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.

Practices


  • 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.


Wednesday, November 25, 2015

Winding Down, Filling Away?

Next Quarter

This shouldn't surprise me anymore but despite it feeling like I just started the first quarter is almost done and I'm about to start registration for the winter. I've decided based on the number of other volunteers that I've found to raise the club size to 18 kids. So hopefully everyone who wants to participate has a chance. If the other dad who was my main assistant can commit for the next quarter I will go all the way to 20. I allow the existing students to preregister so they can continue if they'd like to all year long since Math Club really functions best as a year long activity. So far 11 out of 16 kids have definitely indicated they will keep going.  I have a few kids among those that have not contacted me yet that I really hope rejoin.  Its hard not to take some of these decisions personally when I know objectively many factors beyond me come into play including busy parents who will still register when it opens up.  On the bright side, I'm confident we'll easily fill up and hopefully based on the wait list from last session I'll have a bit better gender parity.  Just to prime the pump I sent the wait list folks a reminder mail which amusingly resulted in an instant email back from one parent asking if they could sign up right now.


Are they Retaining Knowledge?

Every once in a while something happens that makes me take pause and worry how much the kids are remembering. Last week it was the RATS = 4 * STAR problem. Some of the kids had seen the problem at least 2 times previously over a year and yet looked oblivious when I mentioned that fact.  Then last night working on a fun problem with my son I had another similar episode.  During the summer we had visited the Field Museum: http://mymathclub.blogspot.com/2015/07/voronoi-diagrams.html.  Besides the Voronoi diagrams there was also a mirror maze that had various problems in it posted on the walls. One of the problems was interesting enough that we talked about it for 10-15 minutes.

Given a 4x4 grid: how many squares can you find?  Can you generalize this result?

Sure enough this showed up in last night's work. I asked him does this look familiar: blank stares. I hinted: remember this summer in Chicago: more blank stares. Sigh, at least he reworked the problem quickly so I assume the learning has occurred subconsciously in this case.  This is why I've stopped worrying about repeating any material from last year. The kids really don't seem to retain particular problems the way I assume they would.

Tuesday, November 24, 2015

A review of dreambox

This week is Thanksgiving break and there are no Math Club meetings. Instead I thought I'd write down my thoughts about dreambox.com, an online math app. I've experimented a little bit with computer programs with my sons in the past with everything from Splash Math, to Khan Academy. My favorite one so far by far is http://dragonbox.com/ which doesn't attempt to be comprehensive but actually manages to be both entertaining and instructive.  However, two weeks ago my younger son's school started a trial license of dreambox and I've been watching him interact with the it since then.

Structure

This program attempts to walk a student through a complete curriculum and covers the years K-8.   The student gets an avatar and explores among four different themes: pixies, pirates, animal friends and dinosaurs. Each themed area has a a series of quests where usually 6 different items need to be found. By practicing different types of problems the student finishes these stories gaining additional bonus points and character cards along the way. The actual problem sets (at least in second grade) seem fairly mainstream and concentrate on numeracy via number bonds, place values,  regrouping etc.  After each set is finished a more difficult one will eventually be offered until the curriculum goal is mastered. This typically take 4-5  sets of approximately 6 problems interspersed among various curriculum strands.




Overall this is the best curriculum replacement I've seen so far. Its compelling. My son voluntarily asks to play with it and it gets him to practice a fair amount very painlessly. The story framework that they scaffold the exercises with keeps him interested and is a lot better than some of the previous apps I've mentioned . There's also a fairly high quality set of different exercises to work on the various skills.  On the downside, while it usually has enough repetition for mastery, its not particularly adaptive. Once going you work through the sequence of problems regardless of whether you could handle them at a different pace or just skip the easy ones. (I haven't confirmed whether it repeats if too many problems are missed but I suspect that's the case.)  I also find the narrative structure to be a bit repetitive but that doesn't seem to be an issue for my son so take that one with a pinch of salt. It also doesn't require much supervision. For instance, when doing Khan Academy I heavily edited the flow and interjected instruction etc.  That isn't really necessary here although I still like to watch him work on the problems to get a sense of how he's thinking and whether there are any issues to address.  If he keeps up with maybe I'll update in a few months with how the game progresses and whether some of the other topic beyond addition/subtraction are also worthwhile.

Part 2: http://mymathclub.blogspot.com/2016/01/an-update-on-dreambox.html

Tuesday, November 17, 2015

11/17 First Olympiad

Today was a really fun day in the math club. We started with me handing out whoppers to everyone as they arrived since the kids had reached our next goal for completed homework problems. I continued with my idea from last week and picked one of the kids who had turned in his work and not demonstrated in front of the group yet to show the solution. Interestingly this week there were at least 4 different solutions running around in the room (which I told the kids was a record for us).
The problem was from AMC8 and I original saw it referenced on mikesmathpage.wordpress.com

Points ABCD are the midpoints of a square and form a smaller square. If the larger square has area 60, what does the smaller one have?
This one can be determined visually if you notice the four squares the larger one is divided into and the 8 triangle that these are further subdivided by.  That means half the triangles are inside the inner square and half are out so the area is 1/2 of the whole. A lot of kids found the solution in variants on this observation. A few computed the side length of the larger square and then the area of the triangles which is a bit more work.  Better yet when I asked what everyone had observed when trying this out with other quadrilaterals many of the kids had experimented and found the basics of Varignon's Theorem i.e. the inner shape is always a parallelogram and has half the area. I definitely want to do more investigations during further club meetings along these lines.

http://mrhonner.com/archives/17154 has a good proof that could be developed in class.


