Tuesday, March 29, 2016

3/29 Euler Characteristic




And like that we're into the Spring session of Math Club. As I eluded to last week I have 5 new students including four new girls. So I brainstormed a lot about what I wanted to start with this week to draw everyone in. Basically, my goal was to find a game, that would be fun, participatory, and not require too much in the way of supplies. Along the way I found an interesting snap-cube challenge which I think I'll pick up in a future week now that I've found a teacher I can borrow enough of them from. Long term, I'm trying to decide which manipulatives to invest in; dice, checkers, snap cubes etc. for next year.  The problem is that I need bulk amounts for 18 kids and I want to buy something that offers the most possibilities for reuse. In the end I found a fun paper and pencil exercise: Criss Cross to try out.

But before I could start into the fun stuff we had to do introductions. I had everyone go around say their names, teachers and either why they had joined if they were new or their favorite part of math club if continuing. I also went over my rough charter points again: Charter.  The ones I tend to emphasize the most are respecting the classroom and each other, listening when others are working on the whiteboard and what to do if you get stuck. (Hint: its not about giving up)

The Criss-Cross game comes from the http://minerva.msri.org/files/circleinabox.pdf  (See page 77) and involves investigating the Euler Characteristic.  To summarize the rules, you break everyone up into pairs (or triplets) and have the kids draw the 3 corners of a triangle plus some dots in the center. Then they take turns connecting two dots with a straight line (no intersections are allowed and only 1 segment can be added between 2 vertices) until all the open moves are used up. Whomever can draw a line last wins. I picked two kids to do a sample game on the whiteboard and then drew a chart up with vertices, 1st player wins, and 2nd player wins for the columns.  The kids split up and starting playing games and generating data. This was a lot of fun for the room. There was a constant stream of kids coming up to add some results to the whiteboard. However, as the games progressed I noticed our results were particularly inaccurate.  At first, I asked for more games for a particular size (say 6 dots) to compensate for the mistakes. But the errors were making it hard for anyone to spot the pattern about which player always wins.

So I called for a halt and said "I think I can see some mistakes in our data lets try again to redo our data chart as a group." This time I  picked two kids at a time and had them play on the whiteboard with the whole group counting and keeping track of mistakes and we started with 5 vertices and added one more each time.  The second attempt was the charm,  The data was consistent now (the 1st player wins if there is an odd number of vertices and the 2nd if its even)  So I was able to have them also count faces and edges after defining what they meant to look for more patterns.

Very quickly I had one boy notice that the number of faces rose by 2 each time, and the number of edges by 3. We were running out of time so I talked about V - E + F and we checked if the characteristic was constant for the game. I then had the kids try out a square on their own.  By this point all the parents had arrived so I plan to follow up and work on the basic idea about why this works next week in the beginning.

The proof is  recursive and fairly simple. Each new vertex must end up in a pre-existing triangle and
follows the 2nd case above adding 3 new edges and 2 new faces. If I repeat again I'll also remember how easy it is to make mistakes and do more work as a group.

For the problem of the week: I chose another @five_triangles problem: (This involves tiling again which is a good tie-in with some our previous problems)



Friday, March 25, 2016

Looking for a more elegant approach




This is a continuation of my geometry problem solving posts. I spent the last few days thinking about the above problem from @five_triangles. This ends up serving as an example of a brute force approach versus a more elegant one.

My first step was to find that $\angle DBC$ is 45$^\circ$  and $\angle ABD$ is $15^\circ$ through a quick angle chase.  That immediately made me think of  trisecting $\angle DBC$ and  reflecting the triangle $\triangle$ BED across the segment $\overline{BD}$. This can be reflected two more times to produce this:



Note: $\triangle ABH$ is half of the big $\triangle ABC$ and the desired ratio is at least 3:1 since there are 3 reflections of $\triangle EBD$ within $\triangle DBC$ plus some other bits.  

From here I added the parallel line GD to EO which makes a trapezoid and some similar triangles. I was hoping that the remaining similar triangles could be easily combined to form a multiple of EBD but while they all had the same altitude this wasn't the case.

