Wednesday, July 19, 2017

Open Ended Problems

I've been thinking more about open ended problems after reading a couple of different posts recently. Full disclosure: I actually engage in problem solving exercises every week when we meet and believe problems are the core of Mathematics teaching.  But I still don't know if I reached the final idea of how to use them: See for some older thinking.

What started this thought chain off was a post offering that the answer to differentiation for a mixed ability classroom is to use open ended problems. This seems to be a widespread consensus based on how often I've read the same idea. That feels problematic to me from several angles.

1. I'm not sure if there are enough open ended problems that provide different levels of depth and easy entry that span the entire curriculum.  I can see such problems as being useful some time but potentially needing to be interleaved a lot of the time with less flexible material more focused on concepts needing to be gone over.

2. These type problems are almost always prescribed in mixed random groups.  How far can one student pull such a group if they are diving deeper and what kind of experience does this provide for all the participants? For that matter, if a student is behind the others will they more or less rely on the group to peer teach whatever underlying concept is involved? How well does that work?  While the problems themselves are good the overall structure aims at collaboration and creating a common experience for everyone rather than differentiating per se.

In search of more ideas on what an open ended problem / curriculum might look like I was reading through this paper:

"Working in well-facilitated small groups on rich problems that are accessible puts students in the position of differentiating the content, processes, and product of their own work. When students are empowered to make natural choices as they work on rich problems together, there are almost always surprises for teachers and often for the students themselves. One of the most important surprises is who comes up with interesting ideas; it is not always who a teacher would have predicted. In this article we discuss what makes a problem rich enough to allow facilitation of this self-differentiated student work."
I did like some of the offered ideas and was thinking about this one:

"Which positive integers can be written as the difference of two squared integers? For example, 17 = 92 – 82?"

The authors caution that the teacher must be ready with questions to keep students going as they get frustrated. Based on my experiences I wonder a bit of how this would work in practice? I can easily picture kids trying out sample numbers and noticing patterns but I don't see them getting from there to concrete reasoning on why this is generally true without a sufficient bit of number theory/modular arithmetic.  i.e. all odd numbers 2n + 1 squared are 1 mod 4 and all even numbers 2n are 0 mod 4 and the only combinations you can get subtracting them are  0,1,or -1  which implies all odds and multiples of 4 but not the non multiple of 4 even numbers i.e. 2 mod 4.

I also don't quite see the pattern noticing section as being particularly fine grained in its accessibility. The most likely outcome would be some kids noticing it after writing a table of numbers and everyone else going "oh that looks true.."

If I were giving this problem I might start with some modular arithmetic background to provide some tools for the kids. Or perhaps treat this like a "3 act" exercise. Working on it long enough to deduce patterns and try to find reasons for them, providing the info on modular arithmetic as a group and then resuming in groups to see what could be discovered more formally.

So I found something that I might use but I'm still not certain that this is differentiating in any significant manner.  I also need more observation to see how it goes with real kids. My main feeling is that group work on interesting problems remains a good practice but one that aims at collaboratively learning as a group. It still suffers from weakness when the members of the group have different amounts of "ability" or experience especially as the gulf widens.

Friday, July 14, 2017

Double angles

I've been thinking about a generalization of the 15-75-90 construction over the last few days and have realized there are a lot more interesting consequences in it that are fairly pleasing.

First construct a rectangle ABED and divide it into 4 isosceles triangles.  My home made puzzle from this construction is: find an expression for the length of CD in terms of AC and AB.

Some angle chasing will show show that the lower and upper triangles are closely related.

If \(\angle{CAB} = \theta\) then \(\angle{DCH} = 2\theta\). What's more given the lengths of the triangle CDH sides (a,b and c) you can find the lengths of the triangle ACG in terms of them i.e.

Triangle ACG has side lengths of a, c - b and \(\sqrt{a^2 + (c-b)^2}\).

I've split the 30-60-90 before to get the 15-75-90 so I'll demonstrate splitting the 45-45-90 to get the 22.5 67.5 90 ratios.   

