“A string is wrapped around four nails arranged as the corners of a rectangle. The string’s length is a constant 80 centimeters.”
We pick up where we left off on Wednesday. I ask the students for their questions. They had some great ones, including many I hadn’t considered yet:
- What’s the greatest/least possible area of the rectangle?
- What’s the greatest/least possible perimeter?
- What’s the range of lengths possible for the sides of the rectangle?
- What’s the greatest/least possible diagonal of the rectangle? (This was one of the ones I hadn’t considered, and I loved it because my immediate reaction was that the square would be greatest, which of course turns out to be the opposite; the square actually has the least diagonal.)
I wanted to spend most of the time looking at the area question, as that was a good jumping off point for talking about the zeroes/intercepts of the function, how to use the calculator to find the maximum of a function, and then also segue into our lesson for the day: methods for multiplying polynomials.
Today we returned to the textbook to work through several examples multiplying polynomials. My two goals were to continue demonstrating different methods for multiplying the polynomials, and to continue hinting at the patterns students can start to use to both multiply and factor polynomials. The patterns goal will also feed into tomorrow’s lesson on special products.
The three methods we looked at were box method (the abstraction of the area model), FOIL (abstraction of distribution), and stacking (abstraction of stack multiplication). I had students go to the board for several examples, demonstrating the method of their choice. I made sure we had several demonstrations of each method. I emphasized box method, because I think that’s the best one for factoring polynomials, but I told the students the choice was really up to them (in the long run, anyway; in the short run, they had to use each method at least twice on the homework).
To further make my case for box method, we closed with binomial-by-trinomial multiplication and trinomial-by-trinomial multiplication. In both examples, I demonstrated why “FOIL” (especially if you’re married to what the letters stand for) becomes quite unwieldy, while box method scales nicely. I also continued to point out how the degree of the original polynomials affected the degree of the product, and how like terms tend to end up on diagonals, provided the initial polynomials are arranged by order of monomial degree.
Selected problems from the textbook.
Use the patterns we hinted at today as a jumping off point for the difference of squares pattern and perfect square trinomials.