I then experimented with the similar triangles starting from $\triangle{BEO}$  That almost worked I could find BGD and BDC in terms of  BEO.  Using the ratio between GD and EO I  could even break the trapezoid apart to find the remaining piece but was left with a  ratio   $\frac{\overline{GD}}{\overline{EO}}$ embedded in my expression. 

So I went back to the drawing board and started with the top $\triangle{AGD}$ which is 30-60-90.

1. Let AD =  x Its possible to then find DG,  and ultimately the entire side AC in terms of x.
2. AHC is also 30-60-90 and since you have AC. You can find AH and HC in terms of x.
3. OH is in a 45-45-90 triangle so it has the same length as HC which is known.
4. The smallest triangle at the the top AGI is also 30-60-90 and similar to AEO.  Its one edge AG is already expressed in terms of x as well and we know AO from the previous step (AO = AH - OH)
5. From that you can finally find GE using the similar triangles. 
6. Putting all of that together you have altitudes and bases for the two triangles all in terms of AD and it simplifies down to 1:4.

This is mechanical but fairly messy and has a lot of expression with radicals.  

Trig Approach

After looking at my first two ideas I didn't like the complexity and I felt the ratio could be visualized more cleanly i.e. something should be quartered.  So I actually asked my friend Dan to try the same problem.  Here's his clever idea which utilizes some trig to quickly find the altitudes and cleanly shows where 1/4 comes in.
Call the intersection point in the middle Q.
Draw QC.
BQC is a 45-degree right triangle.
So the height of DBC is QC which is also BQ.
Angle QBE is 15 degrees.
BE is BQ * cos 15.
Altitude from E to BD is BE * sin 15.
So altitude of triangle EBD is altitude of triangle DBC * cos 15 * sin 15 = altitude * ½ sin 30
Ratio of altitudes is 4, so ratio of areas is also 4.

Final Improved Version



Playing with Dan's idea a bit more you can remove the need for trig.  

1. First note: $\triangle{EBD}$ and $\triangle{DBC}$ share a common base $\overline{BD}$.
2. Angle chase to find $\angle OBC$ is 45$^\circ$  and $\angle ABO$ is $15^\circ$.
3. Reflect $\overline{BO}$ and you get a right triangle BOC so $\overline{OC}$ is the altitude of $\triangle{DBC}$.
4. Drop another altitude $\overline{EG}$ for $\triangle{EBD}$ 
5. Now the trick is to find the ratio between the two altitudes. This is accomplished through following a series of isosceles triangles.
6. First extend $\overline{EO}$ to F. [I experimented with this auxiliary line because the 2 overlapping 30-60-90 triangles were so useful in my first try. I thought I would directly use its side ratios but as soon as it was there all of the following observations were visible one after another.] $\angle{EFC}$ is 150 so $\angle{FOC}$ is 15 and $\triangle{OFC}$ is isosceles.  Therefore $\overline{FC}$ = $\overline{OF}$.
7. Next angle chase again $\angle{DOF}$ is also 75 so $\triangle{ODF}$ is isosceles and
$\overline{DF}$ = $\overline{OF}$.
8. Then look at $\triangle{AOF}$ which is also isoscleses. So $\overline{AO}=\overline{OF}$
9. $\triangle{AEO}$ is a 30-60-90 so $\overline{EO}$ is 1/2 $\overline{AO}$
10. Its easy to verify then that $\triangle{OEG}$ and $\triangle{ODC}$ are similar 15-75-90 triangles.
11. Since  $\triangle{OEG}$  has a hypotenuse (EO) 1/4 of the hypotenuse of  $\triangle{ODC}$  (DC). The altitudes must be in the same ratio, and also the areas. 




Thursday, March 24, 2016

3/22 Winter Game Day

I had to hold back a laugh this week. While waiting for everyone to arrive in the cafeteria, one of the kids asked if we were going to have pie again. "If you join Math Club next year during the spring, I'll be serving pie for 2017 Pi Day." I replied. Never underestimate the power of  baked goods.

