We were in a grade 8 classroom yesterday and presented them with this problem which Liisa originally saw presented by Nathalie Sinclair at an OMCA conference. She mentioned it was from the Connected Math Project but we have adjusted the numbers slightly for a variety of reasons.

Liisa spent some time discussing the problem with the grade 8′s. She asked them if they knew what orange concentrate was. Then she asked them to guess (there’s power in guessing! Dan Meyer talks about how guessing can make tasks more engaging in this post). Out of the twenty kids in the class, 18 chose Mix A!!!! That surprised me (it didn’t surprise Liisa… she’s done this task before and found the same results). I wondered why they picked it. Did they truly have a gut sense of the proportions? Was it because it was the first one? Was it because everyone else was doing it? The two students who didn’t pick A guessed Mix C and D.

The students set to work with big paper and markers in groups of two and the fun began! Liisa led the lesson but there were 5 other teachers in the room that picked a group and observed without saying anything. The two girls I was sitting with talked about the problem in an interesting way at first:

S1: What do we do?

S2: Well we don’t have to do math or anything. We just, you know, think about it and decide which one is orangeiest.

They then began to write out the reasons that they felt Mixes B, C and D weren’t as orangey: “It wouldn’t be Mix C because there wouldn’t be enough orange concentrate. It wouldn’t be mix B because there’s too much water so it would make it watery and not orangey enough. It wouldn’t be Mix D because there’s still too much water but not as watery as B and C.” (I found that last sentence really interesting… they knew that! They just didn’t have the words to explain how they knew!)

Liisa came by and asked the girls to explain their reasoning. They couldn’t, so she asked them to draw some pictures or find a way to justify their thinking. This is what they came up with:

What a great way to represent the whole!! Even after they had come up with this representation, they weren’t entirely sure how that helped them explain.

Liisa started a congress and had several groups present. There was some great thinking going on! Some groups took the same strategy the girls did and drew a representation of the parts to whole. Others used the strategy of reducing the ratio to a unit rate.

Some used percentages:

One of the students shared his interesting method: He divided Mix A into a unit rate (1:1.5) and then multiplied it by 3 so that he could compare it with the ratio for Mix D (3:5). That way he was able to see which mixture had more water for 3 cups. Cool!

In sum, the lesson brought out some great proportional reasoning and thinking. The students were engaged, asked good questions, and did some great problem solving. These were grade 8′s… but the task could really be used in any grade that is trying to teach proportional reasoning.

A P.S. from Liisa….

Although I have tried this problem several times with teachers and students and am not surprised when many of them guess which solution is the “orangiest” – I am still surprised at the variety of strategies that people use to prove this and how rich the discussion is. In this particular class the common theme in the strategies that the students used was the need to make something the “same” in order to compare. Some adjusted their ratios to find the amount of water needed for 1 cup of orange juice, others turned everything into percents so that they “were all out of 100″ and it would be easier to compare, some drew circle graphs so that their “whole” looked the same….What was nice is that many students recognized this similarity in their solutions even though they looked quite different.