Rates, proportions and derived units
Use proportional relationships and cancel units through single-step and multidimensional conversions.
What you will tackle
- Reason with ratios, unit rates, proportional relationships and scale drawings
- Convert simple, compound, squared and cubed units
- Combine product and quotient units across multistep contexts
Units are part of the calculation
A rate of 18 km in 30 minutes is 36 km/h because 30 minutes is 0.5 h. For an average speed over several legs, divide total distance by total time rather than averaging unequal-duration speeds. A proportional relationship has the form y = kx and passes through the origin.
Apply a conversion in every dimension
Since 1 m = 100 cm, 1 m² = 10,000 cm² and 1 m³ = 1,000,000 cm³. For a speed or density, convert both numerator and denominator as needed. A map scale 1:25,000 compares matching length units; 4 cm represents 100,000 cm, or 1 km.
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
Convert 72 km/h to m/s, 0.4 m² to cm², and a 6 cm map length at 1:25,000 into kilometres.
Show a starting hint
Write conversion fractions so unwanted units cancel.
Compare with the explained answer
- 72 × 1000/3600 = 20 m/s.
- 0.4 × 100² = 4000 cm².
- 6 × 25,000 = 150,000 cm = 1.5 km.
Answer / response criteria: 20 m/s; 4000 cm²; 1.5 km.
Independent practice
Independent 1
A material has density 2.4 g/cm³. Convert it to kg/m³. Then find the mass of 0.003 m³.
Compare with the explained answer
- One gram is 0.001 kg and one cubic centimetre is 0.000001 m³.
- 2.4 × 0.001/0.000001 = 2400 kg/m³.
- Mass = density × volume = 2400 × 0.003 = 7.2 kg.
Answer / response criteria: 2400 kg/m³; 7.2 kg.
Application practice
Application 1
A cyclist travels 30 km at 15 km/h and returns 30 km at 30 km/h. Find the average speed and explain why 22.5 km/h is incorrect.
Compare with the explained answer
- The outward journey takes 2 h and the return 1 h.
- Total distance is 60 km in 3 h, so average speed is 20 km/h.
- A simple average of speeds weights each speed equally, but the slower leg lasts twice as long.
Answer / response criteria: 20 km/h; total distance divided by total time is required.
What can you explain now?
A cyclist travels 30 km at 15 km/h and returns 30 km at 30 km/h. Find the average speed and explain why 22.5 km/h is incorrect.
Compare with the explanation
20 km/h; total distance divided by total time is required.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide