Yeah, baby!

I could not be happier than to see you working on student strategies.

This is really, really lovely.

Now…you raise the question “Where is the unit rate in the proportion?” The question I have always had is this…”If I multiply $2.29 by 7 eggs, I get 16.03 dollar-eggs?” What’s the unit when I cross-multiply?

Of course, I know the answer. It’s not really 7 eggs, nor 2.29 dollars that I’m multiplying by. I’m really using 7 (or 2.29, depending on your viewpoint) as a convenient scale factor for scaling the ratio of eggs:dollars. It’s a different 7. But that’s not how we talk to kids about it. And is it really such a lofty goal to insist that, when working contextually, both teacher and student should be able to talk about the meaning of each and every number they generate along the way?

Students can also consider a discount if purchasing more items or a complete set. i.e. 75 cents for one can of apple juice, 65 cents per can of apple juice if you buy more than two but less than 6 cans, or 54.8 cents per can of apple juice if you buy the full 6-pack. Piecewise-defined functions!

Thanks Dan and Cindy. That Father of the Bride clip is the best! Will be using next week. :)

Good, Tom. Now what does the computation look like? Is it 5*3.29, then divide that by 6? If so, how are you thinking about the meaning of the 16.45 you get along the way? Or rather, do you divide by 6, then multiply by 5? And how do you think about the (approx) .548 that you get along the way?

If I read Dan’s writing correctly here, he is taking seriously a framework that asks teachers to begin with rich accessible tasks, and then to help students build connections among various solution strategies.

Your fraction-as-operator strategy is a valuable addition to the list of possibilities Dan started with. But rather than “skip” the others, kids can learn from seeing how the others relate to this one.

Dan: I have found motivating (eg.) unit rates to be much, much trickier than helping students learn how to use them.

Say more about that.

I think you mean this, It is easier to show students how unit rates will solve a problem than it is to design a problem that consistently leads to students wantingto use unit rates on their own accord.

Do I have that right?

Good. So why not let that go? Why not be happy with a problem such as the one you’ve developed here; one in which unit rates are going to suggest themselves to LOTS of students, but not to all of them? Then your job as a teacher is to make sure that unit rates come up in Act 3. If it’s because many students did it in Act 2, brilliant! If not, you put it on the table yourself.

I don’t think you have to engineer things quite so tightly.

I agree with Luke’s interpretation of (5/6)($3.29). I know that I want 5/6 of a six-pack, and I know that a six-pack = $3.29, so naturally it is 5/6 of $3.29 that I owe. That’s all a student needs to understand about these problems, and if you insist that this problem requires some understanding or use of unit rates or proportions I would like to see the argument. Unit rates and proportions certainly work, but they’re not the simplest approach. Does the problem serve the technique or vice versa?

Christopher: What if a student says: “I don’t stop at the 16.45 or at the .548 along the way, or think about them, because I have confidence in my construction of the problem and I just punch it into the calculator”?

Tom:What if a student says: “I don’t stop at the 16.45 or at the .548 along the way, or think about them, because I have confidence in my construction of the problem and I just punch it into the calculator”?

What a lovely and challenging question. I guess I would have to think about my goals for the lesson. If a major goal is getting to unit rate, then I would have to question whether this is the right day for having out the fraction calculators. Instead, this might be a good day for four-function calculators.

But whatever calculator we have (even the four-function calculator is gonna let the kid convert to a decimal), I also have to think ahead. If I’m not going to be content with this strategy, it ought to be because there’s something I think the kid won’t be able to do with it; some important understanding that’ll be missing. Two such questions come to mind: (1) What if you want to buy seven juices? and (2) What if you want to buy six juices? (and can I throw in What if you want to buy one juice?) If the kid comes through swimmingly on all of those, then I think there’s evidence of a robust understanding of the strategy and I’m a happy teacher. Indeed, this will be one of the strategies I’ll want to draw out in Act 3.

