[WCYDWT] Grocery Shrink Ray

Start with discussion, brainstorming, estimation.

You run a Dollar Tree franchise. Everything costs a dollar.

You sell shampoo for a dollar — a popular brand called “White Rain.”

Problem: times are tough and the people who sell you the shampoo need to raise their prices higher than a dollar. What are your options at this point?

Teacher: write down student suggestions on the board. When someone suggests “sell less shampoo for the same price” resist the urge to declare, “Ahhh … that one’s eeeeenteresting,” thereby cluing your students to the fact that they weren’t really brainstorming, they were playing another game of “guess what the teacher wants to talk about.”

Spend a few seconds talking about, for instance, the student’s suggestion to rename the store. To what? Somewhere Around A Dollar Tree? You’re getting students all across the spectrum to invest in the problem and it’s costing you — what? — a few minutes.

Lower a little more structure onto the problem.

It turns out that “sell less shampoo for the same price” is exactly how that went down.

Teacher: “So what’s an interesting question we could ask here?”

Teacher: “How much extra are you actually paying here? What should the smaller bottle actually cost?”

Teacher: take guesses.

Your location in the scope and sequence of proportions will determine how much direct instruction your students will need here. That’s on you.

Issue extensions.

I dug through the entire Consumerist collection. The most valuable entries included both price and quantity.

Some of the others are great for discussion, though. Notice Dial’s effort to conceal the decrease with a taller, thinner container:

Tropicana’s opaque containers offer them enormous flexibility in the quantity of orange juice they give you:

And, if your class feels like venturing into some 3D geometry, ask your students how Yoplait has managed to shrink the amount of yogurt by 33% in a new container that looks nearly the same as the old.

The answer to that question (along with all the lesson materials) is located in this zipped archive.

[h/t MPG for the idea]

About 
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.

29 Comments

  1. Ice cream is another shrinking product… Breyer’s used to come in 1/2 gal cartons and is now barely 1-1/2 quarts! But the containers sure look aerodynamic now, so that’s a plus.

  2. For the yogurt, part of the issue is not just packaging size, but the actual product. The “larger” container is what one typically thinks of when they think of yogurt. The “smaller” container is “whipped” yogurt. The size of product you highlight is weight, not fluid ounces. The whipped yogurt weighs less for the same volume because it has some sort of gas (I can’t remember what kind off the top of my head) “whipped” into it so that it’s fluffier.

  3. Is it fair to say that 75-80% of the WCYDWT problems on your blog involve rates/ratios/percentages? Is it fair to say that 90% of them involve either rates/ratios/percentages or area/volume?

    There’s nothing wrong with this–this is where you find math in the world, and it’s probably the most natural place for finding it.

    But I guess here’s where I find myself. I teach linear equations, it goes great, lots of examples, wonderful. Ratios/rates/percentages, we’re having a blast. Then we have to learn how to work with exponents, and it’s a long couple of weeks, and now we’re back in area/volume and we’re having a bit more fun.

    Next comes factoring, and I’m dreading it. It’s not rates and volume that leaves my kids wondering why we bother with math–it’s all the stuff in between, and I don’t have much of an answer for that except to try and make it fun. But when I try to make it fun they know that I’m covering up the math in a game; the game isn’t coming naturally from what we’re learning. I suppose that this is something that I have to do a better job with.

    Anyway, I guess the point of this comment is that we can’t WCYDWT every topic.

  4. OK – so I’m thoroughly confused now. Apparently it isn’t good to give kids the illusion of a real context (pseudocontext) but it is good to give kids the illusion of an open ended problem, when it isn’t really open ended at all? The kid that says “change the name of the store” gets a pat on the head, but at the end of the lesson everyone knows that he was wrong and the one who said “put less in the bottle” was right.

    They actually were playing a game of “guess what the teacher wants to talk about” so why pretend otherwise?