More seriously this was our last session for the Winter. We switch over without a break next week to the new term. I'm very pleased that my reaching out to the teachers and every parent I knew seems to have paid off. The number of girls will rise to 1/3 of the club. I will continue to do some recruiting from now on before the enrollment periods.

We started the session with a quick survey of what activities the kids liked during this term. Many of the weeks project were mentioned but by far the winner was: Fold And Cut. From there we reviewed the problem of the week:

The question is what is the area of the shaded region?  My volunteers all found variants on the same idea. If you draw a parallel line through the center dot. Then you can find the area of the two sub parallelograms since its twice the area of the embedded triangles.

The games choices for today were a bit limited by supplies. Right now most of my house is packed up due to a house renovation so I chose dots again, a logic puzzle from http://www.puzzlersparadise.com/article1013.html, the game of 24 cards  and to try out checkers stacks. See: https://mathenchant.wordpress.com/2015/08/12/the-life-of-games/







We didn't really have enough checkers for everyone to try at once so for this session I just had the kids try out the basic rules and think about strategies. My plan is to return on a further day and start talking about the optimization strategy and if that goes well to branch into surreal numbers properly.



Sunday, March 20, 2016

3/15 Pi Day 2016 (more or less)

One of these years Pi Day will actually occur on the day Math Club meets. Until that happens it also serves as a demonstration of  approximation for the kids. Luckily, this year one of my parents volunteered to bring in the pie so the only extra item  I took with me this week was a pie server.  We started with pie in the cafeteria and while the parent handed out the apple pie I frantically tried to get my laptop and video projector working. As it turned out a recent update had broken my wifi so I made an emergency dash up to the fourth floor. Fortunately one of the fourth grade teachers generously let us use her projector.

In the end, I was able to show the following Buffon's Needle video from numberphile:


I personally still find it fairly amazing how pi can show up in a monte carlo simulation just involving sticks. The real advantage of a video here is doing this in club would take half of our time and not be nearly as likely to produce as good of an estimate. Hopefully it was as interesting for the fourth and fifth graders. I ended up cutting it a bit short at the calculus section and just talked about the principles involved in general terms.

We then walked down to our regular space and continued with the problem of the day discussion.  See:  Temple Riddle The most important part of this discussion was bringing out the strategies the kids used. So I emphasized every time kids used various charts and notation to dissect the cases.  I'd definitely like to do some more casework type logic puzzles later this year.

Next I  recapped our circumference discussion from two weeks ago (see here) and asked what other basic formula involving pi could they think of. After one false start, a student volunteered the  area of a circle = pi * r^2.  Like the last session, I broke the kids up into groups and asked them to brainstorm for a few minutes for reasons why this was true. Of all the kid's ideas, my favorite one was to take all the rings from center out to the circumference and add them together. The boy with the idea didn't know how to carry through with that and I ended up telling him that it would definitely work but we'd also need calculus to show it. On reflection while an integral would work, its also not necessary.   You can imagine unrolling the circumferences of the circle from the center outwards. The one by the center would 0 in length and the one at the edge 2 * r * pi.  Effectively if you wave your hands a bit about the rate change being regular you'd find yourself with a triangle with a height of 2 * pi * r and a base of r. So I'll come back to the idea this week (I really want to give credit for some good thinking too).




What I actually used next instead is a visual proof I also really like where you cut the circle into pi slices and approximate a rectangle.


For the back half of the session I ended up handing out a pi worksheet from Math Counts . (Which I unfortunately can't link to). Finally for the problem of the week I used a recent problem from @five_triangles.



The problem is to find the missing triangle's area. This is a great one for my kids since it doesn't need anything more than some logic and knowledge of the triangle and parallelogram area formulas.

Friday, March 11, 2016

3/8 Olympiad #5

We finally reached the last Math Olympiad for the year. Earlier in the week I had received a mail from MOEMS giving some corrections for the third problem. So I tried to explain the ambiguity to the kids at the start of the test and totally flubbed it. I managed to completely mis-describe the problem basically the opposite of what the clarification email indicated and it took a followup question from one of the kids for me to realize what I had done.  I recovered and fixed my mistake but it was a bit mortifying. Moral of the story: not enough prep this week. And the secondary lesson, sometimes things go south and you have to recover and keep going.

