Students considered situations where the net gain was zero. A lot of students suggested a variable like x to represent sand taken out of the whole, and the sand x, being added back into the hole, -x, bringing it back to zero. Then they had to make zero 2 ways. Some students just moved a zero pair from one region to the other after asking them how they could make a zero pair with those same tiles. I made mental notes on two different people with the two different ways.
We came together as a class to discuss this. I used an idea from a former colleague, Ms. Gruner to stress the two ways. One is 2 DIFFERENT tiles, or 2 different shadings (1 positive 1 negative) in SAME regions. The other way I ask, are these the same shaded tiles? No. So, DIFFERENT shading. Are they the same or different regions? Different. So, I think choral response helps here. Also, I reinforce how the students must shade them in their sketches. Is negative shaded or unshaded? Etc... This helps and students love to shout out.
Then students build an expression out of tiles and are instructed to write the algebraic expression that is represented by the model. I stressed to do 1 simplification, then write the new expression. Then simplify one more time, and write your final simplified expression. The book mentions "various amounts of tiles" and I showed that I wanted 3 equivalent expressions. Some skipped, and I challenged them to show me the intermediate step.
Then we only had time to represent -(-2y+1) using tiles. This was similar to one of Friday's expressions.
In accelerated math, we did the Big Race Finals. I videotaped my students when they were working on this. We did this lesson, 2.2.3, before 2.2.2 so tomorrow they will do 2.2.2, Rate of Change. I think this lesson will be a chance to reinforce what they learned today.
The really cool aspect of this lesson is that each group is given 6 cards. 2 cards everyone can look at, but each team member gets an individual card that they must share with their group about.
One common mistake I saw was when students were given 2 coordinates of a racer, they connected those points, and then connected the point on the left to the origin. They reasoned that everyone has to start at the starting line. I then pointed them out the rule of the race is that everyone races at a constant speed. Can it be a constant speed if it starts steep and then flattens out? This was a great learning point.
Students end up with a graph that looks like: