Wednesday, March 21, 2018

3/20 Visible Math

This week started with a walk through of the MathCounts problem that I gave out last week to do at home.

Six standard six-sided dice are rolled, and the sum S is calculated. What is the probability that S × (42 – S ) < 297? Express your answer as a common fraction.

This was the last question in the sprint round at Chapters.  As I remember from the stats almost no one at the entire contest finished it correctly making it the hardest of the set.  I decided this would make for a good communal walk through because so many of the kids had seen it once and it hits a couple of different themes.   However, that's also the weakness of this problem. Conceptually its a bizarre hybrid  of a counting problem and a quadratic inequality neither of which naturally goes with each other.   I actually mentioned this to the kids. The phrase "franken-problem" might have been used.

At any rate, I started with the basics and asked some background questions:

  • What is the range of values for the sum of the dice throws?
  • How many total combinations are there for 6 dice throws in a row?  Why?
  • What is the most common sum / what would a probability graph look like?
This part was very approachable and the kids easily supplied various answers. So it was time for the quadratic inequality.  First I asked how many kids knew how to solve this algebraically? (Some of the room have not covered this at all)  It turns out even those kids with Algebra actually used guess and check anyway. There are only 31 values after all and its not too hard to just plug them in and see what happens.  The risk here is missing there is a range at both ends of the curve which I mentioned.

I had one volunteer who brought the equation into almost standard form but no volunteers to finish the process. So I demoed the formal method myself.
  • Factor  to:  (S-33)(S-9) > 0
  • Do a parity check: both factors are positive in which case S > 33 or both factors are negative in which case S < 9.
  • Notice the symmetry.
This felt new to the room and the work with signs of the inequality also exposed some conceptual weakness. So something to look for more problems to do in another context.

From here the problem becomes more standard and I had the kids do the case work on numbers of combinations for the 2 ranges.  We've been doing small amounts but could also use more combinatorics exposure.

That covered, I was ready for the fun part of today.  I've been looking at George Hart's site and was fascinated by some of the constructions. So I chose the sample one: to try out. 
Over the weekend I tested the templates and built my own ball:

It was a bit tricky, my ball almost fell apart at one time and I misplaced a few triangles leading to a dead end all of which gave me some ideas for how to guide when the kids tried it out.  Its really important to stress being precise when cutting the slots and also to work together when building the ball out to hold it together.

Beforehand I pre-printed the templates at a copy shop on 110 lb card stock paper. I also bought some thicker colored card stock which couldn't go through a copy machine and required tracing. I then mostly followed the lesson suggested on George Hart's site. We worked through discovering combinations of 3, 4 and five triangles first before really working as group. It took the kids the entire rest of the hour to build the balls once in white and then again in a multicolored version. 

This last one above was the most hard fought version. This group was the least focused and sloppiest cutters. So there were a few weakened triangles in their set. I kept coming over for a bit and helping them move forward with advice for kids to help hold the structure in place etc. But then in between when I went to work with others it tended to collapse.  Finally, I decided I really wanted everyone to achieve success and I should stay in place until they finished. I had them substitute in some borrowed extra triangles from the other groups and basically guided them through the tricky middle stage when the ball is most unstable. They finished right at the end and there was a literal cheer from the group. (I was extremely relieved)

The other groups actually made it through the multi-colored version where I had them try to create a symmetry in their use of color:

I was hoping to have enough time to discuss the extension questions about the combinatoric aspects of the colored balls but we ran the clock down.  As usual for me, I worried about the exact opposite case and had printed out the next template for early finishers which no one needed to use.   I'm currently testing this at home.  (Someone has to use the card stock.) Based on that experience the second ball is quite a bit more difficult to assemble and I'd budget much more time for it / prepare for some dexterity challenges.  That said, overall, I highly recommend this project. It was definitely a crowd pleaser!

(Its a bit like the 2nd death star right now)

This one comes from Matt Enlow and is an interesting number theory experiment.

Wednesday, March 14, 2018

3/13 Pi Day - 1

This is my fourth experience with Pi Day or "Pi Day - 1" as I called it since we meet on Tuesdays.



In a nutshell, because there's pie to eat, the kids always have fun.  But I was reminded of another perspective today from @evelyn_lamb

"Pi Day bothers me not just because it celebrates the the ratio of a circle’s circumference to its diameter, or the number 3.14159 … It’s also about the misplaced focus. What do we see on Pi Day? Circles, the Greek letter π, and digits. Oh, the digits! Scads of them! The digits of π are endemic in math gear in general, but of course they make a special showing on Pi Day. You can buy everything from T-shirts and dresses to laptop cases and watches emblazoned with the digits of π."