  5. @MBP

    For the factoring, I’m not sure how deep you go into it, but here is an interesting problem I did for my students (albeit after the lesson as I didn’t find it until then) http://www.ted.com/talks/arthur_benjamin_does_mathemagic.html

    It is a 15 minute TED video, and towards the end, he explains that he is going to do a trick out loud and tells the audience how he does it. However, the same method can be used for the earlier tricks as well. I showed them the first part, then the obvious question is “how did he do that” or “How do I do that”. To square a two digit number in your head, you can treat it asa the square of a binomail. (a+b)^2 = a^2 +2ab + b^2, so 37^2 can be thought of as (30 + 7)^2. take 30 * 7 and double it. The add it to 30^2 and 7^2. All essentially single digit multiplication.

    Another “trick” is multiplying two numbers together who’s tens digits are the same, and ones digits sum to 10 — like 37 and 33, or 86 and 84… The last two digits of the product is the product of the one’s place. The first digits of the answer is the tens digit times the next consecutive digit. So 37*33 = 1221 since 3*7=21 and 3*4 (the next digit)=12. 86*84 = 7224 since 6*4=24 and 8*9 (the next digit)=72.

    I’m sure there are more out there. These are just two I found recently.

  6. @MBP

    You’re right that ratios & proportions (and to a lesser extent, linear functions) are all over the place, and will likely make a disproportionate (!) chunk of any real-world curriculum. That’s definitely true on Mathalicious, too.

    To your question about exponents, though, here are a few lessons that your students might enjoy:

    Who’s Your Granddaddy: Are we related to everyone on the planet?
    I Remember: The math of memory
    iPod dPreciation: How has the iPod depreciated over time?
    XBOX Xponential: Have video game consoles followed the exponential growth of Moore’s Law?

  7. Robert: The kid that says “change the name of the store” gets a pat on the head, but at the end of the lesson everyone knows that he was wrong and the one who said “put less in the bottle” was right.

    You’re operating under a strange, punitive definition of right-ness and wrong-ness. The dollar store actually offered less for the same price. I can’t change that. It doesn’t mean we can’t honor other student suggestions.

    Robert: I should have said that it works more honestly if you start with the picture of the two white rain bottles. See what they can do with that.

    This is a mistake. The context offers you an entry point that requires a lighter mathematical framework. (“Say you run a Dollar Tree franchise.”) You’ll invest more learners there.

    MBP: Anyway, I guess the point of this comment is that we can’t WCYDWT every topic.

    This is fair. WCYDWT isn’t a prescription for all curriculum. WCYDWT is a prescription for application problems. As in, “if we take the premise that having students apply math to the world around them is important, what’s the best way we can do that under classroom constraints.” That, to me, is still an open question, though WCYDWT is my best attempt at answering it.

  8. I was using “right” and “wrong” as a sloppy shorthand for how the students feel. The student(s) who’s suggestion goes on to be the basis of the rest of the lesson is going to feel differently than the student(s) who’s idea does not. The kid who’s idea does not will not feel invested in the problem just because you took a few seconds to play with his/her idea before it ceased to play any further part in the lesson. This kid will almost inevitably feel that their idea was not as good as the idea that got picked up for the rest of the lesson. Maybe there’s a cultural difference here, but I know that this is how it would affect many Scottish kids.

  9. I should also have said (I’m making a habit of this) that this is another brilliant piece of work – hats off to you Dan. Maybe my assumption that it was acceptable to start criticising without saying that first is another cultural difference. Or maybe I’m just rude! Sorry either way.

  10. Robert: The kid who’s idea does not will not feel invested in the problem just because you took a few seconds to play with his/her idea before it ceased to play any further part in the lesson. This kid will almost inevitably feel that their idea was not as good as the idea that got picked up for the rest of the lesson.

    Help me out here. Culturally speaking, does that student only feel validated if we then spend the rest of the period creating new logotypes and business cards for her proposed “Somewhere Around A Dollar Tree” franchise? What of the student who tosses out a joke answer like, “Shut down the store.” Will the student feel invalidated if I just laugh at the line and move on?

    If that’s the case, I’d say your assessment of WCYDWT as culturally incompatible with Scottish math students is right on.

