G14: Units of Measure
Use standard units of measure and related concepts; length, area, volume, capacity, mass, time, money
📋 Key Concepts
Units of measure are standard quantities used to measure physical quantities like length, mass, time, etc.
📝 Length
Metric units: km (kilometre), m (metre), cm (centimetre), mm (millimetre)
1 km = 1000 m
1 m = 100 cm
1 cm = 10 mm
1 km = 100,000 cm
1 m = 100 cm
1 cm = 10 mm
1 km = 100,000 cm
Example 1
Convert 3.5 km to metres.
Solution: 3.5 × 1000 = 3500 m
Example 2
Convert 4500 mm to metres.
Solution: 4500 ÷ 1000 = 4.5 m
📝 Area
Metric units: km², m², cm², mm², hectares (ha)
1 km² = 1,000,000 m²
1 m² = 10,000 cm²
1 cm² = 100 mm²
1 hectare = 10,000 m²
1 m² = 10,000 cm²
1 cm² = 100 mm²
1 hectare = 10,000 m²
Important: Area units are squared, so conversion factors are also squared!
Example 3
Convert 5 m² to cm².
Solution: 5 × 10,000 = 50,000 cm²
Example 4
Convert 30,000 cm² to m².
Solution: 30,000 ÷ 10,000 = 3 m²
📝 Volume
Metric units: km³, m³, cm³, mm³, litres (L), millilitres (mL)
1 m³ = 1,000,000 cm³
1 cm³ = 1000 mm³
1 litre = 1000 mL
1 cm³ = 1 mL
1 cm³ = 1000 mm³
1 litre = 1000 mL
1 cm³ = 1 mL
Example 5
Convert 2.5 m³ to cm³.
Solution: 2.5 × 1,000,000 = 2,500,000 cm³
Example 6
Convert 3500 mL to litres.
Solution: 3500 ÷ 1000 = 3.5 L
📝 Mass
Metric units: tonne (t), kilogram (kg), gram (g), milligram (mg)
1 tonne = 1000 kg
1 kg = 1000 g
1 g = 1000 mg
1 kg = 1000 g
1 g = 1000 mg
Example 7
Convert 0.5 tonnes to grams.
Solution:
0.5 tonnes = 500 kg
500 kg = 500,000 g
📝 Time
Time units: hours (h), minutes (min), seconds (s)
1 hour = 60 minutes
1 minute = 60 seconds
1 hour = 3600 seconds
1 minute = 60 seconds
1 hour = 3600 seconds
Example 8
Convert 2.5 hours to minutes.
Solution: 2.5 × 60 = 150 minutes
Example 9
Convert 90 minutes to hours.
Solution: 90 ÷ 60 = 1.5 hours
📝 Imperial-Metric Conversions
Approximate conversions:
| Imperial | Metric |
|---|---|
| 1 inch | ≈ 2.5 cm |
| 1 foot | ≈ 30 cm |
| 1 mile | ≈ 1.6 km |
| 1 pound (lb) | ≈ 450 g |
| 1 gallon | ≈ 4.5 litres |
🧮 Worked Examples — Exam Style
Unit conversion errors cost marks across the whole paper. Two rules: know the factor, and square it for areas (×100), cube it for volumes (×1000… actually ×10⁶ for m²→cm²).
Convert 2.5 m² to cm².
1 m = 100 cm, but area scales by the SQUARE: 1 m² = 100 × 100 = 10,000 cm²
2.5 × 10,000 = 25,000 cm²
M1 for 10,000 (the squared factor), A1 for 25,000. Multiplying by 100 (the length factor) is the classic error.
Convert 90 minutes to hours, giving your answer as a decimal.
90 ÷ 60 = 1.5 hours
Not 1.90 or 1.3 — minutes are sixtieths, not hundredths.
B1 for 1.5. Time-to-decimal conversions feed compound measures (speed questions).
Exam wording to expect: "Convert …", "Give your answer in …" — check the target unit before calculating.
❓ Practice Questions
Q1: Convert 7500 m to km.
Q2: Convert 0.8 m² to cm².
Q3: Convert 3 litres to mL.
Q4: Convert 2.4 kg to grams.
Q5: Convert 135 minutes to hours.
Q6: Convert 4.5 hours to minutes
Q7: Convert 0.05 kg to grams
✅ Answers
- 7.5 km
- 8000 cm²
- 3000 mL
- 2400 g
- 2.25 hours (or 2 hours 15 minutes)
🧮 Worked Examples — Harder Exam Style
A water tank holds 0.75 m³. How many litres is that?
1 m = 100 cm → 1 m³ = 100³ = 1,000,000 cm³
0.75 m³ = 750,000 cm³
1 litre = 1000 cm³ → 750,000 ÷ 1000 = 750 litres
M1 for the cubed factor, A1 for 750 litres. The m³-to-litres bridge (× 1000) appears in real-world capacity questions every year.