We then jumped into our first MOEMS Olympiad for the year. In my tryout beforehand I noticed a few things. First, by coincidence it has a simple decoding problem in the middle which meshed really well with the examples we had done that last few weeks. Secondly the final problem, arrange the 8 integers 1..8 around a grid (no center) with some specified sums for the rows and columns and then find the sum of the numbers in the four corners looked like it would take the most time.  It was solvable algebraically but as predicted everyone who was successful ended up using a guess and check strategy. Many more kids this year took the full ~30 minutes to do the problems than last year which is probably a function of age i.e. fourth graders vs. fifth graders. Incidentally, the cool part of having some returning students is the ability to see how much they've grown in a year via their performance on these contests.

Secondly on review I still have kids who given a simple sum will plow through it from left to right despite me trying to stress looking for shortcuts via rearranging the numbers. As an aside, with problems like this I'm taking more advantage of class surveys. So I'll start by just asking for a show of hands of kids who added the numbers in order and then ask for anyone who did something differently to bring out the alternative strategies a little more efficiently.  I should definitely stress this point more and do some examples of regrouping in club.

My favorite part of the whole experience by far was the wrap up when I had the kids show their solutions to each other. All our work on listening seems to be paying off. Despite there being 5 problems rather than our normal one, everyone was rapt during the demonstrations and I had lots of volunteers wanting to show how they had found the answers.

Also new this year, I planned for kids finishing early and brought another exercise from "This is not a Maths Book". This time we did the page on Pascal's Triangle which involved building your own triangle and colouring in evens and odds to search for patterns. This proved really fascinating for the 6 kids who were done early and kept them busy (and non-distracting) to the others who were still finishing.  Again this year, I plan to do a full unit around Pascal's Triangle some time in the winter. There's too much good stuff here to not touch again.

Finally, I'm trying a problem from a new source http://www.cemc.uwaterloo.ca/resources/potw/2015-16/POTWD-15-NA-09-P.pdf for the take home problem. I'm hoping again that at least some of the kids will notice that 1001 is a common factor in all numbers formed this way. We'll see what they come up with when everyone comes back from Thanksgiving break.








Friday, November 13, 2015

That article in Atlantic about explaining your work

The Atlantic recently published an interesting article about requiring students to explain their work http://www.theatlantic.com/education/archive/2015/11/math-showing-work/414924/.

"If a student can consistently solve a variety of problems, that student likely has some level of mathematical understanding. Teachers can assess this more deeply by looking at the solutions and any work shown and asking some spontaneous follow-up questions tailored to the child’s verbal abilities. But it’s far from clear whether a general requirement to accompany all solutions with verbal explanations provides a more accurate measurement of mathematical understanding than the answers themselves and any work the student has produced along the way.  At best, verbal explanations beyond “showing the work” may be superfluous; at worst, they shortchange certain students and encumber the mathematics for everyone."


This has produced quite a bit of chatter across the blogosphere. I initially had a strong instinctive reaction.  Personally, I find these written explanations to be tedious and not very useful so its not a technique I would  use with others either. However, I acknowledge that my own feelings are often in the minority and more interestingly I tend to be more progressive in practice when dealing with a room of the kids than I am in private when imagining these type of pedagogy questions.

Thinking about this some more, I think one can disregard the straw man position where these type questions are overused, or not handled effectively. One can find bad teaching any where that misapplies a good tool. So let's imagine, a teacher is only asking these questions occasionally to truly confirm understanding, is generous in reading the answers, provides appropriate feedback and adjusts based on the information discovered and doesn't mandate a particular format.  Reluctantly I think there is probably a place for such a technique. The key is to use it carefully. I also think that a lot of the same effects can often be had just through regular classroom interactions where good questions are asked back and forth and where you listen carefully to what the students are saying. Likewise, where ever possible I think understanding should be demonstrated through problems. Usually these discussion are framed around fairly simple problems that  are mostly straight procedure like:

"A coat has been reduced by 20 percent to sell for $160. What was the original price of the coat?”

I'd much rather see conceptual understanding of percentages pushed through a series of more difficult and interesting problems prior to requiring any written queries.  Likewise, any written explanations ought to be directed first towards proof building skills.  Its a lot more valuable to have students explain how they made a break through, than to parrot back a procedure they've already learned (and this is subject to blind memorization just as much as anything else).

Bringing this back to my experiences with the math club because my first priority is making a fun exploratory environment,  I will never require such explanations.  But I do find myself having many conversations about why a technique works. For example, our recent dive into why the divisibility for nines works which involved me asking a series of questions while working through the basic idea i.e. what does it mean to be a multiple, if you add a multiple of 9 to another multiple of 9 why is it also a multiple of 9 etc.?

