Tuesday, December 22, 2015

Geometry Diversion / corny geometry pun.

I was looking at this one from @five_triangles on twitter and had an interesting search process to find the solution which is worth documenting. All told I probably thought about this over 3 days working 20 minutes each time I came back to it. At points I felt stuck and briefly comforted myself by thinking if I don't get this in a week I'll post for help on stack exchange. What kept me going was the knowledge that finding your own solution is way more valuable as is persisting with  problems that are not immediately obvious. Additionally, I think this shows a good example of using tools to help point a way towards a solution.

My first impression on looking at it was that the interior triangle is a Pythagorean triple and thus the angle $\angle HbHcHa$ is a right angle.  That implies the area of the inner triangle is $\frac{1}{2} \times 9 \times 12 = 54$.  That immediately sent me down an area based solution. I was particularly reminded of the problem from last year : http://mymathclub.blogspot.com/2015/04/simplifying-proof.html.  This time however there were line lengths rather than sub areas.

Nonetheless I divided the triangle into a box with four quadrants and started to play with relationships.  There were an obvious 3 Pythagorean theorem relationships on the outer box to work with plus the relation between the inner triangle and the box.

This seemed fruitful at first despite the messy squares roots:

The key realization from above is that $\overline {AB}$ is the sum of the 2 sides of the box i.e. a + b + c + d and that the 2 smaller triangles are both also isosceles right triangles.

In other words this is a geometric interpretation of $(x +y)^2 = x^2 + 2xy + y^2$ where after multiplying by 1/2 you arrive at $\frac{1}{2}(x+y)^2 = \frac{1}{2}x^2 + xy + \frac{1}{2}y^2$
or in terms of shapes the big triangle is equal to the upper isosceles right triangle + the rectangle + the lower isosceles right triangle.

I even found a few tantalizing areas:

But I was ultimately looking for the area of the entire inner box or some variant of an expression that looked like the side squared and I was stuck at that point.

At this point  I also played around  a little in Geogebra to get a sense of what I was aiming for. A triangle of side length 24 seemed to be about right.

I was trying to fall asleep the next evening when two ideas occurred to me in succession.

1. What if you just rotated the inner triangle on its vertex $Hc$?   The left side would be a bit longer than $\overline{AHc}$ and the right a bit shorter than $\overline{AHb}$. Perhaps the two differences would cancel each other out. This would leave the length at 21 which was a bit short but I drew a model and checked it first before ruling it out.

2.  However, that made me also reconsider the original two outer triangles. They are clearly similar because $\angle HbHcHa$ is a right angle.

Since I was looking for a cutesy solution I wondering what if they were also similar to the interior triangle. If that was the case then they were also 3-4-5 variants which I could find the lengths for immediately. I checked the resulting length for side $\overline{AB} = \overline{AHb} + \overline{HbE} + \overline{EHa}$.  This worked out to exactly 24.  I was about to celebrate when I looked more closely and decided to verify the center point P was at a right angle to the hypotenuse of the outer triangle as required. Nope: even visual inspection showed issues.  This did however make me think about that key last relationship again. The line $\overline{PHa}$ actually forms another right isosceles triangle and its height is equal to its length.

So its easy to check  if $\overline{AD} - \overline{HbA} = \overline{HcE}$.  At which point I realized I had the solution.  I redrew the following diagram:

The 2 triangles are similar and the left one is a 3/4 scaled version of the right one.  So if
I let the y be the height of the right triangle and x its length.    The smaller triangle's height is 3/4 x and its length is 3/4 y (note the inversion since its flipped).

That means the following two relations needed to be solved:

1. The Pythagorean relationship: $12^2 = x^2 + y^2$
2. The length of $\overline{PHa}$ is the hypotenuse of a 45-45-90 and
    $y -  \frac{3}{4}x = x$

Combining those two and you get $x = \frac{48}{65}\sqrt{65}$ and the whole length is $3 \sqrt{65}$ or ~24.

I still had a moment of doubt because I did not expect an irrational number so I double checked exactly in geogebra and sure enough it worked.

I think  the author liked the fact that answer eludes to 365 days in the year but I would reframe to ask for  the area of the outer triangle to remove the pesky square root myself.

