The Invariant in Proportion
How do we know when a relationship is proportional?
The story unfolds
The mathematics beneath the story
Recognize proportional relationships across tables, graphs, and y = kx. Use unit rates with fractional quantities and multiplicative percent changes. A straight graph alone is not sufficient.
A learning breakpoint to notice
Proportional reasoning can be overgeneralized to every increasing table or straight line. Check constant y/x for nonzero x and consistency with the origin. Do not form 0/0.
See the structure
Teacher investigation
Compare y = 2x with y = 2x + 3. Calculate y/x for x = 1 and 2. Explain the role of the starting amount.
Try it before revealing the solution. Name every quantity and unit. Accept several valid representations.
Study a completed solution
For y = (3/2)x, pairs (0,0), (2,3), and (4,6) share the same rate: 3/2 dollars per meter. At x = 0, the equation gives y = 0, although y/x is undefined there. With a 2-dollar fee, the same lengths cost 5 and 8 dollars; their total ratios differ.
Compare approaches
Scaling an entire pair and multiplying a unit rate agree in a proportional relationship. For the fee model, differences still scale with changes in length; total coordinate ratios do not stay fixed.
Analyze an incorrect response
“A 20% increase and a 20% decrease cancel.” Multipliers 1.2 and 0.8 give 0.96: the second percentage uses a changed whole. The result is 4% below the original.
Identify the first invalid inference. Ask what evidence would help the learner revise it.
Complete the reasoning
A proportional cost has y = kx and passes through (4,10). Then k = __ and the cost at x = 6 is __.
Check after trying
5/2; 15. The unit rate is cost divided by quantity.
Transfer independently
A walker travels 3/4 mile in 1/2 hour at constant speed. Write a distance-time equation, state the unit rate, and predict the distance in 2 hours.
Compare your reasoning
1.5 miles per hour; d = (3/2)t from the start of the walk. In 2 hours, 3 miles.
Pause and reflect
Which unit must students keep track of? What would an accurate answer conceal? Choose a different context or representation to test whether the idea transfers.
Bring it to your classroom
Grade 7 · Printable student practice and a separate teacher key. Includes an optional “Think Like a Mathematician” extension.