# 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

### Video Index

Session 1: What Is Geometry?
Describing Geometric Designs
Constructing Triangles
Defining Basic Objects in Geometry

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

Session 9: Solids
Constructing Platonic Solids
Representing 3-D Figures in 2-D

Session 10: Classroom Case Studies

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

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

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)

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)

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)

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)

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)

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)

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)

Geometric Thinking and Geometric Concepts

### Support Materials

Introduction

Session 1: What is Geometry?

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, including Glossary

### 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

Web Site Production Credits

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

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

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

Joe Karaman

Unit Managers
Maria Constantinides

With the assistance of
Geoffrey Dixon
Stephani Roberts Lincoln
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

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

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

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

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
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
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