Older pupils do not need to memorise every calculation independently. A known fact, benchmark or structure can be used to rebuild a less accessible result. Make the relationship explicit, ask the pupil to explain why the adjustment works and revisit it after a gap.

This is strategic reconstruction, not a collection of unexplained shortcuts.

Start with known knowledge

Useful starting points include:

• secure multiplication facts;

• doubles and halves;

• powers of ten;

• common fraction, decimal and percentage equivalents;

• inverse operations;

• familiar algebraic identities.

Check the starting fact before relying on it. If 7 × 6 is not available, using it to derive 14 × 6 will not reduce the demand.

Model useful relationships

Scaling a multiplication fact

7 × 6 = 42
14 × 6 = 84

Explain that doubling one factor doubles the product.

Compensating in subtraction

503 − 198
= 503 − 200 + 2
= 305

Ask why subtracting 200 first requires adding two back.

Building percentages

15% of 80
= 10% of 80 + 5% of 80
= 8 + 4
= 12

Connect 5% with half of 10%.

Using equivalence

0.25 × 64
= one quarter of 64
= 16

The pupil should choose the form that makes the relationship visible.

Write connections side by side

Present the known and derived calculation together. Ask:

• “What stayed the same?”

• “What changed?”

• “How must the answer change?”

• “Why does the adjustment work?”

• “Can you create another example?”

The explanation distinguishes understanding from mechanical imitation.

Practise a connected set

Use several examples of the same relationship, then revisit after a few days. Mix secure facts with one or two derived items.

For compensation:

• 302 − 99;

• 615 − 198;

• 1,004 − 497.

Keep the focus on the shared structure, then vary the context once the relationship is secure.

Balance strategy and efficient access

Allow the pupil to reconstruct rather than waiting indefinitely for recall. With repeated use, some results may become immediately available.

Do not insist on writing every step once the relationship is fluent. Conversely, ask for an explanation if the pupil applies an adjustment but cannot say why.

Connect with current mathematics

Use relationships that support upcoming work:

• fraction benchmarks before probability;

• percentage relationships before growth and decay;

• multiplication facts before factorisation;

• powers of ten before standard form;

• inverse operations before rearranging equations.

This makes retrieval purposeful and improves access to the lesson.

Common mistakes to avoid

• Presenting relationships as magic tricks.

• Starting from an insecure fact.

• Teaching many shortcuts at once.

• Accepting a copied pattern without explanation.

• Preventing a pupil from using a dependable reconstruction strategy.

• Continuing full written steps after access is efficient.

• Practising connections unrelated to current work.

Evidence and further reading

The national curriculum mathematics programmes of study emphasise inverse relationships, efficient methods and maintenance of arithmetic fluency. The Education Endowment Foundation’s Improving Mathematics in Key Stages 2 and 3 provides wider guidance on mathematical strategies and the use of connected representations.