This worksheet applies mathematical modelling to proportion and scaling problems, teaching students that length, area and volume all scale differently under the same enlargement. The worksheet is split into warm-up questions, standard questions and then extension questions to test you.
The concepts it covers:
- A scale of 1:n means every 1 unit on the model/map/drawing represents n units in real life.
- Under a similarity transformation with linear scale factor k: length scales by k, area scales by k², and volume scales by k³.
- If the ratio of corresponding lengths between two similar objects is a:b, the ratio of their areas is a²:b², and the ratio of their volumes is a³:b³.
- Given a volume (or area) ratio, the linear scale factor k can be recovered by taking the cube root (or square root) of that ratio.
- Always convert to consistent units (cm, m, km) before applying a scale factor or comparing measurements.
What your child will practise:
- Converting a length between a scale model/map and the real object, in both directions.
- Applying the area-scales-as-k² and volume-scales-as-k³ rules to find a missing area or volume for a similar object.
- Recovering a linear scale factor from a given area or volume ratio, using a square or cube root.
- Modeling a real-world scaling scenario (packaging, furniture, vehicles) and interpret the result sensibly.
- Converting between units (cm²/m², g/kg, cm/m/km) as part of a scaling problem.
Every section opens with a worked example, and the download includes a full answer key with step-by-step solutions and teaching notes on the mistakes students most commonly make.
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