Wednesday, December 16, 2015

12/15 Olympiad #2

Once they get started, the MOEMS Olympiads come quite frequently. This month's version surprised me because in my first glance I had thought we had only had one prior to winter break.  So despite it being the end of the quarter we did the contest today instead of just a wrap up celebration.  As I mentioned last week, I jiggled my scheduling and did the Math Game Day then to compensate for this.

Before starting even the review of the problem of the week, I had everyone who went to the Knights of Pi contest give a trip report to the rest of the room. Amusingly, the most fascinating piece of information was the fact pizza was served at the end. Hopefully, this can increase interest for the next contests we go to.

This month's MOEM Olympiad was surprising for another reason. When I went through it before hand I thought it would be particularly easy.  Each question took me under a minute and they seemed fairly straightforward except perhaps the last one which was still amenable to a guess and check strategy.  Nevertheless, in practice, the kids didn't do quite as well on this one as last month.   To start off as usual I gave my normal advice before starting.

  1. Sign your name on the answer sheet. 
  2. Always read the questions carefully and make sure to answer what they are asking.
  3. If you have extra time, go back over your answers carefully and try to solve things a second way if you have lots of time to double check.
Sadly, I had one kid forget to sign his name and then a bunch of kids failed #2.  The first problem had a version of find the smallest 3 digit even number that is a multiple of X.  So imagine my surprise when I walked around the room to see a particular odd number as a common answer to the problem.  I mentioned this when we were going over the problems to the room in the hopes that we avoid this on future tests,  I'm not sure exactly why this was so difficult for so many kids to parse. In general, this made me think of mathmistakes.org and I might start an external conversation there to see what people think are the best ways to correct this bad habit. I'm also hoping that this month's experience will be in everyone's mind next time and I think I'll emphasize it again before we start the next round. That said, the great thing about math contests in elementary school is they are basically a low stakes chance to practice and get better.

Also based on a survey only about half the room caught the distributive law / regrouping problem (There's always one of these every time.) The rest continued to do the problem mechanically left  to right.  I've gotten into the habit of telling the kids to keep an eye out for it and that still doesn't quite stick yet. I'm very tempted to sneak some more examples of simple problems that are easier to do with some reordering or factoring. 

And now for something fun. I'm moving forward on my plan to produce a t-shirt for the club. For the post contest activity I had everyone draw ideas for the t-shirt design.

I plan to have the kids vote on the design to actually use when we get back next quarter. Speaking of which after bumping the cap up, I have completely filled the club and will have 18 members next time.

Today was also personally very satisfying. At the very end one of the boys gave me this box of candy which I find touching.  Along the same lines I had another boy beg his Dad to stay until the end rather than leave early when he came to pick him up. Moments like these make me feel like I'm on the right track. Now I just have to make sure to have my kids sign the notes on the gifts we've bought for their own teachers.

Tuesday, December 15, 2015

Knights of Pi 2015

Last Saturday I once again found myself at Newport High School in Bellevue for the Knights of Pi Math Competition. This year I brought one team mostly of fourth graders of which half the kids were at a math competition for the very first time. So I was nervous about how they perceived the whole process. I knew they probably weren't going to win any awards I didn't want this to be discouraging.  From the start I didn't recruit this one very heavily and I messaged that it was going to be mostly about the experience for the kids. I gave a quick talk before the awards ceremony to frame things that we were unlikely to win and to emphasize what we came for which was the experience. I think we were one of the few groups of fourth graders there on top of everything else which gave me an easy explanation.  I feel this tension even when we happen to place because not all the kids do equally well.  Its obviously most important to encourage continuing to work/explore math rather than focusing on winning an elementary math contest.  Its also really hard sometimes to be convincing even though its completely true.

