AQA GCSE Mathematics Paper 1 (Higher), June 2025: Question 11

3 marks · Medium difficulty · Short Answer

Find the translation vector mapping shape A to shape B, and fully describe a rotation mapping shape A to shape B.

Practise this question

Question

A Cartesian coordinate grid showing x-axis from -4 to 4 and y-axis from 0 to 5. Square A of dimensions 2 by 2 is positioned with vertices at (-4, 3), (-2, 3), (-2, 5), and (-4, 5). Square B of dimensions 2 by 2 is positioned with vertices at (2, 1), (4, 1), (4, 3), and (2, 3). Question 11(a) asks to write down the translation vector that maps shape A to shape B. Question 11(b) asks to describe fully a rotation that maps A to B.
Question text

11 (a) Write down the translation vector that maps shape A to shape B.

[1 mark]

Answer

11 (b) Describe fully a rotation that maps A to B.

[2 marks]

Mark scheme

Show the mark scheme Mark scheme for 11(a) awards 1 mark (B1) for the column vector (6, -2). For 11(b), awards 2 marks (B2) for fully describing any valid rotation: 180 degrees about centre (0, 3); 90 degrees clockwise about centre (-1, 0); or 90 degrees anticlockwise about centre (1, 6). B1 is available for correct angle and direction without correct centre, or correct centre with incorrect rotation.

Q Answer Mark Comments

6 +6

11(a) B1 condone or written as a fraction

−2 −2

Alternative method 1: rotation 180°

180° ignore clockwise or anticlockwise

1 B1 180°

10 or turn

B2 or turn

and 2

(0, 3) or

(0, 3) with incorrect rotation

Alternative method 2: rotation 90° clockwise

90° clockwise B1 90° clockwise

or 270° anticlockwise or 270° anticlockwise

or turn clockwise or turn clockwise

B2

or turn anticlockwise or turn anticlockwise

and or

(–1, 0) (–1, 0) with incorrect rotation

11(b)

Alternative method 3: rotation 90° anticlockwise

90° anticlockwise B1 90° anticlockwise

or 270° clockwise or 270° clockwise

or turn anticlockwise or turn anticlockwise

B2

or turn clockwise or turn clockwise

and or

(1, 6) (1, 6) with incorrect rotation

Additional Guidance

Accept counterclockwise for anticlockwise

For the centre of rotation, condone missing brackets on coordinates, but not

written as a vector

Other or compound transformation stated or implied

eg ‘about the line’ or ‘scale factor’ or ‘moved by ’ B0

How to answer it

Transformations: Translations & Rotations on a Coordinate Grid

📌 What this question tests
  • Representing 2D translations using column vector notation correctly.
  • Tracking corresponding vertices between object and image shapes.
  • Fully describing a rotation by giving the angle, direction, and centre of rotation coordinates (x, y) .
  • Exploiting rotational symmetry of squares to find alternative valid descriptions.

Part 11 (a) — Translation Vector Mapping A to B

1 Mark • Assessment Objective: AO1 (Recall & Basic Application)

📐 Step-by-Step Method

  1. 1 Pick one corner on shape A:
    Bottom-left corner of A = (-4, 3) .
  2. 2 Find the matching corner on shape B:
    Bottom-left corner of B = (2, 1) .
  3. 3 Calculate horizontal shift (top number, x):
    From -4 to +2 = +6 (6 units right).
  4. 4 Calculate vertical shift (bottom number, y):
    From 3 to 1 = -2 (2 units down).

✅ Correct Answer

Column Vector:

( 6 -2 )
Mark Scheme:
[B1] for ( 6 -2 ) . Condones ( +6 -2 ) or written as a fraction line.

🧠 Exam Technique: Choose the Centre

If picking corners feels confusing, use the centre of each shape:

  • Centre of A = (-3, 4)
  • Centre of B = (3, 2)
  • Change in x = 3 - (-3) = 6
  • Change in y = 2 - 4 = -2

❌ Common Errors to Avoid

  • Inverting x and y: Writing ( -2 6 ) instead of ( 6 -2 ) . Remember: x is always on top!
  • Sign errors: Writing +2 instead of -2 because shape B is lower down than shape A.
  • Wrong direction: Mapping B to A gives ( -6 2 ) , losing the mark. Always read carefully: "A to B".

Part 11 (b) — Describe Fully a Rotation Mapping A to B

2 Marks • Assessment Objective: AO1 & AO2 (Geometric Reasoning)

✅ Any ONE of these Full Descriptions (2 Marks)

  • Method 1 (Easiest):
    Rotation of 180° (or half turn) about centre (0, 3).
    (No direction needed for 180°)
  • Method 2:
    Rotation of 90° clockwise (or 270° anticlockwise) about centre (-1, 0).
  • Method 3:
    Rotation of 90° anticlockwise (or 270° clockwise) about centre (1, 6).
Mark Scheme Breakdown:
[B1] Correct angle & direction (e.g. 180°, or 90° clockwise)
[B1] Correct centre coordinates (e.g. (0, 3))
Both correct award full [B2].

📐 Finding the Centre for 180°

  1. 1 Why 180° is the easiest:
    A 180° rotation sends any point straight through the centre of rotation to the opposite side.
  2. 2 Find the midpoint of the centres:
    Centre of A = (-3, 4)
    Centre of B = (3, 2)
  3. 3 Calculate midpoint coordinates:
    x = (-3 + 3) ÷ 2 = 0
    y = (4 + 2) ÷ 2 = 3
    Centre of rotation = (0, 3).

💡 Key Knowledge: Fully Describing Rotations

A full description of a rotation must ALWAYS include 3 elements:

  • Transformation type: "Rotation" (already stated in this question prompt).
  • Angle: e.g., 90°, 180°, 270°.
  • Direction: Clockwise or Anticlockwise (not needed if the angle is 180°).
  • Centre of rotation: Given as a coordinate pair (x, y) .

❌ Common Traps & Examiner Commentary

  • Writing coordinates as vectors: Writing the centre as ( 0 3 ) instead of (0, 3) loses the mark for the centre.
  • Forgetting direction for 90°: Simply saying "90° about (-1, 0)" earns only 1 mark because clockwise was omitted.
  • Mentioning multiple transformations: If a student says "Rotate 90° and then translate...", the mark scheme gives 0 marks (B0)! Only single transformations are allowed.

Topics

Geometry and measures · 3.4.1 Properties and constructions · 3.4.3 Vectors

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 1 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.