3x + 2 = 29
6x - 3 = 15
x-squared - 5 = 20
Those kinds of equations.
She doesn't particularly like it that I make her do them out - i.e., for the first one, subtracting the 2 from both sides and then writing 3x=27, then x=9. She wants to do them in her head. David remembers having the same frustration all the way through school, but he agrees with my rationale to her: You need to know the method in order to be able to apply it to more complex problems. Also, I think it's a useful skill to be able to concretize what you're doing in your head so quickly.
And then, just over the weekend in casual conversation, we moved on to graphing X and Y and what a function is.
See, she'd made up her own function in her head, and was busy applying it all the time. So we decided to show her what it looks like in a graph. Just casually, over the lunch table. This is what happens in a mathematician's house.
- Madison's Function:
Take your favorite 2-digit number. To find your SECOND-favorite two digit number, here is the method: Take the two digits and add them together. Then subtract that result from your favorite number (unless your favorite is smaller, then subtract the favorite from that result), and add those together with the original number. That's your second favorite number.
David graphed it on Excel and it looks like this:
While he was working on the function, I gave her a basic introduction to graphing X and Y points, using the simple kinds of equations that she's been working on, but substituting Y for the answer:
3x + 2 = y
I first showed her what an X,Y coordinate was, then started at x=0 and started generating Y results. I told her in passing what a Y-intercept is. After four or so answers were graphed, she saw that it made a line, and was equally fascinated that you can simply extend the line with a ruler and you'll get the answers for all other values of X.
Then, she had enough background knowledge to be able to understand in general what the graph of her own function was all about.
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