As I mined the 6th-grade Ratios and Proportional Relationships standards, I had to develop my own conceptual understanding of ratios and rates. I thought this would be easy; I’ve used ratios and
rates for most of my life. They’re obvious, right? But I found that what was obvious wasn’t necessarily correct or complete.
First, aren’t ratios and rates the same thing? Sort of and sort of not. While every rate is a ratio, not every ratio is a rate.
So, what is a ratio? A ratio is a quantitative comparison of two things or groups of things. That means that we are applying a number value to one thing or group and comparing it to the number value of the other thing or group. Some common examples are 2 boys for 3 girls, 3 red gumballs out of 10 gumballs, 32 mpg, and 3 hamburgers for $5. What do these have in common? Numbers. They are countable or measurable. So, is 20 desks in each classroom a ratio? Yep, we have 20 to 1 even though we don’t make the 1 obvious. The world has 2 poles also is a ratio with the less obvious 1. When we have 8 boys in the class, we could have an 8 to 1 ratio (8 boys to 1 class) or if we know the class size is 12 we have an 8 to 12 ratio (8 boys out of 12 students).
Finding non-examples for ratios was harder than I expected. Several of the examples I came up with had that less obvious 1 – a hug for Grandma – 1:1. Even the example of “There are no dogs in the house,” has a ratio of 0 to 1. And 3 dogs, 2 cats, and a rat have a continuing ratio (a comparison of more than 2 amounts) of 3:2:1. This means that the only non-examples are when you are not comparing at least 2 amounts. 5 cats is a non-example because there is nothing to compare the amount of cats to. Love vs. hate is a non-example because we do not have an amount of love to compare to the amount of hate.
A rate is a special ratio using different measurements. 32 mpg is a rate; it is a comparison of miles per gallon – different measurements. The Venn diagram below shows how rates are inside the ratios circle– a specific group or subset of ratios.
How do we know when our ratio is a rate? We know it is a rate when we see different types of measurements – cups/gallons, miles/hours, number of items/$. A ratio is not the special ratio called a rate when you are comparing things with the same measure – inches/inches, hours/hours,
or number of items/number of items. Sometimes it seems that comparing the amount of 2 very different items that the comparison must be a rate since the items are so different. However, the measurement used is the same for the different items. If we use the ratio 20 stitches per 3 elephants, we are comparing a number counted to another number counted – the same type of measurement.
Now that I have used a whole page to compare ratios and rates, what does that have to do with the Common Core State Standards? NOTHING! CCSS uses the word rate only in reference to unit ratios. The Mariam Webster dictionary also defines rate as a comparison per unit. But a search on National Council of Teachers of Mathematics website returned The Common Core Mathematics Companion: The Standards Decoded, Grades 6–8. In Part 1 page 4 the key vocabulary defines rate as having different units. So that is the definition that I am sticking with.
The standards show any ratio of 2 amounts that has a 1 as the second amount is a unit rate. I would define a ratio with the same unit and a second value of 1 (a:1) as a unit ratio. If they have different units, it is a unit rate.
What about part-to-part and part-to-whole? They are not mentioned in the standards. They are conventions we use to help the students understand that we can use ratios in more than 1 way. We have the ratios that compare one item to another – notice the word to. We can call that part-to-part but we can just call it comparing two items. Then we have the out of or part of the whole. This is an important concept to understand since this is where we get percents. Do we need to teach these concepts? Yes. Do we need to use specific terms for this? Not necessarily. If you use a textbook, use the terms they use. If not, use what makes sense to you and your students.
Now, lets look at the actual conceptual standards for ratios and unit rates. They are 6.RP.A.1 and 2. Number 1 states:
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.”
This standard has 2 parts:
- Understand ratios
- Use ratio language
Sometimes the best way to show understanding is to give a non-example and explain why it is a non-example. I listed a couple of non-examples above. Not all 6th grade students will be able to come up with non-examples on their own, but all should be able to explain why the non-example is not a ratio. This technique can be used as both a formative and summative assessment of understanding ratios, but it does not necessarily work for the second part of the standard – use ratio language. I am a stickler, sometimes, about using precise language (part of Math Practice 6) but what language do we need to be precise about? The standard doesn’t give specifics, but it does give examples. If we base the precise language on the examples all the students need to do is read some ratios aloud or write out the ratio “in English”. However, I again feel that the standard is inadequate. This is where the part-part and part-whole could come into play. But the standard doesn’t say that, so if pressed for time do not go beyond the standard. But understanding the concept of part to whole is important in percents just as it was for fractions.
Next standard – 6.RP.A.2:
Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” Expectations for unit rates in this grade are limited to non-complex fractions.
Again, this is a two-part standard. The understand part is a little different. The student must understand that a unit rate is associated with other ratios. For this to make sense, the students should see how the unit rate can be converted to other ratios and how other ratios can be converted to a unit rate. By knowing how this works and having examples to talk about, the student should be able to explain the association.
This review showed me a few mistakes I have made in the past when creating teaching materials and lessons that were meant to meet these 2 standards. Some of the materials went too far and some didn’t go far enough. That is why I am taking the time to mine these standards. As I develop more materials, I will create some that match the standard as close as I can and some that will go beyond with a note to show that it goes outside of the minimum of the standard.
Resources:
Williams, Lois A.., Harbin Miles, Ruth. The Common Core Mathematics Companion: The Standards Decoded, Grades 6-8: What They Say, What They Mean, How to Teach Them. United States: SAGE Publications, 2016.
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards. Washington, DC: Authors.