In this case,  let a = 1 then b = 1 and c =  \(\sqrt{2}\) and from that you get the 22.5, 67.5 90 triangle is in the ratio  \(1 : \sqrt{2} - 1 : \sqrt{4 - 2\sqrt{2}}\)

Essentially this is the  geometric equivalent of the trig double angle formulas:

  • \(\sin2\theta = 2\sin\theta\cos\theta\)
  • \(\cos2\theta = \cos^2\theta - \sin^2\theta\)
and allows you to calculate the ratios of the \(\theta\) triangle from the \(2\theta\) triangle (or vice versa).

In fact its easy to verify the equivalence:

\( \sin2\theta = 2\sin\theta\cos\theta \)
\( \frac{a}{c} = 2\frac{a}{\sqrt{a^2 + (c-b)^2}} \frac{c-b}{\sqrt{a^2 + (c-b)^2}} \)
\(                    = 2\frac{a(c - b)}{a^2 + (c-b)^2} \)
\(                    = 2\frac{a(c - b)}{a^2 + c^2 -2bc + b^2} \)
\(                    = 2\frac{a(c - b)}{2c^2 -2bc}  \)  (Pythagorean Theorem)
\(                    = \frac{a(c - b)}{c^2 -bc}  \)  
\(                    = \frac{a(c - b)}{c(c - b)}  \)
\(                    = \frac{a}{c}  \)

Some interesting relationships fall out when you play with integer ratios.  For example
the 1:2 triangle double angle cousin is the  3 : 4 : 5 which is consistent with their structure.

My next thought is there is probably another general construction for the triple angle formulas which would be nifty since it would allow you to generate ratios for a lot more common angles but I haven't gone farther down this track yet.

Tuesday, July 4, 2017

And yet more 15-75-90 fun

Continuing on the theme of 15-75-90 triangles (See: Last time and First Time) several interesting riffs on 15-75-90's in a box have come up recently.

Example dividing a square with four 15-75-90 triangles:

As is often the case, finding the relative area of the  triangles and square is straight forward using trigonometry:

Let s be the length of the  sides of the square:

The area of each triangle = \(\frac{1}{2} s^2 cos(15)sin(15) \) and using the double angle formulas
\(sin(30)=2sin(15)cos(15)\) so  after substitution and knowing sin(30) = \(\frac{1}{2}\) out pops the area = \(\frac{1}{8}s^2\)

But why is this happening? As usual a 30-60-90 triangle is usually lurking around that allows a Euclidean explanation.

What's particularly interesting about this is that it hints that dissections exist to transform a 1/4 or 1/8 of the larger square into the triangles and sure enough you slide the 1/4 triangle ABO until it becomes 2 15-75-90'!

But let's return to the original problem. There's another easy explanation of what's occurring that just uses the ratios of the triangle:

1. Note the  area of this triangle is \(\frac{1}{2}(2 - \sqrt{3})\)
2. Squaring the hypotenuse you get \(4(2 - \sqrt{3})\) which is 8 times the triangle area.
3. Or in other words each triangle is 1/8 of the square made on the hypotenuse.

And we've refound our original result.

Further questions: Are there other common triangles that divide the square into a unit or "simple" fraction.

I'll leave it to the reader to decide which problem based on this property is more fun (from @eylem and @sansu-seijin):

Given the square of length 6cm, how large is the shaded region?

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.

Wednesday, May 31, 2017

5/29 Combinations and Pascal's Triangle

This week I decided to hit a bit of combinatorics before the year ends. I know most of the kids understand permutations fairly well but not combinations and that seemed fairly accessible.  I like the connection between combinations and Pascal's triangle and that led me to the idea of warming up with a repeat from "This is not a Math Book".

In probably my favorite page of the book, Anna has a coloring exercise with Pascal's triangle where you search for patterns after coloring all square that are multiples of 2 different numbers, 2 different colors. I've used this before and it still drew the kids in (even the ones who were here last year).

