Here is the next one:
I have to say that my students are rocking this! They really seem to understand how to relate the length and width of a rectangle back to the step number. We looked at different ways of seeing the shape grow for pattern #10:
And this is how we came up with the rule:
Then it was time for pattern #11:
I told my class that I had been unable to figure out a rule for this one algebraically so I had reached out to my math friends and Dave Lanovaz showed me a brilliant way of doing it. Here is the way I showed my class:
The original pattern was in blue. I repeated the pattern in yellow, then put them together thus creating rectangles! <insert angels singing> I asked if they could find the rule for this pattern and they all said yes. When I asked what they had to do to get the rule for the original pattern, they said we'd have to divide by 2.
Here is what we did:
With that, our visual patterns came to an end. At least for now.
I decided to spend the rest of the class multiplying binomials in a somewhat more formal way than what we had done last week. We started by multiplying numbers, beginning with 23 x 51. It was eye-opening to watch them attempt this without a calculator. So many place value issues and *magical* answers with no work and no ability to explain what had been done. Then I showed them how to break up the numbers like this:
One student actually said that this was like a miracle! She has been relying on a calculator for as long as she has been allowed to and I don't think she understood how to multiply properly until today. At the end of class she even said that this was fun. (O.M.G. day made!)
I did explain that the area models should be representative of the size of each number, like this (which I stole from Tina Dittrich):
We went on to repeat the process with variables in the mix:
Then we went over the distributive property again, which I also demonstrated with a trinomial multiplied by some crazy quartic function. The point being that it works regardless of how many terms are in each set of parentheses.
My students now all have a tool they can use to successfully multiply binomials. I then put up four more examples and wrote a student's name next to each. I explained to them that whenever they go to put work up on the board, they can choose to insert mistakes along the way. So we will never know if someone actually made a mistake or if they planted one for the class to find. My goal in doing this is to ensure that no one ever feels dumb for making a mistake in front of the class. They will (hopefully) come to appreciate all the mistakes and learn from them so they won't make them come test day.