Friday, June 30, 2017

First Impressions: A Decade of the Berkeley Math Circle

After seeing a recommendation online, this book arrived at the house in the mail yesterday. I started reading it after dinner and was immediately inspired.  Enough so, that I'm going to write up some first impressions even though I'm only into the first chapter.

The preface described the Berkeley Math Circle's history and a little bit of how it runs. So often discussions of math run between either impassioned defenses of direct instruction vs. constructivism/progressivism.  There's no sense of problem solving in the first camp (students don't have enough knowledge, their working memories get overrun and they only engage in  means-ends analysis that doesn't stick.:David Didau's argument along these lines.  On the other hand, the other side is passionately concerned with equity and breaking down narratives of who is capable of doing mathematics. In the process, the curriculum mostly takes second place to the pedagogy around it.

The paragraph above speaks to the third way I'm always looking for: focused on problem solving yet rigorous and above all fascinating.  The kids in the circle were engaging in really interesting problems and getting exposure to a much richer view of the subject.  Computation was not important but proofs took central place.

At the same time, this model raises a lot of questions for me.  There's a certain elitism present from the start. The Berkeley Math Circle is described as a "top tier" math circle. This is not idle boasting, it took a prestigious university and multiple mathematicians to create this group. Each session drew on a small slice of the Bay Area (20-30 students) able to discover it through the web site or word of mouth and lasted an intense 2 hours.  Here in Seattle which is another tech node we have a similar  Math Circle run out of UW aimed at the Middle School level.   At the same time,  I wonder how can we replicate this model on a larger scale? There are challenges both in finding kids and instructors and covering the full range of  MS through HS.

I think part of the solution probably lies in virtual math circles and the online world since even if  most research universities setup a circle (and that would be a tall order) there still would be huge swaths of kids out of reach. Can you infuse any of this into a classroom?  But this train of thought probably deserves some more work.

The Meat (Inversions in the plane)

After thinking about the preface I finally started the first chapter. At this point my mind was blown away.  

I'm only about 12 pages into a discussion of circle inversions. For me, this is a revelation at the same level as learning about cyclic quadrilaterals (which I hope would be shared by kids).  

The basic idea is you apply a transform based on projecting in or out of a given circle and the geometry problem can be solved in the new space more easily and then translated back.

Example Ptolemy's theorem:

More info:   As I said, its fascinating so far, and I'm going to work through the exercises to see exactly how powerful it can be. As a bonus: geogebra already has a built in inversion tool (reflect about a circle) which I never even noticed.

The book is focused on material as opposed to methods of instruction. So unfortunately, you won't find how this material was presented and the student's reaction to it. For me that's the next frontier: can I build something around geometric circle inversions that will work for the kids I'm going to have?

Tuesday, June 27, 2017

Factoring Proof

One of the fun possibilities next year is that we can do problems with polynomials and factoring (at least by the end of the year.) I was reading a post by @cav:  blog entry link with the following problem:

Can you prove that the product of 4 consecutive odd numbers plus 16 is a perfect square?

This is a nice group exercise to tie factoring into a larger proof. But one of the followup observations offered online was even more interesting. This works for even numbers too.   (As an aside, getting kids to ask questions along these lines "What about evens?" is a huge goal of mine.)

I'm going to guess the question was phrased as chosen because the factoring of odds is easier:

For symmetry let the odds be e 2n - 3, 2n - 1, 2n + 1 and 2n  + 3:

$$(2n - 3)(2n -1)(2n+1)(2n+3) + 16 = $$
$$(2n-3)(2n+3)(2n - 1)(2n + 1) + 16$$
$$= (4n^2 - 9)(4n^2 - 1) + 16$$
$$ = 16n^4 - 40n^2 + 25$$
$$=  (4n^2 -5)^2$$   

One immediate observation is that if we relax the constraints and let n also be k/2 where k is an integer, we can also get a sequence of evens and the end square is still also an integer. But if given the general problem I probably wouldn't have started this way and instead have constructed:

$$n(n+2)(n+4)(n+6) + 16$$
$$= n^4 + 12n^3 +44n^2 + 48n + 16$$

This is harder (but not impossible see: quartic to factor naively.

