83
Math Activity 3.1 Numeration and Place Value with Base-Five Pieces * Grouping
and Number Bases * Ancient Numeration Systems * Reading and Writing
Numbers * Rounding Numbers * Deciphering Ancient Numeration Systems Applet *
Models for Numeration
3.2 Addition and Subtraction 100
Math Activity 3.2 Addition and Subtraction with Base-Five Pieces * Models for
Addition Algorithms * Number Properties * Inequality of Whole Numbers * Models for
Subtraction Algorithms * Mental Calculations * Approximation of Sums and Differences
3.3 Multiplication 122
Math Activity 3.3 Multiplication with Base-Five Pieces * Models for
Multiplication Algorithms * Number Properties * Mental Calculations *
Approximation of Products
3.4 Division and Exponents 145
Math Activity 3.4 Division with Base-Five and Base-Ten Pieces * Models for
Division Algorithms * Division Algorithm Theorem * Mental Calculations *
Approximation of Quotients * Exponents * Order of Operations
Chapter 3 Test 168
v
vi Contents
CHAPTER 4 NUMBER THEORY 171

4.1 Factors and Multiples 173
Math Activity 4.1 Divisibility with Base-Ten Pieces * Models for Factors and
Multiples * Odd and Even Numbers * Divisibility Tests * Prime and Composite
Numbers * Prime Number Test * Sieve of Eratosthenes
4.2 Greatest Common Factor and Least Common Multiple 193
Math Activity 4.2 Factors and Multiples from Tile Patterns * Prime Factorizations *
Fundamental Theorem of Arithmetic * Factor Trees * Factors of Numbers * Greatest
Common Factor * Least Common Multiple * Analyzing Star Polygons Applet
Chapter 4 Test 213
CHAPTER 5 INTEGERS AND FRACTIONS 215

5.1 Integers 217
Math Activity 5.1 Addition and Subtraction with Black and Red Tiles * Integers *
Uses of Integers * Models for Integers * Addition * Subtraction * Multiplication *
Division * Using Calculators for Integer Operations * Inequality * Number
Properties * Mental Calculations * Approximation
5.2 Introduction to Fractions 241
Math Activity 5.2 Equality and Inequality with Fraction Bars * Fraction
Terminology * Fraction Concepts * Equality of Fractions * Simplifying Fractions
with Calculators * Common Denominators * Inequality * Density of Fractions *
Mixed Numbers and Improper Fractions * Mental Calculations * Approximation *
Taking A Chance Applet
5.3 Operations with Fractions 270
Math Activity 5.3 Operations with Fraction Bars * Addition * Subtraction *
Multiplication * Division * Using Calculators for Fraction Operations * Number
Properties * Mental Calculations * Approximation
Chapter 5 Test 299
CHAPTER 6 DECIMALS: RATIONAL AND IRRATIONAL NUMBERS 301

6.1 Decimals and Rational Numbers 303
Math Activity 6.1 Decimal Place Value with Base-Ten Pieces and Decimal
Squares * Decimal Terminology and Notation * Models for Decimals * Equality
of Decimals * Inequality of Decimals * Rational Numbers * Density of Rational
Numbers * Approximation * Competing At Place Value Applet
6.2 Operations with Decimals 328
Math Activity 6.2 Decimal Operations with Decimal Squares * Addition *
Subtraction * Multiplication * Division * Order of Operation * Properties of
Rational Numbers * Mental Computation * Approximation
6.3 Ratio, Percent, and Scientific Notation 353
Math Activity 6.3 Percents with Percent Grids * Ratios * Rates * Proportions *
Percent * Calculations with Percents * Mental Calculations with Percents *
Approximation * Scientific Notation
6.4 Irrational and Real Numbers 380
Math Activity 6.4 Irrational Numbers on Geoboards * Pythagorean
Theorem * Square Roots * Irrational Numbers * Cube Roots and Other
Roots * Real Numbers * Properties of Real Numbers * Operations with
Irrational Numbers
Chapter 6 Test 402
Contents vii
CHAPTER 7 STATISTICS 403

7.1 Collecting and Graphing Data 405
Math Activity 7.1 Forming Bar Graphs with Color Tiles * Bar Graphs * Pie
Graphs * Pictographs * Line Plots * Stem-and-Leaf Plots * Histograms * Line
Graphs * Scatter Plots * Curves of Best Fit
7.2 Describing and Analyzing Data 436
Math Activity 7.2 Averages with Columns of Tiles * Measures of Central Tendency *
Box-and-Whisker Plots * Measures of Variability
7.3 Sampling, Predictions, and Simulations 460
Math Activity 7.3 Simulations in Statistics * Sampling * Skewed and Symmetric
Distributions * Normal Distributions * Distributions Applet * Measures of
Relative Standing * Simulations
Chapter 7 Test 482
CHAPTER 8 PROBABILITY AND COUNTING 485

8.1 Single-Stage Experiments 487
Math Activity 8.1 Experimental Probabilities from Simulations * Probabilities of
Outcomes * Probabilities of Events * Probabilities of Compound Events * Odds *
Experimental Probability * Simulations
8.2 Multistage Experiments 510
Math Activity 8.2 Determining the Fairness of Games * Probabilities of Multistage
Experiments * Probabilities of Events * Independent and Dependent Events *
Complementary Events * Expected Value * Door Prizes Applet
8.3 Counting, Permutations, and Combinations 530
Math Activity 8.3 Counting Arrangements * Counting Questions and Venn
Diagrams * Permutations * Combinations * Combination Patterns * Binomial
Coefficients and the Binomial Theorem
Chapter 8 Test 549
CHAPTER 9 ALGEBRA 551

9.1 Problem Solving with Algebra 553
Math Activity 9.1 Extending Tile Patterns * Variables and Expressions *
Equations * Inequalities * Using Algebra as a Problem-Solving Strategy
9.2 Coordinates, Slope, and Lines 572
Math Activity 9.2 Slopes of Geoboard Line Segments * Rectangular Coordinates *
Slope * Equations of Lines * Systems of Linear Equations * Hunting for Hidden
Polygons Applet
9.3 Functions and Graphs 593
Math Activity 9.3 Sorting by Rules * Functions * Linear Functions * Nonlinear
Functions * Interpreting Graphs
Chapter 9 Test 613
CHAPTER 10 GEOMETRIC FIGURES 617

10.1 Plane Figures 619
Math Activity 10.1 Angles in Pattern Block Figures * Mathematical Systems *
Points, Lines, and Planes * Half-Planes, Segments, Rays, and Angles * Angle
Measurements * Perpendicular and Parallel Lines * Curves and Convex Sets *
Polygons * Triangle and Quadrilateral Definitions
viii Contents
10.2 Polygons and Tessellations 642

Math Activity 10.2 Tessellations with Polygons * Angles in Polygons * Congruence *
Regular Polygons * Drawing Regular Polygons * Tessellations with Polygons
10.3 Space Figures 659
Math Activity 10.3 Views of Cube Figures * Planes * Polyhedra * Regular
Polyhedra * Pyramids and Prisms * Cones and Cylinders * Spheres and Maps *
Cross-Sections of a Cube Applet
10.4 Symmetric Figures 682
Math Activity 10.4 Symmetries of Pattern Block Figures * Reflection Symmetry for
Plane Figures * Rotation Symmetry for Plane Figures * Reflection Symmetry for
Space Figures * Rotation Symmetry for Space Figures
Chapter 10 Test 702
CHAPTER 11 MEASUREMENT 705

11.1 Systems of Measurement 707
Math Activity 11.1 Perimeters of Pattern Block Figures * Nonstandard Units of
Length * English Units * Metric Units * Precision and Small Measurements *
International System of Units
11.2 Area and Perimeter 730
Math Activity 11.2 Areas of Pattern Blocks Using Different Units * Nonstandard
Units of Area * Standard Units of Area * Perimeter * Areas of Polygons *
Circumferences and Areas of Circles
11.3 Volume and Surface Area 754
Math Activity 11.3 Surface Area and Volume for Three-Dimensional Figures *
Nonstandard Units of Volume * Standard Units of Volume * Surface Area *
Volumes and Surface Areas of Space Figures * Irregular Shapes * Creating
Surface Area * Filling 3-D Shapes Applet
Chapter 11 Test 781
CHAPTER 12 TRANSFORMATIONS: CONSTRUCTIONS, CONGRUENCE,

AND SIMILARITY MAPPING 785
12.1 Congruence and Constructions 787
Math Activity 12.1 Tracing Figures from Motions with Tiles * Mappings *
Constructing Segments and Angles * Constructing Triangles and Triangle Congruence
Properties * Constructing Bisectors * Constructing Perpendicular and Parallel Lines
* Circumscribing Circles About Triangles * Constructions With A Mira
12.2 Congruence Mappings 813
Math Activity 12.2 Rotating, Reflecting, and Translating Figures on Grids *
Translations * Reflections * Rotations * Composition of Mappings * Wallpaper
Patterns * Congruence * Mapping Figures Onto Themselves * Using Translations,
Rotations, and Reflections to Create Escher-Type Tessellations * Tessellations Applet
12.3 Similarity Mappings 840
Math Activity 12.3 Enlargements with Pattern Blocks * Similarity and Scale
Factors * Similar Polygons * Similar Triangles * Scale Factors, Area, and
Volume * Sizes and Shapes of Living Things
Chapter 12 Test 865
References R-1
Answers to Selected Math Activities A-1
Answers to Problem Openers A-7
Answers to Try Its A-11
Answers to Problem-Solving Applications A-27
Answers to Odd-Numbered Exercises, Problems, and Chapter Tests A-35
Credits C-1
Index I-1
Preface
The authors of Mathematics for Elementary Teachers: A Conceptual Approach believe
that all students should learn mathematics in a way that fosters conceptual understanding
and that prospective teachers should learn in a manner that generates enthusiasm for
learning and teaching mathematics.
New to the tenth edition is a special emphasis on the Common Core State
Standards and Practices for Mathematics. Each section of the book now highlights
connections between these practices and standards and the concepts and skills dis-
cussed in that section.
This edition also continues to embody the standards and practices developed over
many years by the National Council of Teachers of Mathematics in their Principles and
Standards and Focal Points publications.
OUR APPROACH
Our primary goal is to support future teachers in their journey toward becoming effective
teachers who can help their own students succeed in mathematics.
Effective teachers
• Understand deeply the mathematical concepts they will be teaching, and also under-
stand the larger scope of concepts that precede and follow them.
• Look for ways to help students build bridges connecting to new knowledge.
• Are problem solvers and have a readily available tool kit of problem-solving techniques.
• Are familiar with the Common Core State Standards and Practices for Mathematics.
• Are comfortable using hands-on activities that model mathematical concepts.
• Understand the connections between their own study of mathematics and the elemen-
tary school curriculum.
To get the most from this book you should

• Be an active learner as you seek to acquire a clear understanding of mathematical concepts.
• Use manipulatives as you work through the Math Activity at the beginning of each section.
• Answer each Try It! question as you read through a text section.
• Answer the questions embedded within each Problem-Solving Application as you
work on a section.
• Read the margin notes to help see how the Common Core State Standards for
Mathematics are connected to the topics in each section.
• Reflect about how your learning will impact your future classroom teaching.
NEW TO THE TENTH EDITION

• The Common Core State Standards and Practices for Mathematics are connected to
all chapters and sections of the text.
◦ A new Spotlight on Teaching and Learning opens each chapter connecting its
contents to the Common Core.
ix
x Preface
◦ Margin notes relate specific Practices and Standards to specific concepts and skills.
◦ End-of-section Classroom Connections pose questions for reflection on the
Common Core State Standards and Practices for Mathematics.
• Section 8.3: Counting, Permutations, and Combinations is new to this edition,
expanding our treatment of permutations and combinations, and includes new subsec-
tions on Pascal’s triangle, the Binomial Coefficients, and the Binomial theorem.
• Chapter 9: Algebra is new to this edition and collects and expands our treatment of
Problem Solving with Algebra, Coordinates, Slopes and Lines, and Functions and Graphs.
• New Try It! questions have been added to every section to increase student involve-
ment as they read the text.
• The Problem-Solving Applications have been reformatted so students must answer
intermediary questions before moving to the next step.
• New Technology and Lab Explorations are featured in some sections and in some
Exercises and Problem sets and offer an opportunity to explore an interesting topic
in more depth.
FEATURES OF THE TENTH EDITION

Chapter Level
• Each chapter begins with a new Spotlight on Teaching and Learning, showing con-
nections between the topics and materials in the chapter and the Common Core State
Standards and Practices for Mathematics.
• Each chapter ends with a Chapter Review and a Chapter Test.
Section Level
• Each section begins with a Math Activity that fosters group discussions and uses
manipulatives from the manipulative kit (cardstock and virtual, see Student Resources)
or other commonly available classroom supplies.
• A Problem Opener poses an interesting problem to be solved and serves as a warm-
up to the topics of the section.
• Try It! questions in every section take the place of many examples and put more respon-
sibility on the student to become involved in the learning of important concepts and skills.
• Problem-Solving Applications offer an emphasis on problem solving and Polya’s
four steps and provide opportunities to practice problem solving skills.
• Common Core and NCTM standards relating to the topics under consideration are noted
throughout each section, and questions corresponding to these standards are found in the
Classroom Connections section of the Exercises and Problems at the end of each section.
• Historical Highlights describe the origins and evolution of key mathematical ideas
and provide background on some of history’s outstanding mathematicians.
• Elementary School Text Pages taken from current grade school textbooks show how
key concepts from the section are presented to K–8 students. Questions corresponding
to these pages can be found in the Teaching Questions section of the Exercises and
Problems at the end of each section.
• Technology and Lab Explorations offer the opportunity to explore an interesting
topic in more depth.
• Interactive Math Applets, described in many sections, can be found at www.mhhe.com/
bbne and are designed for interactive explorations of some of the key concepts of the text.
• McGraw-Hill’s Connect is a web-based assignment and assessment platform that
helps students connect to their coursework and prepares them to succeed in and
beyond the course. A full color e-book is available for each Connect user.
Preface xi
STUDENT AND INSTRUCTOR RESOURCES

