Geometric Structures: An Inquiry-Based Approach for Prospective Elementary and Middle School Teachers

Douglas B. Aichele / John Wolfe  
Total pages
April 2007
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Geometric Structures: An Inquiry-Based Approach for Prospective Elementary and Middle School Teachers
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For prospective elementary and middle school teachers.


This text provides a creative, inquiry-based experience with geometry that is appropriate for prospective elementary and middle school teachers.  The coherent series of text activities supports each student's growth toward being a confident, independent learner empowered with the help of peers to make sense of the geometric world. This curriculum is explicitly developed to provide future elementary and middle school teachers with


  • experience recalling and appropriately using standard geometry ideas,
  • experience learning and making sense of new geometry,
  • experience discussing geometry with peers,
  • experience asking questions about geometry,
  • experience listening and understanding as others talk about geometry,
  • experience gaining meaning from reading geometry,
  • experience expressing geometry ideas through writing,
  • experience thinking about geometry, and
  • experience doing geometry.


These activities constitute an “inquiry based” curriculum.  In this style of learning and teaching, whole class discussions and group work replace listening to lectures as the dominant class activity.



·  Topic coverage and approach.

All topics included on mandated state teacher competency exams and in certification requirements are touched upon in text activities.  However, the approach is not to drill on these basics one more time. Activities provide opportunities to use basic relationships and facts to solve problems and make sense of situations.  Grounding experiences with geometry are provided to anchor and stabilize personal understanding and to stimulate individual creation of connections and bridges between basic ideas.


·   Organization of Textbook.

The text is loosely organized into five parts based on the manipulative that is highlighted: i) paper folding, ii) geoboards, iii) straightedge and compass, iv) computers, and v) mira or reflecta and tracing paper. By providing a meaningful representation for most geometry ideas, this emphasis helps keep our focus on grounding all ideas in concrete examples. These models provide motivating examples that are used to “develop” basic geometric relationships in a believable way. Models can also stimulate conjecturing as well as provide a means for checking tentative ideas - higher order thinking skills basic to learning geometry. The instructor, however, plays a critical role as a mediator between the physical models and the underlying geometric ideas.


·  Topic Development through Exploration.

The topics presented in this inquiry-based curriculum are "developed"  by involving students with exploring evidence that can provide both credibility and meaning for basic relationships. Often students are asked to observe, describe and conjecture on relationships that can be apparent in models and examples explored in activities. Also, activities are designed to stimulate group and/or whole class discussion of students' observations to arrive at a social consensus about the truth and meaning of geometric ideas.

An astonishing statistic dealing with "developed" topics vs. "stated" topics (the traditional manner in which mathematics is taught) is reported in the comparative video study of U.S., Japanese and German fifth grade classes in the book The Teaching Gap by James Stibler and James Hiebert (1999). In the U.S., only 21.9% of the topics are "developed". In Japan and Germany, the percentages are 83% and 77%, respectively. Our approach to teachers' preparation addresses this shortcoming.


·  Level of content and level of coverage.

The ideas presented are entirely elementary and can be adapted for elementary or middle schools students.  Even a few juvenile activities are included such as snowflake and symmetric cutouts and recognizing types of quadrilaterals.  This does not mean the material presented is necessarily easy or infantile.3  The activities are designed to give teachers a view of these juvenile activities in a broader context and to highlight their foundational relationship with larger ideas in the geometry curriculum.  Also some modern topics are covered including Pick's theorem, symmetries of borders and Escher tessellations which can all be adapted for use with grade school students.  These topics also provide opportunities for all students in this course, even those generally well prepared in geometry, to experience and learn new ideas.


·  Activity Pages.

The core of the text is a coherent and diverse set of “Activity Pages” which guide the class attention through a full and rich geometry curriculum.  These activity pages are formatted to allow students to write directly on them and to use the perforations to hand them in to the instructor.


·  Both the “How and Why” of problem solving are supported.

A constant and varied stream of geometry problems is provided by individual, group, and class activities.  Student sharing of their thinking and experiences with these problems provides many insights into how and why different methods work for solving problems.  Users have been gratified with the fact that, although there is little drill and practice homework presented in these activities, students seem to become as good at working standard problems as they do in more traditional courses.  Users do find that students in this curriculum are definitely more adventurous in attempting unorthodox problems as well as stronger at explaining why the methods they choose make sense. 


·  Teaching Practices.

