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Polyominoes, Polygons, and Polyhedra:
Discovering Geometry through Exploration
Teresa D. Magnus
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Front Matter
Colophon
Dedication
Acknowledgements
Preface
How to use this book
1
Introduction to Geometrical Thinking
1.1
Invitation to Explore
1.1.1
To The Student
1.1.2
First Exploration
1.1.3
Learning from the Explorations
1.1.4
Exploring to Learn
1.1.5
Exercises
1.2
Invitation to Apply Knowledge
1.2.1
Building a Course Toolbox
1.2.2
Defining Polygons
1.2.3
Exercises
1.3
Invitation to Share our Discoveries
1.3.1
Identifying a Pattern
1.3.2
Conjectures and Explanations in Mathematics
1.3.3
A Sample Explanatory Letter
1.3.4
Exercises
1.4
Invitation to Overcome Challenges
1.4.1
Growing through Persistance
1.4.2
Confronting the Challenge
1.4.3
A Geometric Tool and an Algebraic Tool
1.4.4
Exercises
1.5
References and Suggested Readings
2
Exploring Polygons and Other Figures on the Plane
2.1
Exploring Area
2.1.1
Area of a Parallelogram
2.1.2
The Area of a Triangle
2.1.3
The Area of a Trapezoid
2.1.4
Thoughts on Area
2.1.4.1
Estimating the Area of Curved Regions
2.1.4.2
The Area of a Circle
2.1.5
Exercises
2.2
Exploring Angle Measure
2.2.1
Linear Objects
2.2.2
Defining and Measuring Angles
2.2.3
Angles of a Triangle
2.2.4
Additivity of Angles and Definitions Associated with Angles
2.2.5
Vertical Angles
2.2.6
Exercises
2.3
What Does Parallel Mean?
2.3.1
What do we remember about lines on a plane?
2.3.2
Geometry on Another Surface
2.3.3
Understanding Parallel Lines
2.3.4
Exercises
2.4
Exploring Regular Polygons
2.4.1
Regular Polygons
2.4.2
Regular Tilings
2.4.3
Semiregular Tilings
2.4.4
Narrowing Down the Options
2.4.5
Exercises
2.5
Exploring Length and the Pythagorean Theorem
2.5.1
Right Triangles
2.5.1
Check Your Understanding
2.5.2
The Pythagorean Theorem
2.5.2
Check your Understanding
2.5.3
Proving the Pythagorean Theorem with Picture Puzzles
2.5.4
Reflecting on the Pythagorean Theorem
2.5.5
Exercises
2.6
References and Suggested Readings
3
Exploring Polyhedra
3.1
Regular Polyhedra
3.1.1
Two-Dimensional and Three-Dimensional Objects
3.1.2
Defining a Polyhedron
3.1.3
Defining Regular Polyhedra
3.1.4
What Regular Polyhedra Exist
3.1.5
Nets for Polyhedra
3.1.6
Exercises
3.2
Semiregular Polyhedra
3.2.1
Defining Semiregular Polyhedra
3.2.2
Examples of Semi-regular Polyhedra
3.2.3
Families of Semiregular Polyhedra
3.2.4
Defining Prisms and Antiprisms
3.2.5
Exercises
3.3
Prisms and Cylinders
3.3.1
Definitions
3.3.2
Volumes of Prisms and Cylinders
3.3.3
The Surface Area of Prisms and Cylinders
3.3.4
Exercises
3.4
Pyramids and Cones
3.4.1
Defining Pyramids and Cones
3.4.2
Patterns with Pyramids
3.4.3
Measuring Pyramids
3.4.3.1
Surface Area of Pyramids and Cones
3.4.3.2
Volume of Pyramids and Cones
3.4.4
Exercises
3.5
Measuring in Different Dimensions
3.5.1
Measuring Interior Space
3.5.2
Growth in Different Dimensions
4
Exploring Transformations
4.1
An Introduction to Reflections
4.1.1
Defining a Reflection
4.1.2
Exploring Reflections by Sketching
4.1.3
Reflections and Coordinates
4.1.4
Key Ideas Regarding Reflections
4.1.5
Optional GeoGebra Interactives
4.1.6
Exercises
4.2
Orientation-Maintaining Transformations
4.2.1
What do we mean by orientation-maintaining?
4.2.2
Collinearities and Isomorphisms
4.2.3
Translations
4.2.3.1
Properties of Translations
4.2.4
Rotations
4.2.4.1
Properties of Rotations
4.2.5
Exercises
4.3
Identifying Isometries
4.3.1
What do we know? What do we wonder?
4.3.2
Isometry Between Congruent Figures
4.3.3
Recognizing Isometries by Fixed Points
4.3.4
Defining Objects of Isometries
4.3.5
What have we learned?
4.3.6
Exercises
4.4
Compositions of Reflections
4.4.1
Compositions
4.4.2
The Four Isometries
4.4.3
Exercises
4.5
Mystery Isometries
4.5.1
Review of Isometries
4.5.2
Solving the Mysteries
4.5.3
Exercises
4.6
Symmetry
4.6.1
Two Types of Symmetry
4.6.2
Exploring Symmetry of Polygons
4.6.3
Symmetry Transformations
4.6.4
Exercises
5
Exploring Similarity
5.1
Transformations that Scale
5.1.1
Distance on the Cartesian Plane
5.1.2
Dilating Figures on a Cartesian
\(xy\)
-Plane
5.1.3
Defining Dilation
5.1.4
Similar Triangles and Dilations
5.1.5
Exercises
5.2
Dilations and Similar Figures
5.2.1
A Dynamic Look at Dilations
5.2.2
Dilations and Similar Triangles
5.2.3
Exercises
5.3
The Pythagorean Theorem and the Golden Triangle
5.3.1
A Review of Similarity
5.3.2
Applications of Similarity
5.3.3
Self-Similar Triangles
5.3.4
A Connection to a Regular Pentagon
5.3.5
Exercises
Backmatter
A
Algebra Review
A.1
Algebraic Expressions and Equations
A.2
Field Properties of the Real Numbers
A.3
Properties of Equality
A.4
Functions and Operations
A.4.1
Working with squares and square roots
A.4.2
Other Pairs of Inverse Functions
A.4.3
Order of Operations
B
List of Principles
C
List of Symbols
D
List of Formulas
Index
Colophon
Colophon
Colophon
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