Friday, February 16, 2018

2/13 Farey Sequences

Today was a special occasion for Math Club. Instead of just me or a vicarious video, we once again had a guest lecture from the UW Applied Math department. 

This time, Professor Jayadev Athreya came out to the middle school to give a talk on Farey Sequences. That was fairly propitious, since I had meant to get to this subject during this session:  but quickly realized I didn't have enough time to cover even Egyptian Fractions.  So there was a good thematic fit with some of the other things we've done.

My favorite moment of the day came early on when Jayadev had each of the kids talk about why they came to math club. (I usually do this on the first session too) There were a smattering of "I like competitive math" responses but then we reached a girl who roughly said "I don't know why I came originally but I like it so I keep coming." That's victory in my book!

What's also interesting here is a chance to more closely observe all the kids and another person's teaching style.  Jayadev's basic structure was fairly similar to what I might have done.

  • Start with having the kids map out all the reduced form fractions where the denominator was less than or equal to 10 and then arrange them by size from smallest to largest. He then graphed this on a number line as a group.

      • Closely investigate the numerators of the fractions (in suitable common denominator form) when comparing them to notice a trend: they always differed by one.

      • Do a formula proof that the mediant is always 1 apart from its generators if they are 1 apart.

      A lot of this was structured as group work at the tables with discussions after a few minutes where things were consolidated.  I always find these transitions a bit tricky to time so it was useful watching someone else.  I would also probably have done a few of these parts as group work on the whiteboard itself and then gallery walked for the discussions but there was good work done at everyone's seats.

      Like last time, I also noticed some unexpected hesitancy with operating on fractions. It took a bit more time to draw out the kids and have them explain how to compare fractions with different denominators. Although create common denominators was mentioned as well as variety of numeracy instincts ("for unit fractions, the fraction become smaller as the denominator increases" or "you can also create common numerators to do informal comparisons").   Again, I felt like this was a useful practice/review for some.  Alternate hypothesis: the kids were more reluctant to volunteer at points which was more social rather than indicating any gaps.  If this is the case, I'd like to work on activities to bring out more questions. One idea I have toyed with in the past is, is selecting one student to be "the skeptic" during any demo and come up with at least one question about the logic. If I do go this way, I'll probably start with having them do this with something I discuss and depending on how it goes try it also during all whiteboard discussions.

      Overall I was really pleased. We now have an invitation to visit the Applied Math Center on the UW campus. I have to investigate whether the logistics are workable.

      Thursday, February 8, 2018

      2/6 Olympiad #3 and AMC 10

      I almost cancelled this week's math club due to feeling ill the night before. But in the end I was well enough and the activities were straightforward so I went ahead with the meeting. We started with the candy I had forgotten to bring last week. My wife picked up some red vines for me, the reception of which I was curious to see.  They were all eaten by the end so there may be more licorice in the future.

      Participation in the problem of the week was lighter that I would like but I had enough kids to still demo solutions. In particular with this problem: the key is to count the total number of pips in the set
      of dominos.

      Image result for domino image

      The students demonstrated two different methods which was good. Most approaches end up with a triangular table since when you calculate the combinations you often end up with  n pips | m pips and m pips | n pips which are the same domino.  I'm trying to elicit more questions from the other kids. This time it worked well when I asked "Does anyone have any questions about X's diagram and how X did ...."  I also spent some time modelling asking questions about their strategies and why they had created the triangles to draw this out.

      Once done we participated in the third MOEMS olympiad. My feeling is this was the most unbalanced of the set so far. The starter questions were all fairly easy and they gave a hint that unwound most of the complexity of one of them and then it ended with a really interesting
      Diophantine fraction equation that was quite a bit more difficult to do.  One followup question I have for myself: is even in cases that simplify is it enough to consider just the factors of the denominator of a sum i.e. if   K1 / A + K2 / B = K3/K4 where all the cases are constant.

      Finally I chose another AMC based question for the Problem of the Week:

      Overall everything ran well but it was not my most inventive day which was probably just as well since I felt very low energy at points.