After maybe 10 minutes of coloring, we gathered together to discuss what we had noticed. There were a lot of mentions of symmetry and triangles. I pointed out several examples of Sierpinski's triangle since these keep recurring this year. [Maybe a whole session on fractals is in order at some point.]  I told all the kids that we'd come back to the triangle but we now were moving to our main task.

From there we talked first about permutations with some group questions on the order of how many ways are there to pick a pair of socks to wear  for a week out of a sock drawer with 7 pairs.  Factorials are pretty well known by now so I just reviewed them on the board after one boy mentioned them.

Next we moved to combinations (my main target).  I started with a group question about picking teams emphasizing the order in which you pick was not important. I chose \(4 \choose{2}\) as a starting point. We took predictions on the result and then the kids worked in tables to enumerate and figure out the answer. Predictably some thought at first it would just be 4 x 3 but after a few minutes the truth was discovered and the kids were able to give some informal reasoning about removing duplicates by dividing by 2.  I then expanded and asked what they thought \(10 \choose{4}\)  would be.  Some kids again predicted that it would be like the permutation but divided by 2.  Again I let everyone work on finding the enumerated answer.

At this point I wrote out the formula \( n \choose{m}\) = \( \frac{n * (n-1) * (n-2)... (n-m+1)}{m!}\)
and asked if anyone could figure out why this was happening?

Unfortunately, no one had a good idea about the denominator. I tried some leading questions how would you calculate the number of duplicates given a concrete set of say 4 items. But in the end, I related this back to combinations by saying the top was the total permutations but included duplicates and for each individual class of duplicates the denominator showed the number of permutations i.e. for pairs: duplicates come in 2 but for trios they come in 6 etc.  [If repeating I think I would linger here and try some more examples as group work to see if more intuitions would develop.]

Finally, for my favorite part I had the kids calculate and write all the combinations for 2, 3 and 4 on the board in a pyramid:

           \( 2 \choose{0}\)  \( 2 \choose{1}\)  \( 2 \choose{2}\)

      \( 3 \choose{0}\)  \( 3 \choose{1}\)  \(3 \choose{2}\)  \(3 \choose{3}\)

 \( 4 \choose{0}\)  \(4 \choose{1}\)  \(4 \choose{2}\)  \(4 \choose{3}\)   \(4 \choose{4}\)

What do notice now?  This elicited some wows when Pascal's triangle re-emerged. So again because this is a bit mysterious I went into an informal explanation centered on the there being 2 cases:

  • The new element n is in the set you pick and then there are combinations of the n - 1 elements for the rest of the set i.e. the left parent.
  • The new element n is not in the set and there n combinations with the rest of the set i.e. the right parent.
This also works best with concrete examples. 

Finally, I found a decent problem set that I based  my own problem set off of: My sheet. Going in though I was a bit worried. This is the end of the year, and I wasn't sure how much focus I could count on. So I hedged my bets a bit and brought another set of skyscraper puzzles with the idea that I would offer them to the kids if they started to flag at the end.  This turned out to be prescient. What I hadn't counted on was today was also a standardized testing day and therefore the kids were more drained than usual after several hours of SBAC testing.  If I had known that ahead of time, I think I would have compromised and picked maybe 3 problems to do on the whiteboards instead of at the table.

For the long run, I still have the aspirational goal of being able to have a group of kids spend 20-30 minutes working through a short problem set (10 or less) of interesting problems. I'm not completely certain that's realistic (ok I'm fairly certain that if it is, its not easy) and I've pivoted more towards group white-boarding or providing choices in these scenarios which allow the kids to work on the problem set or a lighter puzzle and switch between tasks.  What I'd really like is some kind of reward/competition that was motivating but not discouraging for the room in these scenarios. Based on the odd fact the kids really liked the Lima beans we used as counters several times, I'm tempted to try spray painting a bag different colors and handing them out as prizes to see what happens.