Based on interviews with kids along the way, I think everyone had a good time. One of my favorite moments was watching the whole team playing board games together after finishing and prior to the awards ceremony. I think they really gelled together. I also had a funny conversation about what's an x and y intercept with the kids on the way home in the car. (Try talking about a graph while driving and having the kids just imagine it in their heads) I unfortunately, just barely made it to the High School in time due to some unexpected traffic delays. So I didn't remind the kids to keep their question sheets. As a result, I have a pile of answer sheets with no questions and I won't be able to analyze anything for a while. If the last years are any guide, the questions tend to be vocabulary heavy and above grade level with not enough time to work through them for most kids.  The most laughable one I had my son try on the practice test was

"What is the name of the 3‐D figure obtained by taking the convex hull of the centers of the faces on a cube?"

That's actually not very hard if you're a nine year old that happen to know what a convex hull is and the names of the various polyhedrons. But as I told him, just skip the ones that you don't understand the vocabulary for and don't worry about it.

The bad traffic actually cost me one team member who was unable to make it to  the Eastside in time. On the bright side I originally had one alternate member whom I had been feeling bad about not being able to try all the events that was able to participate in our team instead.

Finally, I once again spent a moment wondering about my kids versus the ones in the winning teams. I don't spend much time practicing for this competition. We did one practice relay type event and  I encouraged the participants to try out the sample questions at home.  This is mostly because I don't think the format is mathematically very interesting. The vocabulary and knowledge will come anyway in a few years or remain irrelevant . Likewise, the speed computation is more of a novelty than anything else versus the problem solving I am interested in. In my heart, I think if I and the kids were motivated to drill some of the particular types of questions for the entire Fall we could do a lot better. I also think that would be dreadfully dull and not in anyone's best interests in the long run.  What I hope is that I can observe measurable growth over time. Especially for the kids who I will see for 2 years. That's ambitious for a club that meets once a week for 50 minutes.

Tuesday, December 8, 2015

12.8 (Math) Game Day

This will be a shorter post than usual since I left my camera at home, I have a math contest to talk about after Saturday and today was all about play.

I was originally inspired for today's math club by this post: http://3jlearneng.blogspot.com/2015/12/solveme-mobiles-and-blockblobs.html where Joshua was playing the blob game with his kids.  I thought my fourth graders would also likely get a kick out of this.  We have only 2 sessions left and at first  was going to save this for next week. But then MOEMS released the next Olympiad and that will take up most of the time then so I decided to focus on games for the whole session this week instead. What I like to do is provide several stations so kids can find something that resonates with them so I rounded the set out with

I started by going around the room and explaining the rules for each game. I could tell this was going to be compelling because I had to keep corralling kids to move to the next game and not just start on the one I had explained. I then let everyone go and play.  As I thought this was 100% absorbing for the entire room and kids had to be pried out by their parents at the end of the session. My favorite moment was having one of the kids who ordinarily is very hard to motivate volunteering the rules for a variant on Dots that she plays at home. I will definitely plan for an all game session next quarter as well. Ultimately 9-11 year olds are still just kids regardless of their math aptitude and they need to play and see math as fun not just school problems.

For the problem of the week I chose: http://cemc.uwaterloo.ca/resources/potw/2015-16/POTWC-15-DP-12-P.pdf  This one I think will be a bit easier than the last several weeks. 

Monday, December 7, 2015

Blog Anniversary.

Why Blog?

Its been about a year, 75 posts and 3900 hits since I first started blogging.  So it seems appropriate to step back and ask some bigger questions. First and foremost I write because I enjoy doing so. It helps clarify my thinking and its fun to watch other people across the world read your thoughts. I never thought I would enjoy twiddling around with google analytics or join twitter and tweet about a post a year ago. I still get a thrill every time I see a comment show up on the page. I started this project in the hopes I would document enough of the daily process of how to run a math club that my eventual successor would have a good starting point when I handed it off.  I've been  trying to stay mindful of this and create enough structure through pages like activity map or resources that even as the number of posts rise, someone new will be able to find the topics they need.

But its a long ways off until that point and I soon decided its lot more rewarding to interact with real readers rather than a theoretical one a year or two in the future.  So one of my other main goals is that others trying the same process in different schools would also find this site and derive some utility from it.  That still remains largely unrealized but secondarily I want to connect with others in the math world and exchange ideas and learn from each other. To some extent I've made good on that goal and it's definitely part of why I keep writing. Along the ways I've found some really cool people whom I wish could be teacher's for my own kids include @patrick_honner, @henry_picciotto and @mike_lawler.