These activities are designed to minimize the need for traditional lecturing. Classroom discussions and group activities provide a rich and essential resource to motivate student's efforts to make sense of the course material. The instructor's ability to manage these discussions and activities without traditional lecturing is a critical factor in the success of this approach. With this approach, we have found that students arrive at a deeper, more robust, understanding and appreciation for the power, beauty and meaning of geometry.


·  Serves teachers who will teach descriptive geometry.

We believe strongly in our approach for the preparation of prospective elementary teachers.  For middle school teachers, we believe that the experiences of these activities provide a depth and richness to the understanding of geometry that provides strong preparation for teaching the “descriptive” geometry topics included in middle school math classes.  Our emphasis is always on building a foundation by connecting ideas to examples, experiences and well known ideas rather than connecting to more abstract math that will be taken at some time in the future.  This text does thoroughly cover straightedge and compass constructions (Chapter 10), computer constructions (Chapters 12 and 13), and an introduction to proofs (Chapter 11). 


·  Growth in communication skills supported.

The ability to express geometric ideas verbally, physically and in writing and drawing is a valuable communication skill for teachers.  Many activities in this text are designed specifically to provide opportunities to exercise these skills.


·  Making sense of geometry-a guide for the student.

Many students experience this course as different from other mathematics courses they have taken since this style is not common in college level mathematics content courses.  For this reason, an article titled A Guide for You, the Student: Making Sense of Geometry in an Inquiry Class is included.  This guide briefly states the goals, objectives and rational for the curriculum and also contains suggestions for the student on how to make the activities a valuable and rewarding learning experience.  A list of these suggestions is reproduced in the front of the text as an aide for students.


·  Extensively class tested.

This text has been extensively class tested for over six years with preliminary versions used by cooperating faculty at several universities across the US through support from National Science Foundation grants #0231161 and #9752288.2