And by the way, isn’t this a lovely thing about units? I see the unit rate as per can, you see the unit rate as per 6-pack and not only is each right, it’s important to be able to see both.

P.S. Does it matter, Dan, that all of your purchases are less than a single composed unit? Would student activity look different if you bought 2 6-packs of juice AND 3 cans, or equivalently if you bought 3 5-packs (each being a 6-pack minus 1)?

I realize, of course, that it matters to the grocer who was kind enough to let you decompose the packs (presumably without buying them all). But does it matter for the mathematics kids would be doing?

Dan:Why wouldn’t they just set up the proportion or the unit rate the same way?

Because they don’t think like we do.

What we have here is a testable hypothesis. I’m imagining a group of regular seventh graders, or of low-performing high schoolers, or of developmental math/College Algebra level college students (i.e. not Calc students, not advanced 8th graders, etc. In short students who have not had tons of training that stuck in solving proportions).

Give half of that group your original apple juice task and the other half a task where there are 2 and 1/2 six-packs. Same juice, same price. I think the latter group is less likely to go to unit rate, more likely to think about 2 1/2 operating on the price in the way Tom and Luke suggest.

And then give the third half of that class a task where there are 3 five-packs? They’ll be more likely to think like the first group-do whatever they do to find 5/6 of $3.29-and then scale. Even though they have the same number of cans as the second group.

But it’s just a hypothesis right now. I’ll see what I can do to investigate it in the next week or so.

Well, I hope I’m not off topic. It is “correct” to find 5/6 of $3.29. Thus, a student can deduce the correct answer using such a method. A student can also use a proportion to find the unit rate. But, the method does not necessarily imply that the student understands what is happening. Following a method or algorithm is just simply following. I would hope to push students to derive a method for determining any of the above mentioned methods, or something else, such as a diagram or chart. Pushing students to use the reasoning tools inside of their minds, to go beyond the comfort zone, is what brings a clear understanding of the problem they are addressing. I remember assisting with a Kaplan course years ago, and the guy teaching the course kept telling the kids to just plug the numbers into a ratio and proportion. So, that’s what they did, but they made no sense of the problem, and had no idea if the answers made any sense in relation to what the problem was asking. There for, I would be looking for proportional reasoning rather than the technique of using a proportion, or finding the fraction of the cost to determine unit rate. If that was the method a student used, I would push further to get under the hood and find out what they are thinking.

Two thoughts:

1) I would rather see this problem as a comparison between price tags at two different stores that are both offering a “2 for $5” sale: one store that keeps that price for smaller quantities, and one store that doesn’t. (Around me, most grocery stores keep consistent and most convenience stores will have tags like “2 for $2 or 1 for $1.19”.) I feel like that’s a little less forced than a hypothetical situation about splitting.

2) Wasn’t sure whether to mention this, but it’s math-related and you always ask what questions we have after seeing your visual aids. My first question was “What’s the pattern/rule behind the letter omissions on the price tags?”. My answer is that there doesn’t seem to be one; the full, unaltered phrase “APPLE JUICE” would fit numerically and width-wise in the space used for “BLKBRRY SODA”.

But Dan, stores don’t allow this!

Aren’t you just creating a contrived situation here? I’m with Dave on this one. While your photos might be good for provoking some questions about pricing strategies, I think a better motivational photo would be those supermarket ads where they advertise “3 for $5,” “buy two, get one free,” etc. This might provoke the same mathematical lesson, along with a valuable life lesson about the psychology of pricing.

I am doing this tomorrow with my Math 8 students. We have talked about equivalent ratios and used it to find out the best deal for tickets to a state fair. I am curious to see if any of them setup a proportion which is what I will be teaching next week and/or the week after. Setting cross products equal to each other.

I did the lesson. They were super engaged. When they divided the price by the number of items they realized they did it incorrectly without me telling them. Great activity, thanks Dan!!

I had them do a 4 step writeup:
1) restate the problem
2) organize the facts and state how you will solve
3) solve it and show work
4) verify and check your work