Student Resources
• Companion Website The Student Center at www.mhhe.com/bbne offers a variety
of resources, including
◦ Virtual Manipulatives
◦ Interactive Math Applets
◦ Downloadable grid and dot paper
◦ Printable manipulative sheets to make your own manipulative sets for individual
or classroom use
◦ Puzzlers with fun math problems
• Answers to text features Selected answers to each Math Activity (marked with a*),
complete answers to all of the Problem Openers, Try Its! and Problem Solving
Applications, and brief answers to the Odd Exercises and Problems for each section
are found at the back of the text.
• Student Solutions Manual Detailed explanations and diagrams for the odd-numbered exer-
cises are available—see the “Additional Supplements for Students” for more information.
• Cardstock and Virtual Manipulatives A kit with ten colored punch-out-ready cardstock
manipulatives for use with the activities, examples, problems and exercises in this textbook
may be packaged with your text. Printable Virtual Manipulative workspaces with electronic
versions of each cardstock manipulative are available at www.mhhe.com/bbne.
Instructor-Only Resource Center

Available for download at the password-protected Instructor Center (www.mhhe.com/
bbne) or through the Instructor Resource library feature of Connect:
• Daily Planning Guides Ideas for teaching every section of the Conceptual Approach
text. For each section the guide includes
◦ A guide for the material and exercises of the sections promoting time for class
discussions and questions
◦ Comments, tips, and teaching suggestions
◦ Suggestions for integrating Math Activities in your classroom
• Complete Instructor’s Manual Complete solutions to all Math Activities, Problem
Openers, Try Its!, Problem-Solving Applications, Lab and Computer Explorations,
and all Exercises and Problems.
• Editable Chapter Tests Three versions per chapter, with solutions.
ADDITIONAL RESOURCES FOR STUDENTS

Mathematics for Elementary Teachers: An Activity Approach,
Tenth Edition
The Activity Approach contains Activity Sets that correspond to each section of the text
and augment the ideas presented in the sections. Each Activity Set consists of a sequence
of hands-on inductive activities and experiments that enable the student to build an
understanding of mathematical ideas through the use of models and the discovery of
patterns. ISBN 13: 9781259298387, ISBN 10: 1259298388
Manipulative Kit
The Manipulative Kit containing 10 colorful manipulatives commonly used in elementary
schools is available for use with this text and for use with the Activity Approach text. This
kit includes labeled envelopes for each type of manipulative. ISBN 13: 9781259293474,
ISBN 10: 1259293475
xii Acknowledgments
Student’s Solution Manual

The Student’s Solutions Manual contains detailed solutions to the odd-numbered exercises
and Chapter Tests. The introduction offers suggestions for solving problems and for answer-
ing the Teaching and Classroom Connections questions in the text. Additional questions
and comments have been included at the ends of some of the solutions to give students
opportunities to extend their learning. ISBN 13: 9781259294754, ISBN 10: 1259294757
COMMON PACKAGING OPTIONS

Mathematics for Elementary Teachers: A Conceptual Approach, Tenth Edition, and
Mathematics for Elementary Teachers: An Activity Approach, packaged with Manipulative
Kit Mathematics for Elementary Teachers, Tenth Edition. ISBN 13: 9781259542190,
ISBN 10: 125954219X
Mathematics for Elementary Teachers: A Conceptual Approach, Tenth Edition, packaged
with Manipulative Kit Mathematics for Elementary Teachers, Tenth Edition. ISBN 13:
9781259542213, ISBN 10: 1259542211
Mathematics for Elementary Teachers: An Activity Approach, Tenth Edition, packaged
with Manipulative Kit Mathematics for Elementary Teachers, Tenth Edition. ISBN 13:
9781259542084, ISBN 10: 1259542084
For additional packaging options please consult your McGraw-Hill sales representative.
To find your rep, please visit www.mhhe.com/rep.
ACKNOWLEDGMENTS
We wish to thank Albert B. Bennett, III for his assistance in preparing the polydron
graphics used on the cover of this book. We thank the many students and instructors
who have used the first nine editions of this text, along with instructors who reviewed
this text and Mathematics for Elementary Teachers: An Activity Approach and have
supported our efforts by contributing comments and suggestions.
We especially acknowledge the following reviewers who contributed excellent
advice and suggestions for the tenth edition and previous editions:
Reviewers of This Edition

Eddie Cheng, Oakland University
Linda Fitzpatrick, Western Kentucky University
Jerrold Grossman, Oakland University
Jeff Hovermill, Northern Arizona University
Joan Jones, Eastern Michigan University
Klay Kruczek, Southern Connecticut State University
Sean Larsen, Portland State University
Lyn J. Miller, Slippery Rock University
Bethany Noblitt, Northern Kentucky University
Douglas Puharic, Edinboro University
Acknowledgments xiii
Eva Thanheiser, Portland State University

John Wilkins, California State University, Dominguez Hills
Reviewers of the Previous Editions

Paul Ache, Kutztown University
Khadija Ahmed, Monroe County Community College
Margo Alexander, Georgia State University
Angela T. Barlow, State University of West Georgia
Shari Beck, Navarro College
Sue Beck, Morehead State University
William L. Blubaugh, University of Northern Colorado
Patty Bonesteel, Wayne State University
Judy Carlson, Indiana University–Purdue University, Indianapolis
Carol Castellon, University of Illinois–Urbana-Champaign
Kristin Chatas, Washtenaw Community College
Chris Christopher, Bridgewater College
Janis Cimperman, St. Cloud State University
Porter Coggins, University of Wisconsin–Stevens Point
Ivette Chuca, El Paso Community College
Joy Darley, Georgia Southern University
Jean F. Davis, Texas State University–San Marcos
Tandy Del Vecchio, University of Maine
Linda Dequire, California State University, Long Beach
Ana Dias, Central Michigan University
Joyce Fischer, Texas State University–San Marcos
Grant A. Fraser, California State University–Los Angeles
Maria Fung, Western Oregon University
Krista Hands, Ashland University
Karen Heinz, Rowan University
Vanessa Huse, Texas A&M University–Commerce
Kathy Johnson, Volunteer State Community College
Joan Jones, Eastern Michigan University
Kurt Killion, Missouri State University
Gregory Klein, Texas A&M University–College Station
Peggy Lakey, University of Nevada, Reno
Pamela Lasher, Edinboro University of Pennsylvania
Elsa Lopez, El Paso Community College
Sarah E. Loyer, Eastern Mennonite University
Judy McBride, Indiana University–Purdue University, Indianapolis
Nicole Muth, Concordia University–Wisconsin
Linda Padilla, Joliet Junior College
Winnie Peterson, Kutztown University
Kimberley Polly, Parkland College and Indiana University
Michael Price, University of Oregon
Sue Purkayastha, University of Illinois–Champaign
Laurie Riggs, California State University–Pomona
F. D. Rivera, San Jose State University
Kathleen Rohrig, Boise State University
xiv Acknowledgments
Eric Rowley, Utah State University

Thomas H. Short, Indiana University of Pennsylvania
Pavel Sikorskii, Michigan State University
Elizabeth Smith, University of Louisiana–Monroe
Mary Ann Teel, University of North Texas
William N. Thomas, Jr., University of Toledo
Patricia Treloar, University of Mississippi
Hazel Truelove, University of West Alabama
Agnes Tuska, California State University, Fresno
Laura Villarreal, University of Texas at Brownsville
Tammy Voepel, Southern Illinois University–Edwardsville
Candide Walton, Southeast Missouri State University
Hiroko Warshauer, Texas State University–San Marcos
Pamela Webster, Texas A&M University–Commerce
Andrew White, Eastern Illinois University
Henry L. Wyzinski, Indiana University Northwest
Finally, at McGraw-Hill Education, thanks to Ryan Blankenship, Managing

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Supplements xv
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1
CHAPTER
Problem Solving
SPOTLIGHT ON TEACHING AND LEARNING

The Common Core State Standards for Mathematics are grade-specific lists of mathemat-
ics standards and a set of mathematical practices that have been adopted across the
country. These standards grew out of earlier recommendations, including the K–12 math-
ematics and process standards of the National Council of Teachers of Mathematics.
Throughout this book you will find connections to the Common Core State Standards
for Mathematics, which consists of
Content Standards, delineated by grade level.
Standards for Mathematical Practice, eight descriptions of the habits of mind
and attitudes that mathematically proficient students attain from discussions,
explorations, contemplation, and involvement in problem solving.
In mathematics, problem solving stands as a bridge that brings together the learning of
mathematical content knowledge and the use of appropriate mathematical practices.
Perhaps that is why the first of the eight Common Core Standards for Mathematical
Practice is,
Make sense of problems and persevere in solving them.
An excerpt from this standard says
Mathematically proficient students start by explaining to themselves the meaning
of a problem and looking for entry points to its solution. They analyze givens,
constraints, relationships, and goals. They make conjectures about the form and
meaning of the solution and plan a solution pathway rather than simply jumping
into a solution attempt. . . . Younger students might rely on using concrete objects
or pictures to help conceptualize and solve a problem. Mathematically proficient
students check their answers to problems using a different method, and they
continually ask themselves, “Does this make sense?”
THINK ABOUT THIS

Here is a puzzle problem to consider. As you solve this problem, monitor your thinking
to see which of the actions underlined above you used. What tools do you have that help
you to persevere in problem solving and to make sense of mathematics?
You and your friend each have some pennies. If you give your friend 1 penny,
then you and your friend will have the same number of pennies. If your friend
gives you 1 penny, then you will have twice as many pennies as your friend. How
many pennies did you each start with?
2 Math Activity 1.1
MATH ACTIVITY 1.1

Peg-Jumping Puzzle
Virtual Manipulatives Purpose: Use four problem-solving strategies to solve a puzzle problem.
Materials: Color Tiles in the Manipulative Kit or Virtual Manipulatives.
Puzzle: There are four mov-
able red pegs in the holes at
one end of a board, four
movable green pegs in the
holes at the other end, and
one empty hole in the center.
The challenge is to inter-
www.mhhe.com/bbne
change the pegs so that the red pegs occupy the positions of the green pegs and vice
versa, in the fewest moves. Here are the legal moves: Any peg can move to an adjacent
empty hole, pegs do not move backward, and a peg of one color can jump over a single
peg of another color if there is a hole to jump into.
1. Using a model: Sketch nine 1-inch by 1-inch squares and place four red tiles on
the left end and four green tiles on the right. Try solving this problem by moving
the tiles according to the rules.
2. Solving a simpler problem: Sketch three squares and use one red tile and one green
tile to solve this simpler problem. Then sketch five squares and solve the problem
with two tiles of each color.
*3. Making a table: Sketch the following table and record the minimum number of moves
and your strategy when there are three tiles on each side. For example, with one tile
on each end you may have moved the red tile first (R), then jumped that with the
green (G), and finally moved the red (R). So your strategy could be recorded RGR.
Tiles (Pegs) Minimum Number

on a Side of Moves Strategy
1 3 RGR
2 8 RGGRRGGR
3
4. Finding patterns: You may have noticed one or more patterns in your table. List at
least one pattern in your strategies. There is also a pattern in the numbers of moves.
Try finding this pattern and predict the number of moves for four tiles on a side.
Then test the strategy for solving the Peg Puzzle with four tiles on a side.
*5. Extending patterns: Use one of the patterns you discovered to predict the fewest
number of moves for solving the puzzle with five or more pegs on each side.
*Answer is given in answer section at back of book.
Section 1.1 Introduction to Problem Solving 3
SECTION 1.1 INTRODUCTION TO PROBLEM SOLVING
There is no more
signiﬁcant privilege than
to release the creative
power of a child’s mind.
Franz F. Hohn
PROBLEM OPENER
Alice counted 7 cycle riders and 19 cycle wheels going past her house. How many
tricycles were there?
Practices & Processes “Learning to solve problems is the principal reason for studying mathematics.”* This
statement by the National Council of Supervisors of Mathematics represents a wide-
Problem solving is the hallmark spread opinion that problem solving should be the central focus of the mathematics
of mathematical activity and a curriculum.
major means of developing A problem exists when there is a situation you want to resolve but no solution is
mathematical knowledge.
readily apparent. Problem solving is the process by which the unfamiliar situation is
National Council of resolved. A situation that is a problem to one person may not be a problem to someone
Teachers of Mathematics
else. For example, determining the number of people in 3 cars when each car contains
5 people may be a problem to some elementary school students. They might solve this
problem by placing chips in boxes or by making a drawing to represent each car and
each person (Figure 1.1) and then counting to determine the total number of people.
Figure 1.1
You may be surprised to know that some problems in mathematics are unsolved
and have resisted the efforts of some of the best mathematicians to solve them. One
*National Council of Supervisors of Mathematics, Essential Mathematics for the 21st Century.
4 Chapter 1 Problem Solving
Practices & Processes such problem was discovered by Arthur Hamann, a seventh-grade student. He
noticed that every even number could be written as the difference of two primes.*
Doing mathematics involves For example,
discovery. Conjecture—that is,
informed guessing—is a major 25523 4 5 11 2 7 6 5 11 2 5 8 5 13 2 5 10 5 13 2 3
pathway to discovery. Teachers
After showing that this was true for all even numbers less than 250, he predicted that
and researchers agree that
students can learn to make,
every even number could be written as the difference of two primes. No one has been
refine, and test conjectures in able to prove or disprove this statement. When a statement is thought to be true but
elementary school. remains unproved, it is called a conjecture.
National Council of
Teachers of Mathematics
PROBLEM SOLVING AND POLYA’S FOUR STEPS
The first of the Common Core State Standards for Mathematical Practice states that
students should make sense of problems and persevere in solving them. Problem solving
is also an overarching focus within the standards of the National Council of Teachers of
Mathematics. And it is the subject of a major portion of research and publishing in
mathematics education. Much of this research is founded on the problem-solving writings
of George Polya, one of the foremost twentieth-century mathematicians. Polya devoted
much of his teaching to helping students become better problem solvers. His book How
to Solve It has been translated into more than 20 languages. In this book, he outlines the
following four-step process for solving problems.
Understanding the Problem Polya suggests that a problem solver needs to become
better acquainted with a problem and work toward a clearer understanding of it before
progressing toward a solution. Increased understanding can come from rereading the
statement of the problem, drawing a sketch or diagram to show connections and relation-
ships, restating the problem in your own words, or making a reasonable guess at the
solution to help become acquainted with the details.
Sometimes the main difficulty in solving a problem is knowing what question is to

be answered.
Devising a Plan The path from understanding a problem to devising a plan may
sometimes be long. Most interesting problems do not have obvious solutions. Experience
and practice are the best teachers for devising plans. Throughout the text you will be
introduced to strategies for devising plans to solve problems.
Carrying Out the Plan The plan gives a general outline of direction. Write down
your thinking so your steps can be retraced. Is it clear that each step has been done cor-
rectly? Also, it’s all right to be stuck, and if this happens, it is sometimes better to put
aside the problem and return to it later.
*M. R. Frame, “Hamann’s Conjecture,” Arithmetic Teacher.

