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hscscience Maths Std · Y11
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Module 2 · L1 of 22 ~35 min MS-M1 ⚡ +90 XP available

Working With Formulas and Units

Every measurement means something, but only if you use the right units and substitute into formulas correctly. This lesson builds the foundation: unit ladders, area and volume conversions, and the four-step substitution method that earns full marks every time.

Today's hook, A recipe calls for 250 mL of milk but you only have a tablespoon (15 mL). How many tablespoons do you need? What did you have to do before you could answer, and how is that like what we do in maths?
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Recall, your gut answer first

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

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01
Recall, your gut answer first
+5 XP warm-up

A recipe calls for 250 mL of milk, but you only have a tablespoon (15 mL). How many tablespoons do you need? What did you have to do before you could answer, and how is that like what we do in maths?

Without calculating write your gut feeling. We'll revisit this at the end of the lesson.

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02
The key relationships you need to own
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Measurement in Maths Standard starts with two core formulas and a system of unit conversions. Lock these in, every other measurement topic builds from them.

Area is length × length, so area units are always squared. Volume is length × length × length, so volume units are cubed. When you convert a length unit, area factors square and volume factors cube.

Basic area and volume formulas with their units.
Units must always match before substituting
Length: powers of 10
mm ÷ 10 = cm ÷ 100 = m ÷ 1000 = km. Going to a bigger unit: divide. Going to a smaller unit: multiply.
Area: factors get squared
1 m = 100 cm, so 1 m² = 100² = 10 000 cm². The length factor is squared because area is length × length.
Volume: 1 cm³ = 1 mL
This bridge between volume and capacity is exact. 1000 cm³ = 1 L and 1 m³ = 1 kL = 1000 L.
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What you'll master
Know

Key facts

  • The metric units for length, area, volume and capacity
  • Conversion factors between mm, cm, m and km
  • Conversion factors for area and volume units
  • What it means to substitute into a formula
  • The abbreviations ha (hectare) and kL (kilolitre)
Understand

Concepts

  • Why area conversions square the length factor
  • Why volume conversions cube the length factor
  • Why units must match before substituting
  • How to read a formula and identify what to find
Can do

Skills

  • Substitute values into a given formula and evaluate
  • Convert between length, area and volume units
  • Convert between volume and capacity units
  • Identify and correct unit-mismatch errors
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Key terms
FormulaA mathematical rule connecting two or more variables, written using symbols.
VariableA letter in a formula representing a quantity that can change (e.g. $A$, $\ell$, $h$).
SubstituteReplace a variable with its known numerical value.
SI UnitsThe internationally agreed system of units, base unit for length is the metre (m).
Hectare (ha)A unit of area equal to 10 000 m², commonly used for land measurement in Australia.
Kilolitre (kL)A unit of capacity equal to 1000 L or 1 m³. Used for tanks, pools and dams.
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What is a formula? And the length unit ladder

A formula is a rule written in symbols that tells you how quantities are related, substitute numbers in, and it does the work...

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What is a formula? And the length unit ladder
core concept

A formula is a rule written in symbols that tells you how quantities are related, substitute numbers in, and it does the work for you.

In Maths Standard, formulas are always given. Your job is to:

  1. Identify which variable you need to find
  2. Check that all values are in consistent units
  3. Substitute the known values
  4. Evaluate (calculate) the answer
  5. Write the answer with the correct unit
Formula notation: When two letters are written next to each other, like $\ell w$, it means multiply. A fraction bar means divide. Indices (powers) mean repeated multiplication: $\ell^2 = \ell \times \ell$.

Every metric length unit is a power of 10 away from its neighbour. Knowing the ladder lets you convert in one step.

Step up or down the metric length ladder.
Direction matters: Converting mm to m, divide by 1000. Converting m to mm, multiply by 1000. Think: "Going to a bigger unit, divide by the factor."

Formulas use symbols to connect quantities; substitute in a 5-step method, write formula, check units match, substitute values, evaluate, write answer with unit. A number without a unit scores zero in the HSC.

Pause, copy the 5-step substitution method (write formula → check units → substitute → evaluate → state answer with unit) and the length unit ladder (mm ÷10 cm ÷100 m ÷1000 km) into your book.

