Test a Whole Table for Proportionality

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: For candidate multiplier 2, which supplied (x,y) table is proportional? Candidate answers: (A) (0,1), (1,2), (2,4) (B) (1,2), (2,2), (4,2) (C) (1,2), (2,4), (4,8) (D) (1,2), (2,5), (4,8).
• Independent check inputs: For candidate multiplier 3, which supplied (x,y) table is proportional? Candidate answers: (A) (0,1), (1,3), (2,6) (B) (1,3), (2,3), (4,3) (C) (1,3), (2,6), (4,12) (D) (1,3), (2,7), (4,12).

Learn the idea

Proportional relationships compare multiplicative changes rather than fixed additions. Rates connect two labeled quantities, and a proportional graph must be consistent with a zero starting amount as well as a constant ratio. A proportional rule has the form y=kx. Every nonzero input must give the same output-to-input ratio; an isolated matching pair cannot establish the whole table’s structure.

Here is the method on the supplied teaching case. The correct table has y/x=2 for every nonzero x. The other tables either change this ratio or imply a nonzero value at x=0.

Check every supplied nonzero row, and check that a stated row with input zero has output zero. Constant differences alone do not distinguish a proportional line from a line with an intercept.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: For candidate multiplier 9, which supplied (x,y) table is proportional? Candidate answers: (A) (0,1), (1,9), (2,18) (B) (1,9), (2,18), (4,36) (C) (1,9), (2,19), (4,36) (D) (1,9), (2,9), (4,9). Show the mathematical reason, not only the selected answer. Expected result: (1,9), (2,18), (4,36). The correct table has y/x=9 for every nonzero x. The other tables either change this ratio or imply a nonzero value at x=0.

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