I'm always looking at the kids work and questions for evidence of what areas they don't quite understand yet and thinking about how to circle back if necessary. Having said this, of course, I have the immense luxury of not needing to hit any curriculum and if I miss something I can rely on their regular teacher's to fill the gaps.


I also have no illusions that I've mastered the technique of lecturing yet. I tend to minimize the amount of work I do on the whiteboard in front of the kids as a result. Every time I engage in a longer talk I think about it afterwards and worry about whether the kids followed along with me. Asking questions during a talk helps and having kids demo parts of the problem on the board also helps but neither is perfect. In sum, I don't have the perfect answer yet here and its something I will worry about and continue to think about how to improve.





Tuesday, November 10, 2015

11/10 Relay Madness!

After my bad activity complexity estimation last week it was a relief to properly size the planned exercises for math club this time around.  We started by going over the problem of the week:

The number 1,525,354,555,657,585,950 is, as it happens, evenly divisible by 99. Fix all the 5s digits where they are in the number, and rearrange the other digits randomly. What is the chance that a rearrangement of this form is still evenly divisible by 99?

As I posted earlier, I sent home links to divisibility rules to help everyone out. In practice, most of the kids found the solution based on this help even without covering it in club. This time while checking kids in I tried something new.  I scanned everyone's sheets and encouraged a boy who hadn't done so before to do the demo on the board.  The rules themselves seem to be fairly easy to master (compared to applying the distributive law for instance). I even sneaked in a quick demo of why the divisibility rule for 9's worked using the standard regrouping technique because the student proof went so quickly. I'm continuing with my theme of emphasizing the distributive law during such examples.  AoPS standard proof

So for the main task we practiced doing math relays using some old ones from the Knights of Pi competition: old contests   The way these works is you group the kids into teams of 4 and then they work on the four part problems one kid at time. After the first kid finishes, their result plugs into become part of the second problem and so on. This means an early error will invalidate the entire team's work. My main goal was to practice the format for those that are going to do the contest later and hopefully have it be a different and fun way to try out problems for everyone else. I originally had planned to stop after each round and correct the problems as a group but I quickly changed my mind.

1. Relays leave 3 kids without problems to work on. So I quickly changed tacks and had multiple rounds going simultaneously. That meant less down time for everyone.

2.  Secondly, I and my assistant ended up individually helping on problems rather than doing the group corrections since everyone was going at a different pace.

Overall watching the kids work I found a few areas to focus on at a future time.

1. There was a smattering of kids who were unable to easily do operation inverses .i.e if the average is 1.5 and there are 6 items, what is the total? 

2.  One of the problem sets I chose happened to use basic statistics which I don't think anyone has covered much.  In this case, this is exactly what might happen at the contest so I wanted the kids to have that experience. But having just gone over the stats chapter in AoPS pre-algebra I could also definitely plan a themed day around it.

3. Another similar triangles problem came up. I really want to do a day doing an inquiy based investigation of how this works i.e drawing triangle and measuring to watch dilation / expansion in action.

Finally based on some twitter traffic I picked  a problem based on Varignon's theorem for the problem of the week: https://drive.google.com/open?id=0B6oYedIeLTUKMU8weEZEZ3NNSmM
I've added on some open ended experiments for the kids after they finish the concrete example.






















Thursday, November 5, 2015

11/3 Oops I did it again

This week I did a great job planning to ensure continuity, rolled with how things played out in the room and didn't finish nearly as much as I wanted to. *sigh*  My originally thinking was last weeks practice Olympiad ended with the STAR * 4 = RATS decoding problem which was the most difficult part for most of the kids. So I thought we'd warm up with another example and keep working on the strategies to use. So I found

   SEND
+ MORE
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MONEY

I gave this a test before I started and thought it was a bit difficult but I could scaffold the room through the problem and still move onto a divisibility talk with some problems. In reality almost the whole math club finished the problem but it took almost the entire session and I'll have to save the divisibility exercises for another week.   The stickler seemed to be reasoning about where you need to carry and then testing the possible number that could fit. I think taking time to do a harder problem was probably worth it and none of the kids wanted to quit and move onto the second half early so I think were still having fun. We'll see if I bring one these back later this year if a 3rd sample will go quicker.

What was unfortunate about my planning was that I picked a problem of the week that was meant to pair with the second half:

Problem of the Week
Via Dan Finkel:
The number 1,525,354,555,657,585,950 is, as it happens, evenly divisible by 99. Fix all the 5s digits where they are in the number, and rearrange the other digits randomly. What is the chance that a rearrangement of this form is still evenly divisible by 99?

We didn't go over testing for divisibility by 11 and  I'm unsure  if the kids all know it or not. I had preprinted the problem so I couldn't change anything on the fly. So as a result I plan to send out a mail with some hint links to: http://www.artofproblemsolving.com/wiki/index.php/Divisibility_rules/Rule_for_11_proof and
https://www.mathsisfun.com/divisibility-rules.html

and hopefully this will be enough for some of the kids to work through this problem which looks much more complicated than it actually really is.

