Mathematics Standard • Year 11 • Module 2 • Lesson 1

Working With Formulas and Units

Practise HSC Mathematics Standard 2-style writing for unit conversions and formula substitution — multi-mark short answers and one structured extended response.

Master · Past-Paper Style

1. Short-answer questions

1.1 A rectangular storage shed has a footprint of 6.4 m by 4.5 m. 3 marks Band 3
(a) Find the area of the footprint in m². (1 mark)
(b) Convert this area to cm². (1 mark)
(c) State the unit you would prefer for a floor-plan diagram, with one-sentence justification. (1 mark)

1.2 The formula for the area of a circle is A = πr². A circular garden bed has a diameter of 3.4 m. Find the area to 2 decimal places. 3 marks Band 3-4

1.3 A council water tank is a cuboid 5 m long, 3 m wide and 2.4 m deep. 4 marks Band 4
(a) Calculate the capacity of the tank in litres.
(b) The tank loses water at a rate of 18 L per hour through evaporation in summer. How many days will a full tank last before it is empty (assume no refill, give a whole number of days)?

Stuck on 1.3(b)? Convert "L per hour" into "L per day" first (× 24), then divide total litres by daily loss.

2. Extended response

2.1 A NSW farmer is comparing two paddocks for a new orchard.

Paddock X: A rectangular paddock 480 m long and 250 m wide.

Paddock Y: A rectangular paddock 6.5 ha in area, with a length of 325 m.

(a) Calculate the area of Paddock X in (i) m² and (ii) hectares.
(b) Calculate the width of Paddock Y in metres.
(c) The farmer plans 280 fruit trees per hectare. Calculate how many MORE trees the larger paddock can support, and state which paddock that is. 7 marks Band 5-6

Explicit marking criteria

Part (a) — 2 marks

• 1 mark — correct area in m² (480 × 250).

• 1 mark — correct conversion to hectares (÷ 10 000).

Part (b) — 2 marks

• 1 mark — converts 6.5 ha to m² (× 10 000) before dividing.

• 1 mark — correct width by dividing area by length.

Part (c) — 3 marks

• 1 mark — uses the hectare figures, not raw m² (i.e. 280 trees / ha applied to ha, not m²).

• 1 mark — correct numerical difference in tree count.

• 1 mark — explicit conclusion sentence stating which paddock is larger and the additional tree count, with units.

Your response:

Stuck on (c)? Convert both paddock areas into hectares first, then multiply each by 280 trees/ha, then subtract — and write "Paddock ___ supports ___ more trees" in your conclusion.

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Answers — sample responses + marking notes

1.1 — Storage shed footprint (3 marks)

Sample response.
(a) A = 6.4 × 4.5 = 28.8 m².
(b) 28.8 × 10 000 = 288 000 cm².
(c) m² is more appropriate, because the shed is several metres across and "288 000" is unwieldy; m² gives a manageable number and matches the scale of the object.

Marking notes. 1 mark — correct m². 1 mark — correct cm² (must use × 10 000, not × 100). 1 mark — clear justification mentioning scale/usability. A response stating only "m²" without justification scores 2/3.

1.2 — Circular garden bed (3 marks)

Sample response.
Diameter = 3.4 m, so r = 3.4 ÷ 2 = 1.7 m.
A = πr² = π × (1.7)² = π × 2.89 = 9.0792...
A ≈ 9.08 m² (to 2 d.p.).

Marking notes. 1 mark — halves the diameter to get the radius (written as a separate step). 1 mark — correct substitution into A = πr² using the π button (not 3.14). 1 mark — correct rounded answer with unit. Common error: substituting d = 3.4 directly into r² → A = π × 11.56 = 36.32 m² (four times too large).

1.3 — Council water tank (4 marks)

Sample response.
(a) V = 5 × 3 × 2.4 = 36 m³. 1 m³ = 1000 L, so capacity = 36 × 1000 = 36 000 L.
(b) Daily loss = 18 × 24 = 432 L/day. Days = 36 000 ÷ 432 = 83.33...
Round DOWN to 83 days, because on day 84 the tank would already be empty before the end of the day.

Marking notes. (a) 1 mark — correct volume in m³. 1 mark — correct conversion to litres. (b) 1 mark — correct daily loss (hours × 24). 1 mark — correct integer days with justified rounding. Common error: rounding 83.33 up to 84.

2.1 — Comparing two paddocks (7 marks): sample Band-6 response with annotations

Sample Band-6 response.

(a) Paddock X area.

(i) A_X = 480 × 250 = 120 000 m². [1 mark — m².]
(ii) 120 000 ÷ 10 000 = 12 ha. [1 mark — ha conversion.]

(b) Paddock Y width.

6.5 ha × 10 000 = 65 000 m². [1 mark — converts ha to m² before dividing.]
Width = 65 000 ÷ 325 = 200 m. [1 mark — correct width.]

(c) Tree-count difference.

Trees on Paddock X = 12 × 280 = 3360 trees.
Trees on Paddock Y = 6.5 × 280 = 1820 trees. [1 mark — applies 280 trees/ha to the ha figures, not to m².]
Difference = 3360 − 1820 = 1540 trees. [1 mark — correct numerical difference.]

Conclusion: Paddock X is the larger paddock and can support 1540 more trees than Paddock Y. [1 mark — explicit conclusion naming the larger paddock and the additional tree count.]

Total: 7/7.

Band descriptors for marker.

Band 3: Calculates Paddock X area in m² only, or makes a unit-mismatch error in Paddock Y (e.g. 6.5 ÷ 325 = 0.02 m wide). ≈ 2-3 marks.

Band 4: Both areas correct in ha; Paddock Y width found; tree-count attempted but applied to raw m² (giving an absurd number like 3.36 million trees), OR forgets the conclusion sentence. ≈ 4-5 marks.

Band 5: Full numerical solution, correct units throughout, but conclusion is bare numbers ("1540") without naming the larger paddock. ≈ 6 marks.

Band 6: Complete, units correct, explicit conclusion sentence naming the larger paddock AND the extra tree count. 7/7.