Ratios and Proportional Relationships 6.RP.A.1
1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes."
For many students, identifying and maintaining a ratio can be challenging until you make the context personal and, you know, not boring. Sure, Susie might have 3 red apples for every green apple, but students will be much more interested to know she also has 2 red dragons for every 3 green dragons. Dragons need to eat healthy, too, you know.
Students should understand that a ratio is a comparison of basically anything. They should be able to sort ratios into one of three types:
- Part-to-part (using the same unit of measure): For every two red dragons, there are three green dragons.
- Part-to-whole (using the same unit of measure): Out of five dragons, two of them are red.
- Rates (using different units of measure): A dragon eats fifteen apples for every minute.
Under this standard, students are expected to identify rates, but not much else. For more on those bad boys, check out 6.RP.2.
We can write out ratios numerically in three different ways: using a colon, the word "to," or a fraction bar. The ratio of two to three can be written as 2:3, 2 to 3, or ⅔. (It's important to stress that only part-to-whole ratios are true fractions. The other ones want to trick us, but come on. Those Groucho glasses aren't fooling anyone.)
Students should be comfortable multiplying fractions and generating equivalent ones. This will help them see the relationship between two quantities in a ratio. If students are having trouble figuring out which number goes where or what it all means, it might be time to whip out the sketchpad and express these ratios visually.
Encourage students to label the parts and the whole so that their descriptions stay consistent. Green dragons should always be referred to as green dragons, not suddenly basilisks or dinosaurs.
Students might have trouble with the mystical "third" numerical value: the whole. We have two red dragons and three green dragons, but it's the drawing that unveils there are five total dragons. And they're all guarding treasure of unfathomable wealth.
With a bit of practice, they'll begin to see and understand the multiplicative relationship between equivalent ratios as ratios grow (increase in magnitude) and shrink (toward their simplest form). Students are basically discovering proportional relationships, but they won't actually find missing values with proportions until later, so avoid the "cross-multiply and divide" approach when growing and shrinking ratios.
For this standard, language is the key to describing proportional relationships. Think of it like a game of Mad Libs where the templates are:
- For every __________ there are ____________.
- There ___________ for every _____________.
- The ratio of __________ to ____________ is ___ to ____.
If students are really struggling to translate between words and numbers, have them explain what the words mean, and then sketch out the relationship labeling the two parts and the whole. If they're having trouble going from numbers to words, suggest writing out some ratio Mad Libs before filling in the blanks.
This is all super important because it gets students ready for concepts like percent and slope. It's also a great opportunity to promote an introduction to 6.NS.8, where students review constructing a table and graphing ordered pairs to visually express a ratio's linear relationship.