# Transformational Geometry Sequence

Henri Picciotto

Below is the sequence of lessons I used last time I taught geometric transformations and symmetry in my Space course at the Urban School of San Francisco. The course is an advanced geometry high school elective for students who have taken Geometry and Algebra 2. You can find more information about the course, including a bibliography, here.

My course long preceded the Common Core. While in spirit it is well aligned with those standards, it cannot be seen as a guide to their implementation, because it is founded on the idea that students have been introduced to congruence and similarity in the traditional way. (The CCSS, in contrast, suggest that the congruence and similarity criteria, instead of being postulates, should be developed on the foundation of geometric transformations. See the note at the end of this page about Common Core compliance.)

Moreover, the course assumes the students have seen a geometric introduction to complex numbers in Algebra 2. More information about that context: Urban School Math | Proof in High School Geometry | Seeking Depth in Algebra 2.

The outline below is approximate, and does not include homework, quizzes, or the review lessons which happen once in a while. Altogether it spans 10 weeks in a block schedule. Also: the outline, being mostly about transformations and symmetry, does not include most of the work we did on abstract algebra, polyhedra, or the fourth dimension, all of which are part of the Space course. Of those, the most relevant is abstract algebra. You can find those lessons, originally created for elementary school students, here.

This is the chronological sequence of lessons — it is not organized by topic. The topics are interweaved: my scheme is to stretch exposure to each topic by not "covering" the topic in a continuous and compact sequence. Stretching it out makes it easier to include homework, quizzes, quiz corrections as part of the sequence, gives students more time to absorb new ideas, and gives me more flexibility. (For example, if I hit a bottleneck on Topic 1, I switch to Topic 2 while figuring out how to move forward on Topic 1.)

--Henri

### Notes

Lessons focusing on symmetry are in italics.

Lessons using Cabri could be done with Geometer's Sketchpad or GeoGebra. (Some useful GeoGebra files are here.)

Lessons using Cabri 3D can only be done with Cabri 3D.

Lessons using Jovo could be done with Polydron.

The amount of material per week varies due to schedule quirks and whether or not there was a quiz or test that week.

## Week 1

Recognizing some basic symmetries in figures.

Very basic intro to finite groups, using Abstract Algebra activities. Includes hands-on and kinesthetic exploration of symmetry group of the equilateral triangle. Interweaved with the work on transformations over the first few weeks.

Informal introduction to transformations in Cabri (interactive geometry software)

Review of definitions and notation about functions (domain, range, inverse). Pre-image, image. Intro to one-to-one functions.

Introduce geometric mappings informally, including dilations, projections, distortions (stretching by different amounts in the horizontal and vertical direction.)

Define transformation of the plane (one-to-one, domain and range are whole plane).

Define reflection in a line, and half-turn. Use Cabri to reflect in a line without using the reflection tool. Repeat with half-turn.

## Week 2

Define isometry. Set goals:
- find all the isometries
- discover their properties
- figure out what happens when you compose them
- learn to compute outputs
- notice connections with symmetry

Use congruent triangles to prove isometries also preserve angles, parallelism, collinearity.

Define rotation, translation.

Use congruent triangles to prove reflections, rotations, and translations are isometries.

Experiences with mirrors and Miras. Kinesthetic experiences. Why does a mirror reverse left and right but not up and down?

Orientation and handedness. Orientation and isometries.

Mystery isometries (recognizing isometries by looking at pre-image and image). Introduction of glide reflection.

## Week 3

Making a glide reflection in Cabri.

Composition of two line reflections in Cabri (parallel, intersecting.)

Composition of two line reflections: proof.

Informally: properties of the four isometries.
- orientation
- fixed points
- invariant sets
- inverse mappings

Symmetry with pattern blocks (from Geometry Labs)

Symmetry group of the rectangle, the square.

## Week 4

Experimentally on graph paper: how many points, and their images, does it take to determine a unique isometry? (one point: infinite possibilities / two points: two possibilities -- one preserves, one reverses orientation / three points: unique!)

Exploration: how to find
- the vector for a translation
- the mirror line for a reflection
- the center of a rotation
- the mirror line for a glide reflection

Complex number review (students were introduced to complex numbers visually in Algebra 2, with multiplication defined in polar form.)

An isometry is determined by three non-collinear points and their images. Cabri experiment lays the foundation for the proof. Formalizing the proof is challenging.

"Two mirrors" activity (from Geometry Labs), using triangle paper (see below) for part of it.

