Mathematics • Year 8 • Unit 3 • Lesson 18
Reflections and Rotations — Mixed Challenge
Pull everything from Lesson 18 together: the three reflection rules, the four rotation rules, identifying transformations from object and image, and combining two transformations in sequence. Six mixed problems, one "find the mistake", and one open-ended composition puzzle.
1. Mixed problems — choose the right rule
Each question pulls a different transformation from Lesson 18. Decide which rule applies before you start writing. Show your working. 3 marks each
1.1 Reflect the point (−5, 2) over the x-axis. State the rule and the image.
1.2 Reflect the point (3, 7) over the line y = x. State the rule and the image.
1.3 Rotate the point (−1, 4) by 180° about the origin. State the rule and the image.
1.4 Rotate the point T(5, −2) by 270° anticlockwise about the origin (= 90° clockwise). State the rule and the image.
1.5 Triangle ABC has vertices A(1, 3), B(4, 3), C(4, 1). Rotate it 90° clockwise about the origin (= 270° ACW). List A′, B′, C′.
1.6 Point G(−3, 5) is first rotated 90° anticlockwise about the origin, then reflected over the x-axis. Find the final coordinates G″, showing each step.
2. Find the mistake
A Year 8 student has tried to reflect the point (−4, 3) over the y-axis. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — reflect (−4, 3) over the y-axis:
Line 1: Rule for y-axis reflection: (x, y) → (−x, −y)
Line 2: Apply: (−4, 3) → (−(−4), −3)
Line 3: = (4, −3)
Line 4: So the image is (4, −3).
(a) Which line contains the original mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected rule and working in full, including the corrected final image.
Stuck? Revisit lesson § Card 6 — for the y-axis, only x changes sign. The rule (x, y) → (−x, −y) is actually the 180° rotation rule, not the y-axis reflection rule.3. Open-ended challenge — Equivalent transformation pairs
This question has more than one valid answer. 4 marks
3.1 Sometimes two transformations applied in sequence give the same result as a single transformation. For example, a reflection over the x-axis followed by a reflection over the y-axis gives the same result as a single 180° rotation about the origin.
Find two more pairs (or short sequences) of transformations that give the same overall result as a single named transformation from the lesson. For each pair you find:
(i) State the two transformations applied in order.
(ii) Pick a test point (e.g. (3, 1) or (2, 5)) and show the working through both steps.
(iii) State the single transformation that produces the same final image.
Bonus: one of your pairs must NOT include any rotation in the two-step sequence (i.e. both steps are reflections).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — (−5, 2) over x-axis
Rule: (x, y) → (x, −y). Image = (−5, −2).
1.2 — (3, 7) over y = x
Rule: (x, y) → (y, x). Image = (7, 3).
1.3 — (−1, 4) rotated 180°
Rule: (x, y) → (−x, −y). Image = (1, −4).
1.4 — T(5, −2) rotated 270° ACW
Rule: (x, y) → (y, −x). Image = (−2, −5) = T′(−2, −5).
1.5 — ABC rotated 90° CW (= 270° ACW)
Rule: (x, y) → (y, −x). A(1, 3) → A′(3, −1); B(4, 3) → B′(3, −4); C(4, 1) → C′(1, −4). A′(3, −1), B′(3, −4), C′(1, −4).
1.6 — G(−3, 5): 90° ACW then x-axis reflection
Step 1 (90° ACW): (x, y) → (−y, x). G(−3, 5) → G′(−5, −3).
Step 2 (x-axis): (x, y) → (x, −y). G′(−5, −3) → G″(−5, 3).
Final: G″(−5, 3).
2 — Find the mistake
(a) The mistake is on Line 1 (which is then carried through to Lines 2, 3 and 4).
(b) The student wrote (x, y) → (−x, −y) for the y-axis reflection. That rule is actually the 180° rotation rule about the origin. For a y-axis reflection, only the x-coordinate changes sign: (x, y) → (−x, y).
(c) Corrected working:
Rule for y-axis reflection: (x, y) → (−x, y)
Apply: (−4, 3) → (−(−4), 3) = (4, 3).
Sanity check: reflecting over the y-axis flips left/right but keeps the height (y) the same — so the y-coordinate 3 must be unchanged.
3 — Equivalent transformation pairs (sample solutions)
There are several valid pairs. Two good examples:
Pair 1: reflect over y-axis, then reflect over x-axis.
Test point (3, 1): step 1 → (−3, 1); step 2 → (−3, −1).
Equivalent single transformation: 180° rotation about the origin. Check direct: (3, 1) → (−3, −1) ✓. (This is the bonus — both steps are reflections.)
Pair 2: rotate 90° ACW, then rotate 90° ACW again.
Test point (3, 1): step 1 → (−1, 3); step 2 → (−3, −1).
Equivalent single transformation: 180° rotation about the origin. Check direct: (3, 1) → (−3, −1) ✓.
Other valid pairs:
— reflect over x-axis then reflect over y = x: (3, 1) → (3, −1) → (−1, 3). Single = 90° ACW rotation about origin ✓.
— rotate 90° ACW four times = identity (back to start).
— reflect over y = x then reflect over x-axis = 90° ACW.
Marking: 1 mark per valid pair with correct test-point working and identified single transformation (up to 3 marks). 1 bonus mark if at least one pair uses two reflections (no rotation step).