Wednesday, January 28, 2015

1/27 KenKen Collaboration

As originally planned I wanted to work more on collaborating together today. So I had everyone give me their names on a slip of paper and I then  paired everyone off randomly. This was a big change from the normal state of affairs where everyone has a regular seat partner that they tend to work with and the boys and girls almost never work together. I thought we'd do another 8x8 kenken puzzle like the previous few weeks. In theory, everyone already had practiced them a bit and had started to figure out some strategies. I then had a set of problems from the five triangles site some of which I discussed previous and some sample Math Olympiad tests that I  was going to let everyone choose between.

It turns out that about twenty minutes into the process  I decided to let them finish the puzzle first before continuing since I've interrupted them every other time we've done this and I've worried that this has detracted from the experience. This turned into the entire session for the day. In my defense, I usually finish the puzzles in about 15 minutes and I thought that 2 minds would be better than one. However, I never had a chance to time my own son doing these at home and so I'm glad to see their relative difficulty more clearly.

8x8 Kenkens are too difficult for a warm up and general strategies need to be discussed more/brain stormed. I went around a lot giving hints on ideas for forward progress but I'm not sure how many breakthroughs were occurring. I'm going to move onto different warm up exercises for the rest of the year. If I bring these back next year they will be at 4x4 and 6x6 level. 

I did end up seeing several new and interesting partnerships during the hour. So shaking things up partly worked. 

About a 1/4 of the kids were not engaged and needed a lot of prodding to keep on target. Maybe next time I'll let them self-choose new partners so there's less tension if they get someone they don't want to work with.

Anyway, I have all the rest of the planning which I will bring out again next week. I'm also thinking I will put out a call for other parents to help out. More adults/kids looks helpful.

Saturday, January 24, 2015

Can we get from here to there?

I've been looking at the five triangles site recently: and I like alot of the problems there. I'm now considering whether the most recent one will work for the club.

Given a pair of positive integers a, b, the operation ab is defined as 23 × a + 31 × b.
For example, 20∗15 = 23 × 20 + 31 × 15 = 925.

Determine both pairs of positive integers x, y such that xy = 2015.

For me this problem is just a regular Diophantine equation. So I find the standard form (via Euclid's method or just checking some of the simple combinations)

3 * 31 - 4 * 23  = 1

From there you multiple everything by 2015:

6045 * 31 - 8060* 23 = 2015

which gets you a solution but not with 2 positive integers.

You can then adjust both terms by adding and subtracting 31 * 23 or 713 to find other solutions

So add  and subtract 261 * 23 * 31 ==  8091 * 23 ==  6003 * 31

and you get

42 * 31 + 31 * 23 = 2015 and you're found the first pair.
Scale down once again to

19 * 31 + 62 * 23 and you have the next pair

As you can see the next scaling will cause the first term to go negative so there really are only 2 pairs of positive tuples.

However none of the kids know number theory so the question is whether this is really solvable for them?

One thought is that you could notice that 2015 is divisible by 31

31 * 65 = 2015  and then perhaps stumble onto scaling to go down to the 2 solutions. From there if we explored scaling a bit with some easier equations first this might be more obvious. Alternatively,  I could just ask them to find the standard form via trial and error and go from there.

I'm, still brainstorming if I can think a sequence of problems to bridge this gap.

Wednesday, January 21, 2015

1/20 Math Olympiad #3

So this week was our third MOEMS Math Olympiad. I brought cupcakes again since they struck such a deep chord with the kids the first time. In the interest of empowering the kids,  I also let them vote on whether to eat them first or after the contest. (I leave it to the reader to guess what they chose) Before we started I had them warm up on another 8x8 kenken puzzle like the ones I had introduced the previous week. These seem to work pretty well although I have to stop them in the middle in order to fit everything in. My hope is that starting things and letting them bring it home is whetting the appetite and beneficial. I need to figure out how the kids are adapting to this approach.

As usual, a lot of the kids finished the Olympiad early. I haven't graded them yet so it remains to be seen if my pleas to double check your answers will have eliminated careless mistakes.  By the end, we had just enough time to rush through a group discussion of the answers.  In retrospect this was probably a mistake, there was too much time pressure to comfortably answer everything and it worked against the collaborative atmosphere I want to foster.