  11. ProfWhoLovesMath

    December 9, 2010 - 12:04 am -

    MDW: Anyway, I guess the point of this comment is that we can’t WCYDWT every topic.

    I’d accept that. That said, maybe it should make us question: Is teaching factoring the best use of time in the classroom? Is it the best topic we can teach students, at this point in their mathematical learning? WCYDWT can’t answer those questions, but it can raise them and get us thinking. In the case of factoring, I think these questions are fair ones. It’s true that I’m asking some pretty leading questions here; as you might be able to guess, my answer would probably be “no”. In my view, factoring is not terribly critical in real life, and this happens to be correlated to the fact that it’s hard to come up with WCYDWT problems (or real-world problems) to motivate learning factoring.

  12. Clearly the kid with the joke answer feels validated when you laugh. That’s what they were looking for. Your straw man argument is hardly conducive to reasonable debate.

    As to the other two, I think we are at an impasse, as I can’t see how you can’t see that they will feel differently about the relative contributions they have made to the lesson. I can’t see how you can’t see that this has a potential impact on their sense of investment in the problem.

    I was going to have another go at explaining my position, but I don’t think there’s any point -what I said was pretty clear and if you don’t see it you don’t see it.

  13. @Robert: I’m going to stick my neck out and try to interpret what you and Dan are back-and-forth-ing. Forgive me if I misrepresent either of you.

    I was just at a conference where someone spoke about the danger of ill-defined vocabulary. That teachers can be talking about two different things, but since they are using the same words to represent those ideas the result is confusion. I *think* that what’s going on here is that you and Dan are using the word “investment” in two different ways.

    The way I see it is that Dan would claim that the student who has suggested that we rename the Dollar Tree store has _already_ invested in the problem, by virtue of considering what to do with the context presented. This student has expressed some form of interest in finding a solution to the big, open-ended question and therefore is likely to be more interested in what other students might come up with.

    It seems like you see the student as investing in the problem when his/her idea gets taken up mathematically. One interpretation is about the entry point into the problem, and the other is about the process of solving the problem.

  14. It doesn’t seem to me like we need to based our entire lesson around any students suggestions. In fact, this is normally how a solid class becomes a blow off class where everyone is right and all tangents are welcome.

    I think that if you honestly consider a student’s idea, discuss the pros and cons, it is okay to tell them that this isn’t realistic, and someone else’s idea is what we’re going to go with. The point is not to jump on the idea you were looking for so fast that people just wait for someone to say the right answer.

    It’s equivalent to only asking someone to justify their work when they’re wrong. Then justification just becomes synonymous with “you’re wrong.”

  15. @ProfWhoLovesMath

    “In my view, factoring is not terribly critical in real life, and this happens to be correlated to the fact that it’s hard to come up with WCYDWT problems (or real-world problems) to motivate learning factoring.”

    I take “real-word problems” to be “problems that arise for people who aren’t math specialists that math can help answer.”

    If application to the real-world in one of these problems is the sole criterion for justifying inclusion in a curriculum, we’re going to stop teaching math after Algebra 1, and we could probably bring it down to a semester.

    Now, this isn’t necessarily a point against your argument. Maybe we do need to radically revise high school math so that it’s more “real-world” focused.

    But “real world math” isn’t the same as “applied math.” Math is applied by all sorts of specialists in all sorts of fields. Now, the nonspecialist has no need to work with polynomials of degree two or higher pretty much ever. The specialist–depending on her field–has to be VERY familiar with them. And the training that the specialist needs requires familiarity with polynomials. If a girl doesn’t know how to work with polynomials in high school it’s hard to see when we’ll give her the training that will encourage her to pursue a career in engineering.

    Here’s my point: real-world application can’t be the only criteria for inclusion in our curriculum. We’re not just trying to prepare most people for their lives. We’re also trying to train and recruit specialists for society and humanity. Besides for showing students where math shows up for nonspecialists, we need to also make being a specialist seem awesome and accessible.

