## I’m In Love With This Toilet Basically

New city. New home. New toilet. One of those pricey Toto jobs. Here’s the question:

You need a puzzle-piece button – one of those buttons that’s cut neatly into two buttons. One side flushes out more than the other. The best button design gets an extra-credit contract to manufacture one million of them.

I’d leave it to the student to ask how much each side flushes out: .9 and 1.6 gallons. (Be less helpful, etc.)

Here the question for you, the math-curious blog-reader: which of these buttons is the best design? Essential secondary question: in each case, what question do you ask students to nudge them away from an inferior design to something superior?

Here’s Toto’s answer, which prompts the question: did they get it right?

A: Sort of.

And yeah, this is what happens when I leave the classroom. I blog about bathroom fixtures.

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. #### paul thomas

October 13, 2010 - 7:24 am -

I am curious about the behavior of the buttons. Does each button behave independently, or does pushing the “big flush” button also make the “small flush” button go in? This changes my approach. That is, am I dealing with whole-to-whole or whole-to-part ratios?

2. #### Breedeen

October 13, 2010 - 7:39 am -

@ Paul Thomas: I have this toilet too. The buttons operate independently, so pushing down on the big one does not cause the button to go down on the small one. Though for all I know, it may cause the flushing mechanism to activate.

@ dan: I’m not sure about how the whole system works, which is one reason why I am scratching my head over this question. I don’t have a good intuition about why the design of the button would have an impact on the amount flushed. Is there a mathematical relationship between button design and water usage? Or is this more of an aesthetic question? And should we be concerned about how much time you spend pondering toilet design?

3. #### keninwa

October 13, 2010 - 7:51 am -

My initial thought was not to set the ratio of the areas equal to the ratio of .9 to 1.6, but to set it equal to the ratio of the number of times the average person goes #1 per day to the number of times the average person goes #2 per day. That way you are aiming for the big button more often than the small button (important for those with poor eyesight or nighttime bathroom trips). I don’t know what that ratio would be; perhaps it is unrealistic.

4. #### Dave

October 13, 2010 - 8:38 am -

After comments 1-2-3, all I can think about is the amount of time spent mopping if you push BOTH buttons.

5. #### John Pederson

October 13, 2010 - 9:10 am -

I immediately saw the design and though “that toilet looks easy to clean”.

6. #### Scott

October 13, 2010 - 12:14 pm -

In this case, I highly doubt that they ever intended to make the button sizes proportional to the amount of water flushed.

This is an industrial design/ergonomics problem, with a bit of user interface design tossed in. The 0.9L flush button can’t be too small or it’ll be too difficult to press. All they really needed to do is represent one button as larger than the other, which they did. The curve is a nice aesthetic touch, but also serves the purpose of making the small button wider while still maintaining a smaller area. A straight line divison (see F) is too narrow, making it hard for fat fingers to push it. Along with the label, I’d say they pretty much nailed it.

7. #### ExMathMajor

October 13, 2010 - 5:16 pm -

I love it!

That said, I want to agree with Scott. Maybe that makes me a bit of a killjoy. What’s the best design? The best design is one that works well for people: e.g., is easy for people to understand, is easy to use, is natural and intuitive. Thus I suspect the best design has one button noticeably larger than the other (so it’s easy to intuit that one causes a larger flush than the other, and easy to guess which one causes a larger flush), yet is chosen so neither button is too small (to make both buttons easy to press for those). I too think the toilet manufacturer nailed it.

Anyway, I’m classifying this primarily as a design and human factors challenge, rather than a math problem. (See “The Design of Everyday Things”, etc.) Still, it’s a fun one for discussion nonetheless! Thanks for sharing this!

8. #### Chris Sears

October 13, 2010 - 7:55 pm -

I saw a this toilet installed on “Ask This Old House.” Here is a link to the video on installing it.

I remember that they discussed how the toilet works after that segment. I’m pretty sure that there are two valves on the toilet to make it flush. The 0.9 gallon button opens one valve, and the 1.6 gallon button works both. I am going from memory, so don’t hold me to those details.

9. #### Chris Sears

October 13, 2010 - 7:57 pm -

I forgot to mention that with two small children, not needing to reach so far to clean under the bowl is a big selling point for me.

10. #### Aaron

October 14, 2010 - 7:04 am -

I agree with #4. Why make the button sizes relative to the size of the flush? Presumably we want to encourage people to use the small flush when possible… which (I think) means we should make the small flush button larger… now how much larger?

11. #### Michelle

October 14, 2010 - 1:49 pm -

I think the button size would be relative if, for instance, they wanted to push you to conserve water while still allowing that sometimes on those irritating low-flow toilets you have to flush 2 or 3 time.

In that case, the larger button would probably be pushed the most because it’s easier, and should be the lower-gallon flush, and the smaller button should be for the double flush.

12. #### Michelle

October 14, 2010 - 1:53 pm -

(Student teaching right now, so NOT doing the math for this. Y’all enjoy.)

On that note, anyone willing to help/discuss elementary ed? I’m teaching all subjects, and the math is interesting in the new books we have. One student was disturbed that she received a wrong mark for adding the equals sign and the answer when the problem asked for the numerical or algebraic expression (4th grade). We’re actually spending a lot of time and 2 or 3 chapters on these expressions which involve very easy operations. Does anyone have a comment on how/whether this helps with future word problem solving in Alg. 1 or 2?

13. #### Alex

October 15, 2010 - 9:20 pm -

Hi,

I’m a student. I don’t understand your calculations. How did you get the area, the radios and the angle?

Thanks!

14. #### Dan Meyer

October 16, 2010 - 12:49 pm -

Hi Alex, I imported the photo into a program called Geogebra (which is free) and Geogebra gave me the radii and the angles. The area calculations are a little trickier, but they’re on the paper I scanned.