Wednesday, February 24, 2016

2/23 Prepping for Pi Day

Riff on the what I want the group to focus on

I was reading a post on the natural math site: http://naturalmath.com/math-circles-1001-leaders-course/  around designing a math circle. This has made me reconsider my informal focus for the group. Under the taxonomy used there, we definitely fall into the Mathematical Olympiad category. I like working on more complex problem sets (with multiple strategies) as well as throwing in a fair helping of puzzles and games. Recognizing this is my comfort zone and I'm unlikely to design free form activities involving dance for instance I do think I want to rotate in some occasional more open ended explorations.  I've been calling these notice and wonder exercises when I've tried them out. The natural math suggested tasks seem like a fertile area to look into this further. Likewise, I also have flirted with some more art related activities this year. For instance, I've used some pages from "This is not a Maths Book" several times. I think I also want to consciously balance these in the mix going forward as well. I suspect these would also work really well as warm ups.

Planning

The motivation for the activities for today was my informal map of what  I want to do in the next few weeks with the math club.   Among other things I plan to:
  • Celebrate Pi Day
  • Run another game day around checker stacks.
  • Fit in the last Olympiad
  • Prep better for Pi Day
The last item falls out what I learned last year: 2015 pi day. Most kids even in fifth grade while conversant with pi related formulas had very little conceptual knowledge about how they are derived. In a single day I was able to run a light-hearted relay using pi related problems but not cover all the background material as much as I wanted. This year I'm compensating by spending several days prior to Pi day working through some basic  models and more interesting problems around at least the concepts of pi and circumference and area. 

To get ready I threw together a progressive circumference related worksheet: Worksheet link. I'm planning to keep this in mind and add to it as I see other related problems. One issue I  immediately found is that its hard to keep the Pythagorean Theorem out of any such sets. I broke the 2 problems I found that need it into a challenge section at the back.  However, next year it would make sense to squeeze in a Pythagorean Theorem day as well. 


I already expected this topic would take the whole hour so after the kids finished  showing each other the problem of the week solutions I dived right in.  To start I had everyone brainstorm first for a few minutes what the meaning of pi is beyond its numeric value. There were some observations about it being irrational and how it was used in various formulas but no one was very crisp about it so I suggested "lets think of pi as the ratio between the circumference of a circle and its diameter or radius."   That lead naturally to a longer follow up brainstorming activity to think of reasons why this should even be true i.e. why is there a constant ratio?  After a few minutes of heated discussion I surveyed the small groups everyone had broken into. The best answers used some informal reasoning about how the radius needed to get bigger to reach the sides as the circumference. As I pointed then however its not immediately clear that the two quantities need to grow at the same rate. 

Moving forward, I gave a small demonstration of approximating the circumference using regular polygons on the whiteboard. After a small nod to the ancient Greeks I drew the regular hexagon which breaks into 6 equilateral triangles. Each triangle has a side length equal to the radius of the circle. Putting this all together this suggests that pi is approximately 3. I then glossed over how using the Pythagorean theorem we could keep subdividing our polygon and get closer and closer to the the true value. At this point one boy raised his hand to mention the Chinese had done this with a 200-gon. 

Note:  on reflection I had only about a third of the kids actively contributing during this portion. I want to focus on cold calling on different kids  more in these cases.

Hopefully, if nothing else sticks this portion and the basic idea of approximating does. From here I handed out the worksheet and the kids spent the rest of the hour working on problems. Focus was reasonable with some prodding here and there. Most kids were able to do 2-3 problems at the beginning which indicates to me that I want to add a few problems at about that difficulty level or a bit easier in the future.   I also ran into an issue where a few groups immediately went to the challenge problems at the back. I ended up asking them to try the beginning problems first when I thought they looked stuck. Perhaps on repeat, I should only give out the second sheet after the first one is handed in to enforce a rough ordering. 

Finally for the problem of the week I took a cool tessellation problem from a a UK contest :









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