Proportional Relationships

A proportional relationship connects two quantities whose ratios remain equivalent. As one quantity changes, the other changes at a constant rate. For example, if each notebook costs $3, then 1 notebook costs $3, 2 notebooks cost $6, and 5 notebooks cost $15. The ratio of cost to number of notebooks is always 3 to 1.

The constant of proportionality is the constant ratio between the two quantities. It is also the unit rate. In an equation written as y = kx, the letter k represents the constant of proportionality. For the notebook example, the equation is y = 3x, where x is the number of notebooks, y is the total cost, and 3 is the cost per notebook.

A table represents a proportional relationship when every pair of values has the same ratio. Consider the pairs (1, 4), (2, 8), (3, 12), and (5, 20). Dividing each y-value by its corresponding x-value gives 4. Therefore, the constant of proportionality is 4, and the equation is y = 4x.

A proportional relationship can be displayed with a double number line. If a person earns $15 per hour, the values 1 hour and $15, 2 hours and $30, and 4 hours and $60 align. Each corresponding pair represents the same constant rate of $15 per hour.

A graph represents a proportional relationship when its points lie on a straight line that passes through the origin, (0, 0). The origin is included because when one quantity is zero, the corresponding quantity must also be zero. The constant of proportionality appears as the vertical change for each horizontal increase of 1.

A proportional relationship must satisfy both important conditions: it must have a constant ratio, and its graph must pass through the origin. A straight line that does not pass through the origin represents a linear relationship, but it is not proportional.

To determine whether a word problem describes a proportional relationship, identify the two quantities and calculate the unit rate for multiple pairs of values. If the unit rate remains constant and the situation begins with zero of one quantity corresponding to zero of the other, the relationship is proportional.

Proportional relationships can be used to solve problems. If 6 pounds of apples cost $10.50, divide 10.50 by 6 to find a constant of proportionality of $1.75 per pound. The equation y = 1.75x can then be used to find the cost of any number of pounds. For 9 pounds, y = 1.75(9), so the cost is $15.75.

Sometimes the constant of proportionality is a fraction or decimal. If a printer produces 18 pages in 3/4 of a minute, divide 18 by 3/4. Multiplying 18 by 4/3 gives 24, so the printer produces 24 pages per minute. The equation is y = 24x.

Common misconceptions and corrections: Students may assume that every straight-line graph is proportional, compare differences instead of ratios, divide in the wrong order, or confuse the constant of proportionality with one of the original quantities. Check for a constant ratio, verify that the graph passes through the origin, divide in the order indicated by the desired units, and explain what the constant represents in context.