Monday, December 26, 2016


I saw a funny ignite talk "Algebra Inferno" the other day comparing disliked teaching practices to the various circles of hell a la Dante.

Among the sinners list were the survivalists: those who refused to ever allow the use of technology in their classes.  While I chuckled at the clever pun, I don't really totally agree with this point. My contrarian instinct is that there is a huge body of math out there that doesn't need calculators that one need never stray from and quite successfully conduct a lesson. And on the converse most of the math lessons that try to use them often turn out to be more about practice punching buttons than thinking mathematically.

There's some larger existential questions wrapped in this debate. In the era of  ubiquitous cellphones and Wolfram Alpha what parts of elementary school mathematics are relevant and can you skip them without throwing away the ladder to higher level skills?

Leaving that question aside, in practice, as often is the case I'm more pragmatic than my initial instincts. The other day my son was working on a problem and asked if he could use Desmos to plot the graphs of some inequalities. That seemed like a very reasonable usage and I'm happy how much he is excited by desmos having only been recently introduced to it. So feeling like another hand plot wasn't needed, I said "of course."

I don't have that  luxury during Math Club where there are neither calculators or computers around. But were that the case these are just a  few of the uses I  think really are worthwhile:

  • Investigating the patterns of digits in repeating decimal numbers is vastly sped up by just trying them out.
  • In the age of infinite precision calculators its now possible to check all those modular arithmetic stumpers  like which is bigger 63^45 or 33^54  directly in python.
  • I love the use of 2-d and 3-d graphs as long as they are a natural extension of a larger problem.
  • Geogebra makes a nicer version of pen and compass constructions and is very useful in exploring more complex geometry proofs.
  • We were recently doing a comparison problem at home between 2^1/2, 3^1/3 and 5^1/5.  Visualizing the graph of x^1/x in desmos made this much richer. 
  • If you want to practice finding factors for larger numbers like say 2017, calculators help speed things up quite a bit.

Tuesday, December 20, 2016

Virtual Math Club for Winter

Here's the cipher I sent of for the kids to decode while they are on break. I'm offering double homework points towards candy for solutions so hopefully some will give it a shot.

"oifi'y r trgo dcit rme ucji gc chhkdl lckf gzti xiscfi trgo hpkx ygrfgy kd rbrzm. z'ni imhceie zg wzgo r ykxygzgkgzcm hzdoif gc trji gozmby tcfi skm:

r dcit
cm r drbi
xkg goim irho pzmi bfiw
gc goi goi wcfe ykt cs goi dfinzcky gwc
kmgzp z ygrfgie gc wcffl rxckg rpp gocyi wcfey hctzmb wzgo ykho sfiakimhl
xihrkyi ry lck hrm yii, zg hrm xi iryl gc fkm ckg cs ydrhi woim r dcit bigy rpp zxcmrhhz yiakimhl."

Monday, December 19, 2016

Winter break cyclic 3-4-5

This is another exercise in documenting geometry problem solving. I chose this problem because again it has a 3-4-5 triangle within it and the overall setup is very simple.  It also shows how one can circle around the correct solution,  as it were, before coming to it.  

Given cyclic quadrilateral ABCD where  ABC is a right triangle and AB = BC = 5  and 
the diagonal BD = 7 find the area of ADC.

Thought process

1. Angle chasing:  \(\triangle{ABC}\) is right isosceleses. Since  ABCD is a cyclic quadrilateral, that means we can angle chase a bunch i.e. \(\angle{BDC} = \angle{BAC} =45^\circ\).
2. Cyclic also means that the product of diagonal portions is equal i.e. \(AE \cdot EC = DE \cdot BE\)
3. Also there are a bunch of similar triangles: \( \triangle{AED} \sim \triangle{BCE} \) and \( \triangle{CDE} \sim \triangle{ABE} \)
4. Realize 2 and 3 are the same basically saying the same thing.
5. At this point I had a general idea that I'd need to use the length of both diagonals as well as  the similarity of the triangles to find the area.
5. I started the algebra with the Pythagorean theorem and similar triangles to find the lengths of AE and ED.  This looked fairly messy even going in although generally solvable.
6. So I stopped and checked with the angle bisector theorem around \( \angle{ADC} \) and realized that produced nothing new.
7.  I already strongly suspected just on visual inspection that the missing part was a 3-4-5 triangle. So once again I paused and checked in geogebra that my hunch was correct. (it was).
8. Then I started grinding through my first approach:  let a = BE and b = CE then:
\( \frac{BE}{EC} = \frac{AE}{ED}\) or \( \frac{a}{b}=\frac{5 \sqrt{2} - b}{7-a} \)

