https://www.mathcounts.org/resources/problem-of-the-week/fence-me

This sheet revolves around a graph paper and dice based game where you roll the dice and fill in the boxes until someone can't go using either a rectangle with the same perimeter or area as the product of the dice roll. I suspected the kids would really like playing the game before we started talking about the questions. So I brought in my graph paper and dice and we spent the first half trying it out.

Sadly I didn't take any pictures of the finished games. But the kids really enjoyed the experience and several asked if we could keep going. This definitely helped with our white board discussion afterwards. Interestingly for the final question, the best the kids could think of was a 7 square solution that would cover the whole board. I know of at least a 6 square one with an upper bound of 5 so I may return to this problem. For the end of this portion I asked everyone to think about if they could find a better solution.

We then transitioned to a talk about divisibility rules. I made several improvements over last time I tried this. First we spent some time group brainstorming about what happens when you add a multiple of n to another multiple of n, a multiple of n to a non-multiple of n and two non-multiples of n together.

After several minutes reflecting I had the kids report their ideas and why they thought that they worked. Interestingly, I had to introduce the notion of the distributive law here. I skipped focusing on it this year since we did so last year but perhaps I should circle back.

**For next time: On reflection I think a really good followup question would be how does this relate to the rules for adding odd and even numbers. The hope would be to build the connection that this is the same as multiples of 2 and that its a special case where 2 non-multiples add to a multiple.**

With that foundation we talked about the simpler rules for multiples of 2 and 5 and why they must work. This flowed fairly well from the previous ideas. I had kids volunteer solid reasoning for both rules. I ran out of time as we just started to work through the more complicated nines case.

My idea for followup is to walk through nines again and then let groups work on 11's with the starter idea that 10 = 11 - 1, 100 = 11* 9 + 1, 1000 = 11*91 - 1 etc and see if they can follow the logic.

Two years ago I showed the rule for 7's just for fun. I may pull that out again although it serves no practical use because of its inherent interest.

I also had a sheet of problems from AoPS that I planned to use that we didn't get to that I might bring back. Buy coincidence yesterday I saw the following problem:

This would dovetail really well with the divisibility work. I'm thinking about having groups brainstorm about the left hand side to see if they can see if must be a multiple of 6^2 by examining concrete values.

P.O.T.W

https://www.theguardian.com/science/2016/oct/10/can-you-solve-it-the-ping-pong-puzzle?CMP=share_btn_tw

Todo: About 7-8 kids were volunteering answers out of the 12 who were here today. I need to draw out the quieter ones in future weeks,.

For Don't Fence Me In, did you play as a competitive game or collaborative puzzle? Recently, as a family, we've been playing Qwirkle as a collaborative puzzle (no scoring) and it has been very successful. I've been meaning to write up notes on our blog.

ReplyDeleteBTW, for games like this, we have players use two different colored pencils to shade their rectangles. I'm not sure there is a mathematical justification, but it makes the ending pictures visually attractive.