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Mathematics at the Mall 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 Copyright © 2012 The National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM. Copyright © 2012 by THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, INC. 1906 Association Drive, Reston, VA 20191-1502 (703) 620-9840; (800) 235-7566; www.nctm.org All rights reserved ISBN 978-0-87353-722-3. The National Council of Teachers of Mathematics is a public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research. When forms, problems, and sample documents are included or made avaialble on NCTM's website, their use is authorzed for educational purposes by educators and noncommercial or nonprofit entities that have purchased this book. Except for that use, permission to photo- copy or use material electronically from More Rich and Engaging Mathematical Tasks: Grades 5–9, must be obtaied from www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for- profit organization that provides licenses and registration for a variety of users. Permission does not automatically extend to any items identified as reprinted by permission of other publishers and copyright holders. Such items must be excluded unless separate permissions are obtained. It will be the responsibility of the user to identify such materials and obtain the permissions. 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 United States 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, April 2004 Conceptualizing Ratios with Look-Alike Polygons .............................................................. 81 Nancy K. Dwyer, Betty J. Causey-Lee, and Nekeya M. Irby; MTMS, April 2003 A Science Application of Area and Ratio Concepts ..............................................................87 Virginia M. Horak MTMS, April 2006 Locusts for Lunch: Connecting Mathematics, Science, and Literature .................................95 Richard A. Austin, Denisse R. Thompson, and Charlene E. Beckmann MTMS, November 2006 The Giants Project .............................................................................................................. 104 Suzanne Levin Weinberg, Penny L. Hammrich, and Matthew H. Bruce MTMS, April 2003 Are We Golden? Investigations with the Golden Ratio ......................................................111 Stacy Reeder117 MTMS, October 2007 House Proportions ............................................................................................................... 117 NCTM SMN, November 2006 4. Numbers and Number Theory 125 Introduction ......................................................................... 126 A Look at Some Numbers of Old: Perfect, Deficient, and Abundant .................................129 Thomasenia Lott Adams and Katherine Murphy MTMS, February 2005 GCF and LCM on a Geoboard .......................................................................................... 135 Neil J. Jeffrey Activities for Junior High School and Middle School Mathematics, 1981 AT, January 1977 Get “Primed” to the Basic Building Blocks of Numbers .......................................................137 Christiana Robbins and Thomasenia Lott Adams MTMS, September 2007 iv A Crossnumber Game with Factors ...........................................................................................145 Jan Unenge Activities for Junior High School and Middle School Mathematics, 1981 AT, May 1975 Making a Problem of Junior High School Mathematics .............................................................149 Peggy A. House Activities for Junior High School and Middle School Mathematics, 1981 Juniper Green ..........................................................................................................................155 L. Lynn Stallings and Patricia L. Bullock MTMS, April 1999 A Fruitful Crop .......................................................................................................................159 Gladis Kersaint MTMS, November 2003, September 2004 5. Patterns and Functions 167 Introduction ......................................................................... 168 Using Pattern Tasks to Develop Mathematical Understandings and Set Classroom Norms ............. 171 Margaret S. Smith, Amy F. Hillen, and Christy L. Catania MTMS, August 2007 Experiences with Patterning ................................................................................................179 Joan Ferrini-Mundy, Glenda Lappan, and Elizabeth Phillips TCM, February 1997 Walk the Line: Making Sense of y = mx – b ........................................................................ 189 Nancy Canavan Anderson Algebra and Algebraic Thinking in School Mathematics, Seventieth NCTM Yearbook Using Models to Build Middle-Grades Students' Understanding of Functions ................. 199 Kathleen Cramer MTMS, January 2001 Giving Exponential Functions a Fair Shake ........................................................................ 209 Jeffrey Wanko MTMS, October 2005 Using Graphs to Introduce Functions ................................................................................................ 217 Frances Van Dyke MT, February 2003 v vi 6. Linear Equations 231 Introduction ......................................................................... 232 Counting Dots and Measuring Area: Rich Problems from Japan.........................................235 Blake E. Peterson MTMS, November 2006 Creating Multiple Representations in Algebra: All Chocolate, No Change ........................243 Arthur Hyde, Katie George, Suzanne Mynard, Christina Hall, Sharon Watson, and Patrick Watson MTMS, February 2006 Supporting Reform through Performance Assessment .........................................................251 Richard Kitchen, April Cherrington, Joanne Gates, Judith Hitchings, Maria Majka, Michael Merk, and George Trubow MTMS, September 2002 Coordinate Plane
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