  16. Surely the key to it is that we want students to be able to apply the mathematics to any situation. We can hook kids in with interesting, urposeful and useful mathematics. We want them to be ‘functional’ . It is worth looking at what schools like San Lorenzo high are doing. The approach is called Complex Instruction…it follows Dan’s idea of being less helpful.
    Steve

  17. Working backward…

    1. Steve, I want to learn more about ‘complex instruction’. How do I do that? I live near enough that I could visit San Lorenzo High in January (while my college is between semesters); is that possible?

    2. Factoring may not have real-life applications, but it leads into cool deeper math topics. I see it as very important in algebra. (But teaching students who don’t want to be there might make me change my tune. I’d want to give them topics they could identify with, even if they weren’t math fans.)

    3. Dan, I was interested in the point Robert Jones brought up. I struggled with that point when I first read your post (compulsively honest as I am). You did have an agenda, a topic you wanted to talk about.

    I guess your point is to not jump too fast to your own agenda, that it’s not a game of ‘guess what the teacher wants to talk about’ if you value their first discussion. Only after respecting that discussion is it time for the one you were waiting for. Have I got that right, Dan? Robert, what do you think of this way of framing it?

  18. @Breedeen we are certainly not seeing eye to eye, that’s for sure :-)

    I think Dan is seeing this in black and white, or thinks that I’m seeing it in black and white, or something like that!

    Pupils invest in the lessons to varying degrees and their degrees of involvement will fluctuate over the course of a lesson. I’m sure we can all agree on that.

    The purpose of the initial discussion is to draw the students in, if I understand correctly. My contention is that having a discussion which is a variant of “guess what the teacher wants to talk about” in the form “guess what happens next” is structurally similar to asking a class, half way through an equation, “can anyone tell me what the next line will be?”. The latter is just plain bad practice.

    It’s worth mentioning that I’m dealing with classes of 30. Dan may have much smaller classes in mind.

    On a more positive note, I would suggest running the start of a lesson as a think-pair share. Give each student 1 minute to consider a solution to the problem, then give students another minute in pairs to discuss their solutions amongst themselves, then finally take responses from pairs. This will at least mean that every student will have spoken about the problem before you move on to how the store actually dealt with it [although – was it really the store that dealt with it? Wasn’t it the shampoo company that changed the bottles rather than the store?].

  19. Robert, what would it look like to validate the Scottish student who suggests we “change the name of the store?” To avoid, per your formulation, “patting the student on the head.”

  20. @ Sue: Sadly, San Lorenzo High is no longer the model of CI that it once was. District changes forced upon the school resulted in about half of the teachers there leaving. I was once one of them, and I would not in good conscience recommend visiting there anymore.

  21. As I’ve said before Dan, this is not a binary “validated/not validated” situation, and there clearly isn’t a way to make the “change the name” student feel exactly the same about the contribution they have made as the “less in the bottle” student, given the situation you have set up at the start of the lesson.

    Actually I think I’m coming back round to my first thought, that it would be best to tell the story of the dollar store then show the first slide – “look what they did!”. That picture of the white rain bottles is the real hook here. That’s when it gets awesomely cool.

  22. I know I jumped into this conversation late, but I think there’s a really important distinction to be made here.

    I don’t see that “validated” means one student feels exactly the same about their contribution as another. Frankly, while all contributions have value, this isn’t the same as all contributions being equal.

  23. Lobsang Rampa

    June 30, 2012 - 12:19 pm -

    Don’t forget my absolute favorite: Propane Gas!

    Most consumer propane tanks in the US are 20lbs tanks. It is safe to fill those up to 20lbs (they in fact take a bit more). “Swap your tank” companies like AmeriGas and others fill it to 15lbs, citing “safety concerns”. It’s like “Hey, yeah, we completely ignore the fact that these containers can be filled up to 20lbs, and we fill them to 15lbs, but charge you a full 20lbs, but hey, we’re concerned about your safety…”

    Yeah, right…

    Always fill in your tanks at metered stations.