Number synthesis
Objective. Combine exact arithmetic, estimation and bounds to justify numerical conclusions.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
This topic draws together exact arithmetic, estimation and bounds so pupils can justify whether a numerical claim is reasonable. A good strategy is to estimate first using rounded figures to check the scale of an answer, then use exact calculation or bounds to confirm or refute a statement precisely. For example, to check if 203 × 48 could exceed 10000, estimate 200 × 50 = 10000 and then refine with exact values. When a claim involves rounded data, bounds show whether the claim holds for every possible true value, not just the rounded one.
Worked example
A plank measures 150 cm to the nearest 10 cm. Is it certain the plank is long enough to span a 148 cm gap? Justify your number reasoning.
- 1Rounding unit is 10 cm, so lower bound = 150 − 5 = 145 cm.
- 2The plank could be as short as 145 cm.
- 3Compare 145 cm with the gap of 148 cm: 145 < 148.
- 4It is not certain; the plank could be too short.
Answer: Not certain, as the plank could be as short as 145 cm
Concept mastery
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Practice
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Start practisingCommon mistake. Pupils accept a claim as true just because it works for the rounded values, ignoring that the true values could lie anywhere in the error interval. Always test the relevant bound, not just the stated figure.
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