• Introduction to course organization.
• Student Information sheets.
• Web Materials
• Portfolios and grading
• Some topics we will study.
• Measurement and the Pythagorean Theorem [1.1]
• Measuring angles, lengths and areas.
• Squares, rectangles, parallelograms and triangles.
• dissections, cut and paste methods of measurement.
• Discuss Pythagorean Theorem (PT) and proofs.
• Do Pythagorean Activity Sheet
• Distribute Tangram sheets.
• Assignment for Tuesday.
• Read 1.1 and 1.2
• 1.1:  5-8, 11, 12, *13

• Measurement and the Pythaagorean Theorem [1.1] Review in part.
• Measuring angles, lengths and areas.
• Squares, rectangles, parallelograms and triangles. Circles.
• dissections, cut and paste methods of measurement.
• Discuss Pythagorean Theorem (PT) and proofs.
• Show video on PT
• The Square Me Puzzle.
• Puzzles and Polygons [1.2]
• Flatland and the plane
• The triangle, quadrilateral, pentagon, and hexagon.
• More on measurements of angles and areas of polyons.
• Activity: 1.2 Ex. 4, 5, 6
• Tangrams.
• Do Tangram Activity.
• Assignment for Wednesday:
• 1.2: 1-3, 8, 9

Cutting and reassembling polygons. Making puzzles.

More on Dissection Puzzles:  Dissections (Junkyard) and equidecomposable polygons (mef)

• Scissors congruence: A sc= B means figure A can be cut into pieces that can be reassembled to form figure B.

This is also described using the word "equidecomposable".  "A and B are equdecomposable to B."
• SC=  is a reflexive, symmetric, and transitive relation. [like congruence and similarity in geometry and equality in arithmetic]
• Discussion: What kind of motions are used in reassembling pieces in a puzzle?
• A sc= B implies Area(A) = Area(B)
• Area(A) = Area(B)  implies A sc= B !!
• Simple cases:
• Parallelograms with common base.
• Parallelograms with common parallels and same area.
• Parallalograms with same area.
• Triangles and parallelograms with same area.
• Adding parallelograms.
• Adding triangles.

Triangulation.

Film: Equidecomposable polygons.

Rigid Motions in (or about) the plane.

Orientation preserving

Translations

Rotations

Orientation reversing

Reflections

Glide reflections

Tilings Chapter 4

Regular

Activity in class: 4.1  Ex. 3, 4, 5

Semiregular

One polygonal Tile

Two tiles

Friezes

Wallpaper

Symmetry Chapter 5.

Polygonal

Planar

Tilings

4.1: 7, 8, 9

Tilings Chapter 4

Regular

Semiregular

Classification using vetex congruence.

How many polygons meet at a vertex?

What types can meet at a vertex?

How do the polygons fit around a triangle?

Read the "Rules" p86-88

One polygonal Tile - Non-regular:

Quadrilateral Activity

4.1: 14 (based on 7)
5.1: 6 (g,h,i)

More on Planar Tilings.

Polygonal

Card tilings

Symmetry Chapter 5.

Reflectional symmetry

Rotational Symmetry

Translation Symmetry

Glide reflection symmetry

4.1: 10, 24
5.2: 1,2
Plus symmetry of alphabet assignment.

Modifying tilings

In class activity: Modifying tilings

Classification of Isometries

Activity: Miras for reflection- one and two reflections

Video : Isometries

Every plane isometry is the product of at most three reflections.

Two reflections = rotation or translation.

Three reflections = reflection or glide reflection

 

	Preserve Orientation	Reverse Orientation
No Fixed points	Translation	Glide reflection
Fixed Point(s)	Rotation	Reflection

Using Isometries to recognize symmetries of a figure or tiling.

• Platonic (regular convex polyhedra) Solids
• Why are there only 5?
• Regular polygons around a vertex.
• All verteces are "the same".
• Symmetries in the plane compared to those in space- an intorduction:
• Translations
• Rotations: Center point - central axis
• Reflection :  across line - across plane
• Symmetries of the cube:
• Rotations
• reflections
• rotation- reflection
• Side trip:
• products of reflections in space:
• Rotations and translations
• Applications to dance
• Duality
• In the plane: dual tiling
• Vertex - Region
• edge-edge
• In space: Dual polyhedron. Connected by counting and constructions.
• Vertex - Face
• edge-edge
• Cube- octahedron; Dodecahedron, icosahedron; tetrahedron.

FAPP Surfaces.
The hypercube.
Turning a sphere inside out.(?)
Inversion: On a line. In the plane. Orthogonal circles.
Non-euclidean universe
Escher and Hyperbolic geometry.
Not knot (?)
Flatland