For better or for worse, here’s what I tried:
Some ruminating and head scratching later, I came up with this:
It’s not too horrible on its own – my big mistake was dumping a discovery project on them with very little setup. These kids are not problem solvers, at least not problems that they’ve never seen before. They love problems that they have a road map for solving, but faced with something novel they go into all sorts of avoidance behavior. I was hoping they’d be able to figure out the rules on their own, but all I got was rebellion.
Once I caved in and fed them the howto, I still got a couple of Aha moments out of the right side. Still, there was a lot of grumpiness leftover.
The fix in the sheet would be to actually demonstrate how the first two answers are arrived at, rather than asking them to try to figure it out. Also, the three columns in the second section would be replaced by two: One to recap how the answers were found, and the other to combine the second two columns in asking why two different diagrams could give you the same answer.
The third section is an application of the 3rd rule of math: Anything you can do one way, you should be able to do backwards. If they’d figured out how why you could get the same answer from different problems, they should be able to figure out how to make different problems with the same answer.
The big mistake in this section was that they presumed that there were 16 white squares, some of which could be converted to black, rather than 16 blank square templates which could be filled with either or neither. Next time, include an example. Once they clued into that, though, most kids managed to figure out how to complete the sheet.
So, why the convoluted intro?
Because there’s always a hang up when you get to teaching how to subtract negative numbers. Being able to create a representation of a number with an arbitrary number of 0 sum pairs hanging off of it is a necessary skill. This hopefully set them up for that.
The idea is to now use the small squares from algebra tiles to represent the diagrams from the first sheet. There will probably be a 10 minute review of the previous material, using the tiles to create the diagrams rather than drawing it on paper.
Then, on to the worksheet: Adding is simple: you create a representation of each number in the two halves of the circle, and the number you get in the whole circle is your answer.
Subtracting will be a bit more complex, but not much more than the last exercise from yesterday: you construct the subtrahend first (preferably without the ELL boggling terminology) in the right half, and then the minuend in the whole circle, which will include the right half. The difference will then be in the left half of the circle.
Hopefully the problems are constructed so as to reveal the general rules that are used to perform addition & subtraction of negative integers. I’ll find out tomorrow.