I was pleasantly surprised that 3 or 4 kids solved last weeks problems. When going over it in the room rather than deriving an algebraic equation based on similar triangle and the area formula I opted for a visual proof where you divide the triangle into 16 congruent sub-triangles.  My hope was that this would be more accessible. If I repeat it again I think I'll take more time and pre-print the triangle so the congruence is very obvious and perhaps even have the kids measure the triangles with a ruler to confirm they all are the same.


Tuesday, October 27, 2015

10/27 First Practice Olympiad

The most exciting part of this week was I think I've found another father who is willing to help assist me. Today was his first time coming and helping out. This is great on two fronts. First the optimal ratio of kids:adults for me 8:1 and a second person lets two activities occur at the same. Secondly if this works out I'll be able to open up the wait list. At this point I think I have at least 5-6 kids whom I did not have room for this quarter in the math club. I'm really excited to be able to plan more flexible and complex transitions in the upcoming weeks.

So for today we started with candy in the lunchroom. I also had 5 sheets turned in from last week's take home problem. As I've mentioned before its fairly crude, but the naked bribery experiment is producing results. What I'm most excited about from today is the kids produced the three main solutions I was hoping they would find. As  a result I was able to I have them demo all of them on the whiteboard.

There is a curious 5 digit number A which when you add a 1 to the end of it is three times larger than when you add a one to the front. What is A?

The first way is to setup the algebraic equation 10A + 1 = 3(A + 100000) which I wasn't sure was within their skill set. (In full disclosure I think some of the parents are working with their children)

Secondly you can treat this as decoding problem:  

1ABCDE 
x          3
ABCDE1

and working right to left use a little number theory to find each digit. For example E must be 7 since 
only 3 x 7 ends in a 1 and so on.

Finally its also possible to do a brute force search through the number space especially with a bit of intelligent bounding.

I'm super proud of how well they all did this week.Which leads to my other observation. This years kids are really good at working during club as well. They almost always form productive groups and grind a way at the problems. I can definitely see the fourth grade vs fifth grade difference in terms of algebraic awareness. But I'm hoping I'll be able to move them forward quite a bit over the year if this keeps up.

For the main activity I had everyone do a practice Olympiad since the first real one is coming in 3 weeks. This went well. I will report that it contained the STAR * 4 = RATS problem in it yet again. This must be the third or fourth place this problem has showed up. As expected this was the hardest problem for most of fourth graders. In followup when we worked through the problems as a group it looked like the decoding procedures were understandable. Also for the followup I was able to split the club into two sections and had the smaller size groups each with an adult which is really great for focusing on listening to each other.
                             
Finally with the extra time after we finished going over the solutions we did some match stick problems (well actually tooth pick problems in my case since I don't risk burning down the school):

Example:

Form a grid like on the left and then moving only 12 sticks form two big squares: (ignore the right hand picture)




These worked really well I presented 4 challenges which was more than enough for the end of the club. Finally I left the kids with following take home challenge.

In triangle ABC, three lines are drawn parallel to side AC dividing the altitude of the triangle into four equal parts. If the area of the second largest part is 35 (the blue region), what is the area of the whole triangle ABC ?

Tuesday, October 20, 2015

10/20 Grids and Graphs

This week I planned a series of more playful activities. We started with a review of the take home problem of the week: (September from www.moems.org/zinger.htm).  I already knew this was easier than October since 3 kids had told me the answer almost immediately when I handed it out. I had 8 kids complete the problem so I will be bringing M&M's to math club next week.  To wrap it up, we did another white board exercise with 2 kids showing their work. Listening worked better this week. My only concern was one student took fairly long to copy her work onto the board and I fretted unnecessarily that I would lose the club's attention.

After that I decided to do a quick demonstration with the distributive law why a negative times a negative is a positive.

For example:
     Start with - 5 * -6.  Now add -5 * 6 to it.
     -5 * -6 + -5 * 6 = -5 (6 + -6) = 5 * 0 = 0
     Then go back to the original sum -5 * -6 + -5 * 6 = 0. 
     We know the second term is -30 already since its a negative times a positive.  
     So -5 * -6 + -30 = 0   which implies  -5 * -6 =  30.

I had each kid pick their own product and work along. This strategy worked OK but I ended needing to go over several more examples from the kids on the board. Overall this was a stretch and next time I should start with a review of factoring with the distributive law even if we did it the week before.

This entire process took about 20 minutes and we then switched to over to a 3x3 logic problem from http://www.logic-puzzles.org. I went much simpler than last year based on my previous experience and even so this was about 30+ minutes of work for most of the kids. Moral of the story I estimate a 4x4 would probably take a full hour if I want to fill one up.

Because things went longer on the logic puzzle I didn't have as much time for my main selection. I did a quick edit of my original ideas and took out the loop-de-loop exercise from: http://www.amazon.com/This-Not-Maths-Book-Activity/dp/1782402055.  As others have reported on the net this was a big hit with the kids. There were a lot of excited students who wanted to show me the different spirals and loops they had created.  I ended up shorting the time I wanted to spend on trying things out so I think I may revisit this at the beginning of next week and have a wrap discussion about what patterns everyone found.