Looking Back When a result has been reached, verify or check it by referring to the
original problem. In the process of reaching a solution, other ways of looking at the prob-
lem may become apparent. Quite often after you become familiar with
a problem, new or perhaps more novel approaches may occur to you. Also, while solving
a problem, you may find other interesting questions or variations that are worth exploring.
Polya’s problem-solving steps will be used throughout the text. The purpose of this
section is to help you become familiar with the four-step process and to acquaint you
with some of the common strategies for solving problems: making a drawing, guessing
and checking, making a table, using a model, and working backward. Additional strate-
gies will be introduced throughout the text.
Many problems can be approached successfully in a variety of ways. Even when
using the same general strategy, different problem solvers may come up with dif-
ferent variations in how the approach is used. For each of the five problems dis-
cussed in this section, we invite you in a Try It! to work on the problem yourself
first, before reading our approach to a solution. Your approach may be as good, or
better than, ours.
You will find Try It! questions throughout the text. When you come to a Try It! you
should stop reading, grab a pencil, and work through the given question. Answers to the
Try It! questions can be found in the back of the text.
To learn mathematics, and especially to learn problem solving, it is important to
learn by doing. Take your time, think through the problems, and don’t be afraid to try a
variety of ideas. Even wrong answers and unproductive approaches are opportunities for
learning. The joy that comes from seeing persistence pay off in a correct solution to a
difficult problem is very gratifying for learners of all ages.
MAKING A DRAWING
PROBLEM-SOLVING APPLICATION
One of the most helpful strategies for understanding a problem and obtaining ideas for
a solution is to draw sketches and diagrams. Most likely you have heard the expression
“A picture is worth a thousand words.” In the following problem, the drawings will help
you to think through the solution.
Problem
For his wife’s birthday, Mr. Jones is planning a dinner party in a large room. There will
be 22 people, and in order to seat them he needs to borrow card tables, the size that
seats one person on each side. He wants to arrange the tables in a rectangular shape so
that they will look like one large table. What is the smallest number of tables that
Mr. Jones needs to borrow?
TRY IT! 1.1.1

Before reading further, use a method that makes sense to you to work on this
problem. Then read through our Making a Drawing approach, answering ng the
questions along the way.
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From Glencoe 307
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Practices & Processes Understanding the Problem The tables must be placed next to each other, edge to
edge, so that they form one large rectangular table.
Practice 1
Make sense of problems and
persevere in solving them.
Younger students might rely on
using concrete objects or
pictures to help conceptualize
One large table
and solve a problem.
Mathematically proficient
students check their answers to Question 1: If two tables are placed end to end, how many people can be seated?
problems using a different
method, and they continually
Devising a Plan Drawing pictures of the different arrangements of card tables is a
ask themselves, “Does this natural approach to solving this problem. There are only a few possibilities. The tables
make sense?” can be placed in one long row; they can be placed side by side with two abreast; etc.
Common Core Question 2: How many people can be seated at five tables if they are placed
State Standards Mathematics
end to end in a single row?
Carrying Out the Plan The following drawings show two of the five possible arrange-
ments that will seat 22 people. The X’s show that 22 people can be seated in each
arrangement. The remaining arrangements—3 by 8, 4 by 7, and 5 by 6—require 24, 28,
and 30 card tables, respectively.
Question 3: What is the smallest number of card tables needed?
x x x x x x x x x x
x x 10 tables
x x x x x x x x x x
x x x x x x x x x
x x
18 tables
x x
x x x x x x x x x
Looking Back The drawings show that a single row of tables requires the fewest tables
because each end table has places for 3 people and each of the remaining tables has
places for 2 people. In all the other arrangements, the corner tables seat only
2 people and the remaining tables seat only 1 person. Therefore, regardless of the num-
ber of people, a single row is the arrangement that uses the smallest number of card
tables, provided the room is long enough.
Question 4: What is the smallest number of card tables required to seat 38 people?
GUESSING AND CHECKING
Even though guessing can sometimes lead to going down a wrong pathway, many
problems can be better understood and even solved by trial-and-error procedures. As
Polya said, “Mathematics in the making consists of guesses.” If your first guess is off,
it may lead to a better guess. Even if guessing doesn’t produce the correct answer, you
may increase your understanding of the problem and obtain an idea for solving it. The
guess-and-check approach is especially appropriate for elementary schoolchildren because

it puts many problems within their reach.
Problem
How far is it from town A to town B in this cartoon?
Peanuts: © United Feature Syndicate, Inc.
TRY IT! 1.1.2

Before reading further, use the information in the first two panels of the cartoon
to see if you can answer the question. Then read through our Guessing ng and
Checking approach, answering the questions along the way.
Understanding the Problem There are several bits of information in this problem.
Let’s see how Peppermint Patty could have obtained a better understanding of the prob-
lem with a diagram. First, let us assume these towns lie in a straight line, so they can
be illustrated by points A, B, C, and D, as shown in (a). Next, it is 10 miles farther from
A to B than from B to C, so we can move point B closer to point C, as in (b). It is also
10 miles farther from B to C than from C to D, so point C can be moved closer to point
D. Finally, the distance from A to D is given as 390 miles.
Question 1: The problem requires finding what distance?
Practices & Processes
Practice 2 A B C D
Reason abstractly and
(a)
quantitatively. Mathematically
proficient students make sense of 390 miles
quantities and their relationships
in problem situations. . . .
Quantitative reasoning entails
habits of creating a coherent A B C D
representation of the problem at
(b)
hand; considering the units
involved; attending to the
meaning of quantities, not just Devising a Plan One method of solving this problem is to make a reasonable guess
how to compute them; and and then use the result to make a better guess. If the 4 towns were equally spaced, as
knowing and flexibly using in (a), the distance between each town would be 130 miles (390 4 3). However, the
different properties of operations distance from town A to town B is the greatest. So let’s begin with a guess of 150 miles
and objects. for the distance from A to B.
Common Core
State Standards Mathematics Question 2: In this case, what is the distance from B to C and C to D?
Carrying Out the Plan Using a guess of 150 for the distance from A to B produces a
total distance from A to D that is greater than 390. If the distance from A to B is 145,
then the B-to-C distance is 135 and the C-to-D distance is 125. The sum of these dis-
tances is 405, which is still too great.
Question 3: What happens if we use a guess of 140 for the distance

from A to B?
Looking Back One of the reasons for looking back at a problem is to consider different
solutions or approaches. For example, you might have noticed that the first guess, which
produced a distance of 420 miles, was 30 miles too great.
Question 4: How can this observation be used to lead quickly to a correct

solution of the original problem?
MAKING A TABLE
A problem can sometimes be solved by listing some of or all the possibilities. A table
is often convenient for organizing such a list.
Problem
Sue and Ann earned the same amount of money, although one worked 6 days more than the
other. If Sue earned $36 per day and Ann earned $60 per day, how many days did each work?
TRY IT! 1.1.3

Before reading further, use a method that makes sense to you to work on this
problem. Then read through our Making a Table approach and compare it to the
he way
that you thought about the problem.
Understanding the Problem Answer a few simple questions to get a feeling for
the problem.
Question 1: How much did Sue earn in 3 days? Did Sue earn as much in
3 days as Ann did in 2 days? Who worked more days?
Devising a Plan One method of solving this problem is to list each day and each per-
son’s total earnings through that day.
Question 2: What is the first amount of total pay that is the same for Sue and
Ann, and how many days did it take each to earn this amount?
Carrying Out the Plan The complete table is shown on page 10. There are three amounts
in Sue’s column that equal amounts in Ann’s column. It took Sue 15 days to earn $540.
Question 3: How many days did it take Ann to earn $540, and what is the
difference between the numbers of days they each required?
Number of Days Sue’s Pay Ann’s Pay

1 36 60
2 72 120
3 108 180
4 144 240
5 180 300
6 216 360
7 252 420
8 288 480
9 324 540
10 360 600
11 396 660
12 432 720
13 468 780
14 504 840
15 540 900
Looking Back You may have noticed that every 5 days Sue earns $180 and every
3 days Ann earns $180.
Question 4: How does this observation suggest a different way to answer the
original question?
USING A MODEL
Practices & Processes Models are important aids for visualizing a problem and suggesting a solution. The
recommendations by the Conference Board of the Mathematical Sciences (CBMS) in
Practice 4 their document, The Mathematical Education of Teachers, say: “Future teachers will need
Model with mathematics. to connect fundamental concepts to a variety of situations, models, and representations.”*
students can apply the
mathematics they know to
solve problems arising in PROBLEM-SOLVING APPLICATION
everyday life.
The next problem uses whole numbers 0, 1, 2, 3, . . . and is solved by using a model. It
Common Core involves a well-known story about the German mathematician Karl Gauss. When Gauss
was 10 years old, his schoolmaster gave him the problem of computing the sum of whole
numbers from 1 to 100. Within a few moments the young Gauss wrote the answer on his
slate and passed it to the teacher. Before you read the solution to the following problem,
try to find a quick method for computing the sum of whole numbers from 1 to 100.
*Conference Board of the Mathematical Sciences (CBMS), The Mathematical Education of Teachers, “Chapter 7:
The Preparation of Elementary Teachers.”
Problem
Find an easy method for computing the sum of consecutive whole numbers from 1 to
any given number.
TRY IT! 1.1.4

Before reading further, think about different ways you might try Using a Model
to illustrate sums of consecutive numbers. For example, how could you create
reate a
model of 1 1 2 1 3 1 4?
Understanding the Problem If the last number in the sum is 8, then the sum is
1 1 2 1 3 1 4 1 5 1 6 1 7 1 8. If the last number in the sum is 100, then the sum
is 1 1 2 1 3 1 . . . 1 100.
Question 1: What is the sum of whole numbers from 1 to 8?
Devising a Plan One method of solving this problem is to cut staircases out of graph
paper. The one shown in (a) is a 1-through-8 staircase: There is 1 square in the first step,
there are 2 squares in the second step, and so forth, to the last step, which has a column
of 8 squares. The total number of squares is the sum 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8.
By using two copies of a staircase and placing them together, as in (b), we can obtain
a rectangle whose total number of squares can easily be found by multiplying length
by width.
Question 2: What are the dimensions of the rectangle in (b), and how many
Practices & Processes small squares does it contain?
Practice 6
Attend to precision.
students try to communicate
precisely to others. They try
to use clear definitions in
discussion with others and
in their own reasoning. They
state the meaning of the symbols
they choose, including using
the equal sign consistently
and appropriately. 1-through-8 staircase Two 1-through-8 staircases
Common Core
State Standards Mathematics (a) (b)
Carrying Out the Plan Cut out two copies of the 1-through-8 staircase and place them
together to form a rectangle. Since the total number of squares is 8 3 9, the number of
(8 3 9)
squares in one of these staircases is 2 5 36. So the sum of whole numbers from 1
to 8 is 36. By placing two staircases together to form a rectangle, we see that the num-
ber of squares in one staircase is just half the number of squares in the rectangle. This
geometric approach to the problem suggests that the sum of consecutive whole numbers
from 1 to any specific number is the product of the last number and the next number,
divided by 2.
Question 3: What is the sum of whole numbers from 1 to 100?
Looking Back Another approach to computing the sum of whole numbers from 1 to 100
is suggested by the following diagram, and it may have been the method used by Gauss. If
the numbers from 1 to 100 are paired as shown, the sum of each pair of numbers is 101.
101
101
101
101
101
1 + 2 + 3 + 4 + ... + 50 + 51 + ... + 97 + 98 + 99 + 100
Question 4: How can this sum be used to obtain the sum of whole numbers
from 1 to 100?
Question 5: Can this method be used for sums of consecutive whole numbers
that do not begin with 1?
HISTORICAL HIGHLIGHT
Athenaeus, a Greek writer (ca. 200), in his book Deipnosophistae mentions a
number of women who were superior mathematicians. However, Hypatia in the
fourth century is the first woman in mathematics of whom we have considerable
knowledge. Her father, Theon, was a professor of mathematics at the University
of Alexandria and was influential in her intellectual development, which eventu-
ally surpassed his own. She became a student in Athens at the school conducted
by Plutarch the Younger, and it was there that her fame as a mathematician became
established. Upon her return to Alexandria, she accepted an invitation to teach
mathematics at the university. Her contemporaries wrote about her great genius.
Socrates, the historian, wrote that her home as well as her lecture room was fre-
quented by the most unrelenting scholars of the day. Hypatia was the author of
several treatises on mathematics, but only fragments of her work remain. A portion
Hypatia, 370–415 of her original treatise On the Astronomical Canon of Diophantus was found dur-
ing the fifteenth century in the Vatican library. She also wrote On the Conics of
Apollonius. She invented an astrolabe and a planisphere, both devices for studying
astronomy, and apparatuses for distilling water and determining the specific
gravity of water.*
*L. M. Osen, Women in Mathematics, pp. 21–32.
WORKING BACKWARD
Problem
A businesswoman went to the bank and sent half of her money to a stockbroker. Other
than a $2 parking fee before she entered the bank and a $1 mail fee after she left the
bank, this was all the money she spent. On the second day she returned to the bank and
sent half of her remaining money to the stockbroker. Once again, the only other expenses
were the $2 parking fee and the $1 mail fee. If she had $182 left, how much money did
she have before the trip to the bank on the first day?
TRY IT! 1.1.5