Problem of the Week
An algebra one from the mathforum that doesn't really need formal Algebra to work it out:

I just saw a recent video on infinite series using pascal's triangle to look at hypercubes. This may make a cool followup for next week.

Tuesday, May 23, 2017

5/23 Triangle Conundrum

In the middle of last week, the MOEMS awards for the year arrived. So I started handing out patches and medals. I'm fairly happy with our overall performance at the Middle School level. Almost all the fifth graders who were present for all 5 tests was in at least the top 50% of 6-8th graders and we had one boy crack the top 10%,  This was not as high a level of achievement as last year when we used the Elementary level but confirmed that this was providing a good level of challenge and was not too difficult. Week to week, almost everyone could access at least 1-2 problems (often more) and we had good discussions about the entire set. As I said before, the MOEMS format has grown on me over the last 3 years. I think I will bring this with me to the middle school level.

On another note, I thought the MathCounts problem of the week was not super interesting so I took a poll of the kids after we discussed it today. Interestingly, the kids seemed to generally like it. I mean to think about this some more. Was it because these type questions are more straightforward? Its definitely a caution for me to remember to vary activities. My taste in Math is my own (and perhaps a bit quirky) and I want to make sure to try to appeal to everyone over time.

Today's main Math Club activity was inspired by the following tweets:

There was mention of the following problem:

That made me think of the classic Martin Gardiner missing square puzzle:

These problems seemed like a good progression of fishy triangle issues and all seem well suited to group problem solving on the whiteboard. So I had everyone getup and circulate among them during the main part of the hour. I liked the general activity.  The most difficult one turned out to be the Tanya Khovanova "triangle". This was the only one the kids didn't fully solve although it brought out some great questions about the Pythagorean Theorem and experimentation with various triangle configurations. As kids cracked the other ones, there were occasional excited shouts "This isn't really a triangle!" I was particularly happy they also connected the problems back to slopes to prove what they discovered.

To close the day out I wanted another game. This time I turned to one I found on Sara Vandewerf's site: 5x5.

I pretty much followed Sara's format. (I always appreciate time estimates for a game in a writeup) We did 5 founds and the kids were just as engaged as promised.  Beforehand, I had wondered if all the scores would bunch around a few values. Even with 14 players that didn't generally happen except when going for low scores.  As a thought experiment: since all the kids loved the lima beans we used as tokens a few months ago it occurred to me afterwards I could spray paint them gold and give them out as "prizes" in the future.  [Would older kids find this corny or fun?]


Some probability work form Waterloo:

Unused: I actually had some more skyscraper brain teasers and  a little bit of combinatorics in my back pocket.

Thursday, May 18, 2017

5/16 Expected Values

My planning process this week went something like this: after last week's talk I either wanted to do some group white-boarding or find a new game to explore. I also was thinking more about combinatorics. I've never done anything on combinations (n choose m) and I mulled choosing that as a theme. Then in the middle of the week the Math Counts finals occurred. Watching the live stream was fun for me and I thought the kids would like that too. So initially, I thought about showing pieces of the video and then pausing and have everyone do the problems on the whiteboard. But after some more thought, I worried that it would emphasize the speed of the competitors too much and I also wanted to dig into the Chicken problem more deeply.

Finally this was the structure I ended up with:

We warmed up with some individual skyscraper puzzles from I really like doing these and they went over well engaging everyone. I cut this short after everyone had at least finished the first one of the set I provided in the interest of time.

After watching the videos I transcribed a few of the problems I liked and thought would be good to try on the board:

Whiteboard Problems from the video:

  • Caroline is going to flip 10 fair coins. If she flips  heads, she will be paid $. What is the expected value of her payout?

  • Sammy is lost and starts to wander aimlessly. Each minute, he walks one meter forward with probability 1/2 , stays where he is with probability 1/3 , and walks one meter backward with probability 1/6. After one hour, what is the expected value for the forward distance (in meters) that Sammy has traveled?

