Today in Precalculus I went on a bit of a 7 minute digression, talking about continued fractions. You see, a recursive problem showed up (we’re doing sequences): Write out the first five terms of the following sequence:
So obviously they go like: ,, , , and
So great. Awesome. NOT. Booooring. So I showed them the decimal expansions:
WHOA! This is getting closer and closer to 2… Weiiiird…
And then I say I can show them this will continue, and we can find a way to show that [where the pattern continues forever] will practically become 2.
DIGRESSION WHICH IS ACTUALLY WHY I WANTED TO BLOG ABOUT THIS
To do this, I start with something else. I don’t know why, but I really wanted to show them a continued fraction first, to get the point across easier than with the square root. This was the continued fraction.
I went through a frenetic mini-lecture, and I think I had about 40% of the kids along with me for the whole ride. I’m not sure… maybe? But later a kid came by my office, and I thought of a better way to show it. Hence, this blogpost, to show you. (I have seen teachers use this method when teaching substitution when solving systems of equations… but I have never used it myself. I’m dumb! This is awesome!) This is what I did when showing the kid how to think about this in my office.
First I took a small piece of paper and I wrote the infinite fraction on it.
Then I flipped it over and on the back wrote what it equaled… Our unknown that we were trying to solve for.
I emphasized that that card itself represented the value of that fraction. The front and back are both different ways to express the same (unknown) quantity we were looking for.
Then I took a big sheet of paper and wrote where I left the denominator blank. And then I put the small card (fraction side up) in the denominator of the fraction…
And I said… what does this whole thing equal?
And without too much thinking, the student gave me the answer…
Yup. We’ve seen that infinite fraction before. That is !
THAT FLIP IS THE COOLEST THING EVER FOR A MATH TEACHER. That flip was the single thing that made me want to blog about this.
Now you have an equation that you can solve for … and is what you’re trying to find the value of. This equation can easily be turned into a quadratic, and when you solve it you get (yes, the Golden Ratio). And it turns out that is close to what we might have predicted…
Because in class, we (by hand) calculated the first few terms of where … and we saw:
And when I drew a numberline on the board, plotted 1, then 2, then 1.5, then 1.66666666, then 1.6, then 1.625, we saw that the numbers bounced back and forth… and they seemed to be getting closer and closer to a single number… And yes, that single number is about 1.618.
BACK TO OUR REGULARLY SCHEDULED PROGRAM
So after I showed them how to calculate the crazy infinite fraction, I went back to the problem at hand… What is ?
Then we can say
And even simply by inspection, we can see that is a solution to this!
 What’s neat is that yesterday I introduced the notion of a recursive sequence that relies on the previous two terms. So soon I can show them the Fibonacci sequence (1,1,2,3,5,8,13,…). What does that have to do with any of this? Well let’s look at the exact values of where .
Lovely. It’s all coming together!!!