Mathematics • Year 8 • Unit 3 • Lesson 18
Reflections and Rotations
Build fluency with the six coordinate rules: reflect over x-axis, y-axis, y = x, and rotate 90° ACW, 180°, 270° ACW about the origin. One fully worked example, one guided example with blanks, then eight independent problems.
1. I do — fully worked example
Read every line. Each step has a short reason so you can see why we do it, not just what.
Problem. Triangle ABC has vertices A(1, 2), B(3, 2), C(2, 4). Reflect it over the x-axis. Then rotate the original triangle 90° anticlockwise about the origin.
Step 1 — State the reflection rule (x-axis).
(x, y) → (x, −y)
Reason: reflecting over the x-axis flips the y-coordinate, leaves x unchanged.
Step 2 — Apply to each vertex.
A(1, 2) → A′(1, −2)
B(3, 2) → B′(3, −2)
C(2, 4) → C′(2, −4)
Reason: apply the same rule to every vertex.
Step 3 — State the rotation rule (90° ACW about origin).
(x, y) → (−y, x)
Reason: swap the coordinates, then negate the new x (i.e. the original y becomes negative).
Step 4 — Apply to each vertex.
A(1, 2) → A″(−2, 1)
B(3, 2) → B″(−2, 3)
C(2, 4) → C″(−4, 2)
Reason: the rotation swap-then-negate rule applied to all three vertices.
Answers — Reflection: A′(1, −2), B′(3, −2), C′(2, −4). Rotation: A″(−2, 1), B″(−2, 3), C″(−4, 2).
2. We do — fill in the missing steps
Same shape as Section 1, with the working faded. Fill in each blank. 4 marks
Problem. Triangle PQR has vertices P(2, 1), Q(5, 1), R(4, 4). Reflect it over the y-axis.
Step 1 — Rule for reflection over the y-axis:
(x, y) → (______, ______)
Step 2 — Apply to each vertex:
P(2, 1) → P′(______, ______)
Q(5, 1) → Q′(______, ______)
R(4, 4) → R′(______, ______)
Step 3 — Check: the y-coordinate ______ the same; the x-coordinate ______ (changed sign / stayed the same).
Step 4 — Sketch check: the reflected triangle is the same size and shape, just on the ______ side of the y-axis.
3. You do — independent practice
Show all working. The first three are foundation (reflect or rotate a single point). The middle three are standard (apply to a whole shape). The last two are extension (identify the transformation from object and image).
Foundation — single-point transformations
3.1 Reflect (4, −3) over the x-axis. State the image. 1 mark
3.2 Reflect (−2, 5) over the y-axis. State the image. 1 mark
3.3 Rotate (2, 3) by 90° anticlockwise about the origin. State the image. 1 mark
Standard — apply to a whole shape
3.4 Triangle ABC has A(1, 1), B(4, 1), C(2, 3). Reflect over the line y = x. List A′, B′, C′. (Rule: (x, y) → (y, x).) 2 marks
3.5 Rotate triangle DEF with D(2, 1), E(5, 1), F(3, 4) by 180° about the origin. List D′, E′, F′. 2 marks
3.6 Rotate square WXYZ with W(1, 1), X(3, 1), Y(3, 3), Z(1, 3) by 270° anticlockwise (= 90° clockwise) about the origin. List the image vertices. 2 marks
Extension — identify the transformation
3.7 A(2, 5) maps to A′(2, −5) and B(4, 3) maps to B′(4, −3). Which transformation was applied? Justify with the coordinate rule. 2 marks
3.8 P(3, 1) maps to P′(−1, 3) and Q(2, −2) maps to Q′(2, 2). Which transformation was applied to each? (Hint: they're different!) Justify each with the coordinate rule. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (PQR reflected over y-axis)
Step 1: (x, y) → (−x, y).
Step 2: P(2,1) → P′(−2, 1); Q(5,1) → Q′(−5, 1); R(4,4) → R′(−4, 4).
Step 3: y-coordinate stays the same; x-coordinate changed sign.
Step 4: the reflected triangle is on the left/opposite side of the y-axis.
3.1 — (4, −3) over x-axis
(x, y) → (x, −y): (4, −3) → (4, 3).
3.2 — (−2, 5) over y-axis
(x, y) → (−x, y): (−2, 5) → (2, 5).
3.3 — (2, 3) by 90° ACW about origin
(x, y) → (−y, x): (2, 3) → (−3, 2).
3.4 — ABC reflected over y = x
(x, y) → (y, x): A(1, 1) → A′(1, 1); B(4, 1) → B′(1, 4); C(2, 3) → C′(3, 2). A′(1, 1), B′(1, 4), C′(3, 2).
3.5 — DEF rotated 180° about origin
(x, y) → (−x, −y): D(2, 1) → D′(−2, −1); E(5, 1) → E′(−5, −1); F(3, 4) → F′(−3, −4). D′(−2,−1), E′(−5,−1), F′(−3,−4).
3.6 — Square WXYZ by 270° ACW about origin
(x, y) → (y, −x): W(1, 1) → W′(1, −1); X(3, 1) → X′(1, −3); Y(3, 3) → Y′(3, −3); Z(1, 3) → Z′(3, −1). W′(1,−1), X′(1,−3), Y′(3,−3), Z′(3,−1).
3.7 — A(2,5)→A′(2,−5); B(4,3)→B′(4,−3)
The x-coordinate is unchanged and the y-coordinate has changed sign. That matches the rule (x, y) → (x, −y), so this is a reflection over the x-axis.
3.8 — P(3,1)→P′(−1,3); Q(2,−2)→Q′(2,2)
P: (3, 1) → (−1, 3). Apply 90° ACW rule (x, y) → (−y, x): (3, 1) → (−1, 3) ✓. So P was rotated 90° anticlockwise about the origin.
Q: (2, −2) → (2, 2). Only the y-coordinate flipped sign, x unchanged: rule (x, y) → (x, −y). So Q was reflected over the x-axis.