Beautiful Math: Algebra One More on Geometric Sequences

Sunday, September 6, 2015

Algebra One More on Geometric Sequences

I posted earlier this year with my ISN materials on geometric sequences in Algebra One. This post is to flesh things out a little bit. This all takes place over three blocks.

We finished up arithmetic sequences and I introduced geometric sequences by posting two sequences on the board (one with r = 2 and the other with r = 1/2) and asked how these were different from the arithmetic ones we just did. Discussion ensued with question about what the graphs might look like? So we graphed them and noticed they didn't look "linear". (are these supposed to be straight?). Why aren't they "straight"? - because of how quickly they grow. Anyone know a word that describes growth that might start slow but then speeds up very quickly? (exponential). We explored a few different sequences and looked at some "application type" problems (bubbles, bouncing ball).

It was easy enough to create recursive rules for these. It was the explicit rule that was a little harder. So we broke down what actually is happening term to term. Here's what that looks like:

How does the number of factors relate to the term number? (one less, thus (n-1)). So from there we developed the explicit formula.

Then we had a summary with our ISN foldable.

There was some confusion - some scenarios described "starting" with something and then data for after the first, second, third (etc) minute (or some other unit). So the start is really the "0 term" when creating formulas. We explored how the explicit formula with the zero term means the same thing as the formula with the first term. This takes a few examples to recognize. Have students explore this and see how they are the same.

This distinction between the two is pretty important and helps to later relate these easily to exponential functions.

Now we go into class block #2 on this topic. I started with the "birthday problem" as a warm up (see documents below). They compare two scenarios to decide which is better. Discussion takes place when they recognize they are examples of arithmetic and geometric sequences and that geometric is a far better option.

Next we looked at scenarios that involved percentages. Here are two examples we discussed:

o Ex 1 – special hair potion, hair will grow 5% a week. Start with 8 inches.

o Ex 2 – your jeans will shrink 5% in length very time they go through the dryer. Starting length = 32 inches.

They had to learn how to translate percent growth & decay into sequences, finding an "r" value. So this took a little time and exploration. They can do percentage change but do it as a two-step process. The trick was to get them to recognize that "growth" results in something greater than 100% thus in example 1 r = 1.05 (105%) while decay results in something less than 100% thus example 2 r = 0.95 (95%). I don't like to just tell them how this work, I like them to play around with the numbers to try to figure it out first and discuss amongst themselves. I usually end up doing a little of both because of time constraints.

Then students had a bunch of practice problems to do. A good idea here is to have students make up a growth and a decay scenario and create a sequence to go with it. Students get super creative with this.

Then we went into class block #3 on this. Warm up was some practice problems to go in the ISN opposite the foldable pages.

Finally we examined geometric sequences in the context of fractals. Most of my students never heard of a fractal. So we first did a little intro worksheet together of drawing a simple fractal design. From there we went to Sierpinski's triangle. There is always some moaning & groaning here, do we have to draw stage 4? I leave that up to them, if they just want to recognize the pattern and state how many USRs there are, that's fine. (but there are always a good handful who do want to draw it!).