In grade 6 students solved ratio problems by reasoning about scale factors or unit rates. In grade 7 they see the two quantities in a set of equivalent ratios as being in a proportional relationship and move towards using the constant of proportionality to find missing numbers. This is useful in the sorts of tasks they are studying in this unit because the tasks involve repeatedly applying the same number (for example, a unit price) to different amounts. The unit price is a constant of proportionality between the amount purchased and the amount paid. When students describe the proportional relationship behind the repeated operation of finding the amount paid, they are engaging in MP8.
In this lesson students move toward solving problems involving proportional relationships by more efficient methods, especially by setting up and reasoning about a two-row table of equivalent ratios. This method encourages them to use the constant of proportionality rather than equivalent ratios.
- Calculate and interpret (orally) the constant of proportionality for a proportional relationship involving fractional quantities.
- Explain (orally and in writing) how to use a table with only two rows to solve a problem involving a proportional relationship.
- Write an equation to represent a given proportional relationship with a fractional constant of proportionality.
Let’s use constants of proportionality to solve more problems.
- I can use a table with 2 rows and 2 columns to find an unknown value in a proportional relationship.
- When there is a constant rate, I can identify the two quantities that are in a proportional relationship.
A percentage is a rate per 100.
For example, a fish tank can hold 36 liters. Right now there is 27 liters of water in the tank. The percentage of the tank that is full is 75%.
A unit rate is a rate per 1.
For example, 12 people share 2 pies equally. One unit rate is 6 people per pie, because \(12 \div 2 = 6\). The other unit rate is \(\frac16\) of a pie per person, because \(2 \div 12 = \frac16\).