Table of Contents

Part I: Paper Folding

Chapter 0 - Warm Up Activities

0.0 Introduction

0.1 Folding Polygons from a Circle

0.2 Making Squares

0.3 Two Congruent Halves

0.4 Dissecting Figures

Chapter 1 - Polygons and the Angle Relationships

1.0 Introduction

1.1 Parallel Line Grid - Triangle Angle Sum

1.2 Envelope Fold - Triangle Angle Sum

1.3 Triangle and Quadrilateral Angle Sums by Tearing

1.4 Polygon Angle Sums: How many Triangles?

1.5 The Angles of a Polygon

1.6 When Does Erika's Idea Work?

1.7 The Greedy Triangle

1.8 Problems: Angle Sums and Angle Relationships

1.9 Four Kinds of Related Angles

1.10 Figuring Angles and Checking by Measurement

1.11 Parallel Lines: How to Recognize Them

1.12 Measuring Sides and Angles of Triangles

1.14 Convex: Different Ways to Make Sense of It

1.14a Angle Problems - Version A

1.14b Angle Problems - Version B

1.15 Angle Probems - More

1.16 How Do I Know if I Understand?

1.17 Conjecturing ABout Quadrilaterals

1.18 Possible or Not?

1.19 True or False (with Example)

1.20 Under What Conditions?


Chapter 2 - Quadrilaterals and Their Definitions

2.0 Introduction

2.1 Checking Properties of Quadrilaterals

2.2 Properties of Quadrilaterals

2.3 Marking Quadrilateral Properties

2.4 Properties of Diagonals of Quadrilaterals

2.5 Checking Quadrilaterals by Folding

2.6 Read Carefully: Every Word Counts!

2.7 Checking Examples Visually or Physically

2.8 Exploring Medial Quadrilaterals

2.9a Problems: Properties of Quadrilaterals, Version A

2.9b Problems: Properties of Quadrilaterals, Version B

2.10 More Problems: Properties of Quadrilaterals

2.11 A Deeper Understanding of Definitions

2.12 Special Cases of Quadrilaterals

2.13 Definitions: Inclusive or Exclusive

2.14 Problems: Inclusive and Exclusive Definitions

2.15 What Is a Kite? Equivalent Definitions

2.16a Problems: Definitions of Quadrilaterals, Version A

2.16b Problems: Definitions of Quadrilaterals, Version B

2.17 More Problems: Definitions of Quadrilaterals

2.18 How Do I Know if I Understand?



Four Contexts for Geometric Constructions

Prologue to Chapters 3, 10, 12, and 14


Chapter 3 - Constructions by Paper Folding

3.0 Introduction

3.1 Introducing CDs: Two Basic Constructions

3.2 CD Problem: A Parallel Line

3.3 CD Problem: The Median

3.4 CD Problem: An Equilateral Triangle

3.5 CD Problem: A Square

3.6 Circumscribing Circle

3.7 Inscribed Circle

3.8 Balance Point of a Triangle

3.9 Additional CD Problems Using Basic Construction Steps

3.10 Group Problem: Inscribed Circles

3.11 Folding a Six-Pointed Star or a "Snowflake"

3.12 Problems Involving Paper Folding

3.13 How Do I Know if I Understand?


Chapter 4 - Explorations in Three-dimensional Geometry

4.0 Introduction

4.1 Polyhedra (Solids) from an Envelope

4.2 Roll-and-Fold Prism and Pyramid Activities

4.3 Net Project A: Prisms

4.4 Prisms

4.5 Makiing Sense of Volume: A Basic Relationship

4.6 Net Project B: Pyramids

4.7 Pyramids

4.8 Edges, Faces, and Vertices of Polyhedra

4.9 Special Kinds of Polyhedra

4.10 Riddles with Solids

4.11 Volumes Prisms, Pyramids, and Spheres

4.12 Volume of a Pyramid

4.13 What Does Volume Really Mean?

4.14 Volume of Solids: First Try

4.15a Solid-Geometry Problems, Version A

4.15b Solid-Geometry Problems, Version B

4.16 More Solid-Geometry Problems



Unit Origami: An Introduction

4.17 Instructions for the Basic Parallelogram Unit

4.18 Project for the Whole Class: Monster Stellated Icosahedron

4.19 Unit Origami Projects

4.20 Some Geometry of Unit Origami

4.21 Convex Deltahedra: How Many Are There?

4.22 Problems: Unit Origami

4.23 How Do I Know if I Understand?


Part2 GeoBoards and Dot Paper


Chapter 5 - Area

5.0 Introduction

5.1 How Much Space in a Triangle?

5.2 Areas on a Geoboard

5.3 Two Ways: Cut-up and Take-away

5.4 Areas: Parallelograms and Trapezoids

5.5 Area by Julie's Way

5.6 Which Ways Work for These Figures?

5.7 Areas: How Many Ways?

5.8 Area Problems: First Try

5.9 A Sampling of Area Problems

5.10 Making Sense of Common Units for Length and Area

5.11a Area Problems, Version A

5.11b Area Problems, Version B

5.12 More Area Problems

5.13 How Do I Know if I Understand?


Chapter 6 - Explorations with Geoboard Areas

6.0 Introduction

6.1 Areas of Skew Quadrilaterals

6.2 Solid Tile Shapes

6.3 Problems: Tile Shapes

6.4 Areas of Tile Shapes

6.5 Areas by Counting Pets

6.6 How Many Tile Shapes with Five Squares?

6.7 Counting Areas: Pick's Formula

6.8 Skew Figures

6.9 Discovering, Describing, and Using Relationships

6.10 Sean's Idea: Area = Inside Pegs

6.11a Problems: Geoboard Areas,  Version A

6.11b Problems: Geoboard Areas,  Version B

6.12 More Problems: Geoboard Areas

6.13 How Do I Know if I Understand?


Chapter 7 - Similarity and Slope