Before reading further, use a method that makes sense to you to work on this prob-
lem. Then read through our Working Backward approach, answering the questions
estions
along the way.
Understanding the Problem Let’s begin by guessing the original amount of money,
say, $800, to get a better feel for the problem.
Question 1: If the businesswoman begins the day with $800, how much money
will she have at the end of the first day, after paying the mail fee?
Devising a Plan Guessing the original amount of money is one possible strategy,
but it requires too many computations. Since we know the businesswoman has $182
at the end of the second day, a more appropriate strategy for solving the problem is
to retrace her steps back through the bank (see the following diagram). First she
receives $1 back from the mail fee. Continue to work back through the second day
in the bank.
Question 2: How much money did the businesswoman have at the beginning
of the second day?
BANK
Parking Enter
1
fee $2 Send
2
of money
Mail
fee $1 Leave
Carrying Out the Plan The businesswoman had $368 at the beginning of the second
day. Continue to work backward through the first day to determine how much money
she had at the beginning of that day.
Question 3: What was this amount?
Looking Back You can now check the solution by beginning with $740, the origi-
nal amount of money, and going through the expenditures for both days to see if
$182 is the remaining amount. The problem can be varied by replacing $182 at the
end of the second day by any amount and working backward to the beginning of
the first day.
Question 4: For example, if there was $240 at the end of the second day, what
was the original amount of money?
Applet What is the least number of moves to transfer four disks from one tower to another if
Explorations only one disk can be moved at a time and a disk cannot be placed on top of a smaller
disk? In this applet, you will solve an ancient problem by finding patterns to determine
the minimum number of moves for transferring an arbitrary number of disks.
Tower Puzzle Applet, Chapter 1

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EXERCISES AND PROBLEMS 1.1
Problems 1 through 20 involve strategies that were pre- and plot the snail’s daily progress. What is the snail’s
sented in this section. Some of these problems are ana- greatest height during the second day?
lyzed by Polya’s four-step process. See if you can solve
these problems before answering parts a, b, c, and d. Other
20
strategies may occur to you, and you are encouraged to use
the ones you wish. Often a good problem requires several
15
strategies.
10
Making a Drawing (1–4)
5
1. A well is 20 feet deep. A snail at the bottom climbs up
4 feet each day and slips back 2 feet each night. How 0
many days will it take the snail to reach the top of
the well?
a. Understanding the Problem. What is the greatest c. Carrying Out the Plan. Trace out the snail’s daily
height the snail reaches during the first 24 hours? progress, and mark its position at the end of each day.
How far up the well will the snail be at the end of the On which day does the snail get out of the well?
first 24 hours? d. Looking Back. There is a “surprise ending” at the
b. Devising a Plan. One plan that is commonly cho- top of the well because the snail does not slip back
sen is to compute 202 , since it appears that the snail on the ninth day. Make up a new snail problem by
gains 2 feet each day. However, 10 days is not the changing the numbers so that there will be a similar
correct answer. A second plan is to make a drawing surprise ending at the top of the well.
2. Five people enter a racquetball tournament in which 6. Sasha and Francisco were selling lemonade for 25 cents
each person must play every other person exactly once. per half cup and 50 cents per full cup. At the end of
Determine the total number of games that will be played. the day they had collected $15 and had used 37 cups.
How many full cups and how many half cups did
3. When two pieces of rope are placed end to end, their
they sell?
combined length is 130 feet. When the two pieces are
placed side by side, one is 26 feet longer than the other. 7. James bought 16 bolts at the hardware store for a
What are the lengths of the two pieces? total of $6.00. Some were 3-inch bolts that cost
36 cents each and the others were 4-inch bolts
4. There are 560 third- and fourth-grade students in King
that cost 42 cents each. How many 3-inch bolts did
Elementary School. If there are 80 more third-graders
James buy?
than fourth-graders, how many third-graders are there
in the school? 8. I had some pennies, nickels, dimes, and quarters in my
pocket. When I reached in and pulled out some change,
I had less than 10 coins whose value was 42 cents.
Making a Table (5–8)
What are all the possibilities for the coins I had in
5. The 24/7 Truck Rental Company has been charging an my hand?
initial rental fee of $29 plus $2.50 per hour for rental of
their small trucks. After a change in pricing policy, the
Guessing and Checking (9–12)
new fee will be $19 up front plus $3.25 per hour. After
how many hours of renting the truck will the new plan 9. There are two 2-digit numbers that satisfy the follow-
be more expensive than the old plan? ing conditions: (1) Each number has the same digits,
a. Understanding the Problem. Try some numbers (2) the sum of the digits in each number is 10, and
to get a feel for the problem. Compute the cost of a (3) the difference between the 2 numbers is 54. What
10-hour rental under the old plan and under the new are the two numbers?
plan. Which plan is more expensive for a customer a. Understanding the Problem. The numbers 58
who rents a truck for 10 hours? and 85 are 2-digit numbers that have the same
b. Devising a Plan. One method of solving this prob- digits, and the sum of the digits in each number is
lem is to make a table showing the cost of 1 hour, 13. Find two 2-digit numbers such that the sum
2 hours, etc., such as that shown here. How much less of the digits is 10 and both numbers have the
does the new plan cost than the old plan for 6 hours? same digits.
b. Devising a Plan. Since there are only nine 2-digit
Cost for Cost for numbers whose digits have a sum of 10, the problem
Hours Old Plan, $ New Plan, $ can be easily solved by guessing. What is the differ-
1 31.50 22.25 ence of your two 2-digit numbers from part a? If this
difference is not 54, it can provide information about
2 34.00 25.50 your next guess.
3 36.50 28.75 c. Carrying Out the Plan. Continue to guess and
check. Which pair of numbers has a difference
4 39.00 32.00 of 54?
5 41.50 35.25 d. Looking Back. This problem can be extended by
changing the requirement that the sum of the two
c. Carrying out the Plan. Extend the table until you digits equals 10. Solve the problem for the case in
reach a point at which the new plan is more expensive which the digits have a sum of 12.
than the old plan. How many hours must the truck be
10. When two numbers are multiplied, their product is 759;
rented before the new plan is more expensive?
but when one is subtracted from the other, their differ-
d. Looking Back. For people who rent the truck for
ence is 10. What are these two numbers?
only 1 hour, the difference in cost between the two
plans is $9.25. What happens to the difference as
the number of hours increases? How many hours
can the truck be rented before the cost under the
new plan becomes $20 more expensive than the
old plan?
11. When asked how a person can 4-gallon 9-gallon d. Looking Back. Suppose the problem had asked for
measure out 1 gallon of water container container the smallest number of colors to form a square of
with only a 4-gallon container nine tiles so that no tile touches another tile of the
and a 9-gallon container, a stu- 0 9 same color along an entire edge. Can it be done in
dent used this “picture.” fewer colors; if so, how many?
a. Briefly describe what the stu-
14. What is the smallest number of different colors of tile
dent could have shown by this 4 5 needed to form a 4 3 4 square so that no tile touches
sketch.
another of the same color along an entire edge?
b. Use a similar sketch to show
how 6 gallons can be mea- 0 5 15. The following patterns can be used to form a cube. A
sured out by using these same cube has six faces: the top and bottom faces, the left
containers. and right faces, and the front and back faces. Two faces
4 1 have been labeled on each of the following patterns.
Label the remaining four faces on each pattern so that
when the cube is assembled with the labels on the out-
12. Carmela opened her piggy bank and found she had
side, each face will be in the correct place.
$15.30. If she had only nickels, dimes, quarters, and
half-dollars and an equal number of coins of each kind,
how many coins in all did she have?
Left Bottom Back

Using a Model (13–16)
Bottom
13. Suppose that you have a supply of red, blue, green,
and yellow square tiles. What is the fewest number of
different colors needed to form a 3 3 3 square of tiles
16. At the left in the following figure is a domino doughnut
so that no tile touches another tile of the same color at
with 11 dots on each side. Arrange the four single dom-
any point?
inoes on the right into a domino doughnut so that all
a. Understanding the Problem. Why is the square
four sides have 12 dots.
arrangement of tiles shown here not a correct
solution?
b. Devising a Plan. One plan is to choose a tile for Domino doughnut

the center of the grid and then place others around
it so that no two of the same color touch. Why must
the center tile be a different color than the other Working Backward (17–20)
eight tiles?
c. Carrying Out the Plan. Suppose that you put a 17. Three girls play three rounds of a game. On each round
blue tile in the center and a red tile in each corner, as there are two winners and one loser. The girl who loses
shown here. Why will it require two more colors for on a round has to double the number of chips that each
the remaining openings? of the other girls has by giving up some of her own
chips. Each girl loses one round. At the end of three
rounds, each girl has 40 chips. How many chips did
each girl have at the beginning of the game?
a. Understanding the Problem. Let’s select some
numbers to get a feel for this game. Suppose girl A, girl
B, and girl C have 70, 30, and 20 chips, respectively,
and girl A loses the first round. Girl B and girl C will Their sum is 112. What are the numbers?
receive chips from girl A, and thus their supply of
chips will be doubled. How many chips will each
First
girl have after this round?
b. Devising a Plan. Since we know the end result (each
girl finished with 40 chips), a natural strategy is to work Second 112
backward through the three rounds to the beginning.
Assume that girl C loses the third round. How many
Third
chips did each girl have at the end of the second round?
A B C 22. Mike has 3 times as many nickels as Larry has dimes.

Mike has 45 cents more than Larry. How much money
Beginning does Mike have?
End of first round
Number of dimes that Larry has
End of second round
End of third round 40 40 40 Number of nickels that Mike has
c. Carrying Out the Plan. Assume that girl B loses

Number of nickels that Larry has
the second round and girl A loses the first round. (if he trades his dimes for nickels)
Continue working back through the three rounds to
determine the number of chips each of the girls had
at the beginning of the game. Extra 45 cents (9 nickels) 45 cents
d. Looking Back. Check your answer by working for- that Mike has
ward from the beginning. The girl with the most chips
23. At Joe’s Cafe 1 cup of coffee and 3 doughnuts cost
at the beginning of this game lost the first round.
$4.10, and 2 cups of coffee and 2 doughnuts cost $4.60.
Could the girl with the fewest chips at the beginning
What is the cost of 1 cup of coffee? 1 doughnut?
of the game have lost the first round? Try it.
18. Sue Ellen and Angela have both saved $51 for their
family trip to the coast. They each put money in their
piggy banks on the same day but Sue Ellen started with
$7 more than Angela. From then on Sue Ellen added $4.10
$1 to her piggy bank each week and Angela put $2 in
her piggy bank each week. How much money did Sue
Ellen put in her piggy bank when they started? $4.60
19. Ramon took a collection of color tiles from a box.

24. One painter can letter a billboard in 4 hours and another
Amelia took 13 tiles from his collection, and Keiko
requires 6 hours. How long will it take them together to
took half of those remaining. Ramon had 11 left. How
letter the billboard?
many did he start with?
Billboard
20. Keiko had 6 more red tiles than yellow tiles. She gave
half of her red tiles to Amelia and half of her yellow
tiles to Ramon. If Ramon has 7 yellow tiles, how many
tiles does Keiko have now? Painter 1
Each of problems 21 through 24 is accompanied by a 1 hour

sketch or diagram that was used by a student to solve it. Painter 2
Describe how you think the student used the diagram, and
use this method to solve the problem. 1 hour
Together
21. There are three numbers. The first number is twice the
second number. The third is twice the first number. 1 hour
Problems 25 through 34 can be solved by using strate- 32. By moving adjacent disks two at a time, you can change
gies presented in this section. While you are problem- the arrangement of large and small disks shown below
solving, try to record the strategies you are using. If you are to an arrangement in which 3 big disks are side by side
using a strategy different from those of this section, try to followed by the 3 little disks. Describe the steps.
identify and record it.
25. There were ships with 3 masts and ships with 4 masts A r B s C t
at the Tall Ships Exhibition. Millie counted a total of
30 masts on the 8 ships she saw. How many of these
ships had 4 masts? 33. How can a chef use an 11-minute hourglass and a
7-minute hourglass to time vegetables that must steam
26. When a teacher counted her students in groups of 4, for 15 minutes?
there were 2 students left over. When she counted them
in groups of 5, she had 1 student left over. If 15 of her
students were girls and she had more girls than boys,
how many students did she have?
27. The movie club to which Lin belongs allows her to
receive a free DVD for every three DVDs she rents.
If she pays $3 for each movie and paid $132 over
a 4-month period, how many free movie DVDs did 34. The curator of an art exhibit wants to place security
she obtain? guards along the four walls of a large auditorium so
that each wall has the same number of guards. Any
28. Linda picked a basket of apples. She gave half of the
guard who is placed in a corner can watch the two ad-
apples to a neighbor, then 8 apples to her mother, then
jacent walls, but each of the other guards can watch
half of the remaining apples to her best friend, and she
only the wall by which she or he is placed. There is a
kept the 3 remaining apples for herself. How many ap-
maximum of one guard per corner.
ples did she start with in the basket?
a. Draw a sketch to show how this can be done with 6
29. Four people want to cross the river. There is only one security guards.
boat available, and it can carry a maximum of 200 b. Show how this can be done for each of the following
pounds. The weight of the four people are 190, 170, numbers of security guards: 7, 8, 9, 10, 11, and 12.
110, and 90 pounds. How can they all manage to get c. List all the numbers less than 100 that are solutions
across the river, and what is the minimum number of to this problem.
crossings required for the boat?
35. Trick questions like the following are fun, and they can
30. A farmer has to get a fox, a goose, and a bag of corn help improve problem-solving ability because they re-
across a river in a boat that is only large enough for quire that a person listen and think carefully about the
her and one of these three items. She does not want to information and the question.
leave the fox alone with the goose nor the goose alone a. Take 2 apples from 3 apples, and what do you have?
with the corn. How can she get all these items across b. A farmer had 17 sheep, and all but 9 died. How
the river? many sheep did he have left?
c. I have two U.S. coins that total 30 cents. One is not
31. Three circular cardboard disks have numbers written
a nickel. What are the two coins?
on the front and back sides. The front sides have the
d. A bottle of cider costs $2.86. The cider costs $2.60
numbers shown here.
more than the bottle. How much does the bottle cost?
e. How much dirt is in a hole 3 feet long, 2 feet wide,
and 2 feet deep?
6 7 8 f. A hen weighs 3 pounds plus half its weight. How
much does it weigh?
g. There are nine brothers in a family and each brother
By tossing all three disks and adding the numbers that has a sister. How many children are in the family?
show face up, we can obtain these totals: 15, 16, 17, 18, h. Which of the following expressions is correct?
19, 20, 21, and 22. What numbers are written on the (1) The whites of the egg are yellow. (2) The whites
back sides of these disks? of the egg is yellow.
Teaching Questions Classroom Connections