Problem of the week courtesy of  Martin Gardiner:

There is a curious 5 digit number A which when you add a 1 to the end of it is three times larger than when you add a one to the front. What is A?

I told everyone to try some strategies on your own first and give this a little time if you get stuck before looking at the hint. This will definitely be harder than last week and I'm looking forward to seeing what the kids come up with.

Hint:
"∀ ɟo sɯɹǝʇ uᴉ sɹǝqɯnu ɹǝɥʇo ɥʇoq ɹoɟ suoᴉssǝɹdxǝ ǝʇɐlnɯɹoɟ"

Planning

Its time to try out a sample Olympiad next week!

Tuesday, October 13, 2015

10/13 The Distributive Property: not just a good idea


After last week I knew that I needed to start with a review of the problem that I handed out to do at home. The great news was 8 of the kids worked on it over the week! So far naked bribery with candy is working and we will keep going with the homework experiment. At this rate, I'll be bringing in M&M's in 2 weeks. In going over the problem, this was our first group white board exercise so I emphasized listening was important before anyone did any math. To get started I also asked the entire room what the formula for the area of a circle is. As I noticed last year that tends to be a bit shaky despite being covered once a year I believe starting in 3rd grade. I then had 3 kids volunteer to  show parts of the October problem of the month (including my shyest student) which was great. I decided to narrate what the kids were doing in a louder voice to make sure the whole room could hear.

We then transitioned to a warm-up with stations. I found this great equivalent fractions puzzle on twitter from NRICH:  Puzzle  which I pre-cut a few versions of. I also brought Game of 24 cards  for the first time. I then let the kids choose which station to go to and we played around for about 15 minutes.  Generally both activities kept everyone's attention. I may alter this structure next week and do a practice Math Olympiad first but in general I still believe games are really important at this age.


Finally for the main activity I wrote up my own distributive property worksheet last night with a progressive set of problems: http://mymathclub.blogspot.com/2015/10/distributive-law-worksheet.html.   I think I did a bit better than my last attempt at worksheet creation and in general this was reasonably leveled for the kids. As I told the kids, at this stage in their math careers the distributive law is probably the most powerful tool they know.  Some general observations. There are varying degrees of emerging algebraic thinking among the room. This was great for an exercise to get kids comfortable gently manipulating variables. I should also definitely go back one step and review why a negative times a negative is a positive in one of the upcoming weeks. At any rate, this lays the foundation to talk about divisibility rules going forward.

Planning Ahead

At some point soon I need to have the kids try a practice Math Olympiad. I also want to do a pattern finding notice and wonder exercise with a group discussion and I still have a short talk about multiplying negatives to fit in.

Distributive Law Worksheet


The Distributive Law



Multiplication distributes over addition: a(b +c) = ab + ac. We sometimes talk about factoring out a number using the distributive property ie. 81 + 45 = 9 * (9+5)

Problems:

Compute $51 \cdot 9 + 51 \cdot 31$.

What is the value of $17 \cdot 13 + 13 \cdot 51 + 32 \cdot 13$?

What is $a(b + c + d +e)$?

what is -(6 + 8)?

What is $-1 \cdot (5 - 9)$ also written $-(5 - 9)$?

What is $-(x + 1)$?

What is $−1 \cdot(a - b + c - d)$?



Find numbers a, b, and c such that $a + (b \cdot c)$  is not equal $(a +b)\cdot(a+c)$ In other words show we can NOT go the other way and distribute addition over multiplication.

Take a look at the first perfect squares from $1^2$ to $10^2$. Do you see a pattern to what number is in the units place? 

Can you use the use distributive law to show why $8^2$ and $2^2$ for example end in the same digit. Hint: 8 = 10 - 2.  Try the same technique with another pair of numbers.

We can also use the distributive law to show why a negative times a negative is a positive. Pick any two negative numbers x and  y then try using the distributive law to find the sum $x \cdot y + x \cdot -y$.


Factor out x from $6x + 9x$.

Factor out x from $6x + 9x^2$

What is $x\cdot(9 +x)$?

What is $(x+1)\cdot(x + 1)$? (Hint use the distributive property multiple times)

What is $(x - 1)(x - 1)$?

*What is $(x -1)(x + 1)$?

The above answer lets you do a neat trick. Can you multiple $19 \cdot 21$ in your head?  Using the above  formula we can think of it instead as $(20 - 1)\cdot(20 + 1)$. What does that equal?

Try doing some other examples with a partner without writing anything down.

What is $(a + b)(c +d)$?

Multiples/Divisibility

We say a number x is a multiple of another one n if we can find another number such that
$x = y \cdot n$. Example: 20 is a multiple of 5 since $20 = 5 \cdot 4$. Can you use the distributive law to prove that if you add two multiples of the same number n that the result is also a multiple of n.
Example: 20 is a multiple of 5, 35 is a multiple of 5.  20 + 35 = 55 is a multiple of 5. How about if a number is not a multiple? For example 11 is not a multiple of 5. Is there a pattern for what numbers you can add to 11 to get a multiple of 5?

