One of the activities I mentioned to help segue into ratios and rates is to practice making all kinds of comparisons. For example, since we’re close to Valentine’s Day you could start by giving each student in class a small box of those nasty Valentine heart candies. Tell the students you want them to write comparisons about the hearts in their box. I wouldn’t give them too many examples of what you mean ahead of time because you want them to generate the comparisons themselves, but if students aren’t sure what you mean, you might say something like, “In my box there are 6 more yellow hearts than green hearts.” That should get them started.

After students have counted and compared their hearts in a variety of ways, share as a class. Without telling the students, group comparison statements into two groups – additive comparisons and multiplicative comparisons. After the students are done sharing, see if they can determine why the statements are grouped the way they are.

Just in case these terms might be new to you, an **additive comparison** is one that uses addition or subtraction to compare quantities. For example if I have 6 green hearts and 9 red hearts, I would say, “I have 3 more red hearts than green hearts,” or, “I have 3 less green hearts than red hearts.” Additive comparisons are ones students are very comfortable with. They have been making these comparisons since elementary school.

Students are likely less experienced with **multiplicative comparisons**, and they need to develop this understanding as they begin working with ratios. For example, if I have 8 yellow hearts and 4 blue hearts, I would say, “I have twice as many yellow hearts as blue hearts,” or, “I have half as many blue hearts as yellow hearts.” Understanding multiplicative comparisons is important because it is essentially the value of the ratio. If I have an 8:4 ratio, I know there are 8/4 or 2 times as many of the first quantity as the second. If I look at it as a 4:8 ratio, I know there are 4/8 or ½ as many of the first quantity as the second.

If no one comes up with a multiplicative comparison in the initial round of sharing, then you have to take a different approach. Tell the students that they all made one type of comparison, and now you are going to share some comparisons that are different from the ones the students shared. Give a few examples of multiplicative comparisons based on the amount of hearts a few students have. Then ask students if they can make comparisons like these using their own hearts. Ask them how these two types of comparing are different. How do you know when you’re making one type or the other?

To extend the activity, you can have students make more additive and multiplicative comparisons by comparing their box with another person’s box. Maybe instead of comparing colors, this time they compare the messages on the hearts. Or you could calculate the total hearts in the entire class and make comparisons about the total amount. You can even look for equivalent ratios. If the total amount in the room has a comparison of “twice as many yellow hearts as red hearts”, you might ask if any one box has that same relationship.

The key through this whole activity is:

- Build on what students know from before, specifically their strength with additive comparisons (even if they don’t know that term).
- Practice making a fairly new type of comparison (multiplicative) to get students comfortable with the language, especially being flexible in going both ways. If I say there are four times as many of quantity 1 as quantity 2, I need to be able to reverse that and say there is ¼ as much of quantity 2 as quantity 1.
- Using concrete objects to model both types of comparisons. (You might have to ask students to justify how they can see a pile of 12 blue hearts and 4 yellow hearts and know that means there are 3 times as many blue as yellow. Where does the “three times” come from in the candies in front of you?)