Test the Whole Relationship

Call the input x and the output y. A proportional relationship uses one constant k so that y = k × x for every allowed input. For x greater than zero, y ÷ x must be the same k, with units. If zero is an allowed input, the pair must be (0, 0). Never calculate 0 ÷ 0.

A supplied water model

For 0 to 4 minutes, Tank A begins empty and gains 3 liters each minute. Tank B begins with 5 liters and also gains 3 liters each minute. Time is the input; total water, including any starting water, is the output.

For A, 3 ÷ 1 = 3, 6 ÷ 2 = 3 and 12 ÷ 4 = 3 liters per minute. The rule is y = 3 × x and zero time gives zero water. For B, 8 ÷ 1 = 8 but 11 ÷ 2 = 5.5 liters per minute. Its total is y = 3 × x + 5. Equal increases do not make those totals proportional.

Each complete cord piece has length 12 centimeters. Total length includes only these identical pieces. The model allows whole-number counts from 0 to 5 pieces.

Compare the quotients for 1, 3 and 5 pieces. Is total length proportional to piece count? Name k and its units, write a rule, and explain the zero-input pair. What total does the rule give for 4 pieces?

An answer for a fractional piece count is outside this stated whole-piece model. A rule’s allowed inputs matter.

A A tank with a starting amount

For 0 to 5 minutes, a tank begins with 4 liters and gains 2 liters each minute. Use total water, including the starting water.

Is total water proportional to time? Use two positive-time quotients and the starting pair. Write a rule matching the supplied model.

B Two matching pairs are not the whole table

A fictional learner checked only the first two pairs and said y = 2 × x fits the whole table. Test that claim against every displayed pair. Which pair resolves the question? No rule for unseen inputs is supplied.