L4 · Proportional or Not? Tables and Graphs

Learning goal

“I can decide whether a relationship is proportional - from a table by the constant-quotient test, from a graph by the straight-line-through-the-origin test - and I can explain exactly which row or which point breaks a non-proportional relationship.”

Constant-quotient test — the table test for proportionality: compute y/x for every row; the relationship is proportional only if every quotient is the same number (that number is k).

Origin — the point (0, 0) where the axes cross. A proportional graph must pass through it, because y = k·x forces y = 0 when x = 0.

Additive pattern — a table where a quantity grows by the same ADDITION each step (+4, +8...). It can look orderly and still not be proportional - growth by adding is not growth by multiplying.

Non-example — a case that fails a definition on purpose, kept around to sharpen the definition: a straight line through (0, 3) is the standard non-example of a proportional graph.

Worked example

Table P: x: 1, 2, 3, 4 → y: 5, 10, 15, 21. Step 1: Run the constant-quotient test: 5/1 = 5, 10/2 = 5, 15/3 = 5, 21/4 = 5.25. Step 2: Three rows agree and the fourth refuses. Verdict: NOT proportional - there is no k, because a constant that fails once is not a constant. Step 3: The repaired table (y: 5, 10, 15, 20) passes every row with k = 5 and the equation y = 5x. One row's difference - 21 vs 20 - is the whole distance between "proportional" and "not."

Check yourself — explain it

The first three rows of Table P agree on k = 5. Why isn't that enough?

Paired problem — example still visible

Table: x: 3, 6, 9 → y: 2, 4, 6. Proportional? If so, find k. -> Proportional: quotients 2/3, 4/6 = 2/3, 6/9 = 2/3 - all equal, so k = 2/3 and y = (2/3)x.

Faded completion — last step is yours

"x: 2, 5, 8 → y: 7, 17.5, 28. Quotients: 7/2 = 3.5, 17.5/5 = ___, 28/8 = 3.5. The relationship is ___________ with k = ___." -> 3.5; proportional; 3.5

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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