Quick check: To convert 450 mm to centimetres, which operation do you apply?

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Area units (squared) and volume units (cubed)

We just saw the 5-step substitution method and the length unit ladder, where each step is a power of 10.

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Area units (squared) and volume units (cubed)
core concept

We just saw the 5-step substitution method and the length unit ladder, where each step is a power of 10. That raises a question: if 1 m = 100 cm, does that mean 1 m² = 100 cm²? This card answers it → area conversions square the length factor (1 m² = 10 000 cm²) and volume conversions cube it (1 m³ = 1 000 000 cm³), because area is length × length and volume is length × length × length.

Area is length × length, so when you convert the length unit, you must square the conversion factor too.

Think of a square that is 1 cm × 1 cm = 1 cm². How many mm² is that? Each side is 10 mm, so the area is 10 × 10 = 100 mm². The conversion factor for lengths (×10) becomes ×100 for areas.

$$1\text{ cm}^2 = 100\text{ mm}^2 \qquad 1\text{ m}^2 = 10\,000\text{ cm}^2 \qquad 1\text{ km}^2 = 1\,000\,000\text{ m}^2$$
Area uses the squared length factor.
Hectare (ha): 1 hectare = 10 000 m². It sits between m² and km² on the ladder. Land in Australia is typically measured in hectares, a standard AFL ground is about 1.6 ha.

Volume, units get cubed. Volume is length × length × length, so conversion factors are cubed. And capacity is just volume measured in litres.

$$1\text{ cm}^3 = 1000\text{ mm}^3 \qquad 1\text{ m}^3 = 1\,000\,000\text{ cm}^3$$
Memory hook: A standard 1 cm × 1 cm × 1 cm sugar cube holds exactly 1 mL. Scale that up: a 10 cm × 10 cm × 10 cm box (1000 cm³) holds 1 litre.

Volume ↔ Capacity bridge: 1 cm³ = 1 mL · 1 000 cm³ = 1 L · 1 m³ = 1 000 L = 1 kL

Area conversions square the length factor (1 m² = 10 000 cm²); volume conversions cube it (1 m³ = 1 000 000 cm³). Capacity bridge: 1 cm³ = 1 mL exactly; 1 m³ = 1 kL. 1 ha = 10 000 m².

Pause, copy the area squaring rule (1 m² = 10 000 cm²), the volume cubing rule (1 m³ = 1 000 000 cm³), and the capacity bridge (1 cm³ = 1 mL; 1 m³ = 1 kL) into your book.

True or false: To convert 5 m² to cm², you multiply by 100 (because 1 m = 100 cm).

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Worked examples · 3 in a row, reveal as you go

Worked examples · 3 in a row, reveal as you go

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · FORMULA SUBSTITUTION

The formula for the area of a trapezium is $A = \dfrac{1}{2}(a + b)h$. Find the area when $a = 6$ cm, $b = 10$ cm and $h = 4$ cm.

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Write the formula
$A = \dfrac{1}{2}(a + b)h$
Always start by writing the formula, it shows the marker you know what rule to apply.
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PROBLEM 2 · LENGTH & AREA CONVERSION

A fence post is 1850 mm tall. Express this height in (a) centimetres and (b) metres. Also: a bathroom tile has area 400 cm², convert to (c) mm² and (d) m².

a
mm → cm: divide by 10
$1850 \div 10 = 185\text{ cm}$
1 cm = 10 mm, so divide by 10 to go from mm to cm.
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PROBLEM 3 · VOLUME & CAPACITY

A rectangular fish tank measures 60 cm long, 30 cm wide and 40 cm high. Find: (a) the volume in cm³, (b) the capacity in litres, (c) the capacity in kL.

a
$V = \ell \times w \times h$
$= 60 \times 30 \times 40$
$= 72\,000\text{ cm}^3$
All dimensions already in cm, substitute directly. No conversion needed.
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Fill the gap: A box with volume 15 000 cm³ holds litres of water (because 1 cm³ = 1 mL, and 1000 mL = 1 L).

