## Linear Fun #2: Stacking Cups

My favorite lessons build an hour of complicated, engaging mathematics from a simple picture, question, or anecdote. This is one of those lessons.

1. The Question

How many Styrofoam cups would you have to stack to reach the top of your math teacher’s head?

Tell them you’re 200 centimeters tall (if you’re me). Measure a cup in front of them and tell ’em it’s around 10 centimeters tall.

Act like you blew it and overestimated the question’s difficulty. Ask them for a fast answer.

Someone will divide quickly and tell you “twenty cups,” at which point you hold up a stack of 20 cups and let them wonder how they underestimated so grossly.

Let them figure out which math problem they actually solved:

3. Offer Them Materials

Ask them what they need from you. Some will ask for hundreds of cups. Offer them ten.

They’ll want a ruler. Offer that.

Some will chase you around the room, running after your feet with their stack of cups, asking you to hold still so they can eyeball the answer. Don’t offer them that.

4. Let It Go

The rest largely runs itself. Just walk around, ask good questions, and correct faulty assumptions.

5. Good Questions
1. How many parts of the cup are there? Two.
2. Which part of the cup matters most over the long run? The lip. The base only counts once but you count the lip every time.
3. If I asked you to tell me how tall a stack of sixty cups would be, what would you do? Add the height of sixty lips to the height of the base.
4. If I asked you to go backwards and tell me how many cups are in a 200-centimeter-tall stack, what would you do? Subtract the height of the base and then divide by the height of the lip.
5. Does it matter if you round to the nearest centimeter? It definitely does.
6. Get A Graph And An Equation

Kids will solve this pretty well without either – two groups socked the answer right on the nose – but this is pretty meaningful context for graphs and equations. The lip-height is the slope and the base-height is the y-intercept.

7. Actually Stack Them

After you’ve a) taken secret-ballot estimates from each group, and b) written them down on the board in descending order, have one member from each group i) count her cups, ii) stack them by your feet, and iii) call out the quantity for the rest of the class to tally up.

If, just for instance, you’re twice as tall as some of your students, have one student stand on a chair to eyeball the answer. (“One more. Okay, one more. Nope, too much.”)

The winning team receives fabulous cash and prizes.

8. Extend It

This project has legs. My kids ran outta interest at different points after we announced the winnersNote to self: postpone that announcement until after the extensions. *smacks forehead* but these extensions are all gold.

1. Ask them the same question with a different cup. A red Solo cupDon’t pretend like you don’t know the ones I’m talking about., plastic, a thin lip, and tall base.
2. Toss up this graphic.

Have them measure the lip and base of each.

Ask them, “Which will be taller after three cups?” (A: Cup B.)

Ask them, “Which will be taller after one hundred cups?” (A: Cup A.)

And then – respect, if you see what’s coming – ask them, “How many cups does it take stack A to rise above stack B?” Wham. You’re solving three-step equations.

These are my favorite projectsOne, again, to which I can only claim certain flourishes. The rest comes out of ed-school at UC Davis.: easily scaffolded, easily differentiated, easily assessed, and arising completely from a simple question, a simple prop, and a single image.

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. #### Rich

March 17, 2008 - 7:15 am -

I like the overall project, and I particularly like the extension of comparing two different types of cups (as in your silhouetted figure) because that definitely helps to introduce solving a system of two equations. That concept can certainly be rather abstract for students to get a handle on, particularly early on in their algebra experience.

2. #### dan

March 17, 2008 - 8:22 pm -

Damn but these math methods posts slow traffic to a crawl. If I didn’t know you were still around to make something good outta ’em, Rich, I’d ditch ’em completely.

3. #### Mr. K.

March 17, 2008 - 9:40 pm -

these math methods posts slow traffic to a crawl.

No poop. I’ve noticed the same thing.

Stupid thing is, even though I don’t respond to them as much, they are the real reason I started blogging in the first place, both from a reading and writing point of view. Now I find myself drifting away from writing them because I get the impression that others find them uninteresting, and I’m not happy with the overall tone of my writing as a result.

4. #### H.

March 18, 2008 - 6:39 am -

Dan, Mr. K – don’t ever think of discontinuing your math posts!

Of course only a small subset of your readers are math teachers. That doesn’t mean they deserve to be ignored.

5. #### Neal

March 18, 2008 - 6:51 am -

They are by far what I most prefer to read here, even though I probably will never teach math. It’s not so much about grifting your concepts verbatim, but internalizing the out-of-the-box thinking that you’ve applied to a subject too often bereft of originality. If I might flatter you some more (I like your shoes, by the way), I’ve been especially impressed (big time) by your ability to make fresh and fun lessons that avoid the siren’s call of excessive whiz-bang.