Which simplifies to \( 7a - a^2 = 5\sqrt{2}b - b^2 \)
In addition using the Pythagorean theorem: \( \overline{AD}^2 + \overline{CD}^2 = 50 \)
Then applying triangle proportionality \( \overline{AD} = \frac{5}{a} * (5\sqrt(2) - b)) \) and  \( \overline{CD} = \frac{5}{a} * b \)
When combined you finally end up with:
$$ 50 = \frac{25}{a^2}((5\sqrt(2) - b)^2 + b^2)$$
$$ 2a^2 = 50 - 10\sqrt{2}b  + 2b^2$$
$$ a^2 = 25  - (5\sqrt{2}b - b^2)$$
$$ a^2 = 25 - 7a + a^2$$
$$ a = \frac{25}{7}$$

9. I started to  solve for b after which point I could find AD and CD. But this was even messier looking than above and was heading towards an ugly quadratic equation.

$$ \frac{25 \cdot 24}{49}  = (5\sqrt{2} -b) \cdot b $$

10. That looked super messy but I realized what I really wanted was \( 1/2 * AD * DC  \) which
comes out to \(  1/2 \cdot \frac{25}{a^2} (5\sqrt{2} - b) \cdot b \) and you can substitute with the two derived results above to find the answer.

11. I was dissatisfied with the opaqueness and algebra of this method plus it never showed the 3-4-5 clearly so I started from scratch.
12. This time I played with the \(\angle{ADC}\). First I stated thinking about breaking up the 3-4-5 into a square and the various 1:2 and 1:3 triangles. But then I realized that I had two 45 degree angles that  could be extended and we know the lengths of the sides of those triangle  and from there the drawing below immediately fell out.  Framed this way the problem was much simpler.


Friday, December 16, 2016

AMC 8 Results

The wait is finally over. We received the results for the 2016 AMC 8.  Now comes my least favorite part sending them out to the parents.  This is the message I ended up with this year:

The results for the AMC 8 contest finally arrived.  First thanks for participating. I'm proud of how everyone did and this is just the beginning, hopefully, of AMC contests for everyone. I want to stress again also to treat this like a baseline on an above-level test. The contest is meant for Middle Schoolers. Since its nationally normed, keep these scores for future years. I'm hoping everyone will generally see growth over time. 
Some Resources:
This has all the problems and solutions for them. If your child is still interested, it can be valuable to go over the problems. Feel free to email me if you have any questions about the questions. 
MAA will eventually publish national statistics when they're done scoring all of the entries. These will be found here:  I don't
generally think these are too useful in our case since the data is normed for 6-8th grade.

Overall I'm very happy we participated.  Hopefully most of the kids will take this again  in future years. As an aside, the opportunity to participate is very site dependent here. I wish there was a more district wide policy so that all middle schools always offered the chance.

Moving forward I'm feeling like some encoded message fun over the break:

[Update]: the overall statistics were just published. One thing  that stuck out at me was WA state is a bit of a math powerhouse although sadly not much of this is coming from our district. Overall the state had highest mean score, median score and was the only one where you needed a perfect score to be in the top 1%.

Thursday, December 15, 2016

12/13 End of the Quarter

And just like that, another quarter has wound down.  I'm always amazed how fast time flies. Its a lot of work planning and running the sessions of Math Club and yet I'm still always torn wishing I had more time with the kids.  This day was more crowded than usual. I had to fit the second MOEMS Olympiad in, go over the problem of the week and there was the end of the  quarter game day to run. Clearly, something had to give and I ended up planning to sacrifice part of the time I normally would allot to game day. I'm considering running another one on the first meeting in January after we go over all the house-keeping  to make up for cutting it short.

As of yesterday I also finally received the updated rosters. I sadly lost 2 kids to scheduling conflicts but I have one new one joining and the overall numbers are still at 13 which is a good size.  Almost every time a student leaves I still feel a little bad though. There was one enthusiastic boy in this group  who often wanted to mention something one on one whom I really think would have had fun if he could have continued and a girl I lost to a conflicting choir practice last year that I ending up thinking of while I looked over the roster.

