A known fact can reduce the calculation demand within a wider problem, but the pupil must still understand what the calculation represents. Make the relationship explicit, record the adjustment and return the result to the original context.

This differs from isolated arithmetic practice because the fact is being selected and used to solve part of a problem.

Identify a useful relationship

Useful options include:

• doubles and near doubles;

• complements;

• compensation;

• inverse operations;

• multiplication and division fact families;

• scaling by 10 or 100.

Do not introduce a derived strategy if the starting fact is not secure.

Work through an example

Problem: Two classes collect 29 cans each. Another pupil brings 3 cans. How many cans are collected altogether?

Use the known double:

30 + 30 = 60
29 + 29 is 2 less, so 58
58 + 3 = 61

Record what each result means. The 58 is the amount collected by the two classes, and 61 is the final total.

A pupil who writes only 60 − 2 + 3 may calculate correctly but should still explain how the adjustment matches the story.

Use compensation transparently

For 48 + 27:

50 + 27 = 77
77 − 2 = 75

Ask why two must be removed. The strategy should preserve the original value, not become an unexplained movement of numbers.

Use inverse relationships

If 6 packs contain 42 items altogether, a known multiplication fact can support 42 ÷ 6 = 7. Ask the pupil to connect the group size, number of groups and total.

The inverse should help interpret the problem, not simply provide a shortcut.

Return to the question

After deriving the result, ask:

• What does this number represent?

• Is it an intermediate or final answer?

• Which unit is needed?

• What happens next?

• Does it fit the expected size?

This prevents a successful arithmetic strategy from becoming detached from the problem.

Practise transfer

Use the same relationship in different contexts:

• near doubles with money;

• compensation in measurement;

• multiplication facts in equal-group problems;

• complements in change;

• inverse relationships in sharing and grouping.

Vary the story while preserving the mathematical connection. Then ask the pupil to identify when the strategy would not be suitable.

Common mistakes to avoid

• Using an insecure starting fact.

• Presenting compensation as a trick.

• Recording adjustments without explaining them.

• Forgetting what the intermediate result represents.

• Practising the relationship only in one story type.

• Assuming a secure fact guarantees understanding of the context.

• Choosing a derived strategy when a simpler dependable method is clearer.

Evidence and further reading

The national curriculum mathematics programmes of study connect arithmetic facts through inverse relationships, commutativity and derived calculations. The Department for Education’s Primary mathematics guidance provides detailed calculation progressions and representations.