  • A finite geometric sequence of real numbers with more than 5 terms has 1 as both its first and last terms. If the common multiplier is not 1 what is the value of the 4th term?

  • The length of a 45 degree arc on circle p, has the same length as a 60 degree arc on circle q. What is the ratio of the areas of circle p to circle q?

  • The novel Cat Lawyer is 300 pages long and averages 240 wd/pg. The sequel Probably Clause is 60 pages longer and 30 more words per page. Probable Clause has what % more words?

  • Ian is going up a flight of stairs. Each time he takes 1,2 or 3 steps. What is the probability that he steps foot on step 4?

  • How many 6 digit integers are divisible by 1000 but not 400?

But I decided to focus on the final question:

"In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately to its left or right. What is the expected number of un-pecked chicks?"

To build up to it, I started with discussing expected value and used dice questions as starters in a group.

1 What's the expected value of a single roll of a six sided die?
2. What's the expected value of 2 rolls?

After we went over the concepts as a group, out came the blue markers and  I had everyone work on the followup problems up on the various whiteboards.

3. What's the expected value of the product of 2 rolls?
4. What the expected value of the product plus the sum of the rolls?
5. What do you notice about this? 

The kids all worked on these followup questions in groups over about 20-30 minutes and then we gathered together to discuss what we found.  I had to bring out the linearity of expectation relationship and did not go into the proof at this point since its a bit too complex. [This is always a struggle to resolve whether to dig into every observed pattern and find the reason or accept its probably true in order to reach a target for the day.]

Then I posed the chicken problem expecting everyone would work again for a while. I was surprised but several kids almost immediately shouted out the answer. It turns out the scaffolding may have been a bit too much after all. If repeating I might lead off with the final question, work on it a few minutes and then go into the build up.

At any rate for the last 10 minutes we did watch the video communally.  Before showing it I prefaced it with a short talk about speed and emphasizing both the competitors had really trained to build this up and that it wasn't really important outside contest while the problems on the other hand were fairly interesting.  So hopefully, I didn't damage any of the kids self-conceptions. As expected, the room was fairly rapt watching the competitors even many of the parents on pickup stopped and watched it until the end.

Continuing the MathCounts theme I chose this weeks problem from their site: although I'm not completely keen on the questions.

Tuesday, May 16, 2017

MathCounts Final

Since it was fun Last Year to think about the Math Counts final question, here is the 2017 version:

Oddly enough when I saw this year's final question, I almost immediately said out loud 25% of the total number or 25. Somewhere very recently I'd seen this problem (I can't remember exactly where), I didn't recall the reasoning offhand but the answer came to mind instantly.  Math Counts at the national level works a bit like that. Seeing lots of problems and being able to quickly either recall the entire answer or the efficient means to solve it is critical to win where kids are answering questions like above in a few seconds. In fact, I couldn't even "borrow" some of the questions since they were answered before being fully read out or printed on the screen.

That's not super interesting in the long run, but I think its balanced out by what happens when the larger set of kids are preparing for the contest and even when bystanders read an article in the nytimes and spend some time thinking about a problem.

This one is fairly fun to model. Many people eventually came to the reasoning that for an individual chick there is a 25% chance it won't be pecked. But this is not independent of what happens to all the other chicks. Because this chick wasn't pecked 2 other neighbors were.  I like thinking about this as a chain of  Ls and Rs where your counting the number of transitions between letters.

Behind all of this is the somewhat counter intuitive: Linearity of Expectation.  Even though the individual outcomes are dependent the expected value can be had by simply adding them up anyway.

I'm planning to do a session  around this today divorced from time pressures. Building up to the Math Counts question through a series of exercises and observations about expected value should make an excellent white board #vnps activity.

Followup interesting tweet stream on the probability distribution:

Tuesday, May 9, 2017

5/9 Dating for Elementary Students

Today was a special Math Club session. Annie Raymond from the UW Math dept. came and gave a talk to the combined fourth and fifth graders on the topic of combinatorics. So I had the unusual opportunity to act as a photographer more than a facilitator. We started with me asking the kids if they had any initial questions. There were few basic ones like "How do you spend your days?"  Answer: teaching a lot of the time and thinking about research in between.