One immediate simplification is to assume that this the square of some quadratic \(ax^2 + bx + c\).  From there we get:

$$(ax^2 + bx + c)^2 = a^2x^4 + 2abx^3 + (2ac + b^2)x^2 + 2bcx + c^2$$

So if this works:

$$a^2 = 1$$ $$c^2 = 16$$ $$2ab = 12$$ $$2ac + b^2 = 44$$ $$2bc = 48$$
a and c appear to be 1 and 4 immediately and checking all the other equations 6 works for c in all of them.

So: \(n(n+2)(n+4)(n+6) + 16 = (n^2 + 6n + 4)^2\)

But wait ....

What if we applied the original insight about odds to the general problem?  Given n,n+2,n+4, and n+6 ,  n+ 3 is right in the middle and the product can be rewritten:

$$n(n+2)(n+4)(n+6) + 16 =  $$ $$([n+3] - 3)([n+3]-1)([n+3] +1)([n+3]+3) + 16 $$

We can the rework the calculations from the beginning.
Let  m = n + 3 (for clarity)

$$(m - 3)(m -1)(m+1)(m+3) + 16 = $$
$$(m-3)(m+3)(m - 1)(m + 1) + 16$$
$$= (m^2 - 9)(m^2 - 1) + 16$$
$$ = n^4 - 10n^2 + 25$$
$$ =(m^2 -5)^2$$ 
$$ = ([n +3]^2 - 5)^2$$
And we yet again get a general result without descending to solving a quartic!  

Friday, June 23, 2017

Ideas for Next Year

It was a good week for finding inspiration on the internet (and in some books from the library). This is a grab bag of ideas for next year based on what I happened to read recently.

Jim Prop's gam: Swine in a Line:

From a given starting position is there a move you can make so you'll always win? There is a followup video.  

Brenford's law 

(first digits are mostly ones (logarithmic) Easy to collect data and observe.  See:

Problem Stream using the Trivial Inequality

(Great for Whiteboard).
Start with the Trivial Inequality
Prove the AM-GM inequality as a demo.

Unit Fractions in Tessellations:

Wednesday, June 14, 2017

2017 Year in Review

One of the big questions I had going into this year was "Will the Math Club be very different this year? Am I going to continue making changes?"  Its never possible to figure everything out but by the third year one can certainly establish patterns and routines and move towards smaller refinements.  Its fairly normal for improvements to slow down or become more subtle at the same time.

So I went back through my blog, especially through the last 2 year end reviews and spent some time thinking about the year as a whole.

For the most part I still agree with my thinking from past years but on reflection I think this year has been different from the last two and I've made some important changes.

As always my key goal is to generate and sustain enthusiasm among the kids and to get them to engage with as many challenging problems as I can manage without losing them along the way. This year in particular I think I made strides towards engagement.

VNPS (Vertical Non Permanent Surfaces)

This was probably my biggest discovery for the year. See:  In the past while I used whiteboards during student demos or while I was explaining a concept, it never occurred to me that it would work so well for group problem solving.  I found I could keep kids going on a problem set much longer in this format than in paper even while both involved group work.  I think there was also good carry over week to week. A successful session set the stage for more focus the next time.   What I liked best was setting up 2-3 problems on all the boards and letting the kids switch between problems as they wished.  I then could stop by a cluster and watch and ask questions.

2 Things I'll keep working on here:
  • Work on listening carefully and not just floating when this is going on. Its easy to watch the kids working on the boards and be excited about the energy rather than focusing on the work they are producing.
  • Integrate gallery walks at the end, where the group goes through the solutions that were found. I started doing this more consistently at the end of the term.