Wednesday, October 7, 2015

10/6 And we're off to the races

This afternoon was finally the first meeting of the math club for the year! Things started with a rude shock when I checked out my assigned room and found it had no desks or chairs. We'll see if I can get a better site in the future weeks but for now I'm going to ask the kids to bring a clip board or some other hard surface for writing on. On the bright side having everyone sprawled out on the floor worked better than I would have expected. It does make things informal in an interesting way.

My second surprise unfortunately was that an unexpected student that was not on my roster showed up that I had to refer to the after school program coordinators to sort out. The enrollment process was not under my control and this was one of the problems I was really worried about beforehand. This later lead to what my wife called " One of the saddest facebook posts ever."  Unfortunately I'm already at capacity and can't make last minute adjustments but it is really hard seeing someone write that you've shattered their child's dreams. I'll probably have some further conversations with the coordinator about the problems in the signup process.

I decided to start the meeting with a quick introduction from me with why I'm leading the class and what we are going to do over the quarter in some general terms. I then had all the kids introduce themselves and say why they were in the club. Almost everyone said essentially they liked math which is good to have said out loud sometimes by a large group of kids.

Next I had everyone get up and form a circle for a game of Buzz.  Rules   I picked the progressive version where you add rules. First we did fives, then added sevens and ended with prime numbers. This was a great success.  Some of the kids asked about doing an elimination round which I'll have to think about for future games but I do like keeping everyone involved.

From there we did the club charter discussion I decided to add on this year. I've put some of the outlines of the areas we covered here: https://drive.google.com/open?id=18NknKDOhmR5AX09RpLEFpli1IAL9F10W2UdVRT1i9VA   But essentially I'm going to try to come back to several key points particularly around what to do when we encounter hard problems. For next time, I'm going to stress collaborating with the others in the room.

This was put to quick test because I decided to start with the 2 problems I had used on my recruitment flier (link)  although I had some trepidation about whether they were too hard. As expected they did initially stump the kids but I was able to scaffold enough to move the room through at least 1 or both of them. The key was mostly to emphasize simplifying the problems and looking for patterns. I don't think anyone reached the 3rd problem which I added for my own peace of mind to make sure no one finished early.

Finally I gave them the problem of the month from MOEMS: http://www.moems.org/zinger.htm to do at home over the week and stressed I would be giving candy out once I had received enough finished problems. I'm really hoping this proves motivating.

Going forward I'm planning to send an email out about unfinished problems since that's a bad habit of mine. I'm going to have to think some more on the best policy to take. Perhaps I should have an answer site that everyone can look at. Or perhaps I'm overthinking the issue.


Thursday, October 1, 2015

Reversing digits

Moderator Note:  What's better than one semi math-crazy after school coach? How about two of them. My friend Dan has agreed to run our feeder Middle School's Math Club and after patiently listening to me talk about the blog for a year I've convinced him to contribute to it. So without further ado:


At the middle school, the Algebra 2 teacher gave a fun "Problem of the Month" to his students. My daughter and I had such a good time with it that I wanted to share it here.

Start with any 4 digit number where all 4 digits are different.
Sort the digits in descending order as your first number.
Sort the digits in ascending order as your second number.
Subtract the second number from the first number.
Repeat.

What happens?

For instance, consider the number: 1625
Sort the digits in descending order: 6521
Sort the digits in ascending order: 1256
Subtract: 6521-1256 = 5265
Repeat:
Sort the digits in descending order: 6552
Sort the digits in ascending order: 2556
Subtract: 6552-2556 = 3996
Repeat:
Subtract: 9963-3699 = 6264
Repeat:
Subtract: 6642-2466 = 4176
Repeat:
Subtract: 7641-1467 = 6174
Repeat:
Subtract: 7641-1467 = wait a second? That's the same as the last subtraction!

We started this problem just by picking some numbers like the one above and seeing what number we'd get. We quickly found that everything we tried converged to 6174. So we started trying to organize the pattern.

The first thing we noticed is that we made some mistakes in our subtraction as we were doing it without a calculator. I pointed out that problems that involve digits frequently also involve multiples of 9. And in fact our subtractions usually resulted in a multiple of 9. This is easy to check by adding the digits to see if they are a multiple of 9. It's not hard to prove that the result is always a multiple of 9. So any subtractions that weren't a multiple of 9 were wrong.

Then my daughter noticed a pattern. She tried 4321, 5432, 6543, 7654, 8765, 9876 as starting points. Sorting the digits and subtracting turns into 3087 for all of those numbers. Then she tried 5421, 6532, etc. Sorting the digits and subtracting turns into 4176 for all of those numbers.

This suggested we find some equivalence classes. Can we describe a set of numbers that will all get the same output? It was pretty easy to suggest that the sorted digits (a+x)(b+x)(c+x)(d+x) for a particular abcd but any x would form an equivalence class. When you subtract, the xs all cancel out and you have just abcd - dcba.

Once we had that equivalence class we could start to write out the possible values for abcd. We quickly saw a more general form of the class. Any pattern of sorted digits (a+x)(b+y)(c+y)(d+x) for a particular abcd but any x and y would form an equivalence class. When you subtract, both the xs and ys cancel out.