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I'm pretty much in total agreement with above. I've gently ranted in the past about pi digit memorization contests and other such trivialities.  But as her article continues, there was a man behind the holiday, Larry Shaw the recently deceased director of the San Francisco Exploratorium. I think his vision was more than just eating pie but it was also an incredibly whimsical gesture which is why I believe its had as much cultural resonance.

So I take the day partly in that spirit of whimsy and also with the mission to always ground it in circle geometry in some way and as said at the start, the kids always have fun celebrating.  Mathematics doesn't have enough moments like this especially in school.

This year I decided to go back to the  basics. I had initially toyed with talking about the unit circle and the derivation of radians versus degrees but on reflection I found  so much material that I couldn't fit that in.  Instead, I started with a survey of student definitions of pi (while they were eating).  This was surprisingly solid. The phrase "ratio of circumference to diameter" came up almost immediately. I then took a poll of how many kids had already done activities in class where they measured circular objects of some sort and divided them  by their measured diameters to find pi approximations.  Again, almost everyone had done so often several years ago in Elementary School.

So with everyone convinced already pi existed and it had a value it was time for some deeper questions. The first one I posed was "Is measuring a single object a good way to prove pi's existence?"  We chatted a bit about accuracy and sample sizes as well as whether from a mathematical perspective we can ever prove something from samples. My favorite version of this is
"What if only ordinary people sized circles have a ratio around pi and if we could measure microscopic or macroscopic versions we'd find something different?"

One of the kids then suggested approximating the circumference of a circle with polygons so we then did that on the board for the hexagon version.  I cold called in this case which I usually don't do to get a student to sum up the perimeter of the hexagon arriving at pi is approximately 3.

From there we took a quick digression to also do the area of a circle visual proof where you cut the circle up and form a rough rectangle that is pi*r by r in size. Again I had the kids fill in and compute the area.

Finally I noted that we don't actually compute pi to a billion digits using geometry and asked if anyone knew of other ways to get it.  This was a new idea for the room and a good setup for the 2 videos I chose for the day.

The first was this amusing (there were a lot of genuine laughs while watching) video of Matt Parker computing pi by hand using the alternating series 1 - 1/3 + 1/5  - 1/7 ....

But of course this doesn't really explain why this works only that it appears to do so. So I also picked the very ambitious following one by 3blue1brown:

Its about as approachable as its going to get with this amount of background knowledge but still a stretch. I stopped several times to ask questions about some of the background concepts. There are several potential stumbling blocks here:

  1. law of inverse squares
  2. Inverse pythagorean theorem
  3. The general abstraction model used
  4. The number line can be thought of as a curve.

The last one was the one  I chose to focus on the most and I framed it as a thought experiment "What if the number line isn't really a line at all but a curve, we're just at a small portion of it and just like with a curve if you magnify enough it appears to be straight."  My hope is that if nothing else stuck that idea was interesting and thought provoking (hello Calculus in the future)   My informal survey is that most kids found it interesting but I may have had one where this pushed too far.  So I am planning to do a little preamble next week "Its ok to give me feedback if you found anything too confusing and I also sometimes want you to focus on the big ideas in moments like this even if the  details aren't accessible yet"

I gave out the last problem from MathCounts this year now that it was released:  

Its actually a fairly awkward merge of quadratic inequalities and dice counting problem but I wanted to provide a capstone to the kids experience there and dig into how to solve it.

Thursday, March 8, 2018

CoCa Photo Diary - Art Math Intersection

I had a chance during lunch to look at Dan Finkel's brainchild at the Center on Contemporary Art.

Its a small space but they filled it with a lot of math related art. Bonus, I recognized several of the mathematicians who participated.

Wednesday, March 7, 2018

3/6 Infinite Countable Sets or more fractions

It was a big weekend for the Math Club or should I say team. We finally participated in the rescheduled MathCounts chapter contest. I was very lucky the new date worked for me personally since we were out of town the prior two weekends and all of the students who had already signed up amazingly also still came out. Overall, I had a great time and from what I can tell debriefing the kids they did as well. The format was fairly intimate. There were 10 schools participating with around 80 sixth to eight graders. During most of the rounds I hung out in the coaches room and chatted. This was a lot of fun. I met teachers from St. Ann's, Lakeside, Kellog M.S. and Hamilton.  I actually told the kids later when we were talking about the day that this was my favorite part. As the kids finished and burst into the hall, I checked how everything was going and how the difficulty level went.  Finally after a nerve wracking countdown round we had one overall 7th place winner and a 4th place team. That was good enough to let us go to the state competition this weekend! What's gratifying is most of the team members were 6th graders and the teams that placed above them were all eight graders so I think there is headway to grow over the next few years (another message I gave everyone)

During the actual club meeting today I had all the kids talk about their experiences at MathCounts to encourage each other. Again everyone even those who hadn't won anything seemed upbeat. I also gave my long but true speech about focusing on the fun parts of the competition and not the absolute outcomes and finding the joy in the math.  Its true but perhaps hard to see in Middle School that the kids who keep going will ultimately benefit regardless of trophies. So as usual, I still worry about the discouraging aspects of these meets but it seems to have gone well.