Definition of symmetry, based on the concept of invariant sets. Intro to rosette, frieze, and wallpaper symmetries.

Notation for rosette groups.

Intro to Cabri 3D.

Complex numbers game (program I wrote in the BYOB programming environment -- very effective way to reinforce basics of complex arithmetic.)

Project: make a beautiful rosette, in any non-computer medium. (Though the computer can be used in planning.) Each student is randomly assigned a specific group to illustrate. Trades are allowed.

## Week 5

Every isometry is equivalent to a succession of at most three line reflections. Cabri experiment lays the foundation for the proof.

The isometries of the plane form a (non-commutative) group.

"Two mirrors" activity in Cabri.

Proof: slopes of perpendicular lines and converse.

Complex numbers and transformations.

Using complex numbers on the calculator to calculate images: translation, dilation, rotation.

Rotation not around the origin.

Frieze symmetry: using A, C, H, L or Z as elements, try to find every possible frieze group. Or: use L's only, but you'll need to place them in different arrangements.

Notation for the frieze symmetry groups.

There are only four isometries of the plane. Start the proof. (The proof is epic: one, then two, then three reflections. The latter has many cases. It takes days and stretches into Weeks 6 and 7.)

Proof (using similar triangles) that complex multiplication in polar and rectangular forms are equivalent: one particular case.

## Week 6

Proof (using similar triangles) that complex multiplication in polar and rectangular forms are equivalent: general case.

Recognizing frieze symmetry groups.

"Fold-up" polyhedra in Cabri 3D. (Involves substantial use of symmetry and transformations.)

Composing rotations.

Project: make a beautiful frieze, using colored paper, scissors, and rubber cement. Each student is randomly assigned a specific group to illustrate. Trades are allowed.

## Week 7

Intro to matrices and matrix multiplication. Identity matrix.

Matrix multiplication for reflection in x-axis, y-axis.

Intro to the wallpaper groups.

Matrix for reflection in y=x. Matrix for 90° rotation around the origin. Composing those two.

Matrix for a rotation around the origin. (Easy to derive from complex multiplication.)

## Week 8

Reflection in a line through the origin: using matrix composition.

Recognizing wallpaper groups.

Exploration: tiling with triangles and quadrilaterals. All triangles and all quadrilaterals tile the plane.

Exploration: tiling with regular polygons.

Matrices on the calculator (TI-89)

Composition of transformations by matrix multiplication.

Archimedean tilings (using template)

## Week 9

Using 3 by 3 matrices to do translations. 3 by 3 matrices for all the isometries.

Making platonic solids with Jovo. Rotations, mirror planes.

Calculator shortcuts for transformation matrices.

Archimedean solids with Jovo.

Matrices for dilations and half-turns. Note the case where the dilation factor is -1.

Project: make a beautiful wallpaper. Each student is randomly assigned a specific group to illustrate. Trades are allowed. Because this is visually challenging, students are asked to come to class with a draft the day before the actual activity. The latter is led by an art teacher. This year, she had the students do their work on translucent material. The resulting designs were taped on windows.

Using and combining transformation matrices on this site.

## Week 10

Isometries in 1, 2, and 3D. Orientation and handedness in 1, 2, and 3D. (This simultaneously reviews and extends the work we did in 2D.)

Formulas for 1D isometries.

Archimedean solids using Jovo. Symmetries.

Mirror symmetry in 3D in Manet's painting: "A Bar at the Folies-Bergère"

### Common Core Compliance

I support the switch in sequencing proposed by the Common Core State Standards: build congruence and similarity on a foundation of transformations, rather than vice versa. (See comments on this on my Transformational Geometry page.) I am working on developing a meaningful and age-appropriate way to do this. What would make sense to me if I had to teach the Common Core right now:

1. Use transformational geometry informally in middle school and/or at the beginning of high school geometry. Use that to establish basic results such as vertical angles, alternate interior angles, and the like. (I have done something like this without using that terminology for many years. For example, I have avoided formal proof that vertical angles are equal, and just argued that the "amount of turning" being the same, the angles should be the same.)
2. Introduce congruence and similarity on the foundation of the transformations, as recommended by the CCSS. See how the congruence and similarity postulates follow. This is the part that will require the most work to sort out.
3. In the second half of high school geometry, allow both traditional and transformational approaches to proof.
4. Teach some version of the above syllabus to 11th or 12th graders, choosing what to put in a core class vs. in an elective.

Related pages on this site:
Transformational Geometry
Geometry Labs
Abstract Algebra
Space