## Series

# Learning Math: Geometry

## Discover how geometric reasoning can be used as a method of problem solving in this video- and Web-based course for K-8 teachers.

*A video- and web-based course for K-8 teachers; 12 half-hour video programs, course guide, and website.*

*Learning Math: Geometry,* a video- and Web-based course for elementary and middle school teachers, introduces geometric reasoning as a method for problem-solving. In this course, which is organized around the content standards of the National Council of Teachers of Mathematics (NCTM), you will explore the properties of geometric figures; make constructions using pencil and paper, and also using dynamic software; and practice using mathematical language to express ideas and justify your reasoning. Some important geometric ideas, such as symmetry, similarity, and trigonometry, will also be examined. You will begin to explore the basis of formal mathematical proofs and solid geometry. The course material progresses from more visual, intuitive ways of solving problems to more formal explorations of geometric ideas, properties, and proofs.

The course consists of 10 approximately two-and-a-half-hour sessions, each with a half hour of video programming, problem-solving activities provided online and in a print guide, and interactive activities and demonstrations on the Web. The 10th session (choose video program 10 or 11 and 12, depending on your grade level) explores ways to apply the concepts of geometry you’ve learned in your own classroom.

### Course Overview

*Geometry *introduces geometric reasoning as a method for problem solving. In this course, you will explore the properties of geometric figures such as triangles, quadrilaterals, and other polygons. You will make constructions using pencil and paper, and also dynamic software, and you will practice using mathematical language to express ideas and justify your reasoning. Some important geometric ideas such as symmetry, similarity, and trigonometry will also be examined. Lastly, you will begin to explore the basis of formal mathematical proofs and solid geometry. The course material progresses from more visual, intuitive ways of solving problems to more formal explorations of geometric ideas, properties, and, finally, proofs.

The course consists of 10 sessions, each with a half hour of video programming, problem-solving activities provided online and in a print guide, and interactive activities and demonstrations on the Web. Although each session includes suggested times for how long it may take to complete all of the required activities, these times are approximate. Some activities may take longer. You should allow at least two and a half hours for each session. The 10th session explores ways to apply the concepts of geometry you’ve learned in K-8 classrooms. You should complete the sessions sequentially.

### Who's Who

**Content Developer/Facilitator**

**Michelle Manes**

Michelle Manes earned a Bachelor’s degree in mathematics from UC Berkeley and a Master’s of Education from Boston University. She has taught mathematics at all levels, from third grade through college. She is a primary author on the Connected Geometry, Impact, and Mathematical Methods curricula. She has special interests in mathematics history, women and girls studying mathematics, and Deaf education. She is currently working towards a Ph.D. in mathematics at Brown University.

**Onscreen Participants**

Laverne AgardSpecial Needs Teacher, Grades 2-5 |
Lolita MattosMath Teacher, Grades 7-8 |
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Brad AnkneyMath Teacher, Grades 4-10 |
Joy RoyTeacher, Grade 4 |
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Jean BeebeTeacher, Grade 6 |
Catalina SaenzMath Teacher, Grade 6 |
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Javier CastilloTeacher, Grades 4-5 |
Helen SellersTeacher, Grade 5 |
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Kenton G. FindellMath Teacher, Grade 6 |
Vicky SuttonTeacher, Grade 5 |
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Michelle KurchianMath Teacher, Grade 5 |
Constance VanneMath Teacher, Grade 8 |
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Filberto Santiago-LizardiMath Specialist, Grade 6 |
Heidi WeberMath Teacher, Grade 8 |
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Ric MosherMath Teacher, Grades 7-9 |
Thomas ZayaMath Teacher, Grade 8 |
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### Video Index

**Session 1: What Is Geometry?**

Describing Geometric Designs

Constructing Triangles

Defining Basic Objects in Geometry

**Session 2: Triangles and Quadrilaterals**

Rules for making Triangles

Another Rule for Making Triangles

Building Toothpick Towers

**Session 3: Polygons**

Finding Hidden Polygons

Testing Definitions

Demonstrating Sum of Angles in a Triangle

Sum of Angles in a Polygon

**Session 4: Parallel Lines and Circles**

Inscirbed Angle in a Semicircle

Constructing with Circles

**Session 5: Dissections and Proof**

Transforming Shapes

Window Making and Proof

**Session 6: Pythagorean Theorem**

Calculating Area of Squares on a Triangle

Proving Angles and Areas of Squares

**Session 7: Symmetry**

Reflecting with Perpendicular Bisectors

Exploring Angles of Rotation

**Session 8: Similarity**

Comparing Copies

Estimating Height with Mirrors

Similar Triangles and Radiation Therapy

**Session 9: Solids**

Constructing Platonic Solids

Representing 3-D Figures in 2-D

**Session 10: Classroom Case Studies**

**Grades K-2: **Rose Christiansen: 2nd Grade, Lincoln School

**Grades 3-5: **Michelle Kurchian: 5th Grade, Alice M. Barrows Elementary School

**Grades 6-8: **Catalina Saenz: 6th Grade, Roxbury Prep Charter School

Heidi Weber: 8th Grade, F.A. Day Middle School

### Reading List

**Education Development Center, Inc. (2000). Perspective: What Is Geometry? In Connected Geometry: A Habits of Mind Approach to Geometry. Glencoe/McGraw-Hill. (Session 1)**

Reproduced with permission from the publisher. Copyright © 2000 by Glencoe/McGraw-Hill. All rights reserved.

