One Relationship, Four Representations — Words, Tables, Graphs and Equations

Start with a relationship

A fictional taxi-style model has a starting charge of $4 and adds $2 for each kilometre travelled. Let d be distance in kilometres and C be cost in dollars. The equation is C = 4 + 2d.

The equation is one representation. The relationship itself is the dependency: cost starts at 4 and rises by 2 for each additional kilometre under the model.

Representation 1: words

“The cost is four dollars plus two dollars for every kilometre.” Words make the meaning of each quantity visible. They can also hide structure if the reader does not identify which amount is fixed and which repeats.

Representation 2: a table

For d = 0, 1, 2, 3 and 4, the corresponding costs are 4, 6, 8, 10 and 12. The table makes repeated change visible: every increase of 1 kilometre corresponds to an increase of $2.

A table shows selected values, not necessarily every possible input. If distance is continuous in the model, values such as 2.5 km can also be considered even if the table lists only whole numbers.

Representation 3: a graph

Plotting C against d produces a straight line with vertical intercept 4 and gradient 2. The intercept represents the starting cost in this model; the gradient represents the cost increase per kilometre.

The graph makes overall behaviour visible quickly. It can show whether a relationship is increasing, decreasing, constant or curved before a reader calculates many individual values.

Representation 4: an equation

C = 4 + 2d compresses the relationship. It can generate table values, identify the gradient and intercept, and solve reverse questions. If C = 18, then 18 = 4 + 2d, giving d = 7 km.

Translation is a mathematical skill

A student may understand a graph but struggle to write its equation, or solve an equation but fail to recognise the same relationship in words. These are representation transitions, not necessarily failures of the underlying arithmetic.

1. Words → identify quantities and how they depend on one another.
2. Table → inspect changes and starting values.
3. Graph → locate intercepts, direction and rate of change.
4. Equation → encode the relationship symbolically.

A nonlinear example

Let A = s² be the area of a square with side length s ≥ 0. In words: area is the square of side length. A table for s = 0, 1, 2, 3 gives A = 0, 1, 4, 9. The graph is curved rather than a straight line. The equation compresses the rule.

The table’s differences are not constant, warning us that the relationship is not linear. The graph’s shape makes that visible. Each representation contributes something different.

Do not force every graph into a story

A mathematical relationship may be studied abstractly without a real-world context. Conversely, a real context may require restrictions that the bare equation does not show. If d represents distance travelled, negative d may be excluded even though the algebraic line extends to negative inputs.

Worked translation set

A. “Starts at 10 and increases by 3 per unit” → y = 10 + 3x.

B. Table x: 0,1,2,3; y: 5,9,13,17 → constant increase 4 with starting value 5 → y = 4x + 5.

C. y = 20 − 2x → graph has intercept 20 and gradient −2; outputs fall by 2 when x rises by 1.

Continue

Read Input, Output and Dependency, then continue into domain and range and deciding whether a graph represents a function. Return to the Secondary Mathematics Master Index.