      The next day,  one of the teachers at Lakeside graciously let me send a few students over to take AMC10. I couldn't justify the cost to do this on site for so few students. In the future, I'm hoping with more eight graders this might change. At any rate, this was fun for me. I had the three kids take a practice test first to make sure this was a reasonable move. My goal was for everyone to get at least 6-10 answers correct.  What I don't want to happen is for kids to go and get so few questions correct that the entire experience is discouraging.  I've also been feeding more sample questions from AMC10 as problems of the week. Generally, given enough time they often make really good exercises.  The kids reported this year's test was a bit harder than the practice versions so I'm cautiously awaiting the official results.

      Looking forward: next week is going to be real fun. One of the professors from UW, Jayadev Athreya is coming to give a guest talk to the kids on Farey sequences (which by coincidence we didn't quite get to on:

      Resource Investigations

      I just learned about the MoMath Rosenthal prize winners I'm going to look through the sample lessons to see if there is anything that is usable in our context. By that I mean far enough off the beaten curriculum track.

      I also really like this investigation from the Math Teacher's Circle Network:  on countable infinite sets. I have to spend some time thinking about it but it looks quite promising.

      Wednesday, January 31, 2018

      1/30 Math Counts Prep Day

      We're only two weeks out from MathCounts and I've been so busy with various topics and activities  that I haven't really specifically focused on it. For the most part we're doing interesting problems that will overlap anyway and it will all work out but I wanted to spend one day going over the format before the kids go so they know what to expect.

      So I went to the MathCounts site and printed out last year's contest questions:

      I knew I would go over the basic format and rules i..e how many questions, can you use a calculator what do you do as a team?  I also wanted to try out a little bit of everything.  Immediately, I decided that I couldn't really do the countdown rounds. Those are run like a quiz bowl and I have neither the equipment nor desire to to replicate that.  For one, I have a few kids who I think would find it too high pressure and secondly it only allows a few kids to participate at a time which I dislike for  class management reasons as well as on general principle that I want every kid doing math for as much of the scant hour that we have. So hopefully that won't have any impact on the performance at the contest.

      Instead I decided to focus on the individual and team sections. (I printed the target round but knew even going in we wouldn't have time to try those out.)

      Thinking about this ahead of time, I decided to try out a new strategy with the individual round: speed dating.

      Basically I had the kids setup a large row of tables in the center of the room and had everyone face someone else. To start I gave out one of the even numbered problems to each kid. My instructions were: this is your problem, you will solve it and then for everyone else you will be the expert and double check their answer as well as help with any problems. We then rotated every few minutes. Every rotation the kids told each other their respective problems and then worked on them.

      I was worried going in that the rotation timing would be tricky especially since the problems varied in difficulty. That turned out to not be an issue because they were generally "simple enough" that everyone could finish within a few minutes and I just had to survey where everyone was. It also let me point out that the difficulty varied and that different people would take different amounts of time depending on which problem they were on. That had a useful effect on expectations.

      Overall, I would use this format again for easier problems/review.  It seemed to keep kids working over a larger set of problems and I liked how it farmed out answer checking. There are 4 issues to keep in mind

      • In a complete rotation everyone will only see half of the problems. So you need to swap the problems at that point if you want to have everyone to do everything.
      • Timing can be still be tricky.  The problems should be varied in difficult but not by "too much".
      • I didn't stress the ownership as much as I need to initially. If I reuse I will emphasize that role and go around and check for any questions at that point about the problems.
      • I suspect this falls apart the more complex the questions are.

      Coincidentally, one of the teacher's running the yearbook wandered in to take photos in the middle of all this. So we'll definitely be in the yearbook looking studious. As my son remarked afterwards, the club hasn't gotten any school paper mentions and I should work on this in the future.  For one, I'll take a team photo at MathCounts and submit it.

      For the second half, I handed out the team tests and just group everyone based on where they had landed at the end of all the seat rotations.  (coincidental Visible Random Grouping) During this section I floated a lot, asked hopefully helpful questions,  answered any of theirs, and pointed out problems that were not correctly done yet.  I was  actually pleased that this went very smoothly. I didn't really need to do any prompting to keep everyone engaged.

      Finally, because in my excitement  I had jumped in I had to reserve 5 minutes at the end to go over the problem of the week.   Interestingly there were two programmatic solutions submitted this time. If this trend continues I'm going to start handing out explicit problems aimed all the kids who want to program.