On that note I've been watching a talk by Richard Rusczyk this week:

My take home from this is I want to work more on strengthening the peer group especially when I'm having kids explain how they solve problems.  My first thought is a quick talk about respecting and listening to others when they're in the front. I occasionally have an issue with one kid saying something like "But you could do X which is much easier."  That interaction is part of what I want but at the same time I don't want the presenter to get shut down which is a tricky line to walk.

The other two things I've observed. The take home AMC8 questions so far have not been that exciting even with the multiple choice element removed. There just isn't much room for the kids to problem solve together in the first 10. I'll wait to see if we get more meat out of the latter ones.

Secondly, behavior this week was much better. Chaos can breed chaos and hopefully I have a handle on things. Secondly, I'm thinking the sessions run better with more work occurring as opposed to lectures. So I still want to continue working on the concept of square roots but I think I'm going to try out making my own worksheet with some exercises along the way and see if that works better at keep everyone engaged.

Friday, January 9, 2015

1/6 Winter Session Starts. Pythagorean Theorem.

Today was the first session coming back from break.  We first talked about our goals for the quarter:

1. Have Fun.
2. Try out new problems / learn some new concepts.
3. Emphasize proving why things work.
4. Go to more contests.

I then had the kids who went talk about how the KPMT competition went. That actually was fun and  I think I'll have the kids do this after every off site event.  Next we warmed up with some game of 24 cards.  Interestingly I asked how many kids had seen them before and only 1 or 2 of the group had.  I would have predicted beforehand that more of them would have used them for enrichment in regular class.  So I  spent more time talking about the rules and guiding kids to try the easy ones first to get the hang of how they work. I let this go on until everyone had solved at least one card or more and said they could take any of the sheets home that they wanted to think more about.  Overall, this was a success and I'm planning to try out kenken or sudoku sheets as a future warmup.

Next came my big experiment. Previously I had surveyed and found that most kids knew the Pythagorean Theorem but did not know why it worked. So I pre-printed a bunch of pictures of a right triangle surrounded by the 3 squares created by its edges and brought along some scissors. I then gave the kids 3 choices.

1. Try to find a free form way to cut the smaller 2 squares into shapes that would fit into the larger one.

2. Try to find two larger squares of the same size that you could make out of the cutout shapes. The hope here was to independently discover:

From there I planned to walk them through why these picture prove the theorem i.e. the area of the 2 little squares + 4 triangles == area of the big square + 4 triangles.

3. Try to figure out a proof based on the widow's chair diagram i.e. the classical Euclidean proof.

Overall I had several kids find free-form versions and one set find the 2 squares and make it to the realization why those proved the theorem. I also had one boy who from the start knew the algebraic proof using the 2nd square i.e. (a+b)^2 == c^2 + 2ab.  I liked the experimentation that went on while they played with the shapes but I had trouble bringing the whole room to a final aha moment. I think I'm going to review the findings at the beginning of next week in a lecture format. My goal is to spring board from here to some investigations about the square root of 2. 

Finally for homework I gave out a copy of the first 5 questions from the AMC 8 contest this year. Hopefully enough kids will try them that we can discuss in a group next week. 

Classroom Issues

I had one disruptive student today. Repeated commands to stop doing things did not calm him down and he tended to find new outlets while I was walking around the room with the other kids. I particularly didn't like that he started bothering one of the other girls. Besides talking with his parents, I'm going to try an assigned seat next week for him to see if that helps provide focus. While I  want to have everyone who wants to be here have a chance to do so that can't come at the cost of ruining the atmosphere for others.

Monday, January 5, 2015

Rescuing a poorly designed worksheet

So my son brought home a math worksheet/game to do over the winter break. It involved rolling four six-sided dice and adding them into 2 sums and then multiplying the two sums. The goal was to be the first person to get totals close to the first 12 perfect squares excluding 1. I took one look and thought sheesh this is going to take forever and has very little meat to it anyway. As expected all the totals clustered in the center.

So first thankfully we own a wide assortment of dice including the mighty twenty sided ones and small tetrahedrons that we added to the game to speed it up. I had to allow any operations to be used that you wanted in order to accelerate the issue. 

Then I had him think about the only roll that would allow him to  get to 144  (4 sixes). Even one five and you'd be closer to 11^2. That provided a quick transition to calculating the probability 1/1296. As I told him, I'm not playing this for that long.