Distinguish proportional relationships from other patterns

Learning goal: Decide whether a modeled relationship is proportional using a description, table, equation and graph. Identify the constant of proportionality and explain why a straight line alone is insufficient.

Two quantities are proportional when one equals a fixed constant times the other across the modeled domain. Write y = kx, where k is the constant of proportionality. For nonzero x, the quotient y/x equals k. The zero case is handled by the equation: when x is zero, y is zero. Do not compute 0/0 and call it the unit rate; that quotient is undefined.

Consider fictional equipment use billed at three budget units per hour with no starting charge. Let x be hours used and y be total cost in budget units. The equation is y = 3x. A table for x = 0, 1, 2, 3, 4 has y = 0, 3, 6, 9, 12. For each nonzero hour value, y/x = 3 budget units per hour. Scaling hours by a factor scales total cost by the same factor.

The graph is a straight line through the origin for nonnegative usage time. The point (0,0) means no use produces no charge under this model. The point (1,3) means one hour costs three budget units; its second coordinate identifies the unit rate. The point (2,6) means two hours cost six budget units, not six hours cost two. Coordinate order follows the labeled horizontal and vertical quantities.

Now consider another fictional plan that charges four budget units to start plus three per hour. The equation is y = 3x + 4, giving costs 4, 7, 10, 13, 16 for the same hour values. Equal one-hour changes still add three to cost. The graph is still straight, but it crosses the vertical axis at four rather than the origin. The quantities are not proportional because the initial charge does not scale with time.

Compare quotients in the second plan: at one hour, cost per total hour is 7/1 = 7; at two hours it is 10/2 = 5; at four hours it is 16/4 = 4. The hourly charge for additional time is three, but total cost divided by time is not constant. Distinguish the added-hour rate from a constant ratio between the complete quantities. Calling every constant-change pattern proportional would erase this difference.

A table can disprove proportionality with a counterexample, but a few matching rows do not automatically establish an unknown rule for every possible input. In these examples, the stated billing rules define the models. If only sample data are available, say that the observed pairs are consistent with a proportional model and investigate whether the relationship continues. Avoid claiming certainty beyond the domain or evidence.

Origin alone is also insufficient. A relationship such as y = x² contains (0,0), but the nonzero pairs (1,1), (2,4) and (3,9) have quotients 1, 2 and 3. Its graph is not a straight proportional line. Both constant ratio and the appropriate straight-line pattern matter. Conversely, a graph that looks almost straight due to a compressed scale still needs numerical or contextual checking.

Use multiple representations to explain k. In a table, divide corresponding nonzero y by x. In an equation y = kx, identify the multiplier. In a diagram of equal groups, compare the repeated quantities. In words, find the amount per one unit. On the graph, identify the y-value when x = 1 or use another nonzero point's ratio. These are different views of the same relationship, not unrelated procedures.

The domain must fit the context. This model allows fractional hours, so a segment between plotted times represents meaningful intermediate usage. If x instead counted whole rented objects, only whole-number x-values might represent possible purchases. A line can display the general proportional relationship while the actual situation permits only selected points. Explain that restriction when interpreting an answer.

Worked decision: A table has (2,5), (4,10), (6,15) and a stated constant-rate rule with no initial amount. Each quotient is 5/2, so y = (5/2)x fits. The point (1,5/2) represents the unit rate. Check a new input: x = 3 gives y = 15/2. The fractional output is acceptable when the context permits it.

Worked counterexample: A student says the fixed-fee plan is proportional because 7, 10, 13 and 16 increase by three. Test scaling instead: doubling time from one to two hours changes cost from seven to ten, not fourteen. The failure of a common scale factor reveals the starting fee. Equal differences are not the same as equivalent ratios.

Worked graph interpretation: On y = 3x, (4,12) means four hours and twelve budget units. On y = 3x + 4, (0,4) represents a four-unit initial charge in the mathematical billing rule, even before time is added. If a real provider only opens a bill when a rental begins, describe that domain convention separately; it does not remove the fixed charge from positive-time comparisons.

Representation workshop: Create tables and graphs for the two supplied plans on the same labeled axes. Mark the origin and the one-hour points. Explain one similarity and two differences. Then test y = x² at three nonzero inputs to show why passing through the origin does not establish proportionality.

Independent performance: Design one proportional and one nonproportional quantity relationship. Provide a verbal rule, equation, five-row table and accurately scaled graph for each. Include fractional inputs if the situation permits them. Write a justification using quotients and a scaling test, not only visual appearance. A partner selects a new input to test your rule and checks whether your graph's domain matches the real or invented situation.