1. Suppose one of your elementary school students was 1. Common Core (www.corestandards.org/Math)
having trouble solving the following problem and Look again at the Spotlight on Teaching and
asked for help: “Tauna gave half of her marbles away. Learning at the beginning of this chapter. Read care-
If she gave some to her sister and twice as many to her fully the excerpt from the first of the Common Core
brother, and had 6 marbles left, how many marbles did Standards for Mathematical Practice in the middle of
she give to her brother?” List a few suggestions you can the page. What connections do you see there with
give to this student to help her solve this problem. Polya’s four-step approach to problem solving?
2. When an elementary schoolteacher who had been 2. On page 6, the example from the Elementary School
teaching problem solving introduced the strategy of Text poses a problem and solves it by the strategy of
making a drawing, one of her students said that he was Making a Drawing.
not good at drawing. Give examples of three problems a. Answer the question at the bottom of the page about
you can give this student that would illustrate that artis- the height of the fourth bounce if the ball bounces
tic ability is not required. Accompany the problems up 23 as high each time.
with solution sketches. b. Describe the strategy that you used to answer this
3. In years past, it was a common practice for teachers to question.
tell students not to draw pictures or sketches because c. What is another strategy that could also work for
“you can’t prove anything with drawings.” Today it is this problem?
common for teachers to encourage students to form 3. The NCTM Standards say that problem solving should
sketches to solve problems. Discuss this change in ap- “provide a context in which concepts and skills are
proach to teaching mathematics. Give examples of ad- learned.” Explain how the staircase model, page 11,
vantages and disadvantages of solving problems by provides this context.
making drawings.
4. The Historical Highlight on page 12 has some exam-
4. At one time, teachers scolded students for guessing the ples of the accomplishments of Hypatia, one of the first
answers to problems. In recent years, mathematics edu- women mathematicians. Learn more about her by re-
cators have recommended that guessing and checking searching history of math books or searching the
be taught to school students. Write a few sentences to Internet. Record some interesting facts or anecdotes
discuss the advantages of teaching students to “guess about Hypatia that you could use to enhance your
and check.” Include examples of problems for which elementary school teaching.
this strategy may be helpful.
5. The Problem Opener on page 3 of this section says
5. Write a definition of what it means for a question to that “Alice counted 7 cycle riders and 19 cycle wheels”
involve “problem solving.” Create a problem that is ap- and it asks for the number of tricycles. Use one or more
propriate for middle school students and explain how it of the problem-solving strategies in this section to find
satisfies your definition of problem solving. all the different answers that are possible if the riders
might have been using unicycles, bicycles, or tricycles.
20 Math Activity 1.2
MATH ACTIVITY 1.2

Pattern Block Sequences
Virtual Manipulatives Purpose: Identify and extend patterns in pattern block sequences.
Materials: Pattern Blocks in the Manipulative Kit or Virtual Manipulatives.
1. Here are the first four pattern block figures of a sequence composed of trapezoids
(red) and parallelograms (tan).
1st 2d 3d 4th
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*a. Find a pattern and use your pattern blocks to build a fifth figure. Sketch this
figure.
*b. If the pattern is continued, how many trapezoids and parallelograms will be in
the 10th figure?
c. What pattern blocks are on each end of the 35th figure in the sequence, and how
many of each shape are in that figure?
d. Determine the total number of pattern blocks in the 75th figure, and write an
explanation describing how you reached your conclusion.
2. Figures 1, 3, 5, and 7 are shown from a sequence using hexagons, squares, and
triangles.
1st 3d 5th 7th
a. Find a pattern and use your pattern blocks to build the eighth and ninth figures.
*b. Write a description of the 20th figure.
c. Write a description of the 174th, 175th, and 176th figures, and include the num-
ber of hexagons, squares, and triangles in each.
3. Use your pattern blocks to build figures 8 and 9 of the following sequence.
1st 2d 3d 4th 5th 6th
*a. Describe the pattern by which you extend the sequence. Determine the number
of triangles and parallelograms in the 20th figure.
b. How many pattern blocks are in the 45th figure?
c. The 5th figure in the sequence has a total of 7 pattern blocks. Which figure has
a total of 87 pattern blocks? Explain your reasoning.
Section 1.2 Patterns and Problem Solving 21
SECTION 1.2 PATTERNS AND PROBLEM SOLVING
The graceful winding arms

of the majestic spiral
galaxy M51 look like a
winding spiral staircase
sweeping through space.
This image of the Whirlpool
Galaxy was captured by the
Hubble Space Telescope
in January 2005 and
released on April 24,
2005, to mark the 15th
anniversary of Hubble’s launch.
PROBLEM OPENER
This matchstick track has 4 squares. If the pattern of squares is continued, how many
matches will be needed to build a track with 60 squares?
FINDING A PATTERN
Patterns play a major role in the solution of problems in all areas of life. Psychologists
analyze patterns of human behavior; meteorologists study weather patterns; astronomers
seek patterns in the movements of stars and galaxies; and detectives look for patterns among
clues. Finding a pattern is such a useful problem-solving strategy in mathematics that some
have called it the art of mathematics.
To find patterns, we need to compare and contrast. We must compare to find features
that remain constant and contrast to find those that are changing. Patterns appear in many
forms. There are number patterns, geometric patterns, word patterns, and letter patterns,
to name a few.
E X A M P LE A Consider the sequence 1, 2, 4, . . . . Find a pattern and determine the next term.
Solution One possibility: Each term is twice the previous term. The next term is 8.
EX A MP L E B Consider the sequence of figures. Find a pattern and determine the next figure.
Solution One possibility: In each block of four squares, one square is shaded. The upper left,
upper right, lower left, and lower right corners are shaded in order. The next term in this sequence
has the shaded block in the lower right corner.
EX A M P L E C Consider the sequence of names. Find a pattern and determine the next name.
Al, Bev, Carl, Donna
Solution One possibility: The first letters of the names are consecutive letters of the alphabet.
The next name begins with E.
Practices & Processes Finding a pattern requires making educated guesses. You are guessing the pattern
based on some observation, and a different observation may lead to another pattern. In
Practice 7 Example A, the difference between the first and second terms is 1, and the difference
Look for and make use of between the second and third terms is 2. So using differences between consecutive terms
structure. Mathematically as the basis of the pattern, we would have a difference of 3 between the third and fourth
proficient students look
terms, and the fourth term would be 7 rather than 8. In Example C, we might use the
closely to discern a pattern
or structure. pattern of alternating masculine and feminine names or of increasing numbers of letters
in the names.
Common Core
PATTERNS IN NATURE
The spiral is a common pattern in nature. It is found in spiderwebs, seashells, plants,
animals, weather patterns, and the shapes of galaxies. The frequent occurrence of
spirals in living things can be explained by different growth rates. Living forms curl
because the faster-growing (longer) surface lies outside and the slower-growing
(shorter) surface lies inside. An example of a living spiral is the shell of the mollusk
chambered nautilus (Figure 1.2). As it grows, the creature lives in successively larger
compartments.
Figure 1.2
Chambered nautilus
A variety of patterns occur in plants and trees. Many of these patterns are related
to a famous sequence of numbers called the Fibonacci numbers. After the first two
numbers of this sequence, which are 1 and 1, each successive number can be obtained
by adding the two previous numbers.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . .
TRY IT! 1.2.1

After 55, what are the next three Fibonacci numbers?
The seeds in the center of a daisy are arranged in two intersecting sets of spirals,
one turning clockwise and the other turning counterclockwise. The number of spirals in
each set is a Fibonacci number. Also, the number of petals will often be a Fibonacci
number. The daisy in Figure 1.3 has 21 petals.
Figure 1.3
Month
Fibonacci numbers are named after the Italian mathematician
Leonardo Fibonacci (ca. 1175–1250), who posed the follow-
1st ing problem. Suppose that a pair of baby rabbits is too young
to produce more rabbits the first month, but produces a pair
of baby rabbits every month thereafter. Each new pair of rab-
2d
bits will follow the same rule. The pairs of rabbits for the first
5 months are shown here. The numbers of pairs of rabbits for the
3d first 5 months are the Fibonacci numbers 1, 1, 2, 3, 5. If this
birthrate pattern is continued, the numbers of pairs of rabbits
in succeeding months will be Fibonacci numbers. The real-
4th ization that Fibonacci numbers could be applied to the sci-
ence of plants and trees occurred several hundred years after
the discovery of this number sequence.
5th
NUMBER PATTERNS
Number patterns have fascinated people since the beginning of recorded history.
One of the earliest patterns to be recognized led to the distinction between even
numbers
0, 2, 4, 6, 8, 10, 12, 14, . . .
and odd numbers
1, 3, 5, 7, 9, 11, 13, 15, . . .
Content Standards
The game Even and Odd has been played for generations. To play this game, one person
3.OA.9
picks up some stones, and a second person guesses whether the number of stones is odd
Identify arithmetic patterns
or even. If the guess is correct, the second person wins.
(including patterns in the
addition table or multiplication
table), and explain them using Pascal’s Triangle The triangular pattern of numbers shown in Figure 1.4 is Pascal’s
properties of operations. triangle. It has been of interest to mathematicians for hundreds of years, appearing in
Common Core
China as early as 1303. This triangle is named after the French mathematician Blaise
State Standards Mathematics Pascal (1623–1662), who wrote a book on some of its uses.
Row 0 1
Row 1 1 1
Row 2 1 2 1
Row 3 1 3 3 1
Figure 1.4 Row 4 1 4 6 4 1
EX A MP L E D 1. Find a pattern that might explain the numbering of the rows as 0, 1, 2, 3, etc.
2. In the fourth row, each of the numbers 4, 6, and 4 can be obtained by adding the
two adjacent numbers from the row above it. What numbers are in the fifth row of
Pascal’s triangle?
Solution 1. Except for row 0, the second number in each row is the number of the row. 2. 1, 5,
10, 10, 5, 1.
TRY IT! 1.2.2

What numbers are in the sixth and seventh rows of Pascal’s triangle?
Arithmetic Sequence Sequences of numbers are often generated by patterns.

The sequences 1, 2, 3, 4, 5, . . . and 2, 4, 6, 8, 10, . . . are among the first that
children learn. In such sequences, each new number is obtained from the previous
number in the sequence by adding a selected number throughout. This selected
number is called the common difference, and the sequence is called an arithmetic
sequence.
E X A M P LE E 7, 11, 15, 19, 23, . . .

172, 256, 340, 424, 508, . . .
The first arithmetic sequence has a common difference of 4. What is the common dif-
ference for the second sequence? Write the next three terms in each sequence.
Solution The next three terms in the first sequence are 27, 31, and 35. The common difference
for the second sequence is 84, and the next three terms are 592, 676, and 760.
Geometric Sequence In a geometric sequence, each new number is obtained by

multiplying the previous number by a selected number. This selected number is called
the common ratio, and the resulting sequence is called a geometric sequence.
E XA M P LE F 3, 6, 12, 24, 48, . . .