      New P.O.T.W:

      A domino pip problem from UWaterloo.  I've liked these type problems in the past.

      Wednesday, January 24, 2018

      1/20 Fold and Cut II

      Today started with an interesting whiteboard demo for the Problem of the Week.  This is a fairly straight forward combinatorics problem on a small 2^9 total set of possibilities. One of my students just went ahead and wrote a python program to brute force check for the answer.  While this won't work in a contest setting, I really like the use of computational math. If I had access to a computer lab and I knew everyone could program I'd love to do a whole session around the Project Euler. It would also make a really cool class structure to learn programming over a period of time.

      But the other thought experiment this generated was what is the purpose of some of these problems in the age of cheap computing?  This is well trod territory.  Open Middle problems as they are commonly formulated often make me think this is better done as a brute force search.

      My current thinking is that computational math is more interesting if its quicker to write a program than a formal method or if essentially you need to search a wide domain for the answers and there isn't much structure to help out.  Also problems can be modified to make the computational requirements more interesting. But this is obviously a fuzzy standard and I'm not sure how to align this with my general ambivalence about calculators.

      The problem was also an opportunity to hand out some geeky stickers I bought on a lark from    As an aside I went back and forth if the black sticker at the bottom should be read "No change in learning (bad) or peak learning (good)"

      For the main activity, I've been meaning to do another day focusing on the Fold and Cut Theorem since it went so well two years ago.  At this point I only have 3 or 4 kids left from that time and I thought I could provide enough different tasks and/or they had not reached the end the first time that it wouldn't be boring for them.

      This time around I went with a part of Erik Demaine's lecture  @ MIT.

      The choice was motivated by the fact Demaine developed a lot of the algorithms and includes some historical notes on the first examples in Japan.  But also I'm terrible at folding and there are a bunch of great demos in the first 10 minutes which the kids really liked. That saved me from a lot of practice at home.

      I paused at around the 9 minute mark and handed out worksheets I've used before from Joel Hamkins:

      These work great even for older kids.  While circulating I just made sure to periodically have everyone throw out their scrap paper and to emphasize the role of symmetry in any of the solutions.

      (Some handiwork)

      Finally I reserved 10 minutes at the end to go further in the video and watch the explanation of the straight-skeleton method.


      Another slightly modified AMC problem.

      In 1998 the population of a town was a perfect square. Ten years later, after an increase of 150 people,
      the population was 9 more than a perfect square. Now in 2018, with an increase of another 150 people
      the population is once again a perfect square.  What was the population in all three years?


      MathCounts Prep 1/30
      Olympiad #3  2/6
      UW Lecture 2/13

      Friday, January 19, 2018

      1/17 Graphs and Paths

      This week I saw a numberphile video with a fairly charming problem that inspired me:
      Can you find a way to arrange the numbers 1 through 15 in sequence such that every pair sums to a perfect square?

      I decided I wanted to do a graph theory day around this.  This goes well on a whiteboard so I had all the kids work on it for about 10-12 minutes. Most found a solution faster than I expected. In retrospect this seems more difficult than it really is since there are only four square sums to consider 4,9,16 and 25 and its clear  there are lots of pairs that sum to 25 1 + 24, 2 + 23 etc and very few that sum to 9: 1 + 8.  To keep pacing on target I had groups that finished early try adding numbers on. I also asked the kids to consider why was this happening at all.

      [If I repeated I definitely would stress this question: Is it expected that this is possible and why or why not?  What patterns related to the square sums affect the likelyhood?]

      After stretching to allow most kids to find the solution we had a group discussion. No one had considered this in terms of graphs so after all the kids were done explaining I showed Matt Parker's solution. This was a good bridge to do a quick discussion about what is a graph, what is an edge, node and degree.

      Next I introduced the classic Bridges of Konigsberg  problem.

      This was a risk because I assumed some kids had seen it before. So I just outright asked who already had worked on it at the beginning. Interestingly most hadn't so I had everyone satisfy themselves like the townspeople that they couldn't find a path for a few minutes. Then we had a discussion about whether there is a way to prove its impossible.  No one made the jump to counting the degree of the nodes. So I talked through Euler's logic.  I think this could be broken apart more formally by asking the kids to create the graph equivalent themselves and then creating other graphs, classifying them and looking for patterns.