Annie went a bit bold and chose to talk about the Stable Marriage Problem which is right up my alley as a computer developer. I was a little worried we'd end up with a lot of nervous laughter and asymmetric interest from the boys and girls but the kids exceeded my expectations and were very attentive and engaged.

The kids did a great job calculating the total number of combinations was 3! at this point.

In the middle she brought an interactive version of the algorithm to test out. There were ~10 boy and girl preference sheets handed out. She then had the kids work through the algorithm in rounds with the boys going to their next choices and the girls picking the top selection. Much amusement and chatter soon followed.  This would be fun to do as an ordinary combinatorics exercise on its own.

What's very nice is there is a not too complicated proof by contradiction that the algorithm works. That fit really well with our recent session on proofs.

Then we observed that the algorithm is asymmetric (with the kids volunteering if they got a good "match") its much better to ask than to choose partners.

Variants you could build more extensions on. She didn't bring it up but I thought a bit about complexity as a tangent. I think the algorithm is O(n^2) for instance in the worst case where everyone has the same preferences. Could this be optimized?

 Overall I think this was a grand success with the kids and I'm hoping I can continue this relationship with the UW Math folks next year.

Monday, May 8, 2017

Questions for Mathematicians

I've been prepping for our guest talk from the UW Math Dept. One of the tasks I've done is survey the kids to generate questions for the talk. Jayadev Athreya  emailed me back with some answers  which I really like:

1. What's your normal day like i.e. what does a mathematician actually do?

We teach, we think, and we write- but mostly we play with patterns- exploring ones we think we understand to see if there is a deeper pattern hiding behind it. Like you might notice that all prime numbers bigger than 2 are odd, then you notice that all prime numbers bigger than 3 aren't divisible by 3, and so on... that's a series of patterns that all come from the definition of a prime number! We do spend quite a bit of time using computers to find patterns too!

2. What did you have to do to become a mathematician and when did you decide to go down that path. What motivated the choice?
I was very lucky in that my mom is a physicist and my dad taught math. So I had great role models and I saw how exciting math and physics could be!

3. Were you really good at math when you were our age?

I worked hard at it, and I liked learning it and exploring it. I didn't always do well on tests. My dad, who is also a mathematician, was not very good at all as a kid but enjoyed playing with problems, and became a really good mathematician.

4. What do you do when you get stuck on a problem?

I follow the advice of a famous mathematician, Polya, who said that for every problem you can't solve there is a simpler problem that you can't solve! So I look for the simpler problem, try and work out a bunch of examples, and try and and play with patterns to see if I can unlock the problem. Sometimes this takes months, or even years- so patience and hard work are key!

Annie Raymond also sent back some answers:

1. What's your normal day like i.e. what does a mathematician actually do?

It depends on the day!

On Monday, Wednesday and Friday, I teach two different classes, one on multivariable calculus, mostly to engineering students, and one on how to prove things to math students. Outside of the two hours when I actually teach, I meet individually with students who need extra help, grade some of their work, come up with new material for them. On Wednesdays, I often go to a talk about combinatorics, one of the fields that I work in, to hear about new work done by colleagues from all over the world. On two of those days, I also meet with some collaborators to discuss our progress on a common project that we are working on. If I'm lucky, I'll have a couple of hours to do research or work on papers as well, but that's not always the case.

Tuesday and Thursday are the days when I actually do my research and write papers. Doing research for me means sitting down and thinking about some problem I'm hoping to solve. The nice thing about problems in math is, when you solve one, it usually opens up ten new ones, so you never run out of problems to solve. Going to talks also helps with finding out about new problems too. It is very hard to explain how you get the good idea that allows you to solve a problem. Usually, it just clicks all of a sudden after you've spent hours and sometimes weeks or even years playing with it.

