Common Core Standards: Math
2a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
Here's a suggestion: start a lesson off telling your students that you can read minds. Seriously. Chances are, they won't just take your word for it—they'll want some proof. Then give them two quantities and say that they're proportional. If they don't question you about it, they're in trouble. Just like with mind reading, they shouldn't just take someone's word that two quantities are proportional to each other; they've gotta test it out themselves.
Students should understand that they can test whether two ratios are proportional/equivalent in a few different ways. A table of values is always a safe bet—they can check to see whether each set of values has a multiplicative relationship to the next set.
Here's a quick example: in the following table, is the ratio of coffee to sugar proportional across the board?
The ratio of coffee to sugar in the first column is 6:1, and we can multiply that ratio by different values of 1 to get each new set of values: , and etc. That multiplicative relationship is the same for every entry, so yep, everything's proportional here.
Another good test is to graph each ratio as a coordinate point, then see if they all sit on a straight line that passes through (0, 0). In our coffee-to-sugar example, we could graph (6, 1), (18, 3), (30, 5), and (42, 7), and all four points would be part of the line . That tells us if we're rocking a 42-oz cup of coffee in the morning (don't judge us, it was a long night), we'd have to put in 7 sugar cubes to get the same level of sweetness as a 6-oz cup with 1 sugar cube.
Numerically, comparing ratios is as easy as reducing their fractions down to their simplest form. If we're trying to decide whether and are proportional to each other, remind your students to reduce, reduce, reduce. Since the simplest form of both is we can tell they're definitely in a proportional relationship.