A useful representation shows what is known, what is unknown and how the quantities relate. It may be built with objects, sketched, shown in a bar model or recorded as a number sentence.

The representation should guide method choice. It should not be added after the calculation merely to make the page look complete.

Start with the relationship

Ask:

• What quantities are involved?

• Which value is unknown?

• Are quantities being combined, compared, changed, grouped or shared?

• Is the problem proportional?

• Does the order matter?

Choose a model that fits this structure rather than the surface story.

Select one useful representation

Objects or counters

Useful when pupils need to see grouping, sharing or a change in quantity. Keep the set manageable.

Drawing

A quick sketch can show groups, measures or movement without reproducing every detail in the story.

Bar model

Useful for part-whole relationships, comparison, ratio and some multi-step structures. It is not automatically the best model for every problem.

Number line

Useful for difference, change, intervals, time and some measurement problems.

Number sentence

Appropriate when the pupil can express the relationship symbolically without needing a fuller visual model.

A worked example

Problem: Sam has 18 marbles. Priya has 7 more marbles than Sam. How many marbles do they have altogether?

A labelled bar model shows:

• Sam: 18;

• Priya: 18 and an extra 7;

• unknown: combined total.

The representation leads to:

18 + 7 = 25
18 + 25 = 43

Label 25 as Priya’s marbles before using it in the second calculation.

The word more does not tell the whole procedure. The model shows that the problem first compares the amounts and then combines them.

Link every part with the question

Ask the pupil to point to:

• where each number appears;

• which part is unknown;

• what the operation represents;

• where the intermediate result belongs;

• what the final answer means.

If a box, line or object cannot be explained, remove it or relabel it.

Keep one familiar model until it is understood

Introducing several representations at once may overload the pupil. Use one model, connect it with the equation and practise it across related problems.

Introduce another representation when it reveals a different relationship or when the pupil is ready to compare models.

Move towards independent representation

Progress from:

• teacher builds and labels the model;

• teacher and pupil complete it together;

• pupil selects from two possible models;

• pupil sketches and labels independently;

• pupil records a concise equation where that is sufficient.

Do not insist on a diagram once the pupil can reliably represent the structure mentally or symbolically.

Common mistakes to avoid

• Choosing a model from a keyword.

• Requiring a bar model for every problem.

• Drawing without labels.

• Using the representation after the method has already been chosen.

• Including irrelevant story details.

• Introducing several models before one is understood.

• Keeping a visual model when a concise equation is clearer.

Evidence and further reading

The NCETM’s Five Big Ideas in Teaching for Mastery explains the role of representations in making mathematical structure visible. The Education Endowment Foundation’s Improving Mathematics in Key Stages 2 and 3 discusses selecting and connecting mathematical representations.