1, 5, 25, 125, 625, . . .
The common ratio in the first sequence is 2. What is the common ratio in the second
sequence? Write the next two terms in each sequence.
Solution The next two terms in the first sequence are 96 and 192. The common ratio for the
second sequence is 5, and the next two terms are 3125 and 15,625.
TRY IT! 1.2.3

Identify each sequence as arithmetic, geometric, or neither arithmetic nor
geometric. If the sequence is arithmetic, give the common difference. If it is
geometric, give the common ratio.
1. 2, 6, 18, 54, 162, . . . 2. 1, 4, 9, 16, 25, . . .
3. 43, 55, 67, 79, 91, . . .
Triangular Numbers The sequence of numbers illustrated in Figure 1.5 is neither

arithmetic nor geometric. These numbers are called triangular numbers because of the
arrangement of dots that is associated with each number. Since each triangular number
is the sum of whole numbers beginning with 1, the formula for the sum of consecutive
whole numbers can be used to obtain triangular numbers (1, 1 1 2, 1 1 2 1 3, etc.).
We saw the triangular numbers as staircases in a problem in Section 1.1.
Figure 1.5 1 3 6 10 15
Archimedes, Newton, and the German mathematician Karl Friedrich Gauss are con-
sidered to be the three greatest mathematicians of all time. Gauss exhibited a clever-
ness with numbers at an early age. The story is told that at age 3, as he watched his
father making out the weekly payroll for laborers of a small bricklaying business,
Gauss pointed out an error in the computation. Gauss enjoyed telling the story later
in life and joked that he could figure before he could talk. Gauss kept a mathematical
diary, which contained records of many of his discoveries. Some of the results were
entered cryptically. For example,
Num 5 ¢ 1 ¢ 1 ¢
is an abbreviated statement that every whole number greater than zero is the sum of
three or fewer triangular numbers.*
Karl Friedrich Gauss,
1777–1855 *H. W. Eves, In Mathematical Circles, pp. 111–115.
EX A M P L E G The first triangular number is 1, and the fifth triangular number is 15. What is the sixth
triangular number?
(6 3 7)
Solution The sixth triangular number is 1 1 2 1 3 1 4 1 5 1 6 5 5 21.
2
There are other types of numbers that receive their names from the numbers of dots
in geometric figures (see 28–30 in Exercises and Problems 1.2). Such numbers are called
figurate numbers, and they represent one kind of link between geometry and arithmetic.
Finite Differences Often sequences of numbers don’t appear to have a pattern. However,
sometimes number patterns can be found by looking at the differences between con-
secutive terms. This approach is called the method of finite differences.
EX A M P L E H Consider the sequence 0, 3, 8, 15, 24, . . . . Find a pattern and determine the
next term.
Practices & Processes
Solution Using the method of finite differences, we can obtain a second sequence of numbers
Practice 5 by computing the differences between numbers from the original sequence, as shown below. Then
Use appropriate tools
a third sequence is obtained by computing the differences from the second sequence. The process
strategically. Mathematically
stops when all the numbers in the sequence of differences are equal. In this example, when the
proficient students consider the
available tools when solving a sequence becomes all 2s, we stop and work our way back from the bottom row to the original
mathematical problem. . . . sequence. Assuming the pattern of 2s continues, the next number after 9 is 11, so the next number
Proficient students are sufficiently after 24 is 35.
familiar with tools appropriate for
their grade or course to make sound
decisions about when each of these 0 3 8 15 24
tools might be helpful, recognizing
both the insight to be gained and 3 5 7 9
their limitations.
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Practice 8
Look for and express Use the method of finite differences to determine the next term in each sequence.
regularity in repeated
1. 3, 6, 13, 24, 39, . . . 2. 1, 5, 14, 30, 55, 91, . . .
reasoning. Mathematically
proficient students notice if
calculations are repeated, and
look both for general methods INDUCTIVE REASONING
and for shortcuts.
Common Core
The process of forming conclusions on the basis of patterns, observations, examples, or
State Standards Mathematics experiments is called inductive reasoning. The National Council of Teachers of
Mathematics describes this type of reasoning in the following statement.
Identifying patterns is a powerful problem-solving strategy. It is also the essence of
inductive reasoning. As students explore problem situations appropriate to their grade
level, they can often consider or generate a set of specific instances, organize them,
and look for a pattern. These, in turn, can lead to conjectures about the problem.
EX A M P L E I Each of these sums of three consecutive whole numbers is divisible by 3.

4 1 5 1 6 5 15 2131459 7 1 8 1 9 5 24
If we conclude, on the basis of these sums, that the sum of any three consecutive whole
numbers is divisible by 3, we are using inductive reasoning.
Inductive reasoning can be thought of as making an “informed guess.” Although this

type of reasoning is important in mathematics, it sometimes leads to incorrect results.
EX A MP L E J Consider the number of regions that can be obtained in a circle by connecting points on
the circumference of the circle. Connecting 2 points produces 2 regions, connecting 3 points
produces 4 regions, etc. Each time a new point on the circle is used, the number of regions
appears to double.
2 points 3 points 4 points 5 points 6 points

Content Standards
Because many elementary and
middle school tasks rely on induc-
tive reasoning, teachers need to be
aware that students might develop
an incorrect expectation that
patterns always generalize in
ways that would be expected on 2 regions 4 regions 8 regions 16 regions
the basis of the regularities found
in the first few terms. The numbers of regions in the circles shown here are the beginning of the geometric
National Council of sequence 2, 4, 8, 16, . . . and it is tempting to conclude that 6 points will produce
Teachers of Mathematics 32 regions. However, no matter how the 6 points are located on the circle, there will not
be more than 31 regions.
Counterexample An example that shows a statement to be false is called a counter-

