# Discovery

I came to school this morning to find this in my mailbox.  What a lovely way to start a Thursday!

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One of my calculus students came up to me today and said, “Remember last year when we studied fractals and you had us make up our own?  Well, I was experimenting this summer with nesting functions and started doing some research.  I realized that if you compose f(x)=sin(x) with itself over and over, you start to get a square-like function, instead of a smooth function.”

I think I stared at him blankly for a few moments.  So many good words you just used, kiddo!

Then, obviously, I had to take a look at what he had been doing, so I pulled up Desmos and quickly composed sin(x) with itself a few times, and sure enough, this is what I got:

Maybe it’s not interesting to you, but I find it fascinating both in a learning sense and a mathematical sense.  Learning:  This kid started playing with three concepts that tend to scare most of my students:  compositions, trig functions, and infinity.  Mathematical:  How cool that a smooth function appears to get “worse” as you compose it with itself (appears that eventually it will not be differentiable, although it must always be differentiable).

It’s the best when kids take something they learned in class and run with it and find ways to make it interesting for them.  It humbles me greatly.

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In her intro letter on the first day of school, one of my kids told me she had a two-month old daughter. Today I asked how baby was doing and what her name was. Wondering if I had overstepped bounds, I decided to leave it at that for today.

A few minutes later, the student came up to me, phone in hand, and the proudest smile you could imagine on her face. She showed me a picture of her beautiful baby girl. I was so touched. Touched that she would share this with me and touched at how deeply her love runs.

Love knows no age, friends. It knows no age.