To be honest, I’m not convinced some of the more theoretical discussions floating around various edublogs have been worth the electronic paper they’re printed on, although they are often fun to contemplate. I have yet to find a sample lesson plan that hasn’t taught me something worthwhile.

I don’t know about the effect on your overall site traffic, but lesson plans seem less conducive to discussion than inflammatory op-ed pieces. I’m beginning to suspect this may be largely due to a combination of hubris and cowardice, but what do I know.

6. #### Mr. K.

March 18, 2008 - 7:14 am -

Here’s some positive feedback:

I’ve seen this lesson before, but I never used it. The reason is that I was afraid the kids would get hung up on what happens when the stack is 0 cups high (which, according to the graph and equation, should be the base height, but in practice is 0).

This variation, by introducing the “how many cups do you need to match my height” question, pulls the attention away from that end, while preserving the direct variation plus some constant aspects.

Once the how many cups question is answered, the offset question can be addressed as an abstraction, rather than as something concrete. That in itself makes this lesson go from something I’d avoided to something I am now excited to try.

Thanks for that, dan.

7. #### Sarah

March 18, 2008 - 12:26 pm -

Dan, Mr. K, and everyone else who helps me by inspiring lessons, please don’t cut them out.

There’s a lot of stuff that I can’t pull off in the same way. Different resources and all. A lot of the time I don’t use a lesson when I first read it. But honestly, I come back and search your sites when I’m going into a new concept.

You should probably also know that I have several bags of styrofoam cups in my closet–leftover from thinking that I *might* use them from your first day of class plan. I sort of did in one class. I’m still hoping to pull this out and love having more ideas to extend the plan.

8. #### Rich

March 18, 2008 - 12:28 pm -

Hmmm, I like that interesting point, Mr. K (about the zero cups solution).

Dan, you know I’ve been around long enough to have more than a few comments, and you might have observed that I mostly focus on the math methods more than the stuff about Web 2.0 (although your very under-commented post about the flu in Asia was much more 2.0ish than some might realize!) (and I’m reasonably serious about firing up our own school’s Moodle for wide use if something like that ever does come to pass) (sorry for cross-commenting in a different post!).

Maybe it’s that even the math teachers among us aren’t all necessarily focused on those particular topics (e.g., linear functions) although I just happen to be right in the thick of that right now with my Pre-Algebra students. So if you’d posted that back in October, I would have read it with interest but possibly without replying.

I might be only one out of 654, but I definitely read every post every day! And the math methods posts are the ones that I read with the greatest interest — even when I don’t comment back!

9. #### Erin Remple

March 18, 2008 - 12:57 pm -

Fantabulous! I love this. My Consumer kids will love this (unfortunately, it will have to be next year as I’ve already taught the variations unit).

Please don’t stop blogging these kinds of posts. I think we learn so much from seeing other teachers teach. While I’m not actually *seeing* you teach this lesson, I am getting ideas about how you teach.

10. #### JackieB

March 18, 2008 - 3:36 pm -

Hey – please don’t stop the mathy posts, it’s why I started reading in the first place. I mean the other bits are good too, but really, I get more out of the posts on lessons. I need help on how to better develop engaging lessons that meet the mathematical goal I have for my students.

What ever happened to the weekly slide summary? I liked that too.

11. #### dan

March 18, 2008 - 7:18 pm -

Shoot. Readers like y’all around here, who needs the rest of ’em? I’ll stick with it.

(Weekly slide summary just got to be a little cumbersome, though.)

12. #### Paul

March 18, 2008 - 8:43 pm -

Yipes, you’re tall.

Math exclusively in metric, I love it.

No, math teachers aren’t the only ones who read these posts.
This is a really good way of approaching teaching quite a lot to kids and the thoughts behind it are useful to anyone trying to actually teach something (instead of just telling students about it and hoping they soak it up). Particularly interesting to parents…..

Paul

13. #### Rich

March 19, 2008 - 3:19 am -

Speaking of metric — we’re working on the Imperial/customary/English system vs. metric this week in 6th grade. Does anyone know the OTHER two countries in the world that don’t use metric besides the U.S.? No fair peaking online….

14. #### Rich

March 19, 2008 - 3:20 am -

(oops, and I forgot “avoirdupois” in that previous post. I wish we could at least agree on a single name for our system!)

15. #### dan

March 19, 2008 - 5:06 am -

Shoot. Liberia’s one. I think that’s the hardest one to remember too.

[Looks it up.]

Okay, never woulda remembered the other one.

16. #### Kate

April 24, 2008 - 12:03 am -

Hi! I just wanted to say thanks for an awesome post. I’m currently in my second year of studying to be a maths/science teacher, and I’m always looking for new and interesting ideas for a class!

I love this, because it’s so hands-on and something I would never have thought of.

Yay!

=D