My pre-calculus class with 75% seniors was better today. It was on the binomial theorem and Pascal's triangle. I started with having them expand (a+b)^n for n=0,1,2,3 by hand. We then isolated the coefficients, found a pattern and soon had Pascal's triangle. We then segued into naming terms on the triangle by using n and r to stand for the line and term on the line where we start counting from 0 (my explanation was that the first line came from expanding from n=0 so we start from zero). We then moved to the choice function. Give your calculator n and r and it spits out the term you want! Go back to the expansion talking about how each of the terms ascends or descends and how the powers always add up to n (the power to which you rose the binomial).
Now we put it together. Want to expand a binomial? Let the power be n, set up the ascending and descending terms, find the coefficients with the nCr and simplify. Go over how subtracting in the binomial changes it (alternating signs). Go over how to deal with coefficients in the binomial other than 1 (don't forget to raise the whole monomial to a power).
Geometry had a test. Lots of kids still don't understand the idea of an exact answer. I don't blame them but I am tired of hearing, "what is the real answer? I have tan36." I understand to them math is numbers but I have not figured out how to convey that those are numbers...
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