Linda Cooper Foreman and Albert B. Bennett, Jr. COURSE III Math Alive! Visual Mathematics, Course III by Linda Cooper Foreman and Albert B. Bennett Jr. Math Alive! Visual Mathematics, Course III is preceded by: Visual Mathematics, Course I Visual Mathematics, Course II Copyright ©1998 The Math Learning Center, PO Box 12929, Salem, Oregon 97309. Tel. 503 370-8130. All rights reserved. Produced for digital distribution November 2016. The Math Learning Center grants permission to classroom teachers to reproduce blackline masters and student activity pages (separate documents) in appropriate quantities for their classroom use. This project was supported, in part, by the National Science Foundation Grant ESI-9452851. Opinions expressed are those of the authors and not necessarily those of the Foundation. Prepared for publication on Macintosh Desktop Publishing system. Printed in the United States of America. VMCIII DIGITAL2016 ISBN 1-886131-45-7 Acknowledgments Contributing authors from Math and the Mind’s Eye Luise Wilkinson and Lou Saponas for their laughter, project: Gene Maier, L. Ted Nelson, Mike Arcidiacono, enthusiasm, commitment, and encouragement every day Mike Shaughnessy, and David Fielker. in the classroom – for seeing possibilities in ideas that didn't work and for celebrating the ones that did. Book Design: Jonathan Maier and Susan Schlichting The many teachers and administrators who field tested Math Cover Design: Susan Schlichting Alive! Course III, allowed us to observe its implementation in their schools, and/or permitted us to explore new ideas with Layout and Graphics: Ingrid Williams their students. In particular, we thank: Illustrator: Travis Waage Colorado Editorial Consultant: Mike Shaughnessy Aurora; Eaglecrest High School – Larry Linnen Production Editor: Vaunie Maier Louisiana Sulphur; Maplewood Middle School – Darlene Morris Materials Production: Tom Schussman and Don Rasmussen Lake Charles; Calcasieu Parish Schools, Middle School Math Consultant – Judy Vail ........................................... WE GIVE SPECIAL THANKS TO: Missouri Gene Maier and Ted Nelson, whose vision, inspiration, Webster Groves; Webster Groves School District, and support have enabled the development of Math Alive! Mathematics Coordinator K-8 – Cathy Fueglein They have been our mentors in the truest sense of the word. Ohio Centerville; Tower Heights Middle School – Bonnie Ingrid Williams, for her unflagging patience with our Thompson revisions and her commitment to creating a friendly layout Oregon and clear graphics. Eugene; Roosevelt Middle School – Audrey Manning We also wish to acknowledge the many children, parents, Gresham; Clear Creek Middle School – Nicole Miller teachers, and school administrators that have been involved with the field testing of Math Alive! Course III. In particular, Hood River; Hood River Middle School – Trudy Mitchell we wish to thank the following: Portland These young mathematicians (and their parents) at Athey WinterHave Alternative School – Paul Griffith Creek Middle School and West Linn High School in West Binnsmead Middle School – Heather Nelson Linn, Oregon, for their willingness to explore, challenge, Welches; Welches Elementary School – Pam Alexander struggle with, and celebrate new ideas. They have touched our hearts, stirred our minds, and enabled our growth as West Linn, Wilsonville writers. Athey Creek Middle School – Kim Noah, Elaine Jones Lindsay Adams, Katie Alfson, Joel Bergman, Morgan Inza R. Wood Middle School – Maureen Callahan Briney, Matthew Eppelsheimer, Jennie Eskridge, Kyle West Linn High School – Joyce Hedstrom, Laura Lanka, Foreman, Michael Geffel, Briaan Grismore, Malia Nicki Hudson, Jamie LeVeque Jerkins, Julie Locke, Tyler Mackeson, JAlex Meinhard, Linden Parker, Dylan Schmidt, Erica Sexton West Linn School District – Roger Woehl, Mike Tannenbaum, Jane Stickney, Bob Hamm Kathy Pfaendler and Patty Quan for their continued support and encouragement and their willingness