Mathematics Standard • Year 11 • Module 2 • Lesson 12
Scale Drawings and Maps — Past-Paper Style
HSC Mathematics Standard 2-style writing on scale drawings — short answers and one structured extended response with explicit marking criteria.
1. Short-answer questions
1.1 A map of regional NSW has scale 1:20 000. Two towns are 7.5 cm apart on the map. Calculate the actual distance between the towns in km, showing your conversion. 2 marks Band 3
1.2 A floor plan of a single rectangular room uses scale 1:100. On the plan the room measures 4.2 cm × 3.0 cm.
(a) State the actual dimensions in metres.
(b) Find the actual area of the room in m². 3 marks Band 3-4
1.3 On a drawing, a 3 cm line represents an actual distance of 12 m.
(a) Find the scale of the drawing in the form 1:n, showing your unit conversion.
(b) A nearby fence is drawn as 5.5 cm on the same drawing. Find the actual length of the fence in metres. 4 marks Band 4
2. Extended response
2.1 A 1:200 scale plan shows a backyard. The lawn appears as a 6 cm × 5 cm rectangle. A rectangular pool, also on the plan, appears as a 4 cm × 2.5 cm rectangle placed inside the lawn area (so the lawn is reduced by the pool footprint).
Turf to re-lay the lawn costs $18 per m² supplied and laid.
The owner says: "the pool only takes up a small bit of the lawn — I'll need turf for almost the whole 6 × 5 area." Test this claim with your calculations.
(a) Find the actual dimensions of the lawn and of the pool, in metres, and the actual area of each in m².
(b) Find the actual area of the lawn that needs turf (i.e. excluding the pool footprint), in m².
(c) Calculate the cost of turf to the nearest dollar, then write a one-sentence conclusion that compares your answer to the owner's claim — does it support or contradict the claim? 7 marks Band 5-6
Explicit marking criteria
Part (a) — 3 marks
• 1 mark — correct actual lawn dimensions (multiply drawn cm by 200, convert to m).
• 1 mark — correct actual pool dimensions.
• 1 mark — correct area for both lawn and pool with units m².
Part (b) — 1 mark
• 1 mark — correct turf area = lawn area − pool area.
Part (c) — 3 marks
• 1 mark — correct cost = turf area × $18, to the nearest dollar.
• 1 mark — compares the turf area (or its fraction) to the lawn area in numerical terms.
• 1 mark — explicit one-sentence conclusion stating whether the owner's claim is supported or contradicted, with justification.
Your response:
Stuck on (c)? Compute (turf area ÷ total lawn area) × 100% to give a percentage you can use in the conclusion sentence.How did this worksheet feel?
What I'll revisit before next class:
1.1 — Map distance 1:20 000 (2 marks)
Sample response.
Actual = 7.5 × 20 000 = 150 000 cm.
150 000 ÷ 100 000 = 1.5 km.
Marking notes. 1 mark — correct multiplication 7.5 × 20 000. 1 mark — correct conversion to km (÷100 000) with units. A bare "1.5 km" with no working scores 1/2.
1.2 — Floor plan 1:100 (3 marks)
(a) Sample. 4.2 × 100 = 420 cm = 4.2 m; 3.0 × 100 = 300 cm = 3.0 m. So 4.2 m × 3.0 m.
(b) Sample. Area = 4.2 × 3.0 = 12.6 m².
Marking notes. (a) 1 mark — correct dims in metres. (b) 1 mark — uses actual m dimensions. 1 mark — correct numerical area with units m². Common error: computing 4.2 × 3.0 = 12.6 cm² and writing "12.6 cm²" or "12.6" without converting/labelling — area must be in m².
1.3 — Finding scale and applying it (4 marks)
(a) Sample. 12 m = 1 200 cm. n = 1 200 ÷ 3 = 400. Scale = 1:400.
(b) Sample. Actual = 5.5 × 400 = 2 200 cm = 22 m.
Marking notes. (a) 1 mark — converts 12 m to 1 200 cm. 1 mark — correct n = 400 and scale stated as 1:400. (b) 1 mark — applies 5.5 × 400. 1 mark — converts to 22 m with units. Common error: writing the scale as 1:4 (forgetting to convert m to cm) — this loses both (a) marks and propagates a wrong answer to (b).
2.1 — Backyard turf with pool subtracted (7 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Actual dimensions and areas.
Lawn: 6 × 200 = 1 200 cm = 12 m; 5 × 200 = 1 000 cm = 10 m. Lawn 12 m × 10 m. [1 mark.]
Pool: 4 × 200 = 800 cm = 8 m; 2.5 × 200 = 500 cm = 5 m. Pool 8 m × 5 m. [1 mark.]
Lawn area = 12 × 10 = 120 m². Pool area = 8 × 5 = 40 m². [1 mark — both areas with units m².]
(b) Turf area (lawn − pool).
Turf area = 120 − 40 = 80 m². [1 mark.]
(c) Cost and comparison with owner's claim.
Cost = 80 × $18 = $1,440. Total cost = $1,440 (already to the nearest dollar). [1 mark — cost calculation.]
Comparing: turf area / lawn area = 80 ÷ 120 ≈ 0.667 = about 67% of the lawn area (the pool takes up the other ≈33%). [1 mark — numerical comparison.]
Conclusion: the owner's claim is contradicted — the pool takes about one-third of the lawn area, so turf is needed for only about 67% of the 6 m × 5 m on the plan (80 m² of turf, not the full 120 m²). [1 mark — explicit conclusion contradicting the claim with justification.]
Total: 7/7.
Band descriptors for marker.
Band 3: Multiplies one dimension correctly by 200 and may compute the full lawn area, but ignores the pool subtraction. ≈ 2-3 marks.
Band 4: Both actual areas correct (lawn and pool) and subtraction performed; misses the cost or the conclusion sentence. ≈ 4-5 marks.
Band 5: Full numerical solution including cost, but the conclusion is bare numbers without explicitly engaging with the owner's claim. ≈ 6 marks.
Band 6: Complete, correct, with a clear conclusion sentence that uses numbers (≈67% / 33%) to support or contradict the owner's claim. 7/7.