Overall this was the trickiest of the five tests. I haven't checked the results yet but I expect them to be a bit lower. I'm also delayed because a followup email from MOEMS indicated they are doing an appeal process about the problem question and to suspend entering scores until they decide what to do.

For the quicker students I brought another medium kenken which I actually had printed a few weeks earlier and not used yet.  This was accidental but actually is a fairly useful practice. Keeping a few fun extra activities in your back pocket (or math bag in my case) gives you a little insurance if an activity goes faster than expected.

I really liked our whiteboard session afterwards when we discussed the answers. The kids surprised me with 5 different strategies for the first problem, a new record.  I'm also having pretty good luck getting everyone to listen through 20 minutes of the other kids showing how to work the problems. The one thing now is I sometimes see a kid patiently waiting their turn to show their ideas but not really listening to whether someone before them does the same thing. I usually try to link these explanations together afterward. For example, "That's great ___ we've now seen 2 different people using the distributive law to get problem b"  I'm not sure whether there is anything to do about this phenomena.

For the problem of the week I chose a riddle from the recent TED talks: Temple Riddle


After transcribing this it turned out to be quite wordy so we'll see how it goes. Also next Tuesday is the day after Pi Day. So I'm gearing up to celebrate. For sure we'll go over the area formula and how to derive it. I think we'll repeat the amazing race themed activity from last year.  If I borrow a projector again, there's a really fun numberphile video about Buffon's needle.  



By coincidence I found this awesome area related problem this last week: http://seekecho.blogspot.fr/2016/02/searching-for-rings.html

Oh and there will definitely be pie!

Wednesday, March 2, 2016

3/1 Fold and Cut

Even after two years, I'm still experimenting and searching for new material. Case in point, yesterday was notable for two changes.  I had decided after last week: pi day prep  that I didn't want to immediately attack the area of a circle formula and instead needed something really different. Then over the week I came across some rave reviews of fold and cut activities from @mrhonner. I also remembered looking at a numberphile video on the subject a while back and thinking it was fairly compelling. So I decided that I would go that direction.   This involved a few logistical risks.


  1. I don't have enough scissors/hole punches for all the kids. Therefore, I asked ahead for everyone to bring any they had. My parents are great but anytime I do something like this I have a semi-irrational fear that everyone will forget and I will be up the creek.
  2. I also had another parent lend me their video projector. So for the first time I could try out using a video in the middle of the session.  However, I didn't have enough time to test the equipment or confirm that the school's WiFi filtering would allow access to youtube.  So I gambled that I had enough to keep the kids busy while I worked through the kinks and a fallback plan if it failed to work.
  3. Anything with scissors and paper means extra cleanup

So we started up with Gummi bears in the cafeteria to celebrate reaching our completed problem of the week goal. And then while discussing the the kids' solutions to the tessellation problem from last week I was lucky and the video hookup worked almost immediately. 


This was a good whiteboard discussion.  Everyone who demonstrated came up with the idea of finding a tiling pattern that could create the image. We had two competing answers: A hexagon with one black square vs. a hexagon with 2 black squares for the base pattern.  I asked the room for various ideas on which was correct but there was no consensus. I decided to point out/ask that there were 6 black triangles per hexagon and each one was shared between 3 hexagons. Rather than telling them the solution I then asked the kids to test each pattern at home. If the test tiles could be combined to create the floor pattern then they had the correct solution. 

Moving on we turned off the lights and watched the following numberphile video on the fold and cut  theorem. 


After an annoying and slightly off color 5 second ad, as expected the video went really well. I'm hoping I can borrow the projector again for Pi Day and show another one that would pair really well with the day's activities.

