Read a Proportional Graph’s Constant

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: A proportional graph passes through (0,0) and (2,6). What is y when x=4? Candidate answers: (A) 11 (B) 12 (C) 13 (D) 24.
• Independent check inputs: A proportional graph passes through (0,0) and (3,12). What is y when x=5? Candidate answers: (A) 19 (B) 20 (C) 21 (D) 40.

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. The point (1,k) on a proportional graph describes its constant. Another nonzero point gives the same k by dividing its vertical coordinate by its horizontal coordinate.

Here is the method on the supplied teaching case. The constant of proportionality is y/x=6/2=3. Use y=3x, so y=3×4=12.

A rising line is not automatically proportional. The stated graph must pass through the origin, and the roles of input and output must stay consistent throughout the calculation.

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: A proportional graph passes through (0,0) and (9,90). What is y when x=11? Candidate answers: (A) 109 (B) 110 (C) 111 (D) 220. Show the mathematical reason, not only the selected answer. Expected result: 110. The constant of proportionality is y/x=90/9=10. Use y=10x, so y=10×11=110.

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