Mathematics Standard • Year 11 • Module 2 • Lesson 1

Working With Formulas and Units

Build fluency in unit conversions (length, area, volume, capacity) and clean formula substitution — one step at a time.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 Complete each length conversion factor:

mm → cm: ÷ ______ cm → m: ÷ ______ m → km: ÷ ______

Q1.2 Complete each AREA conversion factor:

cm² → mm²: × ______ m² → cm²: × ______ 1 hectare = ______ m²

Q1.3 Complete each capacity equivalence:

1 cm³ = ______ mL 1 L = ______ mL 1 kL = ______ L = ______ m³

Stuck? Revisit lesson § Key Formulas and § Length / Area / Volume Conversion Factors.

2. Worked example — substitute and evaluate

Follow each line. Every step has a reason on the right.

Problem. The area of a trapezium is given by A = ½(a + b)h. Find A when a = 6 cm, b = 10 cm, h = 4 cm.

Step 1 — Write the formula first.

A = ½(a + b)h

Reason: writing the formula earns the method mark even if a later number slips.

Step 2 — Substitute known values.

A = ½(6 + 10) × 4

Reason: replace each pronumeral with its number. All in cm, so units are consistent.

Step 3 — Brackets first (BODMAS).

A = ½ × 16 × 4

Reason: 6 + 10 = 16 inside the brackets before any multiplication.

Step 4 — Evaluate.

A = 8 × 4 = 32

Reason: ½ × 16 = 8, then 8 × 4 = 32.

Step 5 — State the answer with the correct unit.

A = 32 cm²

Reason: area unit = (length unit)². Lengths in cm → answer in cm². A bare "32" loses the final mark.

3. Faded example — fill in the missing steps

The volume of a rectangular fish tank is V = ℓ × w × h. Find V in cm³ when ℓ = 60 cm, w = 30 cm, h = 40 cm, then convert to litres. Fill in every blank. 4 marks

Step 1 — Write the formula: V = ______ × ______ × ______

Step 2 — Substitute: V = 60 × 30 × ______

Step 3 — Evaluate: V = ______ cm³

Step 4 — Convert to L: 1 L = ______ cm³, so V = ______ ÷ ______ = ______ L

Conclusion. The tank holds ______ cm³, which is ______ L.

Stuck? Revisit lesson § Worked Example 4 — Volume & Capacity. Remember 1 cm³ = 1 mL exactly.

4. Graduated practice — unit conversions and substitution

Show your working in the space below each part. Write units on every answer.

Foundation — single-step conversions (4 questions)

Q	Problem	Answer (with unit)
4.1 1	Convert 3.6 km to metres.
4.2 1	Convert 850 mm to centimetres.
4.3 1	Convert 2 500 mL to litres.
4.4 1	Convert 4.8 m to centimetres.

Standard — typical HSC difficulty (6 questions)

Write the formula or conversion line, then substitute, then state the answer with its unit.

4.5 A rectangle is 12 m long and 7 m wide. Use A = ℓ × w to find the area. 2 marks

4.6 The area of a triangle is A = ½bh. Find A when b = 14 cm, h = 9 cm. 2 marks

4.7 Convert 5.2 m² to cm². 2 marks

4.8 Convert 3 500 000 mm² to m². 2 marks

4.9 A storage box is 50 cm long, 20 cm wide and 15 cm high. Calculate its volume in cm³, then convert to litres. 2 marks

4.10 A paddock has an area of 3.6 hectares. Express the area in (a) m² and (b) km². 2 marks

Extension — combine substitution + conversion (2 questions)

4.11 A rectangular garden is 8.5 m long and 6 m wide. Find its area in (a) m² and (b) cm². 3 marks

4.12 A water tank holds 2.5 kL. Express the capacity in (a) litres, (b) cm³, and (c) mL. 3 marks

Stuck on 4.12? Step it: kL → L (× 1000), then L → mL (× 1000), and use 1 mL = 1 cm³.

5. Self-check the easy 3

Tick the first three once you've checked your method works.

For 4.1 (3.6 km) I multiplied by 1000 (km → m goes to a smaller unit, so multiply).

For 4.2 (850 mm) I divided by 10 (mm → cm goes to a larger unit, so divide).

For 4.7 (5.2 m² → cm²) I multiplied by 10 000 (area factor = length factor², so 100² = 10 000), not 100.

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Answers — Do not peek before attempting

Q1.1 — Length conversions

mm → cm: ÷ 10. cm → m: ÷ 100. m → km: ÷ 1000.

Q1.2 — Area conversions

cm² → mm²: × 100 (since 10² = 100). m² → cm²: × 10 000 (since 100² = 10 000). 1 hectare = 10 000 m².

Q1.3 — Capacity equivalences

1 cm³ = 1 mL. 1 L = 1000 mL. 1 kL = 1000 L = 1 m³.

Q3 — Faded example (fish tank)

Step 1: V = ℓ × w × h.
Step 2: V = 60 × 30 × 40.
Step 3: V = 72 000 cm³.
Step 4: 1 L = 1000 cm³, so V = 72 000 ÷ 1000 = 72 L.
Conclusion: 72 000 cm³ = 72 L.

Q4.1 — 3.6 km to m

3.6 × 1000 = 3600 m.

Q4.2 — 850 mm to cm

850 ÷ 10 = 85 cm.

Q4.3 — 2500 mL to L

2500 ÷ 1000 = 2.5 L.

Q4.4 — 4.8 m to cm

4.8 × 100 = 480 cm.

Q4.5 — Rectangle area (12 m × 7 m)

A = ℓ × w = 12 × 7 = 84 m².

Q4.6 — Triangle area (b = 14, h = 9)

A = ½bh = ½ × 14 × 9 = 63 cm².

Q4.7 — 5.2 m² to cm²

1 m² = 10 000 cm², so 5.2 × 10 000 = 52 000 cm².

Q4.8 — 3 500 000 mm² to m²

1 m² = 1 000 000 mm², so 3 500 000 ÷ 1 000 000 = 3.5 m².

Q4.9 — Box: volume in cm³ → L

V = 50 × 20 × 15 = 15 000 cm³.
15 000 cm³ ÷ 1000 = 15 L.

Q4.10 — 3.6 hectares

(a) 1 ha = 10 000 m², so 3.6 × 10 000 = 36 000 m².
(b) 1 km² = 1 000 000 m², so 36 000 ÷ 1 000 000 = 0.036 km².

Q4.11 — Garden 8.5 m × 6 m

(a) A = 8.5 × 6 = 51 m².
(b) 51 × 10 000 = 510 000 cm².

Q4.12 — 2.5 kL conversions

(a) 2.5 kL × 1000 = 2500 L.
(b) 2500 L × 1000 = 2 500 000 mL, and 1 mL = 1 cm³, so 2 500 000 cm³.
(c) 2 500 000 mL. Sense check: 1 m³ = 1 kL, so 2.5 kL = 2.5 m³ — a small backyard tank.