Finally, on Tuesday night, I teach college-level math to inmates at a prison. I believe making education more accessible to everybody is the best way to create a strong and fair society.

I do need to mention that traveling to go to conferences and give talks and meet with other mathematicians from all over the world is a pretty regular thing too. Of course, those days are completely different!

2. What did you have to do to become a mathematician and when did you decide to go down that path. What motivated the choice?

I had to study a really long time: I first got a bachelor's degree in mathematics (and music!), and then I went to grad school to get a phd. The nice thing is that, in science, you usually get paid while you do your phd, so you're not a starving student. I'm now finishing up a postdoc which is something you do after you get your phd to prove that you're ready to be a professor. You do more or less the same thing as a professor, but your job is temporary. Next year, after 4+5+3 years of being at a university, I'll finally be a professor.

I decided to go down that path right before going to college. I went to math camp the two summers prior to college, and I really loved it. Up until then, I knew I liked math, but I didn't have a good idea what more advanced math looked like, and I thought---wrongly---that math was a pretty useless field, and I wanted to do something useful. Math camp opened my eyes on how amazing math can be.

3. Were you really good at math when you were our age?

I was pretty good at school overall---it came easily to me. I did find college very hard however. We all find things very hard at some point. How we deal with that and how we persevere are both more important than how long we found things easy.

4. What do you do when you get stuck on a problem?

Being stuck on a problem is my normal state, and the normal state of most mathematicians. I have spent a few years working on a few problems. But that's normal: there are many problems in mathematics that have been open for 10 years or 100 years! I've learned not to be frustrated if I don't know immediately what to do and I try to enjoy the phase where you play blindly with the problem, where you try to look at it from every possible side. If I ever get too frustrated or don't know what to try next, I move on to a different problem or different task I need to accomplish: often, new ideas come when I am doing something else. Discussing the problem with friends and colleagues also help: it helps make my ideas clearer and combining our ideas together often leads to a winning strategy!

I think this is fairly interesting. Maybe next year I'll have the kids write letters and see if we can get more responses.

5/2 Assessment

This is a short placeholder entry.  I decided to administer the AoPS algebra assessment last week since the 5th graders are currently choosing next year's math course. In our district 6th graders have the option to opt-up to Algebra I. This is determined via a set of opaque measures which often leave parents uncertain about the best choice.

Some notes for future reference.
  1. While this is useful for parents. I think its best done out of club after all. I would just offer links in the future even knowing only  a small portion of parents would take advantage of them.
  2. There is no time limit for the 12 questions. I thought that would be sufficient but most kids only finished about 7.  So it probably takes closer 1:30-2 hours.   (This could be threaded through several weeks.)
  3.  To make up for that, I'm going to grade what I have and give the parents links to the full test so they can finish it they they desire.
  4. It takes a lot of work to keep kids focused on something so "class-like". Another reason why I don't think I will do this again. I did decently since I know all the kids well but it required constantly moving between tables and encouraging them to keep going / checking in.  I was a little surprised how difficult some of the kids found the questions "This makes my brain hurt."

Tuesday, April 25, 2017

4/25 Platonic Solids

This week's inspiration started with a very late school bus. My son's bus driver has been on vacation and the substitute drivers have been really, really tardy. So much so that he missed most of Math the other morning when the class was going over the volume of a pyramid. I checked with his teacher and seeing he had missed the explanation for the formula decided to try out some activities at home to make up the gap.

To start, while I love much of Geometry the introduction of sundry area, and volume formulas in the middle school sequence seem pretty pointless to me. They don't connect with much before or afterwards and are often taught without sufficient explanations. Frankly you can go really far even in pure Mathematics without ever missing the pyramid formula. (Brainstorm topic: where would this fit more naturally? The calculus connection is fairly compelling ...)  The missed experiments in class compared prisms and pyramids and the volume of rice they held. As an experimental process this is not bad but as a mathematical foundation it doesn't totally satisfy me. Its neither universal "How do you know that if the pyramid dimensions shift the relationship stays constant?" nor does it speak to "Why is this happening?"  The question I want to provoke is "Why 1/3 and not 1/4?"

