Saturday, March 21, 2015

Math is Cool 5th Grade Contest

This was my second contest experience. I brought 2 teams of four this time and because of a late withdrawal ended up drafting my son to fill the last spot. So this one was special for being our first experience doing this together. Overall this was another huge event. There were over 30 schools and 600 kids participating. The noise level in the auditorium at the end was deafening and the number of tests to grade was also a bit daunting. Also unlike last time all the coaches graded tests together during  most of the contest. I found this to be really interesting since it let me talk to other coaches at other schools and find out what they are doing.

The downside of this as far as the blog goes is that while I can recite the answers to the test by heart I didn't have a chance to really look at any of the questions. My general observation was that explicit practice would probably raise performance. Especially on the team relay and mental math sections. However, I don't think they mesh with my core values for the club that well. The test in general tend to emphasize speed and precision mostly centered around calculation. The best sections were the challenge problems at the end of the individual test but they gave the kids only 35 minutes to do 40 questions. So having prior knowledge helps a tremendous amount especially if you're lucky enough to have seem similar problem variants.

If I were designing one of these contest myself I'd like to see (much) longer formats with more challenging problems. Give the kids more than a minute per problem and see what they  can actually reason about. Maybe have divisions by geometry or number theory. Also I'd de-emphasize mental math. Perhaps have several divisions and have one for mental calculations that did more problems and more varied ones.  So let those who like doing these type of tasks still have a space but broaden the scope so more kinds of mathematics were present.

That all said, as was the case last time there was a lot of energy in the rooms and I think it was valuable for the kids to be in a space with several hundred other peers who were into math. I'm going to have everyone talk about it when we meet again to see how the kids felt about the experience as well.  And  I even recruited a to-be fourth grade girl for the next year who will be transferring to our school. At the end day we left with one kid winning 4th place overall for the division and the whole team at sixth place.

Note: next time bring earplugs.

Update: As Dan pointed out I made an incorrect assumption in my sequence generation. The better technique is to generate the 2 lowest integers find the third based on the second sum and the first integer but to then find the fourth integer based on the largest sum.

I was looking at the latest problem from the fivetriangles blog last night.

"There are four integers.  The sums of number pairs selected from those four integers are 48, 53, 57, 62, 66, and 71.  Determine the four integers."

Bizarrely enough I don't think there any integral solutions only rational ones.

i.e. (19.5, 28.5, 33.5, 37.5) works.

Here's my messy proof.

To start given six numbers there are  $${4 \choose 2} = 6$$ combinations.  So the given sums are all the possible ones. That also means that all the numbers are distinct since all the sums are also. Note if you add all the partial sums together given the combinatorics you get 3 * sum of all the numbers.  Which means the sum is 119 and that we either have 1 odd number of 3 odd numbers in the set. Given the lowest sum is the sum of the two smallest numbers that means we have 2 cases.

1. The 2 lowest numbers are odd. That means the next number must be even since 53 is odd and the fourth must be even since its the only one left.

That sequence $$O_1, O_2, E_3, 0_4$$ however will produce the following sums

Even, Odd, Even, Odd, Even, Odd

2. The 2 lowest numbers are even. Since the largest sum is odd and involves the 2 other numbers. They must be odd and even as well. There are two subcases here.

2a) The sequence $$E_1, E_2, E_3, 0_4$$ which produces the following sums

Even,  Even, Odd, Even, Odd, Odd

2b) The  sequence $$E_1, E_2, O_3, E_4$$ which produces the following sums

Even, Odd, Even, Odd, Even, Odd

None of these patterns match our target set which is Even, Odd, Odd, Even, Even, Odd

Tuesday, March 10, 2015

3/10 Pi Day (approximately)

Given when Math Club meets I could have done this lesson 4 days early or 3 days late. I chose the former because I've been super excited about attacking Pi head on and I didn't want the in class celebrations to steal any of my thunder.  I also tried a variety of strategies based on my previous sessions that I think worked out really well today.

So first of all, I had a parent volunteer helping out this time which is always super helpful especially when serving a messy treat and trying to do a relay activity that can potentially have multiple kids asking for the next piece simultaneously.

I started by serving pie, (never delay gratification - it just leads to begging). All the kids were surprisingly excited by super market apple pie and there was a lot more bang for the buck than any of my normal snacks. With my volunteer it was really fairly neat and all my worries about cleanup were fortunately not fulfilled.  Just for fun, I had everyone recite as many digits of pi as possible before being served in the line. Its fairly silly but this was also exciting for most of the kids.

We then took a small break while everyone was snacking to brainstorm about activities to run during math night at school which is coming up next month. It turns out pi muddles the brain and I mostly heard variations on "let's do a pie eating contest." So I'll give this a shot again next week and maybe exercise some fiat if I don't hear anything exciting. My main goal is to do something aimed at the 3rd and 4th grade with an eye on recruiting students for next year.

Next I had everyone tell me as many formulas using pi as they knew which produced all the usual suspects. After writing these down on the board I asked "So why is 2 * pi * r == circumference?" As expected there were not many ideas. The best idea was that as the diameter grew the circumference would obviously need to as well.  So I then talked a bit about approximating the circumference with regular polygons. See: http://itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.html

Drawing on the board, I had the kids works out the hexagon case where pi is approximately 3.

We then talked briefly about area as well which required a little of the Pythagorean theorem to find the height of the triangle. Again I asked for answers as I drew the figures on the board. Hopefully, this discussion and the followup on how to continually divide the triangles into finer approximations was accessible to everyone. This could definitely have been expanded into a more student led activity but I would have needed most of the class time to do so.

With the conceptual work done, I moved onto a warm-up packet practicing applying the basic area, circumference etc. formulas. These were not meant to be tricky or fancy, but just to brush up basic skills for a few minutes prior to starting our real task.  Spontaneously the kids ran to the bucket of calculators that I did not even know were present in the classroom. This seemed reasonable for the problem set so I let it be. Interestingly, one kid was still confused on moving from diameter to radius so this was useful in flushing out some basic principles. I walked around and could see that almost everyone was making progress.

Finally we reached my main activity a math relay I took from here:

This worked really well engaging the kids. (And was only manageable with my extra helper) I had  a really high level of completion on the various stages and there was a quiet humming in the room as the various teams worked their way through the problems. I will definitely repeat some more game-like activities like this in the future.