The problem of the week was only lightly participated in and I'm going to need to invigorate it a bit. So again I had kids work on it on the board: sketching out numbers and connecting the factors to see how far they could get. Since I'm finally getting used to the room, I took full advantage of the whiteboards on both sides and had 4 kids working at once.  That works great and sometimes a few extra kids come over and form a group which I like watching. What I didn't do but I will in the future was explicitly hand out paper to have everyone else working at the tables. Otherwise, some number kids don't watch or start working on their own  and sadly don't volunteer that they are missing the paper needed for scratch work. This is one of the areas where I think you just need to bridge the gap. Once  I imagined the group of kids always having a notebook and diligently bringing it. Maybe that works in Middle School but in fifth grade you have to do more to keep everyone on track.

There were a few behaviors I noticed during the Olympiad that I'm hoping to keep working on.  The first was 2 or 3 kids got stuck on one of the problems and basically gave up and turned the contest in with time still remaining. In each case, I encouraged them to take advantage of the time and keep working on the problems but it was a hard sell. (I also had a larger group that didn't figure out the solution to everything but kept working the whole time possible.) Persistence is one of those key attributes that I'm really trying to emphasize. I'll probably talk about it again in January but I'm still thinking about what the best way to handle this is. My first idea is to ask some of those who kept going to talk about what their thinking/strategies were when stuck.  Despite my skepticism sometimes of the value of most of the growth mindset theory that's really what's at play here. How do you get kids to buy into continually thinking about a solution (which they may not arrive at) and not shutting down?  It's this uncomfortable space where I think the most learning occurs.

My favorite moment of the day actually happened right after the contest was done. There was one problem that involved figuring the missing numbers in a consecutive sequence of number given the sum of some of them.  Listening to two boys discuss it one of them said "I solved it using try and fail (guess and check)" and the other replied "I solved it with algebra."  One its always cool to examine the structure of a problem through different strategies. But  more importantly this represents the inflection point many of them are at between informal methods and algebraic thinking.   I find this transition to be fascinating. There comes a point when you see problems and you immediately model them using linear equations and formally solve. Often this comes at a trade-off where the previous conceptual / informal reasoning takes a back seat for a while.  I find this analogous to how standard algorithms usually supplant informal computation strategies after they are initially learned.

Finally, I brought pente, prime climb, trezetto and pentago as well as a deck of "24 cards". These were all great hits and the kids immediately settled in once they had finished the contest.

Wednesday, December 7, 2016

12/6 Catching Up

As I alluded to in my last post I had to miss last week's Math club meeting. This was one of those times when expanding to two instructors really paid off. Thanks to the other parent, I was able to have the kids do a joint session and didn't have to cancel the day.  This was also made a bit easier by the fact that we were scheduled to do the first MOEMs Olympiad. So for the most part the kids were working by themselves on the problems in a proctored setting.

Nevertheless, that meant  I had a lot of items to catch up on when I rejoined everyone this week. To start off I had all the kids talk about their recent experiences with the AMC 8 test and with the first Olympiad.  The general consensus was that the first Olympiad was pretty hard. This was our first time trying out the middle school division so I was unsure what to expect. I always do the contests myself before hand to gauge their difficulty and form ideas about what I expect the kids to have trouble with. I also thought this was going to present a high degree of challenge. Interestingly, I've now looked at the second in the series and its doesn't appear as hard. So I'm going to continue on with the experiment and not switch back to the elementary division for now.

Because they hadn't gone over the solutions as a group I decided we would do so now even though they had been handed out. My general observation is that most kids don't reflect on problems unless you setup the structure to encourage it (even with a solution set).  The drawback was that this was a week later and so there was less excitement than there would have been in the moment but I think it was critical to draw out the discussion and have everyone think more about the problems they hadn't been able to solve.  

I'm unable to directly discuss the problems but several observations I did have were:

  • Comprehension is a bit of a problem. For instance there was one problem that asked the kids to find only cubed numbers. Many of them totally misunderstood and just looked for all the numbers. I'm going to have a discussion about careful reading before the next version and see if that's enough otherwise we may need to do some group practice parsing.
  • Many of the kids are unfamiliar with cubes and exponents in general. That's not completely unexpected given the scope and sequence for the year and I don't usually pick problems that hinge on them. But I may need to make an exception to facilitate with the contest and plan a day around exponents to give the kids the tools they need. The key idea I'd like them to understand is how an exponent is just notation that can always be expanded out into multiplication.