After this talk,  we briefly went over the old problem of the week. I only had one student really work on it so I'm thinking about what to do to refresh. This weeks problem is quite a bit easier  and approachable  which may help. I'm also thinking about different types of problems and to remember  to talk about participation at the beginning and end of each session for the immediate future, I'm hoping to get back to near half of the kids working on this.

For the main task of the day I chose a  topic from the recent Math Teacher's Circle magazine,  counting the set of rationals. In addition to looking interesting, this tied in well with 2 weeks ago on Farey Sequences:

Before starting though I wanted to warm up a bit with a small problem I'd seen on twitter. So I had all the kids work on the whiteboard with a number line to find where to place 1/4 between 1/3 and 1/5. I didn't supply any hash marks or much more than a simple explanation of the problem.  I was gratified this time that almost everyone came up with an accurate answer.  Universally kids chose to convert the fractions into the GCD denominator of 60 and place 1/4 = 15/60 between 20/60 and 12/60.  On review as  a group, I also asked since 1/4 is not there what is the number in the  exact middle which was a good followup question.   Note: for some reason when  I did it myself beforehand I chose to calculate the difference between each endpoint and 1/4 and then find the ratio of the two distances which no one else did.

With that covered we dove in and  I described the hyperbinary system. It uses the binary place values but in addition to  0 and 1 you may also use  2 in every digit.  i.e. 27 = 0x11011 AND 0x2211  This took a few examples to make clear. From there I had everyone start to make a chart of the first 15 numbers in hyperbinary and how many representations each had.

The next step was to get everyone to find a pattern in the chart.  This was only partly successful. The kids eventually identified the left hand rule where  b(n) = b(2n+1)   but finding b(n) + b(2n+1) = b(2n+2) proved more difficult.  So in the interest of time I showed them this one. I'm still brainstorming how to do this a bit better in the future (and I really want to repeat this again since the whole activity is fascinating). One note here: its super important to keep these 2 rules in mind yourself and practicing write beforehand is very helpful.

We then moved onto the big discovery the  Calkin-Wolf Tree

Again I described the rules for the tree and had everyone generate them on the whiteboard. I asked kids who finished to see if they could find a relationship between the tree and the previous hyperbinary numbers chart. We had just enough time for one student to discover they were identical and wrap up a bit as a group. So I pointed out a few more interesting facts about the tree we didn't have time to work on like the presence of every rational reduced fraction.

Overall,  this went well but I could definitely improve the experience. I think I had between 50-75% engagement over the whole session which was a bit too low to my taste. The sustained effort by the time we were working on the tree was definitely at the limit for some students.   On the bright side those who stayed engaged were very excited. I think what might work better would be to do both parts simultaneously and let the kids move between white board stations. At the end we could then look for the patterns between the two parts as a group.

Thursday, March 1, 2018

Carnival of Mathematics 155

Welcome to the 155th Carnival of Mathematics which collects a sampling of interesting math(s) related posts from around the web. This is my first time hosting and as my passion is topics for middle school math clubs you'll see a few of my personal choices. For all those interested in Carnival of Mathematics future and past, visit The Aperiodical  where you can also submit future posts.

"Chain of Circles - Daniel Metrard @dment27"

To start off here's a few facts about the number 155 I found on the wikipedia:

155 is:
There are 155 primitive permutation groups of degree 81. A000019

If one adds up all the primes from the least through the greatest prime factors of 155, that is, 5 and 31, the result is 155. (sequence A055233 in the OEIS) Only three other "small" semiprimes (10, 39, and 371) share this attribute.


Patrick Honner's Favorite Theorem
"In this episode of My Favorite Theorem, Kevin Knudson and I were happy have Patrick Honner, a math teacher at Brooklyn Technical High School, as our guest. You can listen to the episode here or at I rarely have cause to include a spoiler warning on this podcast, but this theorem is so fun, you might want to stop the episode around the 4:18 mark and play with the ideas a little bit before finishing the episode. Parents and teachers may want to listen to it alone before sharing the ideas with their kids or students."
This entire series at Scientific American has been really fun to read/listen to. This month's exploration of Varignon's theorem may be the best one yet.