So that means we can figure out which equivalence class any number fits in pretty easily. Start with the original 1625. Sort the digits to 6521. Use the largest possible x = 1. Use the largest possible y = 2. That leaves us with 5300. 6521-1256 = 5300-0035. Each equivalence class will contain a number with two leading digits and two zeros.

Now we can make a table of all possible pairs of digits and figure out the next number from the operation for each pair. The table can easily include the cases where the digits aren't all different, although 1111, 2222, 3333, ..., 9999 become 0000 in one step.

From the table, you can also make a graph of how each equivalence class transforms to the next class. This graph can help you find the longest chains and how many steps any particular number will take.

Here's one of the longest possible chains:
4100 -> 4086 (8200) -> 8172 (7500) -> 7443 (4000) -> 3996 (6300) -> 6264 (4200) -> 4176 (6200) -> 6175
For each repeat, I listed out the actual result with its equivalence class in parentheses. There are several other classes with equally long chains.

If you look carefully, you can find some additional equivalence classes that narrow the problem down further.

I researched this problem and it is known as "Kaprekar's operation" after the mathematician who first published it. There's a nice write up with a bit more of the algebra at https://plus.maths.org/content/mysterious-number-6174. This is a good general math blog, as long as you don't mind your math with an S at the end!

Friday, September 25, 2015

Demographics

I just received the roster list. So some quick demographics for the beginning of the year.
It looks like as expected the kids skew towards 4th grade this time. Unfortunately I did not maintain gender parity. I may come back to this next quarter and see if I can recruit some more girls with the teachers' help.

13 4th graders
 2  5th graders

11 Boys
4 girls

 And unlike last year I actually know 7 of the kids before starting.  That includes my own son who will be joining me this year now that he's in 4th grade.

Update: There are now 2 girls on my waiting list. I've put out a call for another volunteer to help.

Thursday, September 24, 2015

Procedural Updates


Starting Up

The roster filled up after a week and I have a small waiting list. Unfortunately I don't have the actual names yet so I can't start emailing the parents or looking for volunteers. If a few more students do show up I am going to try to find another volunteer so we can serve more kids.

In the meantime I have registered for MOEMS again, am about to start the AMC8 process and have had an independent email from the Washington Student Math Assoc. about having someone come out and speak to the math club later this fall.

Here's my intro email draft which I'm trying to decide if I can make any more fun:

Thanks for signing up for the Math Club.  I'm excited to start up a new year. In this mail I'm going to go over the basic procedures.

1. Please fill out my own membership form which I've enclosed in this mail and return it to me.

2. We meet after school finishes in the cafeteria.  New this year: there is no food allowed in the classroom. So please bring your own snacks and be prepared to eat them in the 10 minutes before we walk up to the room.  

3. Besides any snack, also bring a notebook and pencils each week. Especially in the beginning I'd appreciate it if you can remind your kids about supplies.  Also, I've found electronics to be very distracting for everyone. So I recommend not bringing any ipads/phones etc. on Math Club days or talking about putting them away for the afternoon beforehand with your kids. 

4. Parent pickup will be at the classroom: N204 at 4:50.  There will be a notebook that you should sign out from. If you haven't arrived by 5:00 I will bring your children down to the KidCo room by the Cafeteria. Be aware: Vanessa has informed me that there will be less leniency on charges this year if that happens.

5. The first day is Tue. 10/6. I've also enclosed the year calendar.

Volunteering:
* I really like having other adults helping out in the room and more adults let me do some different activities. Please email me back if you can come  one of the afternoons. No special math experience is necessary. 

* This year I'm thinking about commissioning a T-Shirt for the club. If you have any graphic design talent and want to help out please also send me a mail.

Fun Stuff: I've re-enclosed my original flyer. For those who are chomping at the bit to get started please take a look at the problems. Then if you have any questions or have a solution send me an email back.

See you soon


I've also been updating the resource page: Resources a bit recently. At some I'm going to reindex by category as well.

Recent Interesting Problems:

I love medians and this problem has a lot of them. This would work great scaffolded with work noticing properties of medians i.e. they divide the area of the triangle in half, and the cool rotations you can do with them by splitting to form another related triangle.   Its a bit of a cheat but if you notice the quadrilateral is not strongly specified in the problem. As that suggests this is a general property. So you can start by considering the case where the quadrilateral is a square and find an easier solution for this subcase which will hold for more complicated versions. (In fact the general proof which involves triangles and medians is a bit less messy to draw in the square version as well.)

This problem showed up again.  Mikesmathpage has a fun treatment of this: https://mikesmathpage.wordpress.com/2015/05/27/what-i-learned-from-grant-wiggins/  
Basically if you  model with some sort of dowel you can find a right triangle and then easily apply the pythagorean theorem.  I really like the idea of the physical modelling and it would fit really well after proving the theorem if we do that again this year.

Tuesday, September 15, 2015

Arthur Benjamin's talk: Exploring the Hidden Magic of Math

Last night I attended Arthur Benjamin's lecture circuit promoting his new book: The Magic of Math: Solving for x and figuring out why.