By this point we had used up more than half of the time and I wanted to switch tacks and do some group work. So I went with the factoring puzzle I had previously found:

As I expected this was compelling and accessible.  As I worked my way around the groups, most of my discussion were around which numbers can you see most immediately i.e. the 5 sticks out first for most kids (and then the 2's) and how do you go forward from there.  I needed to ask a series of leading questions to get some of the kids to think about factoring and structure. I.e. since all these numbers share factors once you've factored one you're only missing one factor in any of the other ones. So it also makes a lot of sense to factor the easiest one 6160 first.  My favorite observation, was one students noticing that once you did have the factors and ordered them, the smallest factor belonged behind the largest value, the second smallest factor belonged behind the second largest value and so on.

By this point I ran a few minutes over schedule. That meant I ended up skipping past the previous problem of the week and the other items I had assembled for the day.  If only we could have kept going for another 30 minutes ... On the bright side, we're well setup for our normal routine next week which will be the last session for this quarter.


Friday, December 2, 2016

Grab bag of problems

Sadly, I  have to miss Math club on Tue. given a family emergency. In the meantime, these are some of the problems I've enjoyed looking at recently that I might use sometime soon:

I've reached that point when I see something like this that I think: huh if I form a  right triangle using B to H as the hypotenuse, the remainder of one side is 3/5 r and the hypotenuse is r to the other side must be 4/5 r so its a 3-4-5 triangle.  There are a 1:3 and 1:2 triangle embedded on the edges of the circles as well.  To actually prove this is true just takes the Pythagorean theorem in my version but I suspect there are several other means to the same end. I'd probably scaffold this a bit.

A lovely factoring and deduction exercise. Not too hard beyond the large numbers.  Actually what might be fun is to adapt this and do it interactively in club.

You only need to find the ratio in one of the eight wedges (I drew in ABU) to find the total answer.
Also you could draw a different wedge by extending the inner triangle IJU but I find the first form a bit easier to reason about. Note: the 45-45-90 triangles and the other similar ones. The answer can be found through a combination of similar triangles and area addition starting from the 45-45-90.

The kids actually did this one. It was my second favorite to discuss. My first solution was a rather tedious exercise in the Pythagorean Theorem. There is a similar triangle version that is much simpler.

Some lovely divisibility rule practice embedded in this puzzle.

Find the smallest possible value of the expression:

$$\sqrt{a^2 + 9} + \sqrt{(b-a)^2 + 4} + \sqrt{(8 - b)^2 + 16}$$

This looks complex but ultimately boils down to the shortest distance between n points is a line! Unfortunately it needs a tiny bit of algebra to recognize the geometric representation and find the line so I'll have to save it for some time in the future.

Monday, November 28, 2016

Photo Diary of my visit to MoMath (Natl. Museum of Math)

The highlight of my recent trip to New York for Thanksgiving was a chance to get into Manhattan and finally checkout MoMath with my sons. I really wish we had something comparable here in Seattle. The exhibits were all very interactive and well suited to grade school kids. Not surprisingly, given its visual nature there was a heavy focus on geometry.

Robot swarm that followed the kids based on the color of the vest they wore. The algorithm was programmable from a station.

Life-size hinged polygon. Now I'm a square, now I'm a triangle.,

Tessellation puzzles.

Harmony of the spheres.

Lifesize rotating maze.In this one you had to cross a red line, blue line and then a white one. (This was one of the few activities I could actually replicate using painter's tape.)

Interactive multiplication parabola sculpture. You chose the 2 numbers to multiply and the connection lit up.

Painting with symmetry. 

Position/elocity interactive game. The kids ran back and forth to make it through a series of gates on the screen.


Galileo's curve. This was adjustable and you timed the cart going down the track.

Square bicycle.

Mobius curve race track. There was a steering wheel and camera view for each of the cars.

Doorway with the obligatory pi handles. 

