Algebra explored what the side lengths are for a rectangle with area x² – y² today. The paper folding required to get there was a lot of work and I am not sure if many of them totally understood what was going on.
Either way it is good for them to see a visual representation of where the heck these things come from.
Geometry practiced Law of Sines and Cosines, which we spent yesterday deriving. I talked to them quite a bit about how parallel these are to calculus in that they are using geometry to set up the problem then algebra to solve for either the side or angle. To me, that seems a lot like how calculus is; use an integral or something to set up them problem but after that it is all algebra to reach a solution.
Algebra explored the expansion of (a + b)² today. Across 100 students, none of them were able to correctly expand (x + 3)² coming into the day. We plugged x = 2 into different expansions. Most students had 13, some had 25. Everyone agreed that 25 was spot on, but couldn’t come up with an explanation as to what (x + 3)² expands to and gives 25 out.
We first talked about the area of a square with side length z and how the are is z². I asked ’em what area meant, they told me length times width. I asked ’em what area meant without using an equation… it took bit but we got here.
We folded the paper and labeled like so
We talked about how the new side length is a + b and to find the area of this square take (a + b)²
They found the length of each piece which makes up the area…
This is where they struggled:
The area of the square calculated from the formula is (a + b)²
But the area is also defied by these squares and rectangles added together.
(a + b)² = a² + 2ab + b²
We used this to expand (x + 3)² to x² + 6x + 9, which gave 25 out for x = 2!!
After a couple more examples students had a good hang of things, I saw a good mix of multiplying out and visual representations.
Tomorrow they are exploring (a + b)(a – b)… it should be fun!