So at home, we started looking at the formulas and  I asked "What does the similarity between the volume of a cube/prism and pyramid suggest to you?" I was lucky that was enough of a prompt for him to suggest "Is there a way to cut a cube into 3 pyramids?" From there we printed out some templates and built some 3-D models to show the trisection.    This was enough fun that I thought I'd build a day out of it for the whole Math Club. To round things out I thought I'd bridge from there to an exploration of platonic solids.   At this point I worried a bit about timing and decided to have some Sudoku puzzles in reserve. But I stayed firm and left them for the end if needed which as it turned out was not the case.

The afternoon began for real with  me handing out spiced gum drops for reaching our problem of the week target. I also left out a sample tetrahedron I had built to see if I could garner any questions. (Nope)  Once we were upstairs I decided to have a short debate about last week's problem. Infinity Link.  I asked everyone to pick a corner of the room. One side for those in favor of Courtier A's offer, the other for Courtier B. A group of students actually remained off to the side and I asked what they supported. Their answer was they thought both offers were equal and since that seemed interesting I setup a 3rd corner for them.  We then went around the room with everyone offering positions on why their side was correct and rebutting the other side's idea. This went on for may 5-6 minutes which was fun . The disadvantage was this format really makes universal participation hard to achieve so I wouldn't rely on it a lot. (To be fair: repeated usage could make it more natural for more kids to speak.)   Secondly, a group of kids really wanted me to rule on the "correct" answer which  I demurred on.  Next time, I should also remember to close this with a final vote.

From there I did a version of my initial process with my son and  we bridged to building the pyramid templates. I used some cutout templates from here: and spread them among the various tables. We eventually cut out 3, folded and assembled them with scotch tape, and confirmed they were identical and formed a cube.

I was hoping that someone would complain that the pyramids weren't exactly the same as the regular ones we started with. That didn't happen so I prompted "Is there anything that doesn't seem quite right in this explanation?" That eventually brought out the idea and I gave a brief hand waving explanation of slicing the pyramids and rearranging them to have the same volume but centered rather than offset to the corner.

Next: I handed out a combination of further platonic solids from the site above (cube, tetrahedron, octahedron, dodecahedron,  icosahedron)  and from  Each table had a different one to assemble. At the same time I brought some pipe cleaners and straws to make companion wire models.

I had everyone work on the models and to tie things together chart the edges, faces and vertices per shape on a communal white board. My hope was to have the kids observe the the Euler Characteristics.patterns and I seeded things a bit by arranging the chart  V / E / F.

Some of the kid's handiwork.

This worked fairly well. Engagement was good among the modellers (which I had to rotate due to limited tape and scissors) but I had to work a bit to keep the other kids counting edges and faces and thinking about patterns. 

At the end I gathered everyone back at the board to discuss the data. There were a few fun observations.
  • All the numbers were even. (Followup for another time: could any characteristic ever be odd?)
  • You could more quickly and accurately calculate edges and vertices than counting by multiplying the number of faces time edges/vertices per shape and dividing by the number of faces that met at an edge/vertex.
I had to have everyone think about a numeric relationship for a few minutes but lucked out and one boy discovered   V - E + F = 2.  At that point my time was almost up so I left with some closing questions:
  • Is there an equivalent relation in 2-D?
  • Why do you think this is happening? 
  • Are there any other platonic solids you can discover?

In the end, this was a lot of fun. I actually had templates for stellated polyhedrons and Archimedean solids we never got to in my back pocket. We could easily do a followup day on the topic although my general style is to zig-zag around subjects.


This one from an old Purple Comet is interesting because its algebraic but linear not quadratic like I first suspected and you don't need to every find the exact dimensions to discover the perimeter.

Related Session: Euler Characteristic