LESSON PLAN

Combinations of Transformations and Tessellations

Apothem Team

Grade 7 · Geometry

LESSON AT A GLANCE

Warm-up

5 min

Explore

15 min

Formalize

10 min

Practice

12 min

Exit ticket

3 min

Warm-up

Show an Escher-style tessellation of interlocking lizards: "No gaps, no overlaps, one repeating creature. What MOVES is the artist secretly using?" The room finds slides and turns (and, in some prints, flips).

Today: transformations compose — moves chain into sequences — and tessellations are the art form where composed rigid motions tile the entire plane. The Grade 6 moves become Grade 7 choreography.

Explore

Composition lab on coordinate grids: (1) Perform "reflect over the y-axis, THEN translate down 3" on a flag; record start/end coordinates; then reverse the order and confirm the destinations differ — composition order matters (recalled from Grade 6, now with coordinates as evidence). (2) The two-reflections theorems, discovered: parallel mirrors compose to a TRANSLATION (double the gap); intersecting mirrors compose to a ROTATION about their crossing (double the angle between them).

Then tessellation studio: test which regular polygons tile alone (triangles ✓, squares ✓, hexagons ✓, pentagons ✗ — angle 108° doesn't divide 360°) and build one Escher-style tile by the nibble-and-slide method: cut a notch from a square's left edge, tape it to the right edge, and the modified tile still tessellates by translation.

Formalize

Formalize composition and the tiling criterion:

$\text{(reflect } \parallel \text{ mirrors)} = \text{translate } 2d \qquad \text{tiles alone} \iff \text{interior angle divides } 360°$

The angle test explains the census: equilateral triangle 60° (six at a point), square 90° (four), hexagon 120° (three) — all divide 360. The pentagon's 108° leaves a gap or forces an overlap. Every tessellation question at a vertex is an angle-accounting question.

Practice

Practice: two composition problems with coordinates (including one order-swap comparison); one "find the single move" (given start and end, name the one transformation that does it); the angle-test table for regular polygons; one nibble-and-slide tile designed and tiled four times.

Exit ticket: why do regular hexagons tessellate but regular octagons don't? (120 divides 360; 135 doesn't — $360/135 = 2.67$ corners can't meet.)

Exit ticket

Exit ticket: why do regular hexagons tessellate but regular octagons don't? (120 divides 360; 135 doesn't — $360/135 = 2.67$ corners can't meet.)

TIP The nibble-and-slide rule (what you cut from one edge must attach to the OPPOSITE edge for translation tiling) is where art projects fail mathematically — police the opposite-edge rule early and the lizards work.

WORKED EXAMPLES

Example 1 — Compose and compare: two orders, two destinations

The flag: vertices $(1,1), (1,4), (3,3)$ . Move A: reflect over the x-axis. Move B: translate up 6.

Step 1: A then B: reflect → $(1,-1), (1,-4), (3,-3)$ ; up 6 → $(1,5), (1,2), (3,3)$ .

Step 2: B then A: up 6 → $(1,7), (1,10), (3,9)$ ; reflect → $(1,-7), (1,-10), (3,-9)$ .

Step 3: Compare: wildly different — one lands in quadrant I, the other deep in IV. Order changed the story completely.

Step 4: The why, in one image: move B slides the flag AWAY from the mirror before A flips it in case 2 — the mirror throws it far. Compositions read RIGHT to left in effect order later in math; for now, "first move first" with the order always stated.

Example 2 — The two-mirror rotation: discovered, then verified

Setup: two mirror lines through the origin — the x-axis, and the line $y = x$ (45° between them).

Step 1: Track $(3, 1)$ : reflect over the x-axis → $(3, -1)$ ; reflect THAT over $y = x$ (swap coordinates) → $(-1, 3)$ .

Step 2: Compare start to finish: $(3,1) \to (-1, 3)$ . Test the rotation hypothesis — 90° counter-clockwise about the origin sends $(x,y) \to (-y, x)$ : $(3,1) \to (-1,3)$ ✓ exactly.

Step 3: Confirm the angle law: mirrors at 45° composed into a rotation of $2 \times 45 = 90°$ ✓.

Step 4: Second point for confidence: $(2, 0) \to (2, 0) \to (0, 2)$ — also the 90° image ✓.

Step 5: Why kaleidoscopes work, in one line: two mirrors at angle $\theta$ manufacture rotations of $2\theta$ — six-fold snowflake symmetry from mirrors at 30°. The lab toy is a composition theorem.

Example 3 — The vertex audit: can squares and octagons tile together?

The pattern in question: the classic bathroom floor — regular octagons with small squares in the gaps. Does the math approve?

Step 1: Interior angles: square 90°; regular octagon 135°.

Step 2: Audit one vertex of the proposed tiling: two octagons and one square meet → $135 + 135 + 90 = 360°$ ✓ exactly. No gap, no overlap — the tiling is legal.

Step 3: Contrast with octagons alone: $135 + 135 = 270$ (gap); three octagons $= 405$ (overlap) — impossible, as the earlier census found.

Step 4: The upgrade this example delivers: MIXED tilings (semi-regular tessellations) open when different polygons' angles sum to exactly 360 at every vertex. Students hunt one more legal combo (triangle + two hexagons: $60 + 120 + 120 + 60$ … care: arrangement matters — $3.6.3.6$ works: $60+120+60+120 = 360$ ✓).

Step 5: The floor under your feet was solving angle equations all along.

MATERIALS

•Grid and tracing paper

•Square tiles for nibbling

•Scissors and tape

•Regular polygon sets

•Practice set (PDF)

WATCH FOR

!Composition assumed commutative — the coordinate evidence from the lab settles it permanently.

!Tessellation judged by "looks like it fits" — the angle test replaces squinting with arithmetic.

!Any pretty repeating pattern called a tessellation — gaps or overlaps disqualify; the definition has teeth.