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Pythagorean Proof 
Loop Space

An interesting twitter thread from above led me to this post. I've experimented with how to teach the Pythagorean therorem in the past several times and like how this approach based on similarity differs from some of the more commonly used algebraic techniques.

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Fun with Fractions—from elementary arithmetic to the Putnam Competition the first 1/2
Dan McQuillan
"Elementary discussions and good questions in grade school can prepare students for far more difficult challenges later. This post provides an example, by starting with simple fraction questions and ending with a Putnam Mathematical Competition Question (intended for stellar undergraduates). It also features atypical ways of comparing fractions. A much shorter discussion of these problems is possible; this discussion reflects an attitude of starting from little and gaining quickly."
We had recently been working with Farey Sequences:  so this article had special resonance for me.  The extension at the end is particularly good.

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Triangulations and face morphing
David Orden
"This post talks about one of the easiest mathematical tools for morphing, using triangulations, and explains recently published results about morphing planar graph drawings."
A very nice overview and perhaps a starting point for further reading.

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Fun Not Competition the story of My Math Club
Dr. Jo Hardin

"For almost three years, I have spent most of my Sunday afternoons doing math with my daughters and a group of their school friends. Below I detail why and how the math club is run. Unlike my day job, which is full of (statistical) learning objectives for my college students, my math club has only the objective that the kids I work with learn to associate mathematics with having fun. My math club has its challenges, but the motivation comes from love of mathematics, which makes it fun, and worth every minute."
This is a lovely personal account of Dr Hardin's experiences working with young children. I'm a very strong believer in the power of Math Circle's to impact students so hopefully this will motivate someone else.

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The many faces of the Petersen graph
Mark Dominus

The Petersen
graph has two sets of five vertices each.  Each set is connected into
a pentagonal ring.  There are five more edges between vertices in
opposite rings, but instead of being connected 0–0 1–1 2–2 3–3 4–4,
they are connected 0–0 1–2 2–4 3–1 4–3.

"The Petersen graph is a small graph that is an important counterexample to all sorts of things. It obviously has a fivefold symmetry. Much less obviously, it _also_ has threefold, fourfold, and sixfold symmetries! You can draw it in many ways and it can be really hard to tell that they are all drawing of the same thing!"

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In game development and 3D image processing it is common to represent 3D rotations not as 3 x 3 matrices but as quaternions. I wrote a somewhat long read at the end of last year describing the relationship between SO(3), the space of 3D rigid rotations, and the unit quaternions. I think readers will enjoy the use of heuristic visualizations to uncover the true 'shape' of SO(3). Also, with SymPy, a wonderful symbolic computation library, I compute representations that give coordinates on SO(3). The calculations are really involved, so SymPy is super helpful; all code is linked within the post.
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DIY Pattern Maker

This is a visual exploration of patterns as well as inventive recycling that looks fun to use in a classroom. 


If you've made it this far and are involved in Mathematics Research I would love it if you would consider contributing some answers to Questions for Mathematicians that I've been compiling for my kids and thanks for reading this post. Either just add a comment on the page or email me.

Monday, February 26, 2018

In praise of the Rational Roots Theorem

First some personal historical background. In my school district, you could do Algebra in middle school but unlike a standard class it only covered linear equations.  When I was in High School after doing  Geometry in 9th grade, I entered a 3 year accelerated math sequence that terminated with AP Calculus BC.  For the first year we did a semester going over quadratics leading up to the derivation of the quadratic formula and a semester of trigonometry.   So for all intents and purposes, I didn't learn anything from the standard Algebra II curriculum. Interestingly, this didn't have any particular consequences and as time went by I learned some of the topics when necessary and required for something else. I remember thinking in College, "I wish I had covered more Linear Algebra/Matrices" but never really "What's Descartes' rule of signs?"

However, over the last few years my affection for two particular tools from there has grown quite a bit: The Rational Roots Theorem and Polynomial division.  First, these are often under attack and dropped (just as in my own experience).  Its not unusual to see people wonder online: what are the real world applications of these or will they ever be used again?  In High School, I might have said you can always graph and use approximation techniques like Newton's Method when these come up. More significantly,  the existence of Wolfram Alpha has made generalized solutions to cubic and quartic equations easily accessible (if not derivable) From my perspective, they are two basic polynomial analysis techniques that offer a gateway to understand higher degree polynomials.  That understanding is valuable in itself but in addition they offer a fairly general technique for a lot of algebraic puzzles that I try out and I find it extremely satisfying to be able to analyze these with  just pencil and paper.