7.0 Introduction

7.1 Slope or Steepness

7.2 Slope: Parallel and Perpendicular

7.3 Slope Problems, Part 1

7.4 Slope Problems, Part 2

7.5 Linear Equations, Tables of Values, and Slopes

7.6 Similar Figures and Their Properties

7.7 Similar Figures and Proportionality

7.8 Measuring Proportionality

7.9 Reasoning withSimilar Triangles

7.10 Similarity and Scale Factors (Length Factors)

7.11 Scaling, Areas, and Area Factors

7.12 Scaling Problems, First Try

7.13 Scaling Problems

7.14 Scaling and Volume of Solids

7.15a Problems: Slope, Similarity, and Scaling, Version A

7.15b Problems: Slope, Similarity, and Scaling, Version B

7.16 More Problems onSlope, Similarity, and Scaling

7.17 How Do I Know if I Understand?


Chapter 8 - Pythagorean Theorem and Perimeter

8.0 Introduction

8.1 RightTriangles of Squares

8.2 Pythagorean Puzzles

8.3 Estimating Perimeters on a Geoboard

8.4 Slant Lengths on a Geoboard

8.5 Geoboard Perimeters

8.6 Three Special Triangles

8.7 Pythagorean Problems, First Try

8.8a Perimeter and Right-Triangle Problems, Version A

8.8b Perimeter and Right-Triangle Problems, Version B

8.9 More Perimeter and Right-Triangle Problems

8.10 How Do I Know if I Understand?


Chapter 9 - Geometry of Circles

9.0 Introduction

9.1 Perimeter (Circumference) of a Circle

9.2 Area of a Circle

9.3 Area and Perimeter of Circles and Sectors

9.4 Area Problems with Circles, First Try

9.5 Problems: Area and Perimeter of Circles

9.6 Inscribed Angles of Arcs of Circles

9.7 The Law of Thales

9.8 Circumscribed or Cyclic Polygons

9.9 Circumscribing Circle for a Cyclic Quadrilateral

9.10 Problems: Inscribed Angles and Circumscribed Polygons

9.11a Problems: Geometry of Circles Version A

9.11b Problems: Geometry of Circles, Version B

9.12 Revisiting Volumes: Cones and Cylinders

9.13 Surface Area of an Orange

9.14 More Problems: Geometry of Circles

9.15 How Do I Know if I Understand?


Part 3 - Straightedge and Compass


Chapter 10 - Straightedge and Compass Constructions

10.0 Introduction

10.1 Basic Straightedge and Compass Constructions

10.2 Straightedge and Compass: Construct a Parallel Line

10.3 Examples: Reasoning in Construction Problems

10.4 Reasoning in Construction Problems

10.5 Making Triangles, I:Side-Side-Side

10.6 Making Triangles, II: Side-Angle-Side

10.7 Making Triangles, III: Angle-Side-Angle

10.8 Making Triangles, IV: Side-Side-Angle (Ambiguous Case)

10.9 Congruence Conditions for Triangles

10.10 How Do I Know I Understand?


Chapter 11 - Congruence Conditions and Reasoning from Definitions to Properties

11.0 Introduction

11.1 Congruence Conditions for Triangles and CPCT

11.2 Problems:Congruence Conditions and CPCT

11.3 Justifications by Congruence Conditions

11.4a Problems: Congruence Conditions, Version A

11.4b Problems: Congruence Conditions, Version B

11.5 More Problems: Congruence Conditions

11.6 FromDefinitions to Properties: Five-Step Reasoning

11.7 Example: Five-Step Reasoning, Problem A

11.8 Five Step reasoning, First Try

11.9 More Problems Using Five-Step Reasoning

11.10 How Do I Know if I Understand?


Part 4 - Computer Constructions and Explorations


Chapter 12 - Computer Constructions

12.0 Introduction

12.1 Getting Started with Computer Construction Software

12.2 Constructing Objects: Midpoints

12.3 Constructing Objects: Bisectors

12.4 Constructing Objects: Altitudes and Medians

12.5 The Euler Line of a Triangle

12.6 The Nine-point Circle of a Triangle

12.7 The Medial Quadrilateral of Quadrilateral

12.8 Problems: Investigating Relationships by Using Geometric Properties

12.9 How Do I Know if I Understand?


Chapter 13 - Computer Explorations

13.0 Introduction

13.1 Triangle Inequalities

13.2 Angle Bisectors: Why the Incenter Works

13.3 Perpendicular Bisectors: Why the Circumcenter Works

13.4 Medians and the Centroid of a Triangle

13.5 Altitudes: The Orthic Triangle

13.6 Angle Bisectors, Medians, and Altitudes: Some Relationships

13.7 Revisiting the Medial Triangle: Perimeter and Area

13.8 Revisiting the Medial Quadrilateral: Area

13.9 Quadrilaterals and Circles

13.10 Circles: Central Angles and Inscribed Angles

13.11 Circles: More on Inscribed Angles and Arcs

13.12 Problems: Investigating Relationships by Using Number Ideas

13.13 How Do I Know if I Understand?


Part 5 - Mira (Reflecta) and Tracing Paper


Chapter 14 - Mira Contructions

14.0 Introduction

14.1 The Mira: What Does it Mean?

14.2 Reflection Lines and Point-Image Segments

14.3 Constructions with a Mira (CDs)

14.4 Altitudes of a Triangle

14.5 Altitudes, Orthocenters, and Trapezoids

14.6 Altitude Constructions with a Mira

14.7 Measuring a Triangle's Three Altitudes

14.8 Where is the Circumcenter?

14.9 How Do I Know if I Understand?


