Common Core Standards: Math
2b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
The constant of proportionality is just a fancy term for the unit rate, with the caveat that we express it as a constant number instead of a rate of change. All we're asking students to do is to track this constant down across the whole kit 'n' caboodle of different proportional representations. Yep. That's it.
If they're looking at a table of values, students should know the constant of proportionality is the quotient of the dependent and independent variables—take any value of y and divide it by its corresponding x. They can also find the spot where the independent variable (x) has a value of 1.
Graphically, this means the constant of proportionality is just the slope of the linear function, a.k.a. the value of r at the point (1, r). Just have your students find the spot on the graph where x = 1, or find the slope using the ol' standby, rise over run.
And that makes it nice and easy to find the unit rate in an equation: just pop that thing into point-slope form, and our unit rate is the m in y = mx + b. It's also handy for students to know that proportional relationships will always have a y-intercept of 0 when we graph them.
For verbal descriptions of proportions, the constant of proportionality is the number that relates the two quantities to each other. If Grandma is baking a dozen cookies per hour, our constant of proportionality is 12 because each hour (the independent variable) translates into 12 cookies (the dependent variable). Too bad our rate of eating cookies is a lot higher than that.