6/10 Penultimate Session

The year is winding down and the weather is ramping up. We’re having a mini heat wave in Seattle and the room today was stuffy and warm today. My plan for next week is to meet outside on the lawn if the heat keeps up.

This was our last group Purple Comet Review problem. Again like the last few times we had a really good flow hypothesizing and group solving the problem.  I went around the room having  the kids notice and wonder pieces of the problem and they were really great about breaking things apart to find the structure and eventually the solution. Three times can’t be a coincidence, I am definitely going to explore some of the protocols I used here again next year.

Multiplication Rules I decided then to  stick in something neat I had seen online from James Tanton. I’ve always justified the rules for multiplying a negative by negative as follows via the commutative/distributive and inverse rules:

$ -a \cdot -b = (-1 \cdot a) \cdot (-1 \cdot b) =  (-1 \cdot -1) \cdot (a \cdot b)  $

So we just need to understand why $ -1 \cdot -1 = 1  $   and that we do as follows:

$ 0 = -1 \cdot 0 =   -1 \cdot (1 + -1) = -1 \cdot 1 +  (-1 \cdot -1)  = -1 + ( -1 \cdot -1) $

In other words  -1 is the inverse of $ -1 \cdot -1 $ and we know that value must be 1.  But I really like James Tantons idea to throw the area/box model in as an organizing principle and I think it does improve the clarity:

What’s $(1 + -1) \cdot (1+ -1)$?  We know that’s the same as $ 0 \cdot 0 = 0 $  Applying the area model we get 1 + -1 + -1 + ?   That missing term must be 1 to balance everything out and end up with zero!

And you can substitute in other numbers to drive home the logic if one example is not enough.  This went over really well. The most interesting observation I had was that most of the room justified the rules via an analogy to double negatives in grammar with a few talking about reversing directions on number lines.  I’m hoping this version will stick with everyone.

More Interesting Stuff

For the main activity I decided to stay with some of the resources I’ve been exploring on Tanton’s site: http://www.jamestanton.com/?p=1072.  I picked the most recent exploration: Double Fractal Sequences  for the group to work on.  As usual I had everyone work on the basic concept as group introducing the sequences and the rules they followed and then split up into table to dive deeper. Overall, I think I should have stayed as a group a bit longer and done some of the table work more communally. I ended up mini-coaching most of the tables through several portions and in retrospect this would have flowed better done as a room. I think the end of the year exhaustion and general heat also started to set in by the end of the day.  For the last week I also do a light game day and bring a cake to celebrate. So I think that will be a good fit.  I’ve also found a cool blacktop game: Flagways from http://www.typp.org/ that I may center the day around.