What I've been up to Recently

Sometimes sessions have a rhythm of their own. This fall seems to be ending up to be about various number theory exercises and centrally the distributive law. I'm working as always hardest to level the days correctly, figure out how to help the kids through the humps and how to bring out their best thinking. 

I think my favorite recent activity was the Pascal's Triangle filler exercise I used after the first Olymiad: first Olympiad. I'm, also fairly please with my first distributive law worksheet. More generally, I've had a lot of success with my problem of the week strategy. I'm looking forward to doing a bit more geometry in the winter session.


  • I was reminded last week of the importance of thinking about how kids might get blocked to prep responsive questions. This was a practice I worked on last spring, that I'm returning to, 
  • I'm always on the lookout for new game, activities to intersperse between problem sets.
  • I'm still developing my ability to foster good mathematical conversations among the students. This is probably my top goal for right now. 
  • As always I want to develop each kids problem solving ability and curiusity about why various mathematical principles work. To paraphrase Arthur Benjamin  the reaction I'm looking for is "Ooh now Why?"

Wednesday, December 2, 2015

12/1 The power of 37

This week I led math club by myself. I decided I would focus on a pair and share exercise where we'd work through 3 problems and spend time explaining answers to the other math club members.  The experience definitely reminded me why I appreciate other volunteers so much.  There seems to often be a 30 minute focus that many kids can sustain on a problem without external reinforcement and this is especially true the less mechanical the process is. With lower rations of students to instructors, you can check in more frequently and unblock kids / encourage them to try new approaches and keep working.  This is also a style of working that I'm trying to improve as a facilitator of.

Because we were coming back from a two week break over Thanksgiving, I had less participation in this week's problem of the week.  This one depends on noticing how you transform computationally from a 3 digit number to a six number.   If you think about this in parts if you multiply XYZ by 1000 you shift the digits over 3 places and end up with XYZ000. You can then add XYZ to end up with XYZXYZ and use the distributive property to realize that the magic number you're looking for is 1001.

XYZ * 1000 + XYZ = XYZ( 1000 + 1) = 1001 * XYZ.

Those who did try the problem either found this relationship after thinking about it for a while or found part of the answer (11 was a common response if the full result wasn't found.) That one's a good answer by itself since it notices all numbers of the form XYZXYZ satisfy the divisibility rule for 11.

We then broke into group, after I encouraged folks to pair with someone different than their normal partners.  Here's the set I wrote up: https://drive.google.com/open?id=0B6oYedIeLTUKeUdWdDdCUHJ6dXM

I'm going to mostly talk about the first one which I found via MathForLove.com.   This asks the students to try adding a digit to itself 3 times i.e. 6 + 6 + 6  and then multiplying it by 37. The goal is to observe the pattern and come up with a reason why its true.  To start I wasn't certain how easy or hard this would be but I suspected it would at least be a good warm up. It turns out, this was quite challenging for the kids.  Assuming they multiplied correctly and there was a few times I pointed out some careless errors, everyone noticed the pattern right away.  (D + D + D)  * 37 = DDD.  But it was really hard without scaffolding for lots of the room to come up with reasons.  I ended up prompting a bit to take a look at the 37 and think about using the commutative and associate properties for those who were stuck for longer periods of time.  Eventually everyone found the key 3 * 37 = 111 but once again there is need here for more practice/play with regrouping. Like the MOEMS problems I really want everyone to sit back and consider a simple computation and how it can be rearranged.  I'm going to keep looking for exercises that require the same principal and see if I can introduce them in future weeks. Also because I was stretched thin, I didn't have as much time as I wanted to observe the kids sharing results back and forth. So I'm looking forward to trying this again with some more help and focusing more narrowly on this aspect.

For the next problem of the week, inspired by mikesmathblog I gave out an old AMC problem (although minus the multiple choices):

Three runners start running simultaneously from the same point on a 500-meter circular track. They each run clockwise around the course maintaining constant speeds of 4.4, 4.8, and 5.0 meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run?

This should pair really well with the fourth graders' current rate and ratios unit.