Partial Product

Imagine you’re at a store that lets you pull products apart and pay for as much or as little of them as you want. What will your total grocery bill be for these three items?

If a student has no idea where to start, you can prompt her to list a price that sounds fair to her for the three sodas or, failing even that, a price that seems unfair to her. (You’re basically asking her to give a wrong answer. It’s a lot easier to give wrong answers than right answers because there are a lot more of them.)

You’ll find students who divide all the way down to the unit rate (ie. each egg costs 19 cents) and then multiply back up. You may also find students who set up a proportion, which will disguise the unit rate in an interesting way.

You’ll find students who set up different but equivalent unit rates. (ie. 19 cents per egg and .05 eggs per cent.) You’ll find students who set up different but equivalent proportions.

One of your many challenges during this activity will be to select students to show work that highlights a) the different ways to find the unit rate, b) the different ways to set up the proportion, c) the equivalence within those methods, and d) the equivalence between those methods (ie. ask your students to help you find the unit rate within the proportions).

The Goods

Partial Product

Featured Comment

Larry Copes:

I’m with Christopher and, I think, Dan here: Toss it out with the understanding that students can use any method that makes sense to them. Then not only share those methods but compare them to see why they yield the same result. Love the word “catharsis” in this context.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.


  1. Reply

    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?

  2. Reply

    This is a brilliant resource. It will be perfect revision for my older students tomorrow, and the first thing I look at the next time I introduce a topic on ratio and proportion. I’m struggling to come up with Acts 2 and 3 but I can get so much out of these three photos.

  3. Reply

    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!

  4. Reply

    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.

  5. Reply

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

    Just a behind-the-scenes note: if my blog takes some consistent flack, it’s for focusing too much on the first act of a problem (using my parlance) rather than the second. Style, not substance, as it goes. Along one line, I find these conversations very, very difficult to facilitate online. Along another, I have found motivating (eg.) unit rates to be much, much trickier than helping students learn how to use them. I should probably mind the balance a little more, though.

    Pete: I’m struggling to come up with Acts 2 and 3 but I can get so much out of these three photos.

    Not every problem is expansive enough to include an act two where students need to determine extra information they’ll need. Not every problem needs an act three that confirms the answer visually. That isn’t always possible. Every problem needs catharsis, though. In this case, the catharsis comes from seeing several different methods all yielding the same result. I considered faking a receipt for act three but I decided I need to hold this three-act math framework with a looser grip.

  6. Reply

    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?

  7. Reply

    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.

  8. Reply

    To me the parenthesis in (5/6)($3.29) imply that the thinking is: a unit is a package, so the unit rate is $3.29, and you are buying 5/6 of a unit. The computation in line with this thinking, and the notation, would be 5 divided by 6 is about .83 units, then multiply by $3.29.

  9. Reply

    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”?

  10. Reply

    I’m with Christopher and, I think, Dan here: Toss it out with the understanding that students can use any method that makes sense to them. Then not only share those methods but compare them to see why they yield the same result. Love the word “catharsis” in this context.

  11. Reply

    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.

  12. Reply

    In the spirit of election season, Danielson nails me with a gotcha question. Isn’t it obvious given the content of the post that I’m happy to find both unit rates or proportions or whatever else?

  13. Reply

    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?

  14. Reply

    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.

  15. Reply

    Chris: I was just giving my opinion on the thinking implied by the notation (5/6)($3.29). I was not endorsing that particular method.

    I agree with the original post that a good goal would be for students to come away with some sort of understanding of how the different methods are interrelated.

  16. Reply

    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.

  17. Reply

    Wow, there was really a lot of great discussion about this topic. As a middle school math teacher, I find that students think in so many different ways about different math topics. Sometimes I think it depends on the students’ previous learning. I have had students who come in from other schools and say, “Wait, I learned it this way before.” I am very frank and tell the students that there is usually more than one way to solve a problem correctly. Teaching the students several strategies for problem solving creates lifelong learners.

  18. Reply

    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”.

  19. Reply

    Not wishing to derail the comments, but just wanted to thank Cindy for the video link. I’m going to use this for a different topic: Lowest (Least) Common Multiple. Seems just as (if not more) relevant to me.

  20. Reply

    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.

  21. Reply

    Dan, my colleague and I have been following your lead with our Algebra curriculum this year and it is fantastic. We each have a class of 7th graders who are taking Algebra 1, so talented math kids. We’ve blended in some work with the new Ti-Nspire CAS handhelds. We find our kids much more involved in thinking about the mathematics they are doing.

    In any case, we used your partial products as the opener one day earlier this week and every student in the class was engaged. Most found the unit rate for each and then the total. We had a great conversation about when to round. One 7th grade boy said that both answers were correct but leaving the entire decimal with the unit rate, then multiplying by however many items you are buying is the more accurate answer. Some kids were not happy with two correct answers. It was a great discussion about mathematics in the real world. Thanks for sharing all you do, it has certainly changed the way I teach.

  22. Reply

    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

  23. Reply

    Thanks a mil for the recap, Marty. I’m struggling lately to figure out what kind of write-up we should ask from students with this kind of work. Your four steps help push my thoughts forward.

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