We then broke out the scissors and punches for some experimentation of our own. I preprinted the following templates pdf file  from Joel Hamkins.  These also were a big hit. We worked on the 8 sheets for about 40 minutes. Most kids completed at least the first 5 pages. The hardest issue here was I had to assist in punching holes for some of the thicker patterns. (Who knew math club was also a test of physical grip strength)





This was actually quite messy and you can probably tell from the photos above. So as each parent arrived for pickup I asked to make sure that there child had cleaned up the holes. Despite my trepidation after a few minutes at the end the room was back to normal.



Tuesday, March 1, 2016

March Mathness Blog Hop

Welcome




Welcome to everyone who's coming here via the Hoagie's "March Mathness" blog hop. I imagine you have some interest in mathematics especially around your children's individual progress. You're looking for interesting resources to enrich at home. Perhaps you're worried about the school curriculum and are considering after schooling etc. In many ways this was the position I found  myself in a few years ago. I'm deeply interested in math and wanted to pass that on. So as my own children began entering school I started discovering what resources existed in my backyard.

And in many ways what I found was that this is a golden age for mathematics education and enrichment.  One of the earliest discoveries I made was a local Math Circle offering enrichment on the weekends, Then there was my introduction to Numberphile videos.  These were great hits with my sons. More formally, there's the innovative curriculum being developed by the Art of Problem Solving. If you look around you can find amazing stories about what kids are experimenting with and discovering.  What summarizes the moment really well for me is a recent article in the Atlantic: http://www.theatlantic.com/magazine/archive/2016/03/the-math-revolution/426855/

This is where  I was two years ago when I made the transition from a family oriented to group oriented focus.  What I realized then was that I could leverage my own passion and affect more kids. It was definitely a bit scary to step out of my comfort zone in front of 15 unknown fifth graders: first impressions. However, I believe it was and still is ultimately very rewarding. As I've said elsewhere, its tremendously gratifying seeing kids get excited about a concept or making a breakthrough or even just engaging enthusiastically with an activity. There's power in creating a peer group of kids who all are interested in the same subject.  For me this goes full circle back to my own fourth and fifth grade experiences where a key teacher made me realize that I too loved to do Math. I'm hoping at least a few readers take a moment and ask themselves the question "Can I do this too?" You don't have to be perfect or an expert on everything. Just like Math itself, running a club is quite creative and can go many directions. It also doesn't have to start very large. But a small effort can have a huge impact on your community.

Getting Started

I inherited my group: background which in many ways made things easier. If you're starting from scratch there are actually a few great places with some checklists.

  1. One of our local high schools has this manual: http://www.wastudentmath.org/content/clubs/plan/StarterPackComprehensive.pdf
  2. Math Circle in a box (one of the first documents I read):  http://www.mathcircles.org/GettingStartedForNewOrganizers_WhatIsAMathCircle_CircleInABox   
  3. Natural math has a really cool quiz to help clarify what your goals are: http://naturalmath.com/math-circles-1001-leaders-course/
In my mind, these are the actually the easy parts. Things become harder when you actually start meeting and going out to lead a group and try planning out activities. That's the area I mostly focus on in this blog.

Here's a few pointers into some of my past writing:
  1. Resource Page: This is where I collect other sites I've found useful: Resources. At the bottom of the page I've embedded links to my activity maps for the year (which are *mostly* up to date)
  2. My mega post on  what I learned over the first year: http://mymathclub.blogspot.com/2015/06/the-year-in-review.html. I'm a big fan of improvement through reflection.
  3. Looking back through the blog most of the posts with a date in the title reflect a log of what I did at that session and how it went. I try to focus on both mechanics. I.e. this took 20 minutes,  evaluation of how well the activities went and thoughts on what I could personally improve on. 

Some goodies for everyone

Finally here are three of my favorite discoveries from this year so far.

  1. The no-rectangles problem: here  and here
  2. Match stick (tooth stick in our case) puzzles: here
  3. "This is not  a Maths book"   here and here


I am joining conversation on math enrichment and math education with Hoagies Gifted Education Page. Check out Hoagies Gifted website and like Hoagies Gifted on Facebook!