Download PDF File:

Perspective: What Is Geometry?

**Steen, Lynn Arthur (1990). Pattern. In On the Shoulders of Giants: New Approaches to Numeracy. Edited by Lynn Arthur Steen (pp. 1-10). Washington, D.C.: National Academy Press. (Session 2)**

Reproduced with permission from the publisher. Copyright © 1990 by National Academy Press. All rights reserved.

Download PDF File:

Pattern

Continued…

**Senechal, Marjorie (1990). Shape. In On the Shoulders of Giants: New Approaches to Numeracy. Edited by Lynn Arthur Steen (pp. 139-148). Washington, D.C.: National Academy Press. (Session 3)**

Reproduced with permission from the publisher. Copyright © 1990 by National Academy Press. All rights reserved.

Download PDF File:

Shape

Continued…

**Cuoco, Al; Goldenberg, E. Paul; and Mark, June (December, 1996). Geometric Approaches to Things. In the paper “Habits of Mind: An Organizing Principle for Mathematics Curriculum.” The Journal of Mathematical Behavior, 5, (4), 375-402. (Session 4)**

Reproduced with permission from the publisher. Copyright © 2002 by Elsevier Science, Inc. All rights reserved.

Download PDF File:

Geometric Approaches to Things

**Senechal, Marjorie (1990). Shape. In On the Shoulders of Giants: New Approaches to Numeracy. Edited by Lynn Arthur Steen (pp. 148-160). Washington, D.C.: National Academy Press. (Session 5)**

Reproduced with permission from the publisher. Copyright © 1990 by National Academy Press. All rights reserved.

Download PDF File:

Shape

Continued…

Continued…

Continued…

**Crowe, D. and Thompson, Thomas M. (Reston, VA: National Council of Teachers of Mathematics, 1987). Transformation Geometry and Archaeology. Learning and Teaching Geometry, K-12, 1987 Yearbook. Edited by Mary Montgomery Lindquist. (Session 7)**

Reproduced with permission from the publisher. Copyright © 1987 by the National Council of Teachers of Mathematics. All rights reserved.

Download PDF File:

Transformation Geometry and Archaeology

**Schifter, Deborah (February, 1999). Learning Geometry: Some Insights Drawn from Teacher Writing. Teaching Children Mathematics, 5, (5), 360-366. (Session 8)**

Reproduced with permission from

*Teaching Children Mathematics.*Copyright © 1999 by the National Council of Teachers of Mathematics. All rights reserved.

Download PDF File:

Learning Geometry: Some Insights Drawn from Teacher Writing

Continued…

**Van de Walle, John A. (2001). Geometric Thinking and Geometric Concepts. In Elementary and Middle School Mathematics: Teaching Developmentally, 4th ed. Boston: Allyn and Bacon. (Session 9)**

Reproduced with permission from the publisher. Copyright © 2001 by Pearson Education. All rights reserved.

Download PDF File:

Geometric Thinking and Geometric Concepts

### Support Materials

The complete guide to the course is available here for download in PDF. You will need a copy of Adobe Acrobat Reader to read these files. You can download it for free from Adobe.

Session 1: What is Geometry?

Session 2: Triangles and Quadrilaterals

Session 3: Polygons

Session 4: Parallel Lines and Circles

Session 5: Dissections and Proof

Session 6: Pythagorean Theorem

Session 7: Symmetry

Session 8: Similarity

Session 9: Solids

Session 10, Grades K-2: Classroom Case Studies

Session 10, Grades 3-5: Classroom Case Studies

Session 10, Grades 6-8: Classroom Case Studies

Appendix, includingGlossary

### Progress Chart

Print out the chart below to keep track of your progress through the *Geometry *online course. It is available in PDF format.

### Credits

*Learning Math: Geometry *is a production of WGBH Interactive and WGBH Educational Programming and Outreach for Annenberg Media.

Copyright © 2003 WGBH Educational Foundation. All rights reserved.

**Senior Producer**

Ted Sicker

**Curriculum Director**

Denise Blumenthal

**Content Developer**

Michelle Manes, Mathematics Teacher and Education Consultant

**Curriculum Developer**

Anna Brooks

**Project Manager**

Sanda Zdjelar

**Special Projects Assistant**

Nina Farouk

**Core Advisors**

Suzanne Chapin

Srdjan Divac

Carol Findell

Bowen Kerins

**Designers**

Plum Crane

Lisa Rosenthal

Christian Wise

**Web Developers**

Joe Brandt

Kirsten Connelly

Bob Donohue

Peter Pinch

**Online Video Segment Coordinator**

Mary Susan Blout

**Business Managers**

Walter Gadecki

Joe Karaman

**Unit Managers**

Maria Constantinides

Adriana Sacchi

**With the assistance of**

Geoffrey Dixon

Stephani Roberts Lincoln

Yasmin Madan

Kate Smyres

Julie Wolf

**Video Series Production Credits**

*Learning Math: Geometry *is a production of WGBH Educational Foundation for Annenberg Media.