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Common errors · the 3 traps that cost marks

Common errors · the 3 traps that cost marks

Common errors · the 3 traps that cost marks

Trap 01
Mixing units in a formula
Calculating area with length in metres and width in centimetres gives nonsense. Always convert to the same unit first. Check before substituting.
Trap 02
Forgetting to square/cube when converting area/volume
Converting 5 m² to cm²: students write 5 × 100 = 500 cm². Wrong, it is 5 × 10 000 = 50 000 cm². Area factor = (length factor)².
Trap 03
Dropping the unit from the answer
"The area is 24", 24 what? mm²? km²? The unit must always be stated. In the HSC, an answer without a unit loses the mark.

Match each conversion: Drag or select the correct factor.

  • cm → mm
  • m² → cm²
  • cm³ to mL
  • kL → L
  • × 1 (equal)
  • × 1 000
  • × 10
  • × 10 000
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Revisit your thinking

Quick-fire practice · 5 calculations

Quick-fire practice · 5 calculations

1

Convert 3.6 km to metres.

2

Convert 850 mm to centimetres.

3

A rectangle is 12 m long and 7 m wide. Use $A = \ell \times w$ to find the area.

4

Convert 5.2 m² to cm².

5

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

Top 3 list: Name THREE different units that can be used to measure area (not volume, not length).

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Revisit your thinking

Look back at what you wrote in the Think First section. For the tablespoon problem: 250 ÷ 15 ≈ 16.7 tablespoons. You had to convert the question into a division, which is exactly what formula substitution does: turns a word problem into a calculation.

What has changed? What did you get right? What surprised you?

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Show what you have learned

Multiple choice, then short answer under exam conditions.

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Multiple choice
+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence. That tells the system what to drill next. Each retry pulls a fresh mix from the bank.

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Short answer
ApplyBand 33 marks

SA 1. The formula for the area of a circle is $A = \pi r^2$. A circular fountain has a radius of 2.5 m. (a) Find the area of the fountain in m², correct to 2 decimal places. (b) Convert the area to cm². (c) Convert the area to mm², expressing your answer in scientific notation. (3 marks)

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ApplyBand 33 marks

SA 2. A rectangular swimming pool is 25 m long, 10 m wide and 1.8 m deep. (a) Calculate the volume of the pool in m³. (b) Convert the volume to litres. (c) Water is sold at $2.30 per kilolitre. Find the cost to fill the pool. (3 marks)

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AnalyseBand 42 marks

SA 3. A student calculates the area of a rectangle: Length = 4 m, Width = 50 cm. $A = 4 \times 50 = 200\text{ m}^2$. Identify the error and write the correct solution. (2 marks)

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ApplyBand 32 marks

SA 4. A paddock has an area of 3.6 hectares. Express this area in: (a) m² and (b) km². (2 marks)

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📖 Comprehensive answers (click to reveal)

Drill 1: 3.6 × 1000 = 3600 m · 2: 850 ÷ 10 = 85 cm · 3: $A = 12 \times 7 = $ 84 m² · 4: $5.2 \times 10\,000 = $ 52 000 cm² · 5: $V = 50 \times 20 \times 15 = 15\,000\text{ cm}^3 = $ 15 L

SA 1 (3 marks): (a) $A = \pi \times 2.5^2 = 19.63\text{ m}^2$ [1]. (b) $19.63 \times 10\,000 = 196\,350\text{ cm}^2$ [1]. (c) $196\,350 \times 100 = 1.9635 \times 10^7\text{ mm}^2$ [1].

SA 2 (3 marks): (a) $V = 25 \times 10 \times 1.8 = 450\text{ m}^3$ [1]. (b) $450 \times 1000 = 450\,000\text{ L}$ [1]. (c) $450\,000 \div 1000 = 450\text{ kL}$; cost = $450 \times 2.30 = \$1035$ [1].

SA 3 (2 marks): Error, width not converted to metres before substituting [1]. Correct: 50 cm = 0.5 m; $A = 4 \times 0.5 = 2\text{ m}^2$ [1].

SA 4 (2 marks): (a) $3.6 \times 10\,000 = 36\,000\text{ m}^2$ [1]. (b) $36\,000 \div 1\,000\,000 = 0.036\text{ km}^2$ [1].