Today in class, we started with a mini-lesson on finding the powers of i, using the "multiplying by i = rotating by 90 degrees about the origin" metaphor. I misjudged - I made it a self-paced worksheet, but it was too complex (no pun intended) for them to handle on their own. When I went over it with them as a class, they started to understand it a lot better, but it ended up taking much more time than I had planned. The circle help students see why the pattern repeats in 4s. If you start at 1 on the real axis, and make a full circle, you will have multiplied by i 4 times. Therefore, any multiple of 4 in the exponent will cause an integral number of circles, landing you back at 1. Then, the remainder indicates how many more rotations you need and where to end up on the complex plane.

I'm at the point in the year where finals are looming and I feel pressed to rush through the curriculum to finish the unit. But what's the point, if they're not going to retain information as I charge ahead? I may need to scale back my learning objectives for this unit, or for the upcoming units.

After the powers of i, I wanted to do a quick review of factoring out the greatest common monomial factor from a polynomial, as well as factoring quadratic trinomials where a = 1. This is basic stuff that they did in algebra 1, but they had forgotten so much of it. By the end of class, most of them were starting to remember having done this before... so I go back to my earlier question about retention. I feel like this is a fundamental algebra 1 concept that should be retained from spring of this year (when they learned it in algebra 1) until now. Maybe going through it again now will help them get it into longer term memory... but how many times do they need to review the same topic until it sinks in?

Instead of giving a worksheet for practice, I wanted to show students how I create factorable trinomials, and for them to create their own and quiz each other. I think that this will be an effective way to help give them a deeper understanding. Unfortunately, we ran out of time because of the earlier problems. I am moving this into tomorrow's lesson, so hopefully it will work out.

A beautiful combinatorics argument

19 hours ago

## 3 comments:

I used "guided instruction" to teach the powers of i using that rotation analogy, and it worked out well.

I'm jumping in in the middle of your story. How do you link the rotation to multiplication by sqr(-1) ?

I'm guessing you set up real and imaginary axes, demo some simple plotting, and then plot a couple of powers until the pattern emerges? Is that close?

Jonathan,

I wrote about that in a previous post.

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