      From there I had less luck creating the problem sequence. So I went with a few problem sets from the chapter on graphs in Jacobs "Mathematics a Human Endeavor".  I liked  the problems in the sets but I knew from experience the format was less than ideal. So I gave a packet to each group and had them focus on finding their favorite problem to show to the group at the next break.  By circulating among groups I was mostly able to keep forward progress going through questions but its hard work.   I'm continually tempted to do a deep dive on a topic but I'm usually still better off creating a coherent problem set stream that come in a few chunks on the whiteboard with discussion interleaved.

      Of the set, there was a Classic Hamiltonian Path problem (find the loop that visits each node below):

      That  I think works well and another maze problem that I would probably break out.

      So overall I think this day was decent but with one or two more Euler/Hamiltonian problems added on (and I'll keep my eyes out for them) I think this could be really tightened.

      Problem of the Week:
      I'm feeling the AMC10 problems more recently so I went with this probability one:

      For the future:

      Saturday, January 13, 2018

      Fun with Pentagons

      I'm in the mood for a geometry walk-through. I'll start out by saying this one has tons of solutions. I've thought of 3 or 4 and seen several additional ones (one of my favorite parts of geometry.)  I tend in this case to prefer the synthetic to trigonometric solutions but  if you add that \( cos(36) =  \frac{\phi}{2} \) or any variant rather than blindly calculating a decimal I'm good.

      This is the second interesting pentagon problem I've seen in a week or so. With this one, I immediately thought I'll be disappointed if the golden ratio is not embedded somewhere in the answer.  When playing around I spent some time angle chasing and looking for similar triangles. This led to several different ways to find the ratio. I've included the simplest one below.

      First I assume a regular pentagon of side length 1 for the rest of this discussion. Secondly, I'm going to briefly discuss how the golden ratio is found within the figure.

      If you look at 4 points on the pentagon (A, C, D and E) its clear they form a cyclic quadrilateral with three sides of length 1. Further all the other sides and diagonals have the same length since they are all in congruent triangles.

      Let \( d = \overline{CE} = \overline{AD} = \overline{AC} \)
      Using Ptolemy's theorem:   \(1^2  + 1\cdot d = d^2\)  Solving you get \(d = \frac{1+\sqrt{5}}{2} = \phi\)  also know as the golden ratio.

      With that result in hand I now did some angle chasing:

      I found three 36-54-90 triangles: DHK, EDI and ACG (which are outlined in red above).  In addition we already know that:

      • \(\overline{EI} = \frac{\phi}{2}\)
      • \(\overline{AC} = \phi \)
      • \(\overline{DE} = 1 \)
      • \(\overline{HI} = \overline{HK} = b\)
      • \(\overline{AG} = 2a \)
      So now we can apply the similar triangles:

      From DHK and EDI:
      $$\frac{\overline{DH}}{\overline{HK}} = \frac{\overline{DE}}{\overline{EI}} $$
      $$\frac{\overline{DH}}{b} = \frac{1}{\frac{\phi}{2}} \text{ or } \overline{DH} = \frac{2b}{\phi}$$

      Then \(\overline{DI} = \overline{DH} + \overline{HI} = \frac{2b}{\phi} + b = b\cdot(\frac{2}{\phi} + 1) \)

      Now look at EDI and ACG:
      $$\frac{\overline{DI}}{\overline{DE}} = \frac{\overline{AG}}{\overline{AC}} $$
      $$\frac{b\cdot(\frac{2}{\phi} + 1)}{1} = \frac{2a}{\overline{\phi}} $$


      $$\frac{a}{b} =  \frac{\phi}{2} \cdot (\frac{2}{\phi} + 1) = \frac{2 + \phi}{2}$$

      Note: there was a fun alternative presented online by  @asitnof using areas rather than similar triangles:

      Again we start with the cross diagonals being phi in length but instead find 2 different expressions for the length of the triangles. One based on the incircle and the second on the base and height.