Walking up to town hall:



Oh wait - math is stuck in the basement again: (The noble prize for literature is totally overrated)



Once in the space it was great to see a full room out on a Monday night for a math talk. This is one of the reasons I love living in Seattle.



It turns out the talk was sponsored by my arch nemesis: Xeno  They still are totally whomping me on google showing up at least 2 pages earlier if you search for "math club blog" despite not even being a blog.  All kidding aside they're a great organization which I haven't had much of chance to interact with in person yet.

The talk itself was a lot of fun. I brought my son and the material was very accessible. There was a bit of warm up mental math tricks for doing squares, two fairly length sections going into Triangle numbers (see my take from last year: http://mymathclub.blogspot.com/2015/02/triangle-number-exercise.html) and  Fibonacci sequences. None of the subjects were particularly new to me but Arthur Benjamin was very entertaining and some of the development paths look like fertile ground..



For instance, I really liked his presentation of how to develop squaring tricks.  One of my favorite quotes: "And who doesn't love squaring a sequence of numbers?" So I plan to  read the book and do a full review some time in the future at which time I expect I will find some ideas to leverage for the math club on a close reading.

Updates

The teacher strike looks like it is coming to close and kids have started signing up for the club. Hopefully I'll have a roster in a week or so.

Friday, September 11, 2015

Using Symmetry: More complex proof walk through

This is a continuation of the last 2 posts exploring using symmetry in proofs. This example is the most complicated yet. Symmetry via reflection produced the base of the proof but it then required recognizing an in-center to complete the process.

I started by looking at problem 45 on gogeometry.



My first instinct here was to increase the overall symmetry in the drawing by reflecting everything.
Once that's done through a variety of different angle chases you repeatedly arrive at 2b + 2a + x = 90.
The easiest place to do this is in corner BAC once the perpendicular bisector is drawn.



I could see the parallel lines in my reflected drawing which gave me a few more x angles but not another triangle equation I could use to solve the problem.

I next tried a few unsuccessful techniques. I back-cracked from the answer 30 degrees and filled in the other angles to see if any other common triangles were there that I had missed on visual  inspection.

I then removed all the x's and replaced them with  90 - 2b - 2a in the total picture to see if this helped.
I looked around to see if there was any way to use the 30-60-90 properties of side length to deduce the angle measure but that looked unlikely given the constraints.

Then I took a break for a day after hitting a wall.  And suddenly on restarting during lunch I noticed that I had the  in-center  of the triangle when redrawing the picture of one side. That meant the third side was also an angle bisector and everything fell into place. [The need for a break before finding a solution on a stumper is pretty normal for me] 



Thursday, September 10, 2015

Using Symmetry: Geometry Walk Though #2

This is another in my series exploring geometric problem solving through rigid transformations (and creating diagrams with geogebra). See: http://mymathclub.blogspot.com/2015/09/ratios-in-square-problem.html

This problem starts with a very simple picture but I want to show my first attempt down a non-productive line of thought.

Given B is the median of AC and angle EAC is 30 and angle ECA is 1 5 find angle BEC.

Two thoughts came to me immediately. Angle EAC is almost part of an equilateral triangle if we reflect the whole triangle. Secondly ECA is half of EAC and it could be divided into two halves so we end up with many 15 degree angles. I pursued the second thought first because it created some attractive symmetry.

With some angle tracing EGF falls out as 30 degrees as well. So it looked like I just had to prove triangle EFG was isosceles (or find angle EFG) and here I became stuck. I started to break the triangle down further to tease out 30-60-90 triangles and side relationships. This didn't quite arrive at a solution. I also tried some more reflections and found a few 45-45-90 sub triangles. But that again didn't quite get me where I wanted to be.

So I went back to my first thought. Reflect the whole triangle AEC and create a giant equilateral triangle out of it.

I connected the two points C and C' and drew the full angle bisector in to where it intersected at D'.
Then I started angle tracing at which point I realized I didn't need the full equilateral triangle the bottom 30-60-90 was interesting by itself.
First I started finding congruent sides. CD was half of AC because of the 30-60-90 triangle. I had forgotten that B was the median by this point and actually started to draw one on for a minute and thought I would have to prove B was a median until I rechecked and realized it was given. I then connected BD to form the interior equilateral triangle since that seemed useful both in terms of symmetry and subdividing angles. For instance ADB is 30 degrees by angle chasing meaning  triangle ABD is isoscleses. By further angle chasing it also turns out that DCE must be 45 degrees (60 - 15) and then the final angle of the right triangle CDE must also be 45 and it is also isosceles. This produced the fifth congruent line DE. At this point I could tell I was very close. I now had proved a new isosceles triangle EBD which gave up EBD was 75 degrees.  This was the last unknown angles besides x itself in the triangle BCE!  x + 15 + 60 + 75 = 180. So x = 30.

This one fell out after trying 2 ideas. With some other problems I go down many more avenues. My general guiding principle is to try out reflections and angle division additions first around the figure always with an aim at more symmetry or new congruences.