Wednesday, February 25, 2015

2/24 Combinatorics

This session I decided to try out some more of the fivetriangle problems. I gave out a double sided sheet with 2 problems on it and let the kids decided whether to start with the easier one or go straight to the more complex problem.

1. Magic Square (easier)
http://fivetriangles.blogspot.com/2015/02/215-integer-table.htm
This one was a general success. Almost everyone finished on their own. Although this is fairly amenable to a guess and check attack so its not surprising to see this happen. I ran a small experiment with a bit of competition that was interesting here.  Five minutes in or so, I announced to the group that 2 of the girls had finished this problem.  It immediately engaged the boys to know the girls were going faster but led to more final answer sharing rather than method sharing. So I need a middle path - engagement without corner cutting. 

2. Geometry problem.
http://fivetriangles.blogspot.com/2015/02/216-three-kissing-circles.html
This problem's tougher and took more time and individual prompting.  You essentially have to figure out the correct order of steps and then:

  • Recognize an isosceles triangle based on it having two sides that are the radii of the same circle.
  • Determine the rest of the angles in this triangle given one and the fact its isosceles.
  • Find the third side of the larger right triangle that was found in the previous step using the Pythagorean theorem.
  • Find a linear system to solve. The first part based on the radii of the largest circle. The second part based on the unknown radii also adding up to the third side of the triangle found above.

On the bright side I had a few aha moments with several of the kids as they worked their way through it. It also was decent practice for some mechanical skills like using the Pythagorean theorem. On the other hand, I exposed some general weaknesses.


1. Several kids did not know how to find the area of circle. This should have been covered given their curriculum especially for the fifth graders. However, I didn't pre-query if they knew it which is my primary take away.  Since I'm brainstorming activities for Pi day, I'm going to bear this in mind. Maybe we should warm up practicing some basic skills like area and circumference calculations and go forward from there.  
2. Isosceles triangles. Also shaky and theoretically covered. This falls under the same thought about basic skill warmups.
3. Applying the Pythagorean theorem. For example: one student didn't fully realize that c^2 had to be the hypotenuse of the triangle. Lesson: when we worked on proving it, a good warm up would have been some sample applications.  I assumed since everyone said they had seen it but just didn't know how it works that this was already true.

After finishing up the above activities I moved kids individually along to a packet I cribbed from AoPS on beginning combinatorics. I've not really touched this topic this year and its been on my pending list. Structurally this meant there was no great place for a central lecture. It didn't appear necessary here and for the most part the kids had few questions about this material. I'm going to return back to it in more depth and I'll discover if this is really the case. On the bright side, I had a few kids actually lingering past pickup time to work out the last few problems and I also had a few asking for my third backup package of AMC8 problems. Score one for engagement.

For the future I'm going to think a lot about whether the followup activity needs to be mostly self driven or require small assists versus the complexity of the warmup. If I expect to do some group work together then the warmup needs to be more uniformly finishable in 10-15 minutes so I can refocus the kids.  One possibility is to move the challenging problems towards the end. I'm a bit reluctant to do so though because I feel like I get the best efforts at the start of the session.


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