example. If you have a general statement, test it to see if it is true for a few special
cases. You may be able to find a counterexample to show that the statement is not true,
or that a conjecture cannot be proved.
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such grateful return as usually falls to mortal mediators. The father
and son were at vulgar loggerheads on the vulgar but important
subject of money. Living together, each wished that the other should
contribute more towards keeping up the household in as much royal
state as could be had for the money. Each also wished the other to
send away the confidential servants that other most wished to keep,
and neither would yield. Subsequently, the London papers tell how
the Cardinal went off in a great huff and princely state, and how he
was received in the ‘Italian cities with guns, like a king’s son,’ as he
was held to be. The ‘King,’ his father, is described as ‘greatly
distressed, having always counted on the affection of his son.’ At
another time came one of those scraps of news which always kept
alive a feeling of hope in the bosoms of Jacobites. ‘The Grand
Pretender’ had been for two hours in conference with the Pope, ‘on
receipt of important despatches from his Eldest Son and Heir,
Edward. The despatches are at present kept a secret.’ They were
supposed to be favourable to something, for the younger son had
promised to return. Probably some tears fell from soft Jacobite eyes
in London, at reading that, as ‘the son tarried, the father stood
patiently waiting for him, in the Hall of his House, and wept over him
when he came.’ The good-natured Pope was almost as much
touched.
All the honours conferred on the Cardinal of York in
ROMAN NEWS
Rome, and all the royal and solemn ceremonies IN LONDON
which took place on the occasion, were duly reported PAPERS.
in the London papers. The father seems to have been
warmly desirous that dignities should be heaped on the younger
son’s head. The cardinal affected, perhaps felt, reluctance. On his
gracefully yielding, the ‘Grand Pretender’ made him a present of a
set of horses.
Reports of the death of Charles Edward had been ripe enough.
The suspense was relieved when, in March, 1753, news reached
London from Rome that the old Pretender had received letters from
his son, with the information that the writer was well; but, says the
‘Weekly Journal,’ ‘the Chevalier de St. George don’t absolutely
discover where his son is.’ That he had known of his son’s
whereabout, from the first, is most certain; but he didn’t absolutely
discover it to every enquirer.
A personage of some note was in London this
A SON OF
year, the eldest son of Rob Roy,—James Drummond ROB ROY.
Macgregor. He seems to have previously petitioned
Charles Edward for pecuniary help, on the ground of suffering from
the persecution of the Hanoverian government, and to have been
willing to serve that government on his own terms. In the introduction
to ‘Rob Roy,’ Sir Walter Scott says that James Drummond
Macgregor made use of a license he held to come to London, and
had an interview, as he avers, with Lord Holdernesse. ‘His lordship
and the Under-Secretary put many puzzling questions to him, and,
as he says, offered him a situation, which would bring him bread, in
the government’s service. This office was advantageous as to
emolument, but in the opinion of James Drummond, his acceptance
of it would have been a disgrace to his birth, and have rendered him
a scourge to his country. If such a tempting offer and sturdy rejection
had any foundation in fact, it probably related to some plan of
espionage on the Jacobites, which the government might hope to
carry on by means of a man who, in the matter of Allan Breck
Stewart, had shown no great nicety of feeling. Drummond Macgregor
was so far accommodating as to intimate his willingness to act in any
station in which other gentlemen of honour served, but not
otherwise; an answer which, compared with some passages of his
past life, may remind the reader of Ancient Pistol standing upon his
reputation. Having thus proved intractable, as he tells the story, to
the proposals of Lord Holdernesse, James Drummond was ordered
instantly to quit England.’
The son of Rob Roy, hated and suspected by the JACOBITE
Jacobites, got over to Dunkirk, but he was hunted PARAGRAPHS.
thence as a spy. He succeeded in reaching Paris,
‘with only the sum of thirteen livres for immediate subsistence, and
with absolute beggary staring him in the face.’
The hopes of the friends of the Stuarts were encouraged by a
paragraph in the London sheets of 1754 stating, that though the
Chevalier was suffering from sciatica, he was well enough to receive
a stranger (in June), ‘who, by the reception he met with, was
supposed to be a person of distinction. Two days later, the banker,
Belloni, had a long private conference with the Chevalier. What
passed was not known, but what followed was; namely, a large sum
of money was advanced by the banker.’ It is easy to imagine how
paragraphs like the above stirred the pulses at the Cocoa Tree and
at St. Alban’s coffee-house.
The Jacobite interest was kept up in 1755 by paragraphs which
showed that the family were well with such a civil potentate as the
King of Spain, and with such a religious one as the Pope. The King
of Spain, it was said, had conferred a benefice on Cardinal York,
worth 6,000 piastres yearly. In the autumn the London papers
announced that ‘The Chevalier de St. George, who enjoyed the
Grand Priory of England, of the Religion of Malta, which gave him an
active and passive voice in the election of Grand Master, had
resigned it, and conferred it on a Commander Altieri. The collation
has been confirmed by the Pope.’
In the same year London was stirred by the HUME’S
publication of Hume’s ‘History of England,’ which was ‘HISTORY.’
denounced as a Jacobite history by the Whigs, and it
was not warmly received by the Jacobites, as it did not sufficiently
laud their historical favourites. ‘It is called Jacobite,’ wrote Walpole to
Bentley, ‘but in my opinion it is only not George-abite. Where others
abuse the Stuarts, he laughs at them. I am sure he does not spare
their ministers.’
But it was still to the news sent from Rome that the Jacobites
looked most eagerly for indications of what might be doing there, and
the significance of it. Under date of January 3, 1756, the paragraph
of news from Rome, the Eternal City, in the ‘Weekly Journal,’
informed all who were interested, that an Irish officer had arrived
there with letters for the Chevalier de St. George, had received a
large sum of money, on a bill of exchange, from Belloni, and had set
out again with the answers to those letters. Again, on January 17th,
the Chevalier’s friends in London were told that two foreigners had
called on him with letters, but that he refused to receive either. ‘He
refused to yield to their most earnest entreaties for an interview.’
‘Read’ communicates a no less remarkable circumstance to the
Jacobite coffee-houses. ‘Tho’ people have talked to him very much
within the last two months of an expedition on Scotland or Ireland, he
has declared that those kind of subjects are no longer agreeable to
him, and that he should be better pleased to hear nothing said about
them.’ Then came news of the Chevalier being sick, and the Pope,
not only sending his own physician, but stopping his coach to
enquire after the exile’s health. Occasionally, the paragraph of news
is communicated by a ‘Papist,’ as, for instance, in an account of the
reception into the Church of Rome of the young son of the Pasha of
Scutari, where it is said that Cardinal York performed the ceremony
of receiving the dusky convert, who had abandoned a splendid
position ‘to come,’ says the writer, at Rome, ‘and embrace our holy
religion.’
For the purpose of reading such intelligence, the AT ROME.
Jacobites opened feverishly the sheet which oftenest
satisfied their curiosity. This had to be satisfied with little. Throughout
’56 and ’57 they learnt little more than that the Pope had been ill, and
that the Chevalier and the Cardinal drove daily from their villas to
leave their names at the dwelling of the Pontiff. Next, that the quaint
Jacobite, Sir William Stanhope, had actually had an audience of the
Pope, to whom he had presented a gold box full of rhubarb; and
reasons were assigned why the contents might prove more useful
than the casket. Then, clever English lords had established
themselves in great magnificence in Roman palaces, or in villas as
magnificent as palaces; and, still more encouraging news for the
Cocoa Tree and St. Alban’s coffee-house, the King of Spain had
increased the income of Cardinal York by 1,200 crowns yearly,
drawn from the revenue of the bishopric of Malaga. On the north side
of Pall Mall, and on the lower terrace of the west side of St. James’s
Street, or beneath the Walnut tree walk in Hyde Park,—places still
much affected by Jacobites, imagination may see them wearing
congratulatory looks on the English lords collecting near the
Chevalier, and the Spanish monarch contributing money to the
Cardinal. If these things were without significance, where should
they look for incidents that would bear cheerful interpretation?
Then ensued long silence, broken only by brief
HOPES AND
announcements of archiepiscopal (and other) INTERESTS.
honours heaped upon Cardinal York, and of splendid
dinners in the Quirinal, with Pope and all the Cardinals, strong
enough to sit up, as the joyous host and guests. Not a word,
however, is to be traced with reference to Charles Edward; nor was it
looked for, at the time, by the ‘quality,’ who were contented with ‘the
happy establishment.’ On Christmas Day of this year, Walpole wrote:
‘Of the Pretenders family one never hears a word. Unless our
Protestant brethren, the Dutch, meddle in their affairs, they will be
totally forgotten; we have too numerous a breed of our own to need
princes from Italy. The old Chevalier ... is likely to precede his rival
(George II.), who, with care, may still last a few years; though I think
he will scarce appear again out of his own house.’
But the hopes and the interest of the London Jacobites had to be
maintained, and, through the London papers, the hopes and the
interest of the adherents of the Stuarts, in the country. The
aspirations of such sympathisers were hardly encouraged by an
incident of which Walpole made the following note, to Conway, in
January, 1759: ‘I forgot to tell you that the King has granted my Lord
Marischal’s pardon, at the request of M. de Knyphausen. I believe
the Pretender himself could get his attainder reversed, if he would
apply to the King of Prussia.’
In the Chatham Correspondence, it is stated that ILLNESS OF
the King of Prussia had said he should consider it a THE OLD
personal favour done to himself. The pardoning of CHEVALIER.
such an able military Jacobite as Keith, Earl
Marischal, indicated that the ‘Elector of Hanover’ considered
Jacobitism as dead, or at least powerless. At the same time, the
more mysteriously secluded Charles Edward kept himself, the more
curiosity there was among ‘curious’ people in London to learn
something about him and his designs, if he had any. The apparently
mortal illness of the Chevalier de St. George, in May 1760, caused
some of the London papers to publish a sort of exulting paragraph,
not over the supposed dying Chevalier, but over the fact, announced
in the words: ‘We shall soon know where the young Pretender is!’ Of
the father’s impending death no doubt was made. Was he not
seventy-two years of age? And had he not for thirty years of the time
been worn out with anxieties caused by his sons? One saucy
paragraph included the saucier remark:—‘He has left his estates,
which may be Nothing, to his eldest son, whom many think is
Nobody.’ But all this was premature. The old prince did not die this
year. George II. did. The grandson of the latter began to reign in
October. The Jacobites laughed at his new Majesty’s boast of being
born a Briton, for ‘James III.’ was more purely British than he; born in
London, and son of a father who was also born in this metropolis, he
was less of a foreigner than George III., whose parents were purely
German. The Jacobites made the most of this difference; and when
such of them as were in Hyde Park saw the king’s horse nearly
break his rider’s neck by suddenly flinging him out of the saddle,
those spectators probably thought of the results of King William’s fall
from horseback, and hoped that heaven was on their side. The
newspapers admiringly recorded the presence of mind of the young
king, who, though shaken, went to the play the same night, to calm
the supposed anxieties of his faithful people.
Much as Jacobites had railed at the late ‘Elector of ACCESSION
Hanover and his bloody son,’ and had devoted both OF GEORGE
of them to eternal perdition in hell, a sort of serio- III.
comic assurance that their malice was ineffective
seems to have been insinuated in the first words of the anthem, set
to music by Boyce, for the king’s (or the elector’s) funeral; namely,
‘The souls of the righteous are in the hands of God, and there shall
be no torment touch them.’ On the first occasion of George III. going
to the Chapel Royal (Sunday, November 17th), the Rev. Dr. Wilson
took his text from Malachi i. 6, where the prophet speaks of the
rebellious spirit and irreligiousness of Israel, a text which Nonjurors,
and especially the Nonjuring clergy, might well take to themselves.
After ‘Chapel’ there was a ‘Court.’ Of the latter, the
KING AND
papers say: ‘By the insolence of the soldiers many PEOPLE.
persons were not suffered to go into the Gallery. All
those that paid for seeing his Majesty were admitted, a practice, it is
hoped, will soon be put a stop to.’ The price of admission is not
stated; but among those who had gathered about the Park were
nearly a thousand tailors, who, rather than stoop to work for five
shillings a day, refused to work at all. The newspapers protested that
it was a thousand pities a press-gang or two had not been in the
Park to sweep these fellows into the ships that lacked men. If they
would not work for themselves on liberal wages, they ought to be
compelled to serve their country on less. There was no doubt about
their bravery, for the London tailors had, not long before, brilliantly
distinguished themselves under Elliot, at Gibraltar. The hint of the
amiable journalists was acted on, on the Coronation day, in 1761.
While the British-born king of a free people, over whom, he said, he
was proud to reign, was being crowned (with his young queen) in the
Abbey, ruffianly press-gangs were making very free with that people
all around the sacred edifice; seizing whom they would; knocking on
the head all who resisted; flinging them into vessels on the river, and
so despatching them to Gravesend, the Nore, and thence to men-of-
war on various stations!
One visitor is alleged to have been present at this CHARLES
coronation, who certainly was not an invited, nor EDWARD AT
would he have been a welcome, guest. This visitor is WESTMINSTE
said to have been Charles Edward himself! As he is R.
also credited with two or three earlier visits to London, the question
as to the truth of the reports may be conveniently considered here.
We will only remark that, in the closing years of the reign of
George II., Jacobites, who had neither been harmless nor intended
to remain so if opportunity favoured them, were allowed to live
undisturbed. As Justice Foxley remarked to Ingoldsby, they attended
markets, horse-races, cock-fights, fairs, hunts, and such like, without
molestation. While they were good companions in the field and over
a bottle, bygones were bygones.
CHAPTER XIV.
(1744 to 1761.)
subject of great interest in the life of Charles Edward
presents itself to consideration in the alleged romantic,
but particularly absurd, incidents of his various
appearances in London, or England. These doubtful
visits commence with the year 1744, and close with the
no longer young Chevalier’s supposed presence at the coronation of
George III., 1761.
In the former year, there was residing at Ancoats, near
Manchester, Sir Oswald Mosley, who had been created a baronet by
the Hanoverian king, George I., in 1720. At the end of nearly a
quarter of a century, if common report do not lie, he seems to have
been a thorough Jacobite, with Charles Edward for his guest, in
disguise! The ‘fact’ is first recorded in Aston’s ‘Metrical Records of
Manchester,’ in the following doggrel lines:—
In the year ’44, a Royal Visitor came,
Tho’ few knew the Prince, or his rank, or his name—
To sound the opinions and gather the strength
Of the party of Stuart, his house, ere the length
Then in petto to which he aspired
If he found the High Tories sufficient inspired
With notions of right, indefeasive, divine,
In favour of his Royal Sire and his line.
No doubt, he was promis’d an army, a host!
But he found to his cost, it was all a vain boast;
For when he return’d in the year ’45,
For the crown of his father, in person to strive,
When in Scottish costume at the head of the clans
He marched to Mancunium to perfect his plans,
The hope he had cherish’d, from promises made,
Remains to this day as a debt that’s unpaid.
A foot-note states that the prince was the guest of CHARLES
Sir Oswald for several weeks, ‘no doubt, to see the EDWARD IN
MANCHESTER
inhabitants of Manchester and its vicinity, who were .
attached to the interests of his family.’
At that time, a girl was living in Manchester, who was about
fourteen years of age. For seventy succeeding years she used to
relate that in 1744, a handsome young gentleman used to come
from Ancoats Hall into Manchester, every post day, to the inn and
post house of her father, Bradbury, for letters or to read the papers
from London, in which papers, as he sat apart, he seemed to take
unusual interest. The girl admired his handsome countenance, his
genteel deportment, and the generous spirit which led him to give
her half-a-crown for some trivial chamber-maid service. In the
following year, when Charles Edward marched past her father’s
house at the head of his troops, the girl made outspoken recognition
of him as the liberal donor of the welcome half-crown. The father, ill-
pleased at her demonstration, drove her in, and silenced her with
threats; but when all danger had ceased to exist, he acknowledged
that the handsome young fellow with the genteel deportment and the
young Chevalier were one and the same.—Such is the substance of
a corroborative story told by a later Sir Oswald Mosley, Bart., in
‘Family Memoirs,’ printed in 1849 for private circulation.
In Miss Beppy Byrom’s Diary, she narrates an MISS
interview which some of the leading Jacobites of BYROM’S
Manchester had with the prince when he was there in DIARY.
the ’45 rebellion. These included her celebrated
father, John Byrom, Deacon, the father of the unlucky young captain
who was afterwards executed on Kennington Common, Clayton, and
others. The day was St. Andrew’s Day, Saturday, November 30th.
Many ladies were making crosses of St. Andrew; Miss Byrom
dressed in white to go and see the prince, who witched her with his
noble horsemanship. The horse seemed self-conscious of bearing a
king’s son. After the review, the lady and others went to church. ‘Mr.
Skrigley read prayers. He prayed for the King and Prince of Wales,
but named no names.’ There was much mild dissipation afterwards,
with too much restlessness to partake of settled meals, but infinite
sipping of wine to Jacobite healths. In the evening, after having seen
the prince at table, the lady and many companions drank more
healths in the officers’ room. ‘They were all exceeding civil,’ she
says, ‘and almost made us fuddled with drinking the P.’s health, for
we had had no dinner. We sat there till Secretary Murray came to let
us know the P. was at leisure, and had done supper; so we all had
the honour to kiss his hand. My papa was fetched prisoner to do the
same, so was Dr. Deacon. Mr. Cattell and Mr. Clayton did it without.
The latter said grace for him. Then we went out and drank his health
in another room,’ &c., &c. This record is quoted in ‘Notes and
Queries,’ May 1, 1869, and as it makes no reference to the alleged
visit of 1744 (only one year before), it may be taken as demolishing
the earliest legend of the legendary visits of Charles Edward to
England.
The next in order of date is a very undefined visit THE VISIT IN
of 1748. In support of it there appears that 1748.
exceedingly, questionable witness, namely,
Thicknesse.
Crazy Philip Thicknesse, in his crazy Memoirs, on the title-page of
which he crazily announced that he had the misfortune to be the
father of George Thicknesse Tuchet. Lord Audley (the son George
had succeeded to the ancient barony, through his deceased mother)
was the man who, on his son refusing to supply him with money, set
up a cobbler’s stall, opposite the son’s house, with a board on which
was painted, ‘Boots and shoes mended in the best and cheapest
manner, by Philip Thicknesse, father of Lord Audley.’ This had the
desired effect. In the farrago, called his Memoirs, Thicknesse says
he knew ‘an Irish officer who had only one arm.’ In a note, the name
Segrave is given as that of the officer; but this editorial addition has
been transferred to the text by all writers who have quoted crazy
Philip’s account. The officer with only one arm assured Thicknesse
that he had been with the Prince in England, between the years
1745 and 1756, and that ‘they,’ Prince and one-armed officer, ‘had
laid a plan of seizing the person of the King, George II., as he
returned from the play, by a body of Irish chairmen, fifteen hundred
of whom were to begin a revolution, in Lincoln’s Inn Fields.’ Philip,
however, with a return of sense, remarks: ‘I cannot vouch for the
truth of this story.’ Yet out of this unfounded story grew a report that
Charles Edward was in London in 1748, which was between the
years above named. Philip Thicknesse was in his 70th year when he
began to put together his book, which was published in 1788. He
reminds his readers, that he ‘never pretended to be an accurate
writer.’ The reminder was hardly necessary.
The next witness, in chronological order, is Dr. THE VISIT IN
King, the Chevalier’s great agent, who gives the year 1750.
1750, as that in which Charles Edward came to
London. This information was first furnished in a book which was
published in 1818, under the title, ‘Political and Literary Anecdotes of
his own time.’
The editor is anonymous. He gives this account of how he came
in possession of the MS. ‘A Friend’ (no name given) ‘who was a long
time a prisoner in France, met with the following work in the
possession of two ladies’ (not named, but who are described as)
‘relations of the writer, Dr. King. From the interesting passages which
he was permitted to extract, the Editor’ (as destitute of name as the
others) ‘conceived that the original might be well worthy of
publication, he therefore desired his friend to procure it, and found,
on a comparison of the hand-writing with that which is well
ascertained to be Dr. King’s, in the account books of St. Mary Hall, in
Oxford,—that there is every reason to suppose this MS. to have
been written by Dr. King himself.’ Four nameless persons, and only
‘a reason to suppose’ among them.
Dr. King’s life extended from 1685 to 1763; and it DR. KING AND
was towards the close of his life, that he collected the THE
anecdotes from the manuscript of which the editor CHEVALIER.
(1818) was permitted to take extracts. Where the
original manuscript is to be found is not mentioned. The only
reference to the young Chevalier of any importance is in the
paragraph in which the writer leads us to infer that the prince was in
England in September, 1750, at Lady Primrose’s house. ‘Lady
Primrose,’ he says, ‘presented me to ——’ Why this mysterious
dash, when frequent mention is made of Charles Edward, in
description of character, as ‘the Prince’ or ‘Prince Charles?’ It is also
stated that the prince was King’s guest, and was recognised by
King’s servants. For a Jacobite, the doctor is as severe a dissector of
the young Chevalier as the bitterest Whig could desire. He speaks ill
of the illustrious visitor, morally and intellectually. As to his religion,
King says he was quite ready to ‘conform’ to the religion of the
country; that he was a Catholic with the Catholics, and with the
Protestants, a Protestant. This was exactly what Lord Kilmarnock
said before he was executed. King further states that Charles
Edward would exhibit an English Common Prayer Book to Protestant
friends; to the Catholics he could not have afforded much pleasure
by letting Gordon, the Nonjuror, christen his first child, of which Miss
Walkenshawe was the mother. Such an easy shifting of livery, from
Peter’s to Martin’s, and back again to Peter’s, was natural enough in
the case of a man, who had been brought up at Rome, but who was
placed under the care of a Protestant tutor, who of express purpose
neglected his education, and who, if King’s surmise be correct, made
a merit of his baseness, to the Government in London, and was
probably rewarded for it by a pension. Dr. King speaks of the prince’s
agents in London, as men of fortune and distinction, and many of the
first nobility, who looked to him as ‘the saviour of their country.’
This visit to London in 1750, if it really was ever
MEMORANDA.
made, is supposed to be referred to, in one of several
memoranda for a letter in the prince’s handwriting, preserved with
other Stuart papers, in Windsor Castle; and first published by Mr.
Woodward, Queen’s Librarian, in 1864. It runs thus: ‘8thly. To
mention my religion (which is) of the Church of England as by law
established, as I have declared myself when in London, the year
1750.’ This memorandum is at the end of a commission from the
writer’s father dated 1743, to which commission is appended a copy
of the ‘Manifesto’ addressed by the prince to Scotland, in 1745. At
what date the memorandum was written there is no possibility of
knowing. If the prince, as was his custom, used only the initial of the
name of the city, it is possible that Liége was meant; and, after the
word ‘when,’ the writer may have omitted the name of one of his
many agents of ‘fortune and distinction,’ who looked to him as the
saviour of their country.
There are other memoranda for letters, supposed
FURTHER
to refer to the above visit. For example:—‘Parted, ye MEMORANDA.
2nd Sept. Arrived to A, ye 6th, parted from thence, ye
12th Sept. E, ye 14th, and at L, ye 16th. Parted from L, ye 22nd, and
arrived at P. ye 24th. From P, parted ye 28th, arrived here ye 30th
Sept.’ In this memorandum the initials are supposed to stand for
Antwerp, England, London, and Paris. There is nothing to prove that
they do; and, it may be said that A and L quite as aptly represent
Avignon and Liége. However this may be, dates and supposed
places are entirely at variance from other dates and places which are
taken as referring to this identical visit of the young Chevalier to
London, in 1750. ‘Ye 5th Sept. O.S. 1750, arrived; ye 11th parted to
D, ye 12th in the morning parted and arrived at B, and ye 13th at P.
R. S. ye 16th Sept. ye 22nd, 23rd, and 24th.’ Here, D and B are
interpreted as signifying Dover and Boulogne, P. is Paris. R. S. have
received no interpretation. It is certain that one of the two records
must be incorrect; and both of them may be.
But, something more definite is reached in a CHARLES
despatch from the British Minister at Florence (Mann), EDWARD’S
which Lord Stanhope published in his ‘Decline of the STATEMENT.
Stuarts.’ The minister, who writes in 1783, describes a
conversation which took place at Florence, between Charles Edward
(then known as Count d’Albany) and Gustavus, King of Sweden, in
the course of which the count told the king that, in September, 1750,
he arrived secretly in London with a Colonel Brett; that together they
examined the outer parts of the Tower, and came to the conclusion
that one of the gates might be blown in by a petard. After which, at a
lodging in Pall Mall, where fifty Jacobites were assembled, including
the Duke of Beaufort and the Earl of Westmoreland, the prince said
to these Jacobites, or rather to Gustavus, that if they could have
assembled only 4,000 men, he would have publicly put himself at
their head. He added that he stayed a fortnight in London, and that
the Government were ignorant of his presence there.
It is to be remembered that this story was told three and thirty
years after the alleged occurrence. The narrator was then an aged
man, whose brains and memory and general health were so
damaged by ‘the drink, the drink, dear Hamlet!’ that not the slightest
trust could be placed in any single word that he uttered in respect to
his past history. He may have dreamed it all, but that any two
gentlemen, the face of one of whom was familiar, from prints and
busts publicly sold, could have so carefully examined the Tower as to
find out where it was vulnerable, without the sentinels having
discovered the same part in the explorers, is surely incredible. The
vaunt of the secret visitor publicly placing himself at the head of an
army of Jacobites, was just such a boast as the brainless drunkard
of 1783 would be likely to make. There is as little reliance to be put
on the statement of the Duke of Beaufort and Earl of Westmoreland
being present at a Jacobite meeting in Pall Mall. The really Jacobite
duke died in 1746. His successor, and also the Earl of Westmoreland
(of the year 1750), may have been often in opposition to the
Government, but no act of their lives would warrant the belief that
they could be insane enough to attend a meeting of half a hundred
Jacobites in Pall Mall, to listen to a project for blowing up the Tower
and pulling down the throne.
Two years after 1750, however, according to the THE VISIT IN
MS. Journal of Lord Elcho, Charles Edward was 1752-3.
again in London, secretly at the house of the very
outspoken Jacobite lady, Lady Primrose. Hume, the historian, says,
in a letter to Sir John Pringle (dated 1773), that he knew with the
greatest certainty that Charles Edward was in London in 1753; his
authority was Lord Marischal, ‘who said it consisted with his certain
knowledge.’ The knowledge was derived from a lady—whom my
Lord refused to name, and whom Hume imagined to be Lady
Primrose. Now, Lady Primrose was the Protestant daughter of the
Dean of Armagh, of Huguenot descent, bearing the name of
Drelincourt. She was the widow of Viscount Primrose who had been
an officer of distinction in the king’s service. Lady Primrose, herself,
was a warm-hearted Jacobite who had given a temporary home in
Essex Street, Strand, to Flora Macdonald, during part of her brief
sojourn in London in 1747. According to this legendary visit of 1753,
Charles Edward, unexpectedly, entered her room, when she was
entertaining a company at cards. He was there unannounced, yet
Lady Primrose called him by a name he assumed! Her object was to
keep him undetected by her friends; but his portrait hung in the
room, and the company identified the visitor. Lord Marischal told
Hume (he thinks, ‘from the authority of the same lady,’ whom Lord
Marischal had refused to name), that the Prince went about the
streets and parks, with no other disguise than not wearing ‘his blue
ribband and star.’ Some years after, Hume spoke of this visit, to Lord
Holdernesse (who in 1753 was Secretary of State). This minister
stated that he received the first intelligence of Charles Edward’s
presence in London from George II.; who may have been
misinformed, and who is reported to have said, ‘When he is tired of
England, he will go abroad again!’ A very unlikely remark. Another
story resembled that of the Lincoln’s Inn Fields’ chairmen, namely,
that in 1753, Lord Elibank, his brother Alexander Murray, and five
dozen associates, were to be employed in carrying off this very
good-natured monarch!
As to the credibility of this story, it is only CREDIBILITY
necessary to remark that, in 1753, Dr. Archibald OF THE
Cameron was hanged in London for being present in STORIES.
Scotland, where mischief was intended; and that, if
the Ministry were so well served by their spies, such as Sam
Cameron was, through whom the Doctor was arrested and executed,
Charles Edward could not possibly have escaped; and his capture
was of great importance at the moment. Moreover, the king was
powerless. It belonged to the Administration to decide whether the
undisguised Prince should be captured or allowed to go free.
Assuming that he was so allowed, he is again CONFLICTING
found in London in 1754. At least, crazy Thicknesse STATEMENTS.
says: ‘that this unfortunate man was in London, about
the year 1754, I can positively assert. He was “at a lady’s house, in
Essex Street;” was recognised in the Park, by a Jacobite gentleman
who attempted to kneel to him, and this so alarmed the lady in Essex
Street, that a boat was procured the same night, in which he was
forthwith despatched to France. Tonnage of boat and captain’s name
not registered.
Later, the date of this last visit is given in a letter, addressed by
Lord Albemarle, British ambassador in Paris, to Sir Thomas
Robinson, namely, May 1754. The writer, in August, 1754, states that
he had been ‘positively’ assured by a discontented Jacobite, that ‘no
longer ago than about three months,’ Charles Edward had been in
London, ‘in a great disguise as may be imagined;’ that the prince had
received friendly notice, at Nottingham, that he was in danger of
being seized, and that he immediately fled. As to the authority, Lord
Albemarle writes:—‘The person from whom I have this, is as likely to
have been informed of it as any of the party, and could have had no
particular reason to have imposed such a story upon me, which
could have served no purpose.’ The ambassador is mistaken. The
purpose of such stories was to keep warm the hopes—fading hopes
—of the Jacobites, and it was not the last story invented with that
purpose in view.
Lastly, there is the story of the prince’s presence at AT THE
the coronation festival of George III., in 1761. CORONATION.
According to some authorities, it was without any
stirring incident. Others say, that very stirring matter indeed sprang
from it, and that much confusion was the consequence.
Walpole, describing the illustrious people, state officers, and
others at the coronation-banquet of George III., September 1761,
pauses at sight of the son of the unhappy Lord Kilmarnock. ‘One
there was ... the noblest figure I ever saw, the High Constable of
Scotland, Lord Errol’ (he had succeeded to this title through his
mother), ‘as one saw him in a place capable of containing him, one
admired him. At the wedding, dressed in tissue, he looked like one of
the Giants in Guildhall, new gilt. It added to the energy of his person
—that one considered him acting so considerable a part in that very
Hall, where, so few years ago, one saw his father, Lord Kilmarnock,
condemned to the block.’ In 1746, Lord Errol, then Lord Boyd, had
fought at Culloden, against his father.
They who were still of that father’s way of thinking
AT THE
were for long afterwards comforted by a story that BANQUET.
when the King’s Champion proclaimed George III.
king, and challenged all who questioned the right of him so
proclaimed, by throwing down his glove, a Champion of James III.
boldly stept forward, took up the glove, and retired with it
unmolested. The story, so to speak, got crystalised. It is still partially
believed in. It may have arisen out of an incident chronicled in
‘Burke’s Peerage.’ It is there said that, officiating at the coronation as
Constable of Scotland, Lord Errol, by accident, neglected to doff his
cap when the king entered; but on his respectfully apologising for his
negligence, his majesty entreated him to be covered, for he looked
on his presence at the ceremony as a very particular honour.’ This
wears an air of absurdity. However that may be, Scott has made use
of the alleged challenge of the king’s right to his crown.
It occurs in ‘Redgauntlet,’ where Lilias swiftly passes through the
covering lines of Jacobites, takes up the gauntlet, and leaves a
pledge of battle in its stead. But contemporary accounts take no note
of any such occurrence. Walpole, an eye-witness, merely records:
‘The Champion acted his part admirably, and dashed down his
gauntlet with proud defiance. His associates, Lord Talbot, Lord
Effingham, and the Duke of Bedford were woful. Lord Talbot [the
Lord High Steward] piqued himself on backing his horse down the
hall and not turning its rump towards the king; but he had taken such
pains to address it to that duty, that it entered backwards; and, at his
retreat, the spectators clapped, a terrible indecorum.’ This
indecorous clapping, as the Champion (Dymoke) and his knights
backed out of the hall may have been taken by those who were not
aware of the cause as some party expression. Out of
GEORGE AND
it the story of the Jacobite taker-up of the glove may CHARLES
have arisen. The story was told with a difference. A EDWARD.
friend (who is anonymous) informed the Earl
Marischal that he had recognised Charles Edward among the
spectators at the coronation banquet, and had spoken to him. The
prince is said to have replied: ‘I came only out of curiosity; and the
person who is the object of all this magnificence is the one I envy the
least.’ Scott, in a note to the incident in ‘Redgauntlet,’ remarks,
—‘The story is probably one of the numerous fictions that were
circulated to keep up the spirits of a sinking faction. The incident
was, however, possible, if it could be supposed to be attended by
any motive adequate to the risk.... George III., it is said, had a police
of his own, whose agency was so efficient that the Sovereign was
able to tell his Prime Minister, on one occasion, to his great surprize,
that the Pretender was in London. The Prime Minister began
immediately to talk of measures to be taken, warrants to be
procured, messengers and guards to be got in readiness. “Pooh!
pooh!” said the good-natured Sovereign, “since I have found him out,
leave me alone to deal with him.” “And what,” said the Minister, “is
your Majesty’s purpose in so serious a case?” “To leave the young
man to himself,” said George III., “and when he tires, he will go back
again.” The truth of this story does not depend on that of the lifting of
the gauntlet, and while the latter could be but an idle bravado, the
former expresses George III.’s goodness of heart and soundness of
policy.’
Altogether it is very clear that dates, persons, and A
places have been inextricably mixed up in the DISQUALIFICATIO
Jacobite legends of the Chevalier’s visit to London. At N.
the same time there seems to be but one opinion
among all writers, without exception, who have dealt with this subject
hitherto, namely, that the alleged visit of 1750 actually occurred.
Perhaps the best evidence is furnished in the ‘Diary of a Lady of
Quality’ (Mrs. Wynne). The writer’s grandson states that his
grandmother had frequently told him that she had had, from Lady
Primrose herself, full particulars of the visit of Charles Edward to
London in 1750. A few questions, however, might easily break down
even this assertion. After all, the decision must be left to the reader’s
judgment.
Although no overt act answered the Champion’s challenge in
Westminster Hall, the right of George III. to succeed to the crown
was vigorously denied in very High Church coteries. Soon after the
king’s birth, in 1738, he was baptised by Secker, Bishop of Oxford.
Now, Secker was born and bred a dissenter, and did not enter the
Church till after he had been a medical student, and had run a not