As I've mentioned before, my thinking on warmups and how to layout a session has evolved a bit this year. My old structure was often a 10-15 minute "warmup" and then a main activity. Over time what I've come to realize is that Math is not baseball.  Warming up is the wrong metaphor for the process. In practice, after doing my first welcome speech for the day when I talk about what we're going to do and having kids go over their solutions to the problem of the week, I've usually provided enough routine for all the kids to transition to thinking about Math.  Instead, what I usually find is the kids benefit from breaks in the middle and the end of the problem solving process.   The need for  pauses is especially evident with problems that are challenging and not quickly solved. So nowadays I usually assume an average focus length of 30 minutes and I'll take something like a kenken puzzle and hold it in reserve for those moments. When kids flag, I'll have them switch gears for a little while or perhaps the remaining time. 
This doesn't mean I don't sometimes do a 2 part activity.  This can still be valuable if the first part directly relates to the second. For example: coloring in Pascal's triangle for patterns was a great setup for looking at combinations.  But if I have a great main activity, my first inclination is just dive in and let it take the whole time if necessary.  As a result, I've been printing and saving puzzles over multiple weeks much more often.


This was an early focus for me this year. I didn't end up using games every week but whenever I found one that I thought was mathematically interesting I consciously tried to build a session around it.  I think  my favorite 2 for the year were Rational Tangles  and Attack of the Clones.  The challenge here is to continue finding new ones when you've worked with the same kids for several years.  When I do repeat activities, I usually find kids don't mind/don't remember the first time nearly as clearly as I do/fear and I do reuse several categories of puzzles like the skyscraper puzzles this spring.

I also am still working on connecting the games explicitly to underlying Mathematics and making the time to talk about the games after we play them.  (This dovetails with my structural shifts. When you just dive in, there is more time left for a post-discussions and you run out of time early less often.)


I was very happy with my experiment getting the kids to try out AMC8. These also provide a little structure in the beginning of the year.  Competition brings out focus that a normal sheet of problems would never elicit. Next year, I'm going to try to leverage that a bit and use practice tests as a way to motivate kids to try and discuss problems more explicitly.

Video Integration

This year I chopped videos up a lot more than I did previously. Wherever I saw opportunities to discuss or try something out I would just pause.  For instance, during the Infinite Series video on proofs (Blog Entry Link) every time a problem was introduced we turned on the lights and tried it out as a group first before hearing the solution. I'm also really happy with the serendipitous alignment of my random topic choices and the final "Infinite Series" video on slicing a n-dimensional cube.

One experiment I'm still evaluating in my mind, was showing the MathCounts final. Kids are still asking to see more clips from MathCounts. So it definitely was popular. I'm not sure if it sent the right message about speed and ability though.

Guest Speaker

Going through the work to arrange a guest speaker was definitely worth it. I will try to maintain my relationship with the UW Math dept. going forward. In my ideal world, I could have a roster of speakers through the whole year.  Since that's not possible video clips act as a surrogate for this experience letting the kids see and hear more mathematicians. Also I'm toying with the idea of having the kids expand on my: I'm hoping we could get a lot more responses.


The biggest one for me is trying to ensure that the club transitions beyond my departure. I'm currently working hard to recruit folks for next year.  At the same time, I'm also starting to brainstorm about the differences in running a M.S. vs an E.S. club. I  expect to have to make many changes next year with older kids.

Topic Map:

Tuesday, June 13, 2017

6/13 And its a wrap

I was really heartened after sending an email out last week that all of the kids made it to Math Club today despite the transportation difficulties posed by a class field trip. Since this was the last session for the year I planned a mostly celebratory day.

First up there was a Math Club cake I ordered from the supermarket.  One unexpected hitch, on examination there were almonds in the cake ingredients. Fortunately I had brought some candy to give out for anyone with a dairy allergy that I also used for those who were allergic to nuts.  I'm still not convinced where the almonds were hidden but I'll be more careful before ordering next time.

We then went upstairs for the rest of the afternoon. I gave a quick speech thanking everyone for their hard work and then went around to survey what the kids liked this time. There was no overwhelming favorite unlike last year. Beside food which is always popular, the kids mentioned the Rational Tangles from last week, Pascal's triangle, some of the videos, Attack of the Clones, etc.  Overall, I think I executed pretty well on my goal of trying lots of different things to draw the different kids in.

Today I brought the projector in so I could show the following video:

Its incredibly cool and  I've been waiting to show this since by happy coincidence this tied into 3 of the topics we explored this quarter:

After that was done we finished with a game day as is traditional. I brought in my normal assortment of board games as well as one printed grid logic puzzle since these have been a great hit.