to Vermont consider and experiment with new ideas. Montpelier; Main Street Middle School – Sue Abrams And finally, we thank our spouses, Jane and Wally, and our families for their patience and encouragement. © 1998 The Math Learning Center Math Alive! Visual Mathematics Course III / iii Contents INTRODUCTION vii LESSON 1 Exploring Symmetry 1 LESSON 2 Introduction to Isometries 25 LESSON 3 Measurement — Inventing Formulas 53 LESSON 4 Arithmetic Sequences 81 LESSON 5 Extended Counting Piece Patterns 105 LESSON 6 Fraction Concepts 135 LESSON 7 Properties, Operations, and Algorithms 161 LESSON 8 Experimental and Theoretical Probability 189 LESSON 9 Reasoning and Radicals 209 LESSON 10 Constructions and Mappings 243 LESSON 11 Introduction to Quadratics 279 LESSON 12 Continuous Graphs 307 LESSON 13 Modeling Situations 335 LESSON 14 Analyzing Graphs 367 LESSON 15 Data—Variability and Spread 397 LESSON 16 Counting and Probability Diagrams 423 LESSON 17 Simulations and Probability 449 APPENDIX Materials Appendix-1 INDEX Index-1 © 1998 The Math Learning Center Math Alive! Visual Mathematics Course III / v . . . . . . . . . . . . . . . . Introduction . . . . . . . . . Math Alive! is a series of four comprehensive, NCTM Standards-based, one-year . . courses for students in the middle grades. This curriculum is in development with . . support from the National Science Foundation and is the grades 5-8 portion of a new . . Math Learning Center seamless K-8 curriculum. . . . . The first two courses in the Math Alive! middle grades series were originally published . . under the names Visual Mathematics, Course I and Course II. Subsequent publications . . of those courses will be renamed Math Alive! Course I and Course II. This book, . . Math Alive! Course III, is the third course in this series. The fourth course is now in . . development. Math Alive! Course III, is designed for use by teachers in grades 7 or 8 . . whose students have completed Course II, or by teachers in grades 7-9 whose students . . are exploring Math Alive! for the first time. To support the teacher whose students may . . need additional background, there is extensive cross referencing to Courses I and II. . . . . Throughout the 17 Math Alive! Course III lessons, each averaging about two weeks . . of class time, there are many implementation suggestions. In addition, the teachers’ . . resource book, Starting Points for Implementing Math Alive!, provides an overview . . of the philosophy and goals of the Math Alive! courses, together with extensive . . suggestions for: organizing materials; planning, pacing, and assessing lesson activities; . . working with parents and the community outside your classroom; finding support as . . you seek changes in your teaching practices; and creating a classroom climate that . . invites risk-taking and nourishes the mathematician within each student. The ideas in . . Starting Points are based on our own classroom experiences and comments we have . . received from many other teachers field testing Math Alive! courses. . . . . Teaching Math Alive! ourselves has affirmed our beliefs in the potential within each . . student, enriched our views of mathematics and the art of teaching mathematics, and . . reinforced our commitment to support teachers in their efforts to change the way . . mathematics is learned and taught. It is our hope that teaching Math Alive! will be . . equally fulfilling for you. . . . . . . . . . . . . . . . . . . . . . . . . . . . . © 1998 The Math Learning Center Math Alive! Visual Mathematics Course III / vii Exploring Symmetry Lesson 1 Exploring Symmetry Lesson 1 . . . . THE BIG IDEA . CONNECTOR . . Actions such as draw- .. OVERVIEW MATERIALS FOR TEACHER ACTIVITY . . Students draw frames for 2- 4 Connector Masters A-C, 4 Butcher paper, 1 large ing, cutting, tracing, . . dimensional figures to iden- 1 copy per group and 1 sheet per class. . framing, rearranging, . tify the figures’ reflectional transparency. 