Free Year 10 Maths Approximation vs Exact Representations Worksheet

This worksheet explores the difference between exact and approximate representations of real numbers, teaching students to compare "round-early" and "round-late" calculation strategies and see how rounding error can build up. The worksheet is split into warm-up questions, standard questions and then extension questions to test you.

The concepts it covers:

• Real numbers like √2, √3, and π are irrational — their exact value can only be written as a surd or symbol, never as a finite or repeating decimal.
• Rounding a value introduces a small error; if that rounded value is then used in further calculations, the error carries through (and can grow) rather than disappearing.
• "Round-early" means rounding at every intermediate step; "round-late" means keeping full precision throughout and rounding only the final answer — round-late is generally more accurate.
• The absolute size of a rounding error can matter a lot even when the relative (percentage) error looks small — especially when the numbers involved are very large (e.g. Earth-scale distances) or repeated many times (e.g. compound growth).
• Comparing an approximate result to an exact one lets you quantify exactly how much precision was lost, using absolute difference or percentage error.

What your child will practise:

• Calculating the exact value of an expression involving a fraction, surd or π, and compare it to a rounded/approximate version.
• Tracing a multi-step calculation using a "round-early" strategy (rounding after every step) and compare it to a "round-late" strategy (rounding only the final answer).
• Quantifying the absolute and percentage error introduced by an approximation.
• Applying exact vs approximate methods to compounding/repeated-growth contexts (e.g. financial growth, repeated squaring or halving).
• Explain,ing using calculated evidence, why rounding error tends to grow as the number of repeated calculation steps increases.

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.