So in two classes (algebra II, if that matters, mostly juniors) I did some problem solving this week instead of “course material” because I like problem solving, but also because we were down 20% due to the AP Spanish exam, and their attention spans were down about 60%, and their nerves were frayed to a similar extent, by the upcoming (today! hooray!) US History AP.

They attacked

Find x:  $x = \sqrt{2 + \sqrt{2 +\sqrt{2 +\sqrt{2 +\sqrt{2 + ...}}}}}$

and in each class some of the kids evaluated some nested root 2s (easy with the calculator, once you think of it) and could tell that the sum was either 2, or close to it. And I teased each class by saying that that was good enough, we would do something else, and each class objected to moving on without knowing. This much I told in the previous post.

So what else did they do with this thing?

1. I refreshed Polya’s problem solving model for them. I was pleased that many recalled it on cue, even though I’ve been racing through “material” in this damned course and haven’t been taking big chunks of time to solve problems. The little challenge problems are nice for playing math, but we haven’t been trotting out Polya’s model.

(Note: this problem was first shared with me with a purpose:  to be confronted with an answer that was so lovely that the student (me) would be encouraged to “look back” to investigate why the answer was so nice.)

2. I introduced limits. I didn’t formalize them. I talked about getting as close as we like. It was a nice chat. It will come back to them next year in precalc. I believe in foreshadowing. I do.

$\sqrt{2 + \sqrt{2 + \sqrt{2 +\sqrt{2 +\sqrt{2 + ...}}}}} = 2$
did that mean we could generalize. Each class tested the idea and found
$\sqrt{3 + \sqrt{3 + \sqrt{3 + \sqrt{3 + \sqrt{3 + ... }}}}} \neq 3$
The teacher rather than confirm their suspicions encouraged them to keep extending the expression until they got close to 3.

4. I asked if they’d ever solved anything similar. As I was intentionally vague, first responses were about sums of geometric sequences. Good, but not what I was going for. Had they solved an equation that looked anything like this?  (recall, I started with x =). And yeah, they mentioned radical equations. I asked them to suggest some, and we solved a couple of quickies, until a kid interrupted to suggest that we go back to our problem and square both sides.

5. So the next line down goes:  $x^2 = 2 + \sqrt{2 + \sqrt{2 +\sqrt{2 +\sqrt{2 +\sqrt{2 + ...}}}}}$. And I congratulate whomever on the clever step, and it could have helped, but it seems to have left us with something even more complicated. (forgive the misdirection. I feel guilty for pointing them too much towards the solution I want). So I stood next to the thing, and some kids asked for explanations of what happened to the outer square root, and others suggested squaring again, or raising to a higher power, and I happily blathered on in each class, until…

6. In each class a kid noticed that we had $x^2 = 2 + x$, or better in the second class, $x^2 - 2 = x$. Why better?  Because I know they transposed the 2 so that they could square again. They made the discovery not just by staring, but by trying to perform difficult algebra. Effort rewarded. I like that.  They looked annoyed when I asked them about the extraneous root.

7. Going back to $\sqrt{3 + \sqrt{3 + \sqrt{3 + \sqrt{3 + \sqrt{3 + ... }}}}} \neq 3$, I had each class search for other integers that would give us integer values. Solved fairly rapidly.

8. Pushing further, I had them calculate $\sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + ... }}}}}$, I named the number $\phi$, and I had them solve “Find a number that is one more than its reciprocal” and “find a number that is one less than its square” and I had them square $\phi$ and divide 1 by $\phi$ and everyone was duly impressed by those long strings of matching decimals.

Another note:  One kid, really growing on me, wanted to know why the TI doesn’t have an “elipsis button.” Good question, eh?

In both classes I did a little golden ratio riff at the end. In the first I pointed out the ratio of successive Fibonacci numbers…  But that was fluff. The good stuff was earlier.

1. May 7, 2010 am31 9:08 am 9:08 am

This is awesome. I love the extension “for what integers under the radicals does the whole thing come out to an integer?” This is the mathematician’s natural next question. (How did they approach this btw?)

May 7, 2010 pm31 12:38 pm 12:38 pm

There’s a nice path to the nested-roots-involving-twos that you started with, gotten to by iterating the half-angle formula for cosine….

3. May 7, 2010 pm31 3:51 pm 3:51 pm

That’s a good one – and it’s really really nice to see you talking about teaching math again :). You do it well and I’ve missed it.

4. May 7, 2010 pm31 10:07 pm 10:07 pm

Thanks Doc. And 2π, I was sitting in my office, door open, when I read your comment, and played with it for a few seconds. “Oh ####” I said – apparently loud enough for people outside my office to check what was wrong.

Ben, by the time they were looking at $x = \sqrt{n + \sqrt{n +\sqrt{n +\sqrt{n +\sqrt{n + ...}}}}}$ we had already solved $x^2 = 2 + x$. Some watched their neighbors, but active students trial and errored (one kid started with 4), and a good number of those, after one trial, examined $x^2 - x - n = 0$ and declared that n needed to be the product of consecutive integers. This took maybe a minute, maybe 2 in each class. It was fast.

5. May 8, 2010 pm31 1:02 pm 1:02 pm

Naturally, having forgotten this stuff, I went and plugged it into a spreadsheet, which was surprisingly quick and easy once I remembered the row() function.

But, of course, cranking out all the “n” that converge to whole numbers in a matter of seconds didn’t prevent me from staring at the numbers looking for the pattern.

sigh. I wish I could do stuff like this in my classroom. I have a few kids who might enjoy it.

6. May 9, 2010 pm31 11:06 pm 11:06 pm

You probably went way past all this, but I once wrote some notes about the iterated roots and posted them here http://pballew.net/iteroots.doc maybe some of your kids could use them…

nice blog..glad to see you back in the math thing…

pat

• May 9, 2010 pm31 11:18 pm 11:18 pm

No, we didn’t get nearly that far. I want to share this with the kiddies; all of them can read the first two pages, though with effort. We don’t usually ask them to read real math.

I have a few who will struggle further, because it is different, and interesting.

But none will finish it. That’s ok. And thanks.