4 Butcher paper strips (4-5" . flipping, turning, imag- .. and rotational symmetries 4 Connector Master D, 2 long), 8-10 per group. .. and order of symmetry. copies per student and 4 Scissors, 1 pair per stu- ining, and discussing .. 2 transparencies. dent. . shapes build spatial . 4 Note (file) cards, 1 un- 4 Tape and marking pens . sense and promote in- .. lined card per student. for each group. .. 4 Plain (unlined) paper, 1 4 Quartered blank trans- sights and intuitions .. sheet per student. parencies (optional) for . about symmetry con- . use at the overhead. . . cepts. Investigations . . . that involve forming . FOCUS . . polygons which satisfy . . OVERVIEW MATERIALS FOR TEACHER ACTIVITY . certain symmetry con- . Students draw symmetric 4 Focus Master A, 1 copy 4 1-cm triangular grid pa- . . figures on square grids and per student and 1 trans- per, 4 sheets per student ditions prompt conjec- . . on triangular grids and note parency. and 1 transparency. . tures and generaliza- .. the different symmetry 4 Focus Student Activities 4 Scissors, 1 pair per stu- . types possible. They inves- tions. Such activities . 1.1-1.2, 1 copy of each dent. . tigate all orders of symme- provide a rich context .. try possible for 3-sided to 8- per student and 1 trans- 4 Butcher paper strips, 8 .. sided polygons and make parency of each. for the teacher, 16 for for experiencing the .. conjectures and generaliza- 4 “We conjecture…/We each group, and several mathematical process. .. tions based on their obser- wonder…” poster from for use by the class, as .. vations. Finally, students the Connector activity, needed during the les- .. reflect on how these activi- 1 for each class. son. .. ties relate to the goals of 4 1-cm squared grid paper, 4 Marking pens and tape . . the class. 4 sheets per student and for each group. . . 1 transparency. . . . . . . FOLLOW-UP . . . OVERVIEW MATERIALS FOR STUDENT ACTIVITY . . . Students form conjectures 4 Student Activity 1.3, . . and generalizations about 1 copy per student. . . the order of symmetry for . . regular n-gons. They iden- . . tify the symmetries of flags . . and logos and create a logo . . with symmetry. They inves- . . tigate and generalize about . . situations involving sym- . . metry. . . . . . . . . Math Alive! Course III / 1 Lesson 1 Exploring Symmetry LESSON IDEAS . STUDENT INVESTIGATIONS dents complete a Follow-up resource book, Starting . QUOTE . Investigations in this lesson at home. Rather than dis- Points for Implementing . Symmetry in two and . provide a rich context for cussing and showing “how- Math Alive! . three dimensions provides . “doing mathematics.” You to” solve problems, during . rich opportunities for stu- . might take time during the class discussions of the Fol- PACING . dents to see geometry in . Focus activity to discuss the low-up, suggest that stu- On average, each lesson . the world of art, nature, . fact that, like for profes- dents seek and share “clues” in this course is designed . construction, and so on. . sional mathematicians and to jump start each other’s to take about 2 weeks. . Butterflies, faces, flowers, . scientists, “successful” in- thinking. Some teachers This may vary according . arrangements of windows, . vestigations may leave the select problems from Fol- to your students’ back- . reflections in water, and . students with more ques- low-ups for in-class assess- grounds, your familiarity . some pottery designs in- . tions than answers. ment activities. with the curriculum, your . volve symmetry. Turning . In our classroom, we ask school schedule, and stu- . symmetry is illustrated by . FOLLOW-UP that students revise incor- dent- or teacher-generated . bicycle gears. Pattern . Keep in mind that Follow- rect work on Follow-ups extensions you choose to . symmetry can be ob- . ups require extended time before a grade is entered in explore. . served in the multiplica- . for students to investigate the grade book and provide . tion table, in numbers ar- . and