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’m thinking, “Will the entire contents of the can fit?” But that’s just because I know tall/skinny containers are deceiving in how little they will hold. Not sure a kid would think to wonder that.

  2. Could you please explain how you placed the centimetre ruler into the pic? I would love to create similar pics for my math students when we study volume of objects such as cylinders, cones etc..

  3. I would give this pic to my students and ask them to develop 3 mathematical questions they would like answered about the pic. They would paste their questions into Etherpad so that as a whole class they could collaborate and see what everyone wants to know about the pic.

    As a class, I would project the Etherpad doc onto our smartboard, we would eliminate duplicate questions and then their assignment for the remainder of the period and that night is to come up with answers to all of the class questions.

  4. @Terry, I took another shot from the same camera position (fixed to a tripod, natch) with a ruler in it for reference. I built a simulated ruler on top of the actual ruler and pasted it on top of the club soda photo.

    I reckon Kate has it closest. Whatever question slash activity you’re going for, it needs to begin with a simple question that relies heavily on intuition.

    “Will the cup overflow?” Simple. Anybody can answer it and defend it from any angle on the basis of good intuition.

    But: “How high will the soda go?”

    Now we’ve taken WCYDWT into calculus.

  5. The kiddos and I have had some fun with this thing. It simulates pouring fluid from one container to another. You guess how high the fluid will rise in the new container. It has rectangular prisms, cylinders, rectangular pyramids, and cones.

  6. Sure, do Geometry. This here is, what, a truncated cone? A frustum? I tend to think it’s an asset that the same (relatively) uncontrived problem can be solved in Geometry and Calculus but, regardless, do you see where this is going? How we’re (quickly) going to rule out Geometry? The bizarre, challenging territory this sort of problem will scale to while maintaining the same simplicity of the original question, “How high?”

    NB. My calculus, apparently, sucks. I’d expect some error due to estimation, the thickness of the glass, etc., but my margin of error is on the order of 600%. Can someone check my work?

  7. Dan, you’re right that there’s a mistake and you’ll kick yourself. You actually forgot to integrate. Take a look – you expanded the bracket and accidently dropped the integral sign.

  8. Crud. Thanks. In a previous draft I just found the area under the curve, didn’t even use solids of revolution. I’m kind of insecure here. Years of teaching remedial algebra has basically put a knife to my higher math skills. Don’t tell anyone.

  9. Don’t worry, they’ll come back. It’s like riding a bike, I think…

    When I read “do you see where this is going? How we’re (quickly) going to rule out Geometry?” I first thought you meant this one couldn’t be solved without calculus. But considering the glasses you bought the other day, I now see that you mean there will be other how-high problems where we no longer have a cone.

    Anyway, here’s me no-calc solution:
    In your equation, y = 1/12 x +1.75, the x is the height and the y is the radius. It seemed easier to me to imagine the whole cone involved (go down or back to the point), so I found the x-intercept of this line, which is x= -21, and then said from there the radius is always 1/12 the height.

    V=pi/3 * r^2*h. But r=1/12 h, so V = pi/432 * h^3.

    But the bottom part can’t have liquid, so
    V =354.88 =pi/432*h^3-pi/432*21^3.
    Soling for h gives 37.5, and subtracting 21 to get back to the original problem gives 16.5cm, which is not to the top.

    Does this count as geometry?

  10. Try and measure the depth of the disturbed water while pouring (meaning how deep is the water visually disrupted/no longer perfectly clear). The goal can be to predict what will happen with a larger experiment at the end of class that they won’t be able to recreate at their desk. To be accurate and scientific students would need to take a picture or freeze frame a video and measure the distance. They would vary the height the water is poured from, the angle of the can(which controls flow), and maybe the time to pour.


  11. Getting the wacky curved glasses out would indeed make an interesting calculus problem, especially with the students figuring out an equation that matches the curve in the first place.

  12. My wife used to drink a combo of club soda and cranberry juice, and a favorite glass of hers was roughly a cone, like the one in the picture. One day while mixing her this drink I wondered: If she wants half cranberry, half soda…

  13. @Northrup, complicated, fun, though the discussion is as far as I would feel confident taking it. (ie. “what variables matter here?”, “what happens when you use a wider glass?” etc.)

    @Tom fun stuff. But seriously, let’s unpack the ellipsis. How do we put that question to a class, aiming at clarity and concision? I’m thinking, “if you fill the club soda up to the 6 cm marker, how high will the drink go after you add the cranberry juice?” Which I dig only because it’d be fun to talk about the obvious wrong answer.

    @Sue, I think that’s what Jason means when he suggests a geometric solution, though California certainly doesn’t include analytic geometry of that rigor in its standards.

    PS. 16.5 cm is extremely accurate, keeping in mind that we start measuring the soda 3 cm up the scale. Here is the answer photo. (Does it go without saying that WCYDWT media-based questioning is better when you have “answer media” that students can contrast with their own work?)