A representation can expose the unknowns, relationships, units and constraints before a method is chosen. Useful options include equations, tables, annotated diagrams, graphs, ratio tables and bar models.

Choose the representation that fits the mathematical structure. A diagram is not automatically better than a concise equation.

Identify the structure first

Ask:

• What is known?

• What is unknown?

• Which quantities vary together?

• Which values are fixed?

• What constraints apply?

• Is there a sequence, comparison, proportion or geometric relationship?

The surface context may hide the same structure found in a very different problem.

Select an appropriate representation

Equation or expression

Useful when relationships can be stated precisely and the pupil understands the notation.

Table

Useful for rates, sequences, proportional relationships and organised data.

Annotated diagram

Useful for geometry, measures, forces, probability or spatial constraints.

Graph

Useful where change, comparison or intersection matters.

Ratio table or double number line

Useful for proportional scaling and unit rates.

Bar model

Useful for selected comparison, part-whole and ratio structures, but not every Secondary problem.

Work through an example

Problem: A taxi charges £3.20 plus £1.80 per mile. A journey costs £17.60. How far was the journey?

Represent the relationship:

17.60 = 3.20 + 1.80m

Label:

• 17.60: total cost;

• 3.20: fixed charge;

• 1.80: cost per mile;

• m: number of miles.

The representation leads to:

17.60 − 3.20 = 14.40
14.40 ÷ 1.80 = 8

The journey was 8 miles.

A table could also work, but the equation states the structure directly.

Test whether the representation is complete

Ask:

• Does every relevant quantity appear?

• Are units included?

• Is the unknown clear?

• Have all constraints been represented?

• Does the model suggest the calculation sequence?

• Is any information irrelevant?

If the pupil cannot link a mark or symbol with the problem, revise the representation.

Move into calculation and interpretation

The representation should lead to a sequenced method. Record intermediate values with labels, then return the final result to the context.

Do not leave the model once calculation starts. Use it to check whether each step preserves the intended relationship.

Build independence

Progress from:

• teacher selects and builds the representation;

• pupil completes labels;

• pupil chooses between two models;

• pupil constructs a concise representation;

• pupil moves directly to an equation or table when appropriate.

Avoid requiring a diagram after the pupil can represent the structure accurately in another form.

Common mistakes to avoid

• Choosing a representation from a keyword.

• Drawing the story rather than the mathematical structure.

• Leaving units and constraints unlabelled.

• Requiring a bar model for every problem.

• Creating a diagram after the method has already been selected.

• Introducing several representations at once.

• Failing to connect the model with the calculation sequence.

Evidence and further reading

The NCETM’s Five Big Ideas in Teaching for Mastery explains how representations make mathematical structure visible. The Education Endowment Foundation’s Improving Mathematics in Key Stages 2 and 3 covers selecting, comparing and connecting representations.