In the usual nontrivial direct-proportion model, y = kx for one fixed constant k. For nonzero x, the ratio y/x stays equal to k. If x is multiplied by a factor, y is multiplied by the same factor wherever those values lie in the model’s domain.

The algebraic graph is a straight line through the origin. A practical domain may show only a segment or positive values, so the origin need not be displayed in the plotted window. Use the equation or stated conditions to establish the relationship rather than insisting that every graph must visibly include zero.

The gradient is three, so each extra unit of x adds three to y. But the fixed five remains even when x changes. At x = 1, y/x = 8; at x = 2, y/x = 11/2 = 5.5. The ratios are not constant.

The line crosses the y-axis at five rather than zero. This offset is the reason that equal additions work while proportional scaling does not. Calling the graph linear in the usual school sense is appropriate, but calling y directly proportional to x would assert a different equation.

Suppose a mathematical model charges a fixed $5 plus $3 per unit: C = 5 + 3n for positive integer quantities n. Two units cost $11 and four units cost $17. Doubling the quantity does not double total cost because the fixed charge is applied once in each calculation.

The variable part C − 5 equals 3n and is directly proportional to n. Total C is not. This shows why identifying the quantities matters: a proportional relationship can exist after subtracting a fixed component, without making the original total proportional. These are invented rates for illustration, not current service prices.

For a proposed direct-proportion table with nonzero x-values, calculate y/x. The pairs (2, 6), (5, 15) and (8, 24) all have ratio three and fit y = 3x. By contrast, (2, 11), (5, 20) and (8, 29) fit y = 3x + 5 and have differing ratios.

Matching several ratios supports a candidate rule for the displayed data. It does not prove that an otherwise unspecified process follows that rule for every possible input. If a question states direct proportion, use that condition to determine k. If it supplies only observations, distinguish fitting a model from establishing a universal law.

A relationship explicitly given as a straight line through the origin has the required form y = kx, subject to the stated domain and convention. But a rough sketch or a scatter cloud is not the same as an exact straight-line relationship. Check what kind of graph the task provides.

A curve can also pass through the origin without expressing direct proportion between y and x. For example, y = x² passes through zero but doubles x to produce four times y. Two conditions belong together here: the straight-line form and the zero intercept, or equivalently the defining constant-ratio equation.

‘Both quantities increase’ is not enough. The relationships y = x² and y = x + 10 increase for positive x but are not direct proportion between y and x. Nor is a constant numerical difference between paired values enough. The required comparison is multiplicative, not simply additive.

Even ‘doubling works once’ needs care if no general model is supplied. A relationship can match selected input-output pairs and differ elsewhere. Use the full stated equation or assumptions. A counterexample can disprove a proposed universal rule, while a small collection of successes does not by itself prove it.

Question 1: is y = 7x direct proportion between y and x? Question 2: is y = 7x − 2? Question 3: y is directly proportional to x and y = 18 when x = 6. Find y when x = 10. Question 4: is y = x² direct proportion between y and x for positive x?

Answers: 1) Yes, with k = 7. 2) No; the intercept is −2. 3) k = 3, so y = 30. 4) No; y/x = x is not constant. It is proportional to x², which is a different stated relationship. Ask your child to justify each decision rather than giving yes or no alone.

Bring a graph or table where the straight-line appearance led to the wrong classification. A useful small-group 3-pax lesson compares y = 3x, y = 3x + 5 and y = x² using both equations and paired values. Each learner should explain which test applies and what the axes represent.

For parents considering Secondary 2 Mathematics tuition in Punggol, ask ‘Is the ratio constant, or only the rate of change?’ That question exposes the distinction without handing over the answer. Once it is clear, change the intercept or the domain and check that the child still reads the actual relationship.