Eyes on Math. A Visual Approach to Teaching Math Concepts

247 Pages·2016·7.94 MB·English

by  Small M.

Eyes on Math
A Visual Approach to Teaching Math Concepts
Eyes on Math
A Visual Approach to Teaching Math Concepts
MARIAN SMALL
Illustrations by Amy Lin
Teachers College
Columbia University
New York and London

1906 Association Drive,
Reston, VA 20191 www.nelson.com
www.nctm.org
Published simultaneously by Teachers College Press, 1234 Amsterdam Avenue, New York, NY 10027, and National Council of
Teachers of Mathematics, 1906 Association Drive, Reston, VA 20191; distributed in Canada by Nelson Education, 1120 Birchmount
Road, Toronto, ON, Canada M1K 5G4.
Published by Teachers College Press, 1234 Amsterdam Avenue, New York, NY 10027
Copyright © 2013 by Teachers College, Columbia University
All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical,
including photocopy, or any information storage and retrieval system, without permission from the publisher.
Library of Congress Cataloging-in-Publication Data
Small, Marian.
Eyes on math: a visual approach to teaching math concepts / Marian Small; illustrations by Amy Lin.
pages cm
Includes bibliographical references and index.
ISBN: 978-0-8077-5391-0 (paper: acid-free paper)
1. Mathematics—Study and teaching—Audio-visual aids. 2. Pictures in education. I. Lin, Amy. II. Title.
QA18.S63 2012
510.71—dc23
2012030002
ISBN: 978-0-8077-5391-0 (paper)
NCTM Stock Number: 14573
e-ISBN: 978-0-8077-7236-2
Contents
Preface
Acknowledgments
1.  Background
Visualization in Math
Focusing on the Important Math
Building Mathematical Communication
2.  Grades K–2
Counting Up by 1s
Counting Back by 1s
Counting Up by 2s
Counting Back by 2s
Counting Up by 5s
Counting Back by 5s
Counting Up by 10s
Counting Back by 10s
Comparing Numbers by Matching
Benchmark Numbers: All About 5
Benchmark Numbers: All About 10
Ordinal Numbers
Addition as Combining
Addition to Describe Part-Part-Whole Situations
Adding 0 and Adding 1
Addition: Commutativity
Addition: Changing Addends, But Not the Sum
Adding or Subtracting 10
Subtraction as Taking Away
Subtraction to Compare
Relating Addition and Subtraction
Naming Two-Digit Numbers
Naming Three-Digit Numbers
Place Value: Grouping in Tens
Place Value: Positions
Comparing Sizes of Numbers
Fractions: Halves
Fractions: Quarters
Measurement: Meaning of Length
Measurement: Effect of Unit Size
Measurement: Standard Units of Length
2-D Shapes Versus 3-D Shapes
Comparing 2-D Shapes
Comparing 3-D Shapes
Composing Shapes
Shape Puzzles
3.  Grades 3–5
Multiplication: Equal Groups
Multiplication: Commutativity
Multiplication: The Distributive Principle
Multiplication: 2-Digit by 2-Digit
Division as Equal Groups or Sharing
Division: Remainders
Rounding Numbers
Place Value: Multiplying and Dividing by Powers of 10
Place Value: Renaming Numbers
Factors: What They Are
Factors Come in Pairs
Fractions: Representing
Fractions: Equivalence
Fractions: Comparing
Fractions: Mixed Number/Improper Fraction Relationship
Fractions: Common Denominators
Adding Fractions
Multiplying Fractions
Fractions: Multiplying as Resizing
Fractions as Division
Decimals: Relating Hundredths to Tenths
Decimals: Equivalence
Decimals: Adding and Subtracting
Measurement: Time Intervals
Measurement: Area of Rectangles
Perimeter versus Area
Measurement Conversions
Graphs with Scales
Coordinate Grids
Classification of Shapes
Parallel and Perpendicular Lines
Lines of Symmetry
Patterns versus Non-patterns
Algebraic Thinking: Growing Additively
Algebraic Thinking: Shrinking Additively
Algebraic Thinking: Growing Multiplicatively
4.  Grades 6–8
Common Factors
Common Multiples
Square Roots
Fraction Division
Ratios: Multiple Ratios Describe Any Situation
Equivalent Ratios
Equivalent Rates
Solving Rate Problems
Describing Percent
Uses of Integers
The Zero Principle
Subtraction of Integers as a Directed Distance
Multiplication and Division of Integers
Area of a Parallelogram
Area of a Triangle
The Pythagorean Theorem
Pi
How Measurements Are and Are Not Related
Mean
Variability
Sampling
Probability: What It Means
Unpredictability
Rotations, Reflections, and Translations
Scale Drawings
Dilatations
Angles with Parallel Lines
Equivalent Expressions
Equation as a Balance
Different Types of Equations
What Is Linear?
Role of the Slope in the Equation of a Line
Systems of Equations
Function Rules
Appendix: Connections Between Topics and Common Core Standards
References
Index
About the Authors
Preface
A COLLEAGUE who works in the area of high school mathematics pointed out how well her
students responded to the presentation of a simple picture that evoked mathematical thinking—no
textbook or complicated activity was necessary. This got us thinking about the potential for a resource
that focused on presentation of important mathematical ideas through pictures at all levels, not just for
secondary grades.
ORGANIZATION OF THE BOOK
Chapter 1 provides an introduction that addresses some of the research on the value of visual
representations of mathematical ideas. It also discusses the importance of questioning that evokes a
deeper understanding of mathematical concepts.
Chapters 2, 3, and 4 provide visuals and questions for discussion of selected mathematical
topics in grades K–2, 3–5, and 6–8, respectively. The intention is to help the teacher build a rich
mathematical environment by using carefully constructed visuals to stimulate student thinking about
mathematical concepts. Although, in certain instances, the visuals may seem similar to what educators
have seen elsewhere, coupling the images with the carefully constructed questions that are provided
is likely to be much more useful in developing robust understanding of the essential ideas underlying
the concept the picture embodies than using either approach in isolation.
The online component of the book provides vibrant color pictures that can form the basis of
valuable mathematical conversations. These can be downloaded (as either PDF or JPEG files) and
projected onto a classroom smart board or screen. The printed book provides teachers with support
for these conversations by
Clarifying the main idea that the visual is intended to bring out and specifying where that concept
appears in the Common Core State Standards for mathematics. The Appendix provides a
convenient listing of topics sorted by standard.
Explaining why the idea is an important one to develop in a mathematics classroom
Explaining how the particular mathematical idea relates to other mathematical learning
Specifically explaining why the provided visual is likely to evoke the idea
Providing a series of additional questions to use in a conversation with students about the
picture and the underlying mathematical idea(s), along with an explanation for teachers about
how each question develops a particular key understanding
Providing extensions. These extensions challenge students to create a drawing, use
manipulatives, or otherwise create a new and related example. The extensions can be used to
help students reinforce the new learning, to help teachers assess understanding, or both.
USING THIS BOOK WITH OTHER RESOURCES
Most teachers will find that the activities in this book nicely supplement whatever textbook or other
materials they currently use. Teachers might use the Appendix or table of contents to explore useful