Check For Understanding

Which of the following five images would best drive a rigorous, analytical, mathematical discussion between a teacher and students, leaving the students best prepared to interpret the world outside their school? Defend your choice.

A. [larger]

B. [larger]

C. [larger]

D. [larger]

E. [larger]

[If you have never found a standard-issue measuring cup utterly mesmerizing, it’s possible you haven’t satisfied some of the prerequisites for this course.]

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


  1. I like the top view of B, but its not centered. If you could take C or D and shoot it from a higher angle, and then switch to E to show the measuring scale …

    We could discuss–conversions, how one would measure, error in measurement, compare liquid v dry (do you have a small scale for weight?)

    Of course we could also just make coffee–How is is that living in Brazil I miss the neighborhood coffee shops?

  2. I can’t tell.

    I get the idea of wanting to have distracting stuff in there, so you’re not prompting to much. But the picture with the interesting bits on it also screams “Hey look at this part!”

    I’m beginning to think that maybe my disenamoredness (I’m a math teacher – I’m allowed to make up words) with digital media isn’t so far off base.

    Looking at these, my inclination is to turn off the projector and buy a half dozen ratio rites. Toss them, instructions and all, out to the crowd, along with rulers and construction paper, and see where it goes.

    Then again, part of the issue is: which standard are you trying to hit? I get something for 6th grade (and the prealgebra general math), algebra 2, and geometry.

  3. Dunno what ratio rites are but the point is worth making: the real-world typically trumps simulations of the real-world.


    a) when the real-world is prohibitively expensive. (eg. purchasing a class set of measuring cups.) Or

    b) when you’d rather control certain unwieldy variables. (eg. I want my kids to watch someone shoot a free throw, watch just the first half of the arc, and tell me if it will go in. That’s a tough order if they’re out on the court watching me.)

    In those cases (and in this case) a carefully controlled digital media simulation is the order.

    The standard I’m aiming at here isn’t 6th grade, Algebra 2, or Geometry. It’s calculus. Whatever the standard, there are certain reasons why [redacted], [redacted], [redacted], and [redacted] will help my kids less than [redacted], though that option is, itself, wanting.

  4. A Ratio Rite is a measuring cup that is used to mix two-stroke engine fuel in different proportions. It’s an inverted cone, so your spacing effect is even more pronounced.

  5. So you should just compare (by doing a lab) that fancy Oxo cup you have there with one of the standard issue Pyrex cups. The home ec department probably has lots of them. You could further stretch this by comparing those results to what you get from the appropriately sized laboratory beaker.

    I am not sure why the brown sugar is in the picture, You wouldn’t measure that with a liquid measuring cup anyway. Is it just to bring up the discussion about why you pack brown sugar? while everything else you don’t?

    I did a little research on the net, and Cook’s Illustrated suggests the Cuisipro Deluxe Liquid measuring cup. $7.99.

  6. C and D are the two best options because you can see and read the measurements given on the glass. I’d vote for C because I can read both 16/14 and 500/400 on the scale. The rest (to me at least) is not clear. Students would have to assume/calculate the scale and then apply it to the question.

    Nevermind. On a second look, I really like picture A.

    Picture A is the best picture.

  7. I just remembered something I read by Steve Martin, comedian. Today on Sunday Morning I heard it again and I thought of your project, deconstructing math instruction. He talks about putting comedy out there for the audience, unstructured, so the audience had to choose for themselves where to laugh. Thereby offering the audience an opportunity for authentic laughter. So just placing the measuring cup on the counter, not artificially adding other stuff, seems to be the right choice for what you are trying to achieve. Allowing students opportunities for authentic learning, the kind that will stick, is always a challenge, even for primary teachers. I think picture a is the best also.

  8. What if you showed pictures A and B and generated discussion around why there is a glare in B (on the countertop) but not in A…Do you think someone could draw a diagram of it?

    I like A too, it just feels more natural.

  9. Why not just bring in one measuring cup to pass around?

    I never thought my Pyrex measuring cup could feel less than adequate. Still, anyone who actually uses these things would see the Oxo as the more difficult to clean–or maybe I’m not seeing it right.

    I have to agree, though, that staring at a measuring cup can be “utterly mesmerizing.”

  10. I would start with D but have close up on both the flour bag and the measuring cup (like E but at least one where we can se that other scale as well). We need focused pictures to take the data but more wide shot to be able to talk about what data we want…

    When I was teaching from the Swedish syllabus then I did a lot of problems with a combination of to much and to little information, my students had to decide what was relevant, what could be estimated and what extra information they might need. Now I teach the IB syllabus and I don’t feel IB really support that type of problems so I have toned that down a bit.