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Mathematics for elementary teachers : a

conceptual approach Tenth Edition.

(eBook PDF) Mathematics for Elementary School Teachers:

(eBook PDF) A Problem Solving Approach to Mathematics

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(eBook PDF) Mathematics for Elementary School Teachers

Mathematics for Elementary Teachers with Activities 5th

(eBook PDF) A Problem Solving Approach to Mathematics

Using & Understanding Mathematics: A Quantitative

(eBook PDF) Mathematical Reasoning for Elementary

BENNETT BURTON NELSON EDIGER

Albert B. Bennett, Jr.

This book is printed on acid-free paper.

Library of Congress Cataloging-in-Publication Data

2.1 Sorting and Classifying Attribute Pieces 40

3.1 Numeration and Place Value with Base-Five Pieces 82

4.1 Divisibility with Base-Ten Pieces 172

7.1 Forming Bar Graphs with Color Tiles 404

8.1 Experimental Probabilities from Simulations 486

9.1 Extending Tile Patterns 552

10.1 Angles in Pattern Block Figures 618

11.1 Perimeters of Pattern Block Figures 706

12.1 Tracing Figures from Motions with Tiles 786

CHAPTER 1 PROBLEM SOLVING 1

CHAPTER 2 SETS AND REASONING 39

CHAPTER 3 WHOLE NUMBERS 81

CHAPTER 4 NUMBER THEORY 171

CHAPTER 5 INTEGERS AND FRACTIONS 215

CHAPTER 6 DECIMALS: RATIONAL AND IRRATIONAL NUMBERS 301

CHAPTER 7 STATISTICS 403

CHAPTER 8 PROBABILITY AND COUNTING 485

CHAPTER 9 ALGEBRA 551

CHAPTER 10 GEOMETRIC FIGURES 617

10.2 Polygons and Tessellations 642

CHAPTER 11 MEASUREMENT 705

CHAPTER 12 TRANSFORMATIONS: CONSTRUCTIONS, CONGRUENCE,

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SPOTLIGHT ON TEACHING AND LEARNING

THINK ABOUT THIS

MATH ACTIVITY 1.1

Tiles (Pegs) Minimum Number

SECTION 1.1 INTRODUCTION TO PROBLEM SOLVING

Sometimes the main difficulty in solving a problem is knowing what question is to

*M. R. Frame, “Hamann’s Conjecture,” Arithmetic Teacher.

TRY IT! 1.1.1

the ball is drop