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.