Compensation, scaling, percentage relationships and inverse operations can simplify a stage of a multi-step problem. The pupil must show why the adjustment preserves the required value and return the result to the original units, constraints and next step.

This is strategic problem-solving, not a collection of isolated calculation tricks.

Select a relationship that fits

Useful relationships include:

• scaling up or down;

• compensation;

• inverse operations;

• percentage benchmarks;

• equivalent fractions or decimals;

• proportional relationships;

• factorisation.

Check that the underlying relationship is understood before using it inside a complex task.

Make the adjustment explicit

For 19% of £250:

20% of £250 = £50
1% of £250 = £2.50
19% = £50 − £2.50 = £47.50

Ask why subtracting 1% preserves the intended value.

For 998 × 24:

1,000 × 24 = 24,000
2 × 24 = 48
998 × 24 = 23,952

The adjustment must remain visible until the pupil can explain it.

Use the result within the problem

Suppose a £250 item is reduced by 19%, then delivery of £8 is added.

Discount = £47.50
Reduced price = £202.50
Final cost = £210.50

Label each value and return to the question. Stopping at £47.50 would give the amount of discount, not the final cost.

Connect with units and constraints

After simplification, ask:

• What does this number represent?

• Is the unit unchanged?

• Does a conversion follow?

• Is this an intermediate result?

• Does a minimum, maximum or integer condition affect the next step?

• Is the result in the expected range?

The relationship should reduce arithmetic demand without obscuring the problem structure.

Practise across topics

Use the same reasoning in varied contexts:

• compensation with money and measures;

• scaling in ratio and similarity;

• percentage benchmarks in finance and statistics;

• inverse operations in equations and formulae;

• factor relationships in area and algebra.

Ask the pupil to identify when the relationship is helpful and when a standard method or calculator would be more dependable.

Avoid isolated shortcuts

Do not present rules such as “add a zero” or “move the decimal point” without place-value meaning. The pupil should be able to justify the transformation and detect when it does not apply.

A strategy that cannot be explained is unlikely to transfer safely into unfamiliar problems.

Common mistakes to avoid

• Starting from an insecure relationship.

• Teaching many shortcuts at once.

• Hiding the adjustment.

• Losing the unit or meaning of an intermediate value.

• Stopping after the simplified calculation.

• Using the relationship where a standard method is clearer.

• Practising only one surface context.

Evidence and further reading

The national curriculum mathematics programmes of study emphasise applying arithmetic, scaling, estimation and inverse relationships in problems. The Education Endowment Foundation’s Improving Mathematics in Key Stages 2 and 3 provides wider guidance on strategy use and transfer.