Chapter 15 - Symmetry

15.0 Introduction

15.1 Miniproject: Fold-andCut Paper Figures

15.2 Fold-and-Cut (Symmetric) Shapes

15.3 Orientation: One or Two Sides?

15.4a Problems: Symmetry, Version A

15.4b Problems: Symmetry, Version B

15.5 Fold and Cut: Three Symmetry Lines

15.6 Fold and Cut: Fivefold Symmetry

15.7 Problems: More on Symmetry

15.8 How Do I Know if I Understand?


Chapter 16 - The Four Symmetries

16.0 Introduction

16.1 Four Actions: Slide, Flip, Turn, and Glide-Flip

16.2 Four Symmetries

16.3 Translations and Coordinates

16.4 Problems: Four Actions or Symmetries

16.5 Combinatons of Reflections

16.6 Actions: Which of the Four Types?

16.7 Rotations and Glide-Reflections: Point-Image Segments

16.8 How Do You Get from One to the Other?

16.9 CD Problem: Find the Center of Rotation

16.10 CD Problem: Find the Glide-Reflection Line

16.11 An Experiment with the Four-Kinds Principle

16.12 Marking Symmetries on Wallpaper Designs

16.13a Problems: Four Types of Symmetry, Version A

16.13b Problems:Four Types of Symmetry, Version B

16.14 More Problems Involving the Four Types of Symmetry

16.15 How Do I Know if I Understand?


Prologue: Symmetries of Decorative Art

Prologue to Chaptes 17, 18, and 19


Chapter 17 - Symmetries of Mandalas

17.0 Introduction

17.1 Symmetries of Mandalas

17.2 Classifying Mandalas, First Try

17.3 Classifying Mandalas

17.4 Mandalas: One or Two Sides?

17.5 Template Design Mandalas

17.6 Template Design Problems

17.7 Express Yourself with a Mandala

17.8 The Symmetry Classification of Mandalas

17.9 Problems: Mandalas

17.10a Problems: Mandalas, Version A

17.10b Problems: Mandalas, Version B

17.11 How Do I Know if I Understand?


Chapter 18 - Symmetries of Borders

18.0 Introduction

18.1 Glide-Reflectional and Half-turn Symmetry

18.2 Classifying Borders, First Try

18.3 Borders: What Is Their Symmetry Type?

18.4 Generating Borders

18.5 Borders: Make Your Own Display

18.6 The Symmetry Classificaton of Borders

18.7 Problems Classifying Borders

18.8a Problems: Borders, Version A

18.8b Problems: Borders, Version B

18.9 How Do I Know if I Understand?


Chapter 19 - Escher-Style Tessellations

19.0 Introduction

19.1 Escher Tessellations, Type TTTT

19.2 How to Make a Type TTTT Tessellation

19.3 Cut and Tape: Make Your Own Tessellating Shape

19.4 Miniproject: Recognizability

19.5 Four Moves for Tessellating Squares

19.6 What Are the Possible Heesch Types?

19.7 What is the Heesch Type?

19.8 Project: Making Escher-Style Tessellations

19.9 Checking Understanding of Heesch Types

19.10 Marking Symmetries on Escher Tessellations

19.11 Do These Tessellations Work?

19.12 How Do I Know if I Understand?



A.1 A Guide for You, the Student: Making Sense of Geometry in an Inquiry-based Class

A.2 GeoSET Website: Internet Resources for Students

A.3 Construct/Describe Problems

A3.1 Hints for Doing CD Problems

A3.2 Shorthand Comments for CD Problems

A3.3 Catalogue of CD Problems

A.4 Dot Paper Template for Copying









John Wolfe, Oklahoma State University: John recently was promoted to professor emeritus at Oklahoma State University. After graduating from the University of California at Berkeley in 1971 he was active in mathematics research including several publications and National Science Foundation research grants in Banach Space Theory. His enthusiasm for teaching has been recognized by awards from both the Mathematical Association of America and the Regents of Oklahoma State University. Over the past several years educational issues with a special passion for geometry have been the focus of his professional life. Woodworking, travel, camping and grandkids are becoming increasingly important in his life.



Douglas B. Aichele, Oklahoma State University: Douglas grew up in Great Neck, NY. He received his undergraduate and graduate education in mathematics at the University of Missouri-Columbia. He has been a faculty member at Oklahoma State University for many years and was appointed Regents Professor in 1989. He is currently serving as Associate Head of the Department of Mathematics. Good teaching of mathematics has always been important to him and he has been recognized over the years by such institutions/organizations as the OSU Blue Key Alumni Association, Mathematical Association of America, and the University of Missouri-Columbia.His other interests reside mainly in the outdoors - particularly spending time with his family in the mountains of Colorado at a log cabin that he built by hand. He is an avid backpacker and hiker; he has climbed most of Colorado's Fourteeners (mountains with elevations greater then 14,000 ft.) and several mountains and glaciers near Juneau, Alaska.


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