I was struck at the end at how smoothly the day went. (Granted I completely stacked the deck) Kids went excitedly from games to the puzzle and organically interacted with each other. I'm also really happy with classroom culture that had developed.

Wednesday, June 7, 2017

6/6 Rational Tangles

This Math Club  was a growth exercise for me. I had decided a few week's ago that I wanted to do John Conway's rational tangle game: in a future session.  It seemed great for a couple of reasons.

  • The problem was posed in a game format that didn't require a lot of supplies.
  • The game was physical (Good ones in this class are always hard to find.)
  • The connection between the game and rational numbers had a lot of depth. 

I also really wanted to stay hands off and maximize the kids own thinking as much as possible. So my challenge to myself was to allow the time for experimentation but keep the kids going all the while sticking to asking questions rather than telling answers.

This is the structure I chose.  First I outlined the rules and had a demo set of kids try out twists and rotates just to make sure everyone understood what we were doing.

For the next 10 minutes or so I had kids in each group create their own tangles and unknot them through experimentation. I mostly observed through this point.  The one exception I made was that its fairly easy to do a twist / rotate / twist combo that gets you back to the starting point. If the kids fell into this path, I'd ask them to add more twists at once i.e. twice 2 or 3 times.

At the end of this phase, I had members from each group make the tangle for the other ones and I asked them to try to make them as challenging as possible.   By this point the kids had developed a reasonable set of strategies that revolved around studying the loops and intuiting the sequence of steps to untangle bit by bit. What was particularly noticeable was they would often rotate through all 4 configurations to find 1 that would improve the tangle if twisted.

Next: I introduced the idea that we were going to map the moves to arithmetic operations. Everyone quickly came up with the idea that twists were  a  +1 operation. Rotations  remained mysterious.  After playing around a few more minutes I added the suggestion that they should try simple configurations and record all the moves they made.

Several ideas developed over the next phase: including are rotate/twists -2? I asked them to try doing 2 twists and seeing if the rotate/twist combo reversed it. (No)  One boy also jumped to the idea of infinity so I was able to ask questions about if we had any states that behaved like infinity i.e. if you twisted them they stayed the same.

Finally, the kids were starting to flag so I intervened  more directly by asking the kids to come up with ideas for what the rotate could mean and had them conduct experiments on simple tangles (usually double twists) to see if they would work. We did this as a group with one of the kids recording the results and tried out 3 or 4 options like multiple by -1.

At this point I was just about out of time so for the last 5 minutes we switched and I  let them do sequences of moves and I would call out the actual state values. Then they were finally able to discover that rotations were a negative reciprocal and that let me do a quick wrap up for the day.

All in all, we worked on the problem for most of the hour and while there were points when the kids were ready to give up, I was able to draw them back into the problem and re-establish flow. Hopefully, I'll have chance to try this again and see if my facilitation can improve further.

Further Highlights

In going over the problem of the week: MathForumProblem  a mostly standard linear system story problem with a small twist I had expected a blend of informal and formal solutions.   In past years, I'd get different strategies from bar charts or guess and check all the way to fully symbolic answers. This time around, I had 3 different kids demoing on the whiteboard all using substitution.  I'm impressed how many kids have already made the algebraic transition prior to middle school.

Finally, the most touching moment for me had to do with next week.  I realized after a parent question that a large group of the kids would be absent next Tuesday on a field trip. The excursion is downtown and finishes without transportation back to the school.  This will be our last session for the year and I assumed most of these kids wouldn't be able to attend. I've sent out an email and so far it looks like the parents will arrange carpools to bring the kids back especially to come to Math Club. The fact they were willing to do this makes me feel really happy.

Sunday, June 4, 2017

15 - 75 Triangle Redux

I came up with this problem after looking at the original one from @five_triangles (Find the area of the trapezoid ABCD) That's a lot of fun but along the way while modelling the solution in geogebra I noticed AF is also on the diagonal of the trapezoid.

Note the 15-75-90 triangles at the bottom. Nowadays when I see them I also think of the following construction:
which allows one to find the ratio of the sides without trigonometry:  1 : \(2 - \sqrt{3}\) for the legs.

Bonus: Another problem with one of these in it: The \(2 - \sqrt{3}\) in the expression is a dead giveaway.