Example 1:

$$x^2 - 13 =\sqrt{x + 13}$$

This looks not to hard at first until you square both sides to get rid of the radical and realize its a quartic in disguise:

\(x^4 - 26x^2 + 169 = x + 13\) => \(x^4 - 26x^2 -x + 156 = 0\)

A common strategy at this point is to look for clever factorizations.  But its often really hard to see where to start. In fact, I find these are often easier to derive backwards after you know the roots anyway.

So let's start with a quick graph of the  functions. This could be done by hand but I'll use geogebra here. The left hand side is a parabola with vertex at (0,-13) while the the right hand side is half of the rotated 90 degree version of the same parabola with a vertex at (-13,0).  If you're looking for factorization this symmetry is something that provides an avenue of attack. But for our purposes it also shows us there are only 2 real roots in the quartic and approximately where they lie.

Here's where the Rational Roots test comes in. Since 156  = \(2^2\cdot 3  \cdot 13\) It says that if there is rational root its going to be either: \( \pm 1, \pm 2, \pm 3, \pm 4\, \pm6, \pm 12, \pm 13, \pm26,  \pm 39, \pm 52, \pm 78\) or \( \pm156 \)  That's a bit daunting but looking at the graph or the behavior of the functions indicate we really only need to test smaller values and 4 is probably the most promising.

I just plugged that back into the original problem rather than doing the quartic and indeed -4 works out. (This is a bit of a cheat since not all the quartic solutions are also solutions to the original problem due to sign issue with the radical but if it does work then you're golden.)

At this point we know know x+4 is a factor of the original quartic and we can divide it out to get a simpler cubic equation.

Apply polynomial division \(\frac{x^4 - 26x^2 - x + 156 }{x+4} = x^3 -4x^2 - 10x + 39\)  to get the remaining cubic part of the equation.

Now once again we can apply the rational roots test but on the much smaller set {1,3,13,39}.  Its clear from the graph that none of these are going to be a solution to the original problem and that again the smaller ones are more likely. So starting at 1, I find that 3 works out  (27 - 36  - 30 + 39 = 0).   That mean x -3 is another factor.  Interestingly you can see why it doesn't work:  9 - 13 = -4 while the square root of 13  + 3 = 4.  So the inverse sign changes have interfered (but if the bottom of the sideways parabola were present that would be an intersection point).

Once again apply polynomial division  \( \frac{x^3 -4x^2 - 10x + 39}{x - 3} = x^2  - x +13 \)   Having factored the quartic down to an approachable quadratic we can now  apply the quadratic formula to find two more solutions:  \( \frac{1 \pm \sqrt{53}}{2} \).   Either by testing or looking at the graph we can see \( \frac{1  - \sqrt{53}}{2} \) is the second solution while its converse again lies on the intersection of missing bottom half of the sideways parabola.

Extension for another time:  We have 3 and 4 wouldn't it be nice if 5 also showed up (and this is tantalizingly close to the generator function for pythagorean triples in the complex plane)?  Is there a general form to the intersections of this type i.e. a parabola and its rotated counterpart?

Example 2:

Find the integer solutions to: \(x^3y^3 - 4xy^3 + y^2 + x^2 - 2y - 3 = 0\)

This again looks fairly complex and of degree 6 on first glance. But lets try experimenting with values of x and see what falls out:  [I'm going to only consider the x >= 0 for simplicity here but somewhat similar logic applies for the negatives.]

if x = 0 this simplifies to:

$$y^2 - 2y - 3 = 0$$ which has 2 integer roots.

if x  = 1 this simplifies to:
$$-3y^3 + y^2 -2y - 2 = 0$$
We can applies the rational root test and check the possible integers \(\pm1, \pm2\) with no hits.

if x = 2 this simplifies to:
$$y^2 -2y + 1 = 0$$ which has one integer solution.

Note the constant term flipped from negative to positive at this point and now something interesting happens

if x = 3 this simplifies to
$$15y^3 + y^2 -2y + 6 = 0$$
The rational roots test now is only going to give positive candidates and the higher degree terms start to dominate making it impossible for this to reach 0 with an integer. I.e. \(15y^3 + y^2  > 2y\) for all integers > 1.

Continuing on the same trend continues
if x = 4 this simplifies to
$$48y^3 + y^2 -2y + 13 = 0$$  For the same reason this is even worse  \(48y^3 + y^2  > 2y\) for all integers > 1.

So we can infer with this logic that no integer solutions exist above x = 2.

Wednesday, February 21, 2018


I'm on mid winter break with the kids for the week in Oahu.

In the meantime check out my collected problems which are almost at 30 in total:

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