      Wednesday, January 10, 2018

      1/9 Olympiad #2


      By today, I was up to 11 boys and 6 girls. So I'm beyond my target size of 15. I also had a bit of a challenge in that 2 kids hadn't shown up the previous week when I focused on introductions and I was primed to do an Olympiad today.  My main strategy here was to be honest with the newcomers via email and send them a practice Olympiad ahead of time as well as stressing that we'd be more "math circle" oriented in future weeks.

      New Largest Prime

      To start up, I decided to start this session with a quick mention of the recent discovery of a new largest prime:  277,232,917-1 which has 23,249,425 digits. My main point was to reinforce that new mathematical discoveries are occurring all the time and the field is evolving.  But in the ensuing discussion one student brought up the  factoring in public / private key encryption.  (As an aside, someday I'd love to do a numerical computing activity like implement some of the RSA algorithm.) This was a great coincidence since I had been planning to talk about that anyway.

      Stealthy Skills Practice

      Thinking about the new prime and the connection between factoring and encryption beforehand I came up with the following quick activity.

      1. Breakout into pairs. I had everyone choose someone they didn't know well as a partner.
      2. One person is the encoder and picks two numbers less than 200 and multiplies them together.
      3. He or she then gives the result to their partner.
      3. The second person then is "the hacker" and has 5 minutes to see if they could find a way to non-trivially factor this product.

      This was meant to serve several purposes. One I wanted the kids to build relationships especially with the two new students. Secondly, it was a great quick demo of the difficulty of factoring and why its so useful for encryption (I had I think only 3 pairs crack the code out of the group).  This led to a few interesting followup conversations.  But also equally important just like last week with some of the 2018 problems this was a chance to practice factoring/multiplication/division in disguise.   Watching kids work through basic computations, I'm always looking for more chances to practice skills which in theory they know but in practice could use a little reinforcement. If I were running a real class I might buckle down and use a review worksheet like those on  But in this context I worry about keeping the kids engaged and maintaining the separation between recreation and school.

      The main activity for today was the 2nd round of the MOEMS Olympiads. If you've been following along, you'll remember I've been moving these around quite a bit to fit our schedule and am about one test behind the official schedule.  Overall on first glance, I believe the kids did a bit better than the first time even though most of them took longer to complete.  My only disappointment was that after going through a speech about reading the directions and making sure to answer the question that was asked I still had a few kids still answer a question asking for a whole number less than 1000 with values that were (much) larger than it.  On the bright side when I had everyone demo answers on the board, I had tons of volunteers and was able to have almost everyone of the new students come up to the whiteboard.  So we're already on a great start.

      As usual, I'm not allowed to directly discuss the problems but I by coincidence saw a very similar problem in AMC10 to my favorite one from the set today that I'm going to discuss instead.

      2016 AMC10B problem 18:

      "In how many ways can 345 be written as the sum of an increasing sequence of two or more consecutive positive integers?"

      What I find interesting in these problems is the different behaviors for odd and even numbers.

      First for odd series with 2n + 1 members, if you write the  sum as:

      (x - n)  +  (x - (n + 1)) +  ...  + (x - 1) +  x   + (x + 1) .... (x + n) its easy to see the sum is just (2n+1)x

      That implies for all the odds (2n+1) saying that such a sum exists is equivalent to saying that number is  a multiple of 2n+1.

      What's also fun is that looking at the series another way you get:

      x + (x + 1) + (x + 2) ....   (x + n - 1) which is equivalent  to   nx + T(n -1) where T is the  triangle number function. So putting that together you have an informal proof that all the odd triangle numbers are also divisible by their index.

      Then looking at the evens  (2n) which are bit more tricky:

      (x - (n - 1)) +  ...  + (x - 1) +  x   + (x + 1) .... + (x + (n - 1)) +  (x + n)   you get  2nx + n or  n(2x +1)
      In other words the even series (2n) are always a multiple of n and some odd number.

      Returning to the original question, this all means its really at heart a question of factoring!


      I saw a similar problem online somewhere in the last few weeks and although I couldn't find the original, I decided to construct my own version.  This is actually a fairly straightforward linear system once you deal with the fact the lines continue on beyond the page so I'm hoping for a lot of participation.