More Rich & Engaging Mathematical Tasks Grades 5–9

Articles amd Activities from Teaching Children Mathematics Mathematics Teaching in the Middle School Mathematics Teacher Student Math Notes; NCTM Yearbook

Edited by

Glenda Lappan Michigan State University

Margaret Schwan Smith University of Pittsburgh

Elizabeth Jones Lansing Public Schools, Lansing, Michigan

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ISBN 978-0-87353-722-3.

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The publications of the National Council of Teachers of Mathematics present a variety of viewpoints. The views expressed or implied in this publication, unless otherwise noted, should not be interpreted as official positions of the Council. Printed in the of America Table of Contents

Preface ...... ix

Acknowledgments ...... x

1. Designing and Enacting Rich Instructional Experiences 1

Introduction ...... 2 Making the Right (Discourse) Moves: Facilitating Discussions in the Mathematics Classroom ...... 3 G. T. Springer and Thomas Dick MT, September 2006 Mathematical Tasks as a Framework for Reflection: From Research to Practice ...... 9 Mary Kay Stein and Margaret Schwan Smith MTMS, January 1998

2. The Meaning of, and Operations on, Rational Numbers 19

Introduction ...... 20

Creating, Naming, and Justifying Fractions ...... 23 Daniel Siebert and Nicole Gaskin TCM, April 2006 Using Alice in Wonderland to Teach Multiplication of Fractions ...... 31 Susan B. Taber MTMS, December 2006/January 2007 Dividing Fractions: Reconciling Self-Generated Solutions with Algorithmic Answers ...... 39 Marcela D. Perlwitz37 MTMS, February 2005 A Conceptual Model for Solving Percent Problems ...... 45 Albert B. Bennett, Jr., and L. Ted Nelson MTMS, April 1994 Making Sense of Percents ...... 53 Mary Lou Metz MTMS, September 2003 Mathematics at the Mall ...... 55 Francis (Skip) Fennell TCM, January 1998 3. Ratios, Rate, and Proportional Relationships 63

Introduction ...... 64 Rename It: Teaching Notes ...... 66 Susan J. Lamon Making Sense of Fractions, Ratios, and Proportions, 2002 NCTM Yearbook Part-Whole Comparisons with Unitizing ...... 71 Susan J. Lamon Making Sense of Fractions, Ratios, and Proportions, 2002 NCTM Yearbook Can You See It? Teaching Notes ...... 74 Susan J. Lamon Making Sense of Fractions, Ratios, and Proportions, 2002 NCTM Yearbook How Long Is Its Projection? ...... 77 Sandra Davis Trowell and Anne M. Reynolds MTMS, A