Ratios and Rates 5 (#MSMathChat)

Final post on ratios. I had a game idea pop in my head during the Twitter chat tonight. People were saying that fractions are a topic students should revisit before starting ratios, especially equivalent fractions. You could make up some kind of game where students practice equivalent fractions. During the game, students earn points for answering questions correctly.

I’m not sure if I would do this at the beginning or the end of the game, but I would have the students also select a rate to convert their points to money. The winner of the game is not the person with the most points, but the person with the most money. Or maybe I would randomly give them their conversion rate. I’m making this up as I go, so I’m not sure at the moment which would be ideal. Basically I would want rates like this:

• 3 points for every $1
• 5 points for every $3
• 2 points for every $1
• 6 points for every $3

If I let students choose the rates, some might figure out the best ones immediately, and that might be fine. If I assign them their rate randomly, that would also work. It just depends on whether I want some kids in the class to have the chance to figure it out right away.

Either way, at the end of the game, we would look at how much money people had and compare it with the number of points they had. I’m hoping that by reviewing a prior knowledge skill, many students would be earning the maximum number of points. The interesting discussion occurs when we see how students with the same number of points ended up with different amounts of money. Or a student who had less points than someone else ended up with more money.

Off the top of my head, I might have students graph their rate and a partner’s rate to see who would come out ahead based on different point values. Just because you won with 100 points, would you still win if you both had 50 points? 25 points? 200 points? How many points would you each need to end up with the same amount of money?

Having all the students graphing would also help us draw some conclusions about what graphs of rates look like. I’d ask them if they could play the game again, which rate of the four rates they would choose and why. After they choose, I might throw out a fifth rate and ask them if they would keep their choice or pick the new rate I just offered. To bring in more equivalent ratios, I might ask what’s another rate that would yield the same dollar value as 2/1 and 6/3.

Whew. I’m done talking about ratios and rates for tonight. The chat just got so many ideas going, and even more came out as I started writing. Brain dump complete!

Ratios and Rates 4 (#MSMathChat)

I was thinking that analyzing a different candy could also make for a fun way to discuss ratios and begin to think about why they matter.

Let’s say each student in the class receives a bag of M&Ms. Before opening them, I might ask the students if they think that the ratio of colors is the same in every bag. Do you think they should be the same? Why or why not?

Have students open their bags and come up with a compound ratio showing how all their colors are related. Compare these ratios across all the bags in the class. To be honest, I have no idea what you’ll find. My gut says you’ll see that the ratio of colors varies per bag. I do wonder if there might be some generally consistency to the ratio though. For example, I doubt you’ll randomly find a bag that’s all or mostly red.

If the ratio of colors isn’t the same in all bags, think about why. Think about a factory and why different ratios of each color are ending up in each bag. What’s going on when the candies are packaged that results in different ratios?

Think about it if they did all have the same ratio. Think about what that factory would look like to ensure this ratio is maintained. Do you think there is an economic reason why they are or are not all the same ratio in each bag? What other factors could affect it?

Can you think of products (not just food) that ensure some kind of a consistent ratio and products that do not? Here are some examples off the top of my head:

• Soda cases generally have a consistent ratio – Basically the ratio is 0 to whatever flavor is labeled on the box. We generally don’t have random flavors of soda in a box.
• An 8-pack of markers has a consistent ratio of 1:1:1:1:1:1:1:1 because you get one of each color.
• I would hazard a guess that packs of multi-color balloons do not have the same ratio of colors from package to package. Or maybe they do. In order to be a multi-color pack, they probably have to guarantee some variability in color. You wouldn’t want to be the person who ends up with 1 green, 1 blue, and 9 red balloons. Maybe they do have a consistent ratio. Or maybe it’s just roughly consistent somehow. Hmm, I wonder.

Ratios and Rates 3 (#MSMathChat)

One thing students need to understand is that they can take any two quantities and make a ratio out of them.

They can.

It may not mean anything useful, but they can do it. For example, if I have 3 potted plants in my classroom and 28 students, I could say the ratio of potted plants to students is 3:28. Does that mean anything? Do you care?

I can’t say I care a whole lot about this ratio.

However, I can still talk about the relationship between these two quantities. With this ratio, I see that there are nearly 9 times as many students as there are potted plants. Another way to put it would be for every 1 potted plant there are nearly 9 students. Again, is there much meaning to that? Do you care that this relationship exists?

Probably not. I can’t say I really care.

Although, that gets me thinking. I might ponder with my students what our classroom would look like if that relationship were reversed. What if we had nearly 9 times as many potted plants as students? What would that look like? Going back to the original ratio, what if we looked in other classrooms? Would we see a similar ratio? Why or why not?

In the end though, we should realize that just because we can make a ratio, doesn’t mean that we should. Often we analyze ratios because the relationship between the quantities matters to us in some way. This is why teachers wrack their brains coming up with “real-life” examples of ratios. And we should do that!

But let the kids play, especially early on in their learning of ratios.

Let them come up with ratios based on objects in the classroom. If a student sees the ratio of girls to boys in the class is 9:16, ask the students to find another ratio of 9:16 in the class. If they can’t, ask them to find a relationship that’s close to the same. The ratio 9:16 tells me there are nearly ½ as many girls as boys, so look for another pair of objects in the room where there is nearly ½ as many of one as the other.

Give students small objects like race cars and other small toys. Ask them to model a 2:1 ratio using those objects. Challenge them to do it with more than 3 objects. Be sure to ask them to model relationships using different language. Instead of giving a ratio like 3:2, say something like, “For every 3 pencils, put out 2 paperclips.” If a student puts out 3 pencils and 2 paperclips, challenge them to keep that relationship but use more pencils and paperclips.

The more students can understand how this is all about relating quantities, the more they can appreciate why we talk about the real-world examples that we use. Teachers choose these examples because the relationships in them matter. And it’s fun to discuss what may happen if that relationship isn’t maintained. Let’s say some kind of dough requires a 1:5 ratio of wet to dry ingredients. What’s going to happen if we goof up and make a 1:5 ratio of dry to wet ingredients? Even better, bring in some flour and water and model what happens!

Ratios and Rates 2 (#MSMathChat)

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?)

Ratios and Rates 1 (#MSMathChat)

Tonight I happened to catch #MSMathChat for the first time in a long time, and the topic was one near and dear to my heart – ratios and rates. The funny thing is that ratios and rates are a topic I loathed up until a couple of years ago.

I loathed them mostly because I didn’t understand them that well. When I was in school growing up, I was one of those kids who was great at math as long as I could be shown a procedure and then follow it. Meaning rarely entered into the equation. Unfortunately, ratios are really all about meaning – about relationships between quantities – and I never got that. I just saw them as two numbers separated by a colon, and that was about it.

A couple of years ago, I had to start writing math lessons on ratios and rates, and I felt like a fish out of water. Since I didn’t have much choice about writing the lessons, I started digging in to the topic. I’m particularly thankful for this book for challenging me to think and reason more than I ever had to growing up. Some of the problems in the book were flat out hard, but I was so proud of myself whenever I came up with the correct answer.

Another challenge in the book was understanding how students had solved the exact same problems I solved. I loved reading over their solutions, trying to figure out what they were thinking and what they were trying to say. I felt just as proud when I would finally piece together how some of these students solved problems completely different than me, but in creative and elegant ways.

I don’t know if I’d go so far as to say that I’m a pro at ratios and rates now, but I do understand them finally, and I love what they represent. During our Twitter chat tonight, I suggested some ideas that I couldn’t really elaborate in bursts of 140 characters, so I’ve taken to my blog to try to flesh them out a bit more.