what is the “right” way?

Today a friend emailed this math problem to me.

What *is* the “right” way for math anyway? Is technology inherently less “right” than algebra? Solving it graphically by Desmos (or by TI-84) still solves the problem, but maybe that doesn’t feel as elegant or satisfying. I did want to share the joy of Desmos (since my non-math teacher peers aged mid- to late-thirties did not grow up with it), so I sent the following screenshot (before I eventually solved it algebraically).

The response:
“Ha ha, neat toy.

Since I forgot all the tricks to resolve mixes of square roots and variables, I looked at it as, it must be an integer since what awful brain teaser would have 4.87645372 as an answer, and it had to be a number with an integer square root, and that square root had to be less than half of 15, and the square of that number is 15 less than the square of (15 – that number). So I tried 7 first, 7 + 8 = 15, square of 7 is 49 which is 15 less than square of 8 (64) so that was it. 7 squared, 49. Had that not worked I would have tried 6, 5, 4 etc.

In the meantime, I’ve filled the backs of two envelopes with desperate attempts to solve it algebraically, going nowhere.”

Poor friend! Sent this to him:

I then sent it to my math colleagues (and my boss, who is a former math teacher).

Boss’ response, which I will ask to see in its original form, since email apparently translated it into gibberish.

Alternate Hint: Try putting the square roots on different sides of the equation and canceling 🙂
I took a pic of my work but don’t want to spoil it… email me if you want a look-y-loo. Thanks for the Thursday PM pick-me-up!

I emailed her to trade answers and she solved it this way. It made us both happy to solve it algebraically, but differently!

So there you have it:
1) logical way
2) graphical way
3) algebraic way #1
4) algebraic way #2

…how many more ways?

One Comment

  • Jon Dreyer

    I did it pretty much the same as “algebraic way #2”, since having a single square root on each side makes squaring easier. A few points: First, since we square both sides we have to make sure we don’t have an extraneous solution, since x^2=y^2 does not imply that x=y. But in this case we’re good. Second, I always discourage my students from “canceling” by crossing things out. In my experience, when they do that, they often don’t think about the math; they just cross things out that look the same. A favorite is “canceling” in fractions, so (x+2)/(x+3) becomes 2/3 after canceling!

    Regarding technology vs algebra, I sometimes wonder if computers had been invented first, whether people would ever have invented most of math, since in a sense it’s no longer “needed.” For example, who needs to integrate or differentiate symbolically if we can let machines do it numerically? On the other hand, I think we’d have a much shallower understanding of math. And in any case it’s purely hypothetical since computers could never have been developed without boatloads of math (and science)!

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