communicate their opportunities before and . rayed in charts, and in . ideas. They are not de- after school to discuss stu- . Pascal’s triangle. . signed to be “due the next dents’ questions. Grades on . NCTM Standards day.” It is not necessary to Follow-ups are determined assign every problem, and according to criteria on the it is reasonable to move on Follow-up Assessment to the next lesson while stu- Guide given in the teacher SELECTED ANSWERS 1. triangle: 3 reflectional symmetries, 3 rotational symme- 3. Here are possible expressions (others are also possible), ( ) tries listed in the order of the chart: 2n; n; n; (cid:107) (cid:51)(cid:54)(cid:48)(cid:176) for k = 1, 2, 3... n; ((cid:110)- (cid:50))(cid:49)(cid:56)(cid:48)(cid:176) . (cid:110) square: 4 reflective, 4 rotational (cid:110) pentagon: 5 reflective, 5 rotational 9. a) True b) True hexagon: 6 reflective, 6 rotational heptagon: 7 reflective, 7 rotational octagon: 8 reflective, 8 rotational Measures of the angles of rotational symmetry for the following regular polygons: triangle: 120(cid:176), 240(cid:176), 360(cid:176) square: 90(cid:176), 180(cid:176), 270(cid:176), 360(cid:176) pentagon: 72(cid:176), 144(cid:176), 216(cid:176), 288(cid:176), 360(cid:176) hexagon: 60(cid:176), 120(cid:176), 180(cid:176), 240(cid:176), 300(cid:176), 360(cid:176) heptagon: 513⁄7(cid:176), 1026⁄7(cid:176), 1542⁄7(cid:176), 2055⁄7(cid:176), 2571⁄7(cid:176), 3084⁄7(cid:176), 360(cid:176) octagon: 45(cid:176), 90(cid:176), 135(cid:176), 180(cid:176), 225(cid:176), 270(cid:176), 315(cid:176), 360(cid:176) 2 / Math Alive! Course III Exploring Symmetry Lesson 1 Connector Teacher Activity OVERVIEW & PURPOSE MATERIALS Students draw frames for 2-dimensional figures to identify 4 Connector Masters A-C, 1 copy per group and the figures’ reflectional and rotational symmetries and order 1 transparency. of symmetry. 4 Connector Master D, 2 copies per student and 2 transparencies. 4 Note (file) cards, 1 unlined card per student. 4 Plain (unlined) paper, 1 sheet per student. 4 Butcher paper, 1 large sheet per class. 4 Butcher paper strips (4-5" long), 8-10 per group. 4 Scissors, 1 pair per student. 4 Tape and marking pens for each group. 4 Quartered blank transparencies (optional) for use at the overhead. 4 Coffee stirrers (optional), 1 per student. ACTIONS COMMENTS 1␣␣Arrange the students in groups and give each student 1␣␣Leave as little space as possible between the card and a plain (unlined) rectangular note (file) card and a plain its frame: sheet of paper. Ask them to label the corners of their note cards A, B, C, and D. Hold a note card against a 1 2 A B sheet of plain paper mounted on the wall, and draw a frame around the card. Label the corners of the frame 1, D C 2, 3, and 4. Ask the students to also draw and label 4 3 frames for their note cards. Then give each group a copy of Connector Master A (see next page) and have the students carry out the instructions. Discuss, inviting volunteers to demonstrate their methods and observa- tions at the overhead. Note that ready to copy masters for all Connector and Focus Masters and Student Activities are contained in Blackline Masters. In addition, Student Activity Packets are available from The Math Learning Center (MLC); each packet contains a one-student supply of masters and student activities needed by individual students for this course. A one-student supply of grid paper, Student Activity Grids, is also available from MLC. Students who have difficulty reading may need some help here. Encourage students to support their groupmates as they interpret the instructions in a)-c). There are an infinite number of points around which the card can be rotated 360(cid:176) (or 0(cid:176) ) to exactly fit back into its frame, since a 360(cid:176) (or 0(cid:176) ) rotation about any point on the card replaces the card in its original posi- tion. A point about which a shape is rotated to fit back (Continued next page.) Math Alive! Course III / 3