LESSON PLAN

Transformations: Translations, Rotations, and Reflections

Apothem Team

Grade 6 · Geometry

LESSON AT A GLANCE

Warm-up

5 min

Explore

15 min

Formalize

10 min

Practice

12 min

Exit ticket

3 min

Warm-up

Project a wallpaper pattern (repeating birds, some mirrored, some rotated) and ask: "The designer drew ONE bird. What moves produced all the others?" Students point and name informally — slid, flipped, spun.

Formal names incoming: translation, reflection, rotation — the three rigid motions. The wallpaper's secret is that all of design's repetition comes from exactly these three moves, and today they get coordinates.

Explore

Coordinate transformation lab on grid paper, one shape (a flag pentomino) through three stations: (1) Translate 6 right, 2 down — record every vertex before/after and hunt the coordinate rule $(x, y) \to (x+6, y-2)$ . (2) Reflect over the y-axis — rule: $(x,y) \to (-x, y)$ ; notice the flag now waves the OTHER way (orientation flipped). (3) Rotate 90° clockwise about the origin with tracing paper — rule discovered by comparing points: $(x, y) \to (y, -x)$ .

Each station's exit question: what stayed the same? (Side lengths, angles, size — everything but position/orientation.) These are the congruence-preserving moves.

Formalize

Formalize the three rigid motions and their coordinate signatures:

$\text{translate: } (x,y) \to (x+a,\, y+b) \qquad \text{reflect}_{y\text{-axis}}: (x,y) \to (-x, y) \qquad \text{rotate}_{90°\text{cw}}: (x,y) \to (y, -x)$

The orientation fingerprint tells the moves apart when the answer sheet is missing: translations and rotations keep a shape's handedness; reflections reverse it. A flipped flag that "slid" is lying — somewhere a mirror was involved.

Practice

Practice: perform one of each motion on given figures with coordinates recorded; identify the transformation (or sequence) mapping figure A to figure B in three mystery pairs; one composite — reflect then translate — with a prediction first.

Exit ticket: the point $(3, -2)$ is rotated 90° clockwise about the origin. Where does it land, and how do you check? ( $(-2, -3)$ ; tracing paper or the rule.)

Exit ticket

Exit ticket: the point $(3, -2)$ is rotated 90° clockwise about the origin. Where does it land, and how do you check? ( $(-2, -3)$ ; tracing paper or the rule.)

TIP Tracing paper is not a crutch — it's the verification instrument. Rule first, trace second, and when they disagree, the trace wins and the rule gets debugged.

WORKED EXAMPLES

Example 1 — Identify the mystery move

The evidence: triangle $A(1,1), B(4,1), C(1,3)$ became $A'(-1,1), B'(-4,1), C'(-1,3)$ .

Step 1: Check the orientation fingerprint: walking $A \to B \to C$ turns left in the original, but $A' \to B' \to C'$ turns right — handedness flipped. A reflection is involved.

Step 2: Hunt the mirror: each point's $y$ survived; each $x$ negated: $(x, y) \to (-x, y)$ — the y-axis reflection signature.

Step 3: Verify one point by distance: $A$ sits 1 unit right of the axis; $A'$ sits 1 unit left ✓ equidistant.

Step 4: Report the move completely: reflection over the y-axis. ("It flipped" is a clue; the LINE is the answer. Transformations are named with their full data: which mirror, which centre, which angle, which slide.)

Example 2 — Compose moves: reflect, then translate — does order matter?

The shape: flag at $(2,1), (2,4), (4,3)$ . Move A: reflect over the y-axis. Move B: translate 5 right.

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

Step 2: B then A: slide → $(7,1), (7,4), (9,3)$ ; reflect → $(-7,1), (-7,4), (-9,3)$ .

Step 3: Compare: completely different destinations. Order MATTERS for mixing reflections with translations — transformation composition is not commutative (the day's biggest phrase, cashed with coordinates).

Step 4: The follow-up worth a minute: two translations in either order DO agree (slides just add). So the non-commutativity came from the mirror, not from composing per se. Which pairs commute is a genuinely deep question — here it's enough to know you must READ the order.

Example 3 — The video-game sprite: which single move replaces two?

The situation: a game sprite was reflected over the y-axis, then reflected again over the vertical line $x = 4$ . The artist wants ONE move with the same effect.

Step 1: Track a test point through both mirrors: $(1, 2) \to (-1, 2)$ (first mirror) $\to (9, 2)$ (second mirror: $-1$ is 5 left of 4, so image is 5 right of 4).

Step 2: Compare start to finish: $(1,2) \to (9,2)$ — slid 8 right, no flip (two mirrors cancel the handedness change).

Step 3: Test a second point to trust the pattern: $(3, 5) \to (-3, 5) \to (11, 5)$ — also 8 right ✓.

Step 4: The single move: translate 8 right — and 8 is exactly TWICE the distance between the mirrors ( $2 \times 4$ ). Two parallel mirrors always compress into one slide of double their gap: wallpaper designers and game engines both lean on this daily.

MATERIALS

•Grid paper

•Tracing paper

•Flag pentomino templates

•Wallpaper pattern printouts

•Practice set (PDF)

WATCH FOR

!Rotation direction confused (clockwise vs counter-) — always state direction AND centre; both are part of the move's name.

!Reflection performed as translation (copy slid across, not flipped) — the orientation fingerprint catches it.

!Coordinate rules memorized without the grid: students apply

(y, -x)

but can't sketch where the shape went. Rule and picture must travel together.