    As a (not US) physics teacher I would try to make sure the discussion cover the strangeness that US use oz. for both a measurement of volume and mass. I love to make my students make a few problems in American units each year so they appreciate that they live in a country that use the metric system.

    When it comes to calculus, picture E gives us decent data for V(h), I would use that to talk about A(h) and dV/dh, we could also discuss how h(t) would be given dV/dt is constant.


  11. Sorry to be completely off the topic (I have no well-reasoned opinion), but this just made me think of your blog:

    Verizon money fail:

  12. I think E the Larger view would be the best to use because it shows the measuring cup in an up close view. It allows you to see the measuring cup from a better angle. For a student this photo would make it very apparent that you are working measurement and conversions . I think this could spark some real world situations and lessons as well. It would get students thinking about all the uses of a measuring cup and how they apply to mathematics.

  13. I’m going to have to go with Michael Doyle on this one. If the stated goal of this digital math business is to bring the outside world into the classroom in simulated form, then why- when there’s an easy way (as there is here) to bring the actual object of interest into the classroom- use the simulated version?

    My ideal activity would be to bring in a few measuring cups as pictured and a few traditional measuring cups (and augment with a digital images for easy-to-see reference), have students compare them and come up with questions, observations that lead to the varying distances between equal intervals on the one cup compared to the other. Then take a regular cone-shaped container (like those cheap paper cone cups) and have students create their own measuring cups. They can then test their own measuring cups’ accuracy and re-try, re-think, etc. if needs be.

    To answer the question you actually asked, I’d go with (C). It uses natural lighting, presents the situation in the format it’s most likely to be seen in real life, shoots a wider angle including extraneous objects, and the markings on the cup are still visible.

  14. Like I said before, digital media is a useful stand-in for the real world only when a) the real world is too costly, or b) you need to control certain unwieldy real-world variables. If you have an extra $80 on hand to bring in ten measuring cups, knock yerself out.

    I’m not extremely proud of my own work in this (impulsive) exercise but I know which answers definitely will not suffice.

    [B] isn’t shot parallel to the plane of action which distorts the measurement lines.

    [D] is shot with a flash which puts an extra layer of artifice between the student and the simulated scene.

    [E] basically hangs a neon sign in front of the measuring cup saying “HEY CHECK OUT THE MARKINGS” which will certainly get us into the problem faster but won’t do anything for my students ability to see that for themselves outside of the classroom.

    Between [A] and [C], I’d pick [C] though this reflects my preference to include more noise for my students to filter out rather than less.

    Vicky’s remark about Steve Martin is extremely relevant here. Martin, in his book Born Standing Up, described stripping his act of all the usual indicators that HEY, IT’S TIME TO LAUGH NOW, which forced the audience to pay very. close. attention. to subtle indicators. Assuming the comedian is smart enough to make your close attention worthwhile (and Martin, his dopey recent comedies notwithstanding, is certainly that) that’s the kind comedy (or “instruction” for our purposes) that makes you smarter.

  15. I love this. I think I’d actually start with [C] but keep [E] close at hand for when the discussion moved that way.

    I think I’d rather see some kind of liquid container instead of a brown sugar bag, though. Big bottle of oil or a jug of milk, maybe. I can’t visualize a constant flow out of the bag given how clumpy brown sugar tends to be.

    So many good conversations (and calculations!) dying to come out of this…

    The distraction for me, though, is the magically appearing Krups coffee maker. Not there in [A], then BAM! there it is…

  16. If you were skillful, you could coax a lot of Calculus out of this. After all, Calculus is essentially about changing rates of change…

    Is the level of [stuff] in the measuring cup going up as quickly when you’re at the bottom as when you’re at the top?

    I don’t keep many of these in my kitchen, but Alton Brown does.

    How would things change if I poured [stuff] into one of those?

  17. OK, so I’m not really suggesting you go buy 8 measuring cups, but you’re bringing in the ones you do have right? Maybe borrowing a couple from your friendly neighbors, teachers?

    Perhaps it’s my visual/tactile side that drives this, but I’d feel like this activity would be missing something if the students weren’t creating a measuring cup of their own out of an unmarked vessel whose radius increased or decreased. Maybe that’d more time than you’d like to spend on it, but I think it’d be a valuable experience to solidify the idea.

    @Scott: Alton Brown is a. maze. ing. The Bill Nye of the Food Network. Perhaps Bill Nye is the Alton Brown of PBS. I’ve often thought of using clips of Good Eats to illustrate various concepts in class. Good stuff.

  18. Or you could introduce some cross-curriculum info with a more fun version of the traditional measuring cup seen here:

    My kids bought me this for christmas this year (along with a pi-plate from the same site). I’m not sure if they think I’m just that nerdy or if I really am because this stuff is sooooo cool.