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. ÷ 1000: 7500 ÷ 1000 = 7.5 km
Q2. Area factor 10,000: 0.8 × 10,000 = 8000 cm²
Q3. 1 litre = 1000 mL: 3000 mL
Q4. 1 kg = 1000 g: 2.4 × 1000 = 2400 g
Q5. 135 ÷ 60 = 2.25 hours (= 2 h 15 min)
Q6. 4.5 × 60 = 270 minutes
Q7. 0.05 × 1000 = 50 g
📌 Key Facts to Memorise
- Length: 1 km = 1000 m · 1 m = 100 cm · 1 cm = 10 mm
- Mass: 1 kg = 1000 g · 1 tonne = 1000 kg
- Volume: 1 litre = 1000 mL = 1000 cm³ · 1 m³ = 1,000,000 cm³
- Area: 1 m² = 10,000 cm² (square the length factor)
- Time: 1 h = 60 min · 1 min = 60 s · 1 h = 3600 s
- Imperial: 1 inch ≈ 2.5 cm · 1 kg ≈ 2.2 pounds · 1 mile ≈ 1.6 km
🗺️ In the Exam
Conversions appear inside other questions (speed, density, area) more often than standalone. 1 mark each; the squared/cubed factors are where marks are lost on Papers 2/3.
Links to: compound measures (R1/R11), area and volume (G16), estimation (N14).
🧭 Board Focus — OCR
Spec J560. Three papers of 100 marks each: Paper 1 non-calculator, Papers 2 and 3 calculator, at Foundation/Higher tiers. OCR marks problem-solving explicitly, and its papers tend to front-load multi-step worded contexts — practise extracting the mathematics from prose before calculating.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Learn the conversion factors by heart
- Remember: area units are squared, volume units are cubed
- 1 cm³ = 1 mL is very useful for volume problems
- For time, use fractions: 45 min = 3/4 hour = 0.75 hours
- Check your answer is sensible - should it be bigger or smaller?
- Practise both metric-metric and imperial-metric conversions
🧠 Problem-Solving Strategies
Problem-Solving
Learn key conversions: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm, 1 litre = 1000 ml, 1 kg = 1000 g, 1 tonne = 1000 kg. For area: 1 m squared = 10,000 cm squared. For volume: 1 m cubed = 1,000,000 cm cubed. Always convert to the same unit before calculating. Use dimensional analysis to check answers.
Multi-Step Problem
A garden measures 12 m by 8 m. Fertiliser is applied at 25 g per square metre and costs £3.50 per kg. Calculate the total cost of fertiliser for the garden.
Solution: Area = 12 x 8 = 96 m squared. Fertiliser needed = 96 x 25 = 2400 g = 2.4 kg. Cost = 2.4 x 3.50 = £8.40.
⚠️ Common Errors
Watch Out!
1. Wrong: Converting 2 m squared to cm squared by multiplying by 100 Correct: 2 m squared = 2 x 10,000 = 20,000 cm squared. For area, multiply by 100 squared = 10,000, not 100.
2. Wrong: Converting 3 m cubed to cm cubed by multiplying by 100 Correct: 3 m cubed = 3 x 1,000,000 = 3,000,000 cm cubed. For volume, multiply by 100 cubed = 1,000,000.
3. Wrong: Mixing imperial and metric units without conversion (e.g. adding feet and metres) Correct: Always convert all measurements to the same unit system before adding or comparing. 1 foot = 30.48 cm, 1 inch = 2.54 cm, 1 mile = 1.609 km.
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A swimming pool is 25 m long, 10 m wide and 2 m deep at the deep end, sloping to 1 m at the shallow end. (a) Calculate the volume of water in the pool in m cubed. (b) Convert this to litres. (c) Water costs 0.2p per litre. Find the cost to fill the pool. The pool loses 2 cm of depth per week through evaporation. How many litres evaporate per week?
(a) Average depth = (2 + 1)/2 = 1.5 m. Volume = 25 x 10 x 1.5 = 375 m cubed.
(b) 375 m cubed = 375 x 1000 = 375,000 litres.
(c) Cost = 375,000 x 0.2p = 75,000p = £750.
Evaporation: 2 cm depth = 0.02 m. Volume per week = 25 x 10 x 0.02 = 5 m cubed = 5000 litres.
Mark scheme: M1 average depth, A1 volume 375, M1 conversion, A1 375000 litres, M1 cost, A1 £750, M1 evaporation volume, A1 5000 litres
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A recipe requires 2 lb of flour and 1 pint of milk. A student has 1 kg of flour and 1 litre of milk.
(a) Does the student have enough flour? (1 lb = 454 g)
(b) Does the student have enough milk? (1 pint = 568 ml)
(c) The student says "I can just use 2 kg of flour instead of 2 lb since they're roughly the same." How far off would this be? Is this an acceptable approximation?
Answers: (a) 2 lb = 908 g = 0.908 kg. Yes, 1 kg is enough (92 g spare). (b) 1 pint = 568 ml. Yes, 1 litre (1000 ml) is enough (432 ml spare). (c) 2 kg instead of 2 lb = 2000 g instead of 908 g. That is 2.2 times too much flour. This is NOT an acceptable approximation — it would completely ruin the recipe. The student is confusing kg with lb.
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🎬 Video Resources
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