Mathematics • Year 8 • Unit 3 • Lesson 17
Translations in the Real World
Use column vectors where they actually show up: a chess piece sliding on a board, a game character moving on screen, a ship navigating a grid, treasure-hunt directions, and a graphic-design layout shift. Then explain your thinking in your own words.
1. Word problems
Each problem hides a translation on a coordinate grid. Set up the column vector, then apply the rule (x, y) → (x + a, y + b). Show working — a single final answer with no working earns only half marks.
1.1 — Chess move. A knight is at square (3, 2) on a chessboard (treat squares as grid coordinates). The knight makes an "L-shaped" move: 2 squares right and 1 square up.
(a) Write the move as a column vector.
(b) Find the knight's new square.
(c) The knight then makes the same move again. Where does it end up? 3 marks
1.2 — Treasure map. A pirate stands at coordinate (5, 2) on a treasure map. The clues say: walk 4 paces west, then 7 paces north.
(a) Write the journey as one combined column vector (single shift).
(b) Find the treasure coordinates. 3 marks
1.3 — Game character move. A character in a video game is at sprite position (12, 8). When the player presses "right", the character moves by vector (3 over 0). When they press "jump", the character moves by (0 over 5). The player presses right, then jump, then right.
(a) Find the character's final position.
(b) What single vector would have got them there in one move? 3 marks
1.4 — Logo shift. A graphic designer has placed a logo with corners at L1(2, 5), L2(8, 5), L3(8, 9), L4(2, 9). The client says: "Please slide the whole logo 4 to the right and 3 down."
(a) Write the shift as a column vector.
(b) Find the new coordinates of all four corners of the logo. 3 marks
1.5 — Ship navigation. A cargo ship is at grid coordinate (−6, 4). After steaming for an hour it is at (2, −1) (1 grid unit = 10 km).
(a) Find the column vector of the ship's hour of travel (in grid units).
(b) Convert each component to kilometres.
(c) Describe the direction in words (e.g. "east and south"). 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate has been asked to find the translation vector that maps A(1, 6) to A′(4, 2). They write "Vector = object − image = (1 − 4 over 6 − 2) = (−3 over 4)". In your own words, explain (i) what mistake they have made, (ii) what the correct vector is and how to get it, and (iii) how they could verify their vector is right by translating A and checking they land on A′. Use the phrase "image minus object" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Chess knight
(a) Vector = (2 over 1).
(b) (3, 2) → (3 + 2, 2 + 1) = (5, 3).
(c) Apply again: (5, 3) → (7, 4). New square = (7, 4).
1.2 — Treasure map
(a) Combined vector = (−4 over 7) (west 4, north 7).
(b) (5, 2) → (5 − 4, 2 + 7) = (1, 9).
1.3 — Game character
(a) (12, 8) right→ (15, 8) jump→ (15, 13) right→ (18, 13).
(b) Single vector = (18 − 12 over 13 − 8) = (6 over 5).
1.4 — Logo shift
(a) Vector = (4 over −3).
(b) L1(2,5) → (6, 2); L2(8,5) → (12, 2); L3(8,9) → (12, 6); L4(2,9) → (6, 6). New corners: (6,2), (12,2), (12,6), (6,6).
1.5 — Ship navigation
(a) Vector = (2 − (−6) over −1 − 4) = (8 over −5).
(b) Horizontal: 8 × 10 = 80 km. Vertical: 5 × 10 = 50 km.
(c) Positive x = east; negative y = south. So the ship travelled 80 km east and 50 km south.
2.1 — Explain your thinking (sample response)
The classmate has subtracted in the wrong direction. The rule is always image minus object, not object minus image. The correct vector is (4 − 1 over 2 − 6) = (3 over −4), which means "3 right and 4 down". To verify, translate A(1, 6) by this vector: (1 + 3, 6 + (−4)) = (4, 2) = A′ ✓. The classmate's incorrect vector would have moved A in the opposite direction, landing at (−2, 10) instead of A′(4, 2).
Marking: 1 mark for spotting "subtracted in the wrong direction"; 1 mark for the correct vector (3 over −4); 1 mark for the verification by translating A; 1 mark for clear full-sentence explanation using the phrase "image minus object".