## Toilet Paper Math

by

Really, is there a more appropriate follow-up to yesterday’s featured theorem?

Last night young Quentin, age 4½, went to get some toilet paper to clean toothpaste out of the sink after brushing his teeth (because — get this — he likes to clean up after himself. I can hardly believe it.). As he pulled off a strip of TP, he suddenly held it against himself and got all excited: “This is as big as my belly!” I pointed out that his belly was three squares big, and asked how long his arm was. He measured, and exclaimed, “My arm is three squares long!” When he tried to measure his leg, it fell short so I suggested he might need one more square. He immediately went to the roll, counted off a strip four squares long, and held it against his leg. Yup, four squares worked.

The sink stayed dirty for a while after that while he went around measuring his hand (one square), our arms, etc. The nice thing about toilet paper is that he could take strips of various sizes and just pick the one that seemed best. His measurements weren’t exact (I’m not going to hire him to build a bookcase, for example) but he did seem to have the basic idea of measurement and that’s a topic that several K-6 teachers I’ve talked to say is the one that students need the most help with after number sense. (Speaking of which, Denise on Let’s Play Math had a great post Tuesday about helping kids learn number sense.) And I think non-standard measurement is one of the NYS math standards. [Quick check -- yup, it's 1.M.2, 1.M.11, 2.M.1, 2.M.10, and 3.M.10. I spent a while last year putting all the NYS math standards into Excel worksheets for easy searching and posted them here if anyone would find that useful.]

Thinking about blogging this, I googled “Toilet Paper Math” and found some other interesting ways to use toilet paper to do math. You can determine the least expensive choice of TP at the grocery store. You can fold it in half twelve times. You can find the thickness of a sheet of TP (although it seems like density — aka fluffiness — might make that inexact). You can calculate how much text you can print on a roll of toilet paper.

And finally, you can read about how Sir Roger Penrose sued the Kimberly Clark Corporation back in 1997 because one of the designs that was printed on Kleenex quilted toilet paper looked like Penrose tiles (see Wolfram’s Mathworld or this more detailed summary from Professor Richard H. Stern’s Computer Law Class at George Washington University.)

Update 5/10: I think these are the kinds of pictures that mbork was referring to below. In the first two examples the triangle has a right angle, but in the third the angle a bit larger.

### 4 Responses to “Toilet Paper Math”

1. mbork Says:

And a nice post again:)
BTW, this can lead to a non-trivial geometry problem, appropriate for olders kids: if you use 3 strips of TP, one 3, one 4 and one 5 squares long, to form a right angle, what is the maximum error, if you join any two consecutive strips at some _random_ point at their shorter sides (not necessarily vertices or midpoints)? (Hope I expressed myself clearly;).)

2. Ξ Says:

mbork, I’m unsure what is being measured for the error: are the sides being jointed at, say, 1″ in and you’re measuring how far from a right angle the angle is, or are you keeping the right angle and measuring how far off from 5 squares the hypotenuse would be? [I suppose either way would work, but I'm wondering which you're thinking of.]

Incidentally, your comment made me think of another idea for toilet-paper math: you could pick two lengths and talk about which lengths could be made from those two lengths (so strips of size 6 and 9 could make 6, 9, 12, 15, and every multiple of 3 from then on, while strips of size 3 and 7 could make 3, 6, 7, 9, 10, 12, 13, 14, 15, and every length from then on.)

3. mbork Says:

I knew it. My English is not good enough .
What I meant was something like this:
* you pick 3 stripes of lengths 3, 4 and 5
* you pick 6 points, two for each stripe – each of them somewhere on each shorter edge
* you form a triangle by joining the strips at the points selected
* you measure the angle which was supposed to be right
* the question is: what is the maximum error?
As for your idea: I quite like it, it is obviously connected with ideals in rings, but I guess it is very nice for younger kids to train their arithmetic .
And this makes me think about the following problem: given a mathematical object or theorem X, explain X in terms of TP . Most probably I will be teaching first year students “introduction to set theory”, I’ll have to think about it .

PS. Did I tell you I like your blog a lot? I’ll have to link it from my page. (Sorry, it’s mostly in Polish, but a small fraction is in English – I used to teach Math Studies SL in an IB school and I’ve put some English materials there. Feel free to use them if you find anything useful there .)

4. Ξ Says:

OK, I get it now! (When I get a chance, probably tomorrow morning, I’ll add a picture to the bottom of the post to demonstrate what I envision.) I’ve looked at your blog a few times and read what I could of it; I’ll take a look at the Math Studies stuff!