**Executive Producer**

Michele Korf

**Senior Project Director**

Amy Tonkonogy

**Producer**

John Browne

**Content Developer/Facilitator**

Michelle Manes, Mathematics Teacher and Education Consultant

**Advisors**

Nicholas Branca, San Diego State University, CA

Miriam Leiva, University of North Carolina, Charlotte

Carol Malloy, University of North Carolina, Chapel Hill

Bill Masalski, University of Massachusetts, Amherst

Arthur Powell, Rutgers-Newark College of Arts and Sciences, NJ

**Content Editor**

Srdjan Divac, Harvard University, MA

**On Location Consultants**

Kenton G. Findell

Bowen Kerins

**Editor**

Miguel Picker

**Additional Editing**

Dickran H. Manoogian

**Associate Producers**

Irena Fayngold

Pamela Lipton

**Project Manager**

Sanda Zdjelar

**Production Manager**

Mary Ellen Gardiner

**Post Production Associate Producer**

Peter Villa

**Production Manager**

Mary Ellen Gardiner

**Post Production Associate Producer**

Peter Villa

**Location Coordinator**

Mary Susan Blout

**Teacher Liaison**

Lisa Eure

**Set Assistant **

Elena Graceffa

**Camera**

Bill Charette

Lance Douglas

Larry LeCain

Steve McCarthy

Dillard Morrison

David Rabinovitz

**Audio**

Steve Bores

Chris Bresnahan

Charlie Collias

Jose Leon

Dennis McCarthy

Gilles Morin

**Interns**

Timothy Barney

Ravi Blatt

Joe Gudema

Justin Hartery

Aimee Jones

**Design**

Gaye Korbet

Daryl Myers

Alisa Placas

**Content Graphics**

Mark Geffen

John O’Brien

**On-line Editors**

David Eells

John O’Brien

**Sound Mix**

John Jenkins

Dan Lesiw

**Original Music**

Tom Martin

**Additional Music**

Concerto in G Major, J.B. de Boismortier

**Soloist**

John Tyson

Recorder

Frances Conover Fitch

Harpsichord

Jane Starkman, Kathryn Shaw

Violins

Jann Cosart

Viola

Alice Robbins

Violincello

Tom Coleman

Double Bass

**Narrator**

Jeff Loeb

**Business Manager**

Joe Karaman

**Unit Manager**

Maria Constantinides

**Office Coordinators**

Justin Brown

Laurie Wolf

**Special Thanks**

**Session 2: Triangles and Quadrilaterals: Boston’s New Bridge**

Kirk Elwell

Lead Field Engineer

Bechtel/Parsons Brinkerhoff

**Session 3: Polygons: Three-Dimensional Printing**

David Russell

Director of Engineering

Z Corporation

Marina Hatsopoulos

CEO

Z Corporation

Tom Clay

President

Z Corporation

**Session 4: Parallel Lines and Circles: Three-Dimensional Maps**

Seth Teller

Associate Professor

Lab for Computer Science, MIT

Hilary Karls

Student, MIT

Priya Agrawal

Student, MIT

**Session 5: Dissections and Proof: Window Making and Proof**

Jim Ialeggio

Architectural Detail In Wood

**Session 6: The Pythagorean Theorem: The Pythagorean Theorem and 18th-Century Cranes**

Rick Brown

Massachusetts College of Art

Wyly Brown

**Session 7: Symmetry: Wooden Boats and Line Symmetry**

Nat Benjamin

Gannon and Benjamin Marine Railway

**Session 8: Similarity: Similar Triangles and Radiation Therapy**

Dr. Anita Mahajan

Radiation Oncologist

Tufts – New England Medical Center

Maxon J. Buscher

Chief Dosimetrist

Tufts – New England Medical Center

Jason K. Talkington

Medical Dosimetry Student

Tufts – New England Medical Center

Sandra John-Baptiste, R.T.T.

Medical Dosimetry Student

Tufts – New England Medical Center

Kevin O’Leary

Tufts – New England Medical Center

Sean O’Leary

Tufts – New England Medical Center

**Session 9: Solids: Architecture and Solid Geometry**

Ann Beha

Principal

Ann Beha Architects

**Session 10: Classroom Case Studies**

Michelle Kurchian, Alice M. Barrows Elementary School, Reading, MA

Catalina Saenz, Roxbury Preparatory Charter School, Roxbury, MA

Heidi Weber, F.A. Day Middle School, Newton, MA

**Site Location**

Ferryway School, Malden, MA