Wednesday, November 16, 2016

11/15 AMC8 and some Topology

What a difference a week makes. I left yesterday feeling very upbeat. To start off, I drove to the school early to help escort nine of the kids over to our sister middle school where they participated in the AMC 8 math contest. I'm super excited about making this happen after mulling it over for 3 years. This year we finally had a large enough group of fifth graders to reach critical mass.  Despite some trepidation, it was not hard assembling everyone in the office and walking over.  I had one parent volunteer who stayed during the contest so I could go back and lead Math club for everyone else. I didn't anticipate how the fifth graders would react to the field trip. It turns out it was very exciting seeing the middle school that many will go to next year and had never been inside.  It's harder to arrange but cross-grade experiences like this seem effective. If I switch over to running a club at the middle school I'm going to remember this and try to reach out.

A lot of the conversation was also over the format of the test, whether there would be prizes (no), what does it mean etc. My main message was to stress that this is a baseline and they will have a chance to try it out over several years and see how they grow over time. I don't expect super high scores since everyone's young but I'm hoping that the kids come out excited to try again next year.

Also one boy remarked that the math we do in the club is more difficult than what happens in class. "Its more like 8th grade math." I smiled at the time especially at what the kids think the the standard is for difficulty.  I'm hoping that was meant in a good way and its something I think I'll follow up on when I survey everyone. Ultimately, what I'm aiming at is for them to say instead "I really had to think - that was different and interesting."

During the regular session with the remainder of the kids, I only had eight students left. This was actually also fun. Working in larger groups, you forget how much more time you can spend individually and how much easier it is to manage flow at the smaller sizes.  I started by stressing that last week was unusual but I expected a return to our normal behavioral norms.  We then looked at the P.O.T.W. I went with one from MathCounts:  This was Veteran's day themed and I was stretched for time so I didn't look for more alternatives. It turns out I wasn't so happy with the choice on  consideration. The problem basically was a simple version of calculating percentages and didn't have much meat to it. So it went fairly quickly and this week I spent more time coming up with a better choice.

For our main focus I turned to a really cool topology experiment I found from Mike Lawler/James Tanton.

I didn't have time to take pictures of our own work so here's the video of Mike's kids exploring:

To manage the scrap paper issues I had everyone gather together on the carpet area. This kept the mess to a small portion of the room. I was surprised that several kids had never heard of a Mobius strip. So I started by cutting strips out and letting the kids tape together a simple mobius strip. They then confirmed via tracing with a pencil there was only one side. From here we went through each of the more complex cutting projects. Before each one we made guess as a group about the outcome and afterwards I also had everyone characterize the results, paying special attention to trace how many sides the new shapes had.  Overall, I agree with Mike, this was a really engaging investigation that's worth repeating.

Finally for the back half I printed out a sample MOEMS contest to practice on. We're off for Thanksgiving but next session I plan to administer the first real round of MOEMS. This is going to involve a bit of creative rearranging to make all the rounds fit.

P.O.T.W: I chose this one from nrich which looks to generate much more interesting white board conversations:

Thursday, November 10, 2016

11/8 Election

This was a pretty raucous session. I didn't really anticipate how much the kids in the Math club would be invested in the election. I was peppered with requests for updates on the results during which I mostly satisfied via my phone. "Nothing has changed in the last 5 minutes ...."  I even had a few try to use the computers in the room to go to election sites (which has never happened before).  The moral of this story is mostly if for some reason four years from now you're running a Math Club on a presidential election night, be prepared.

In fact, based on a conversation with my friend Dan I had already scoured the web for election math based activities and had ended up with the following one from NCTM:

My plan was to go through the communal starter, and then run the worksheets in groups and reassemble back together to discuss after 5-10 minutes.

I then committed an error in judgement that hopefully didn't bother anyone. Given the excitement in the room and some requests from the kids rather than doing a synthetic election tally  I decided to do a full blown mock election and to tally the results on the board for analysis. Without thinking about it I asked the kids to just add their own selections to the tallies on the board. It wasn't until 5 minutes later I thought about it and realized that I should have maintained secret ballots for this. With any luck, this wasn't significant but I inadvertently violated a core principle with respect to politics. If I'm ever in this situation again I will be more mindful.

This mistake weighed on me the rest of the session along with the need to keep everyone focused on the math rather than the actual election. So overall I'm looking forward to a more normal next week.

The worksheets were pretty good but it took a lot of effort on my part with each table to keep everyone one track. I'm probably also going to go over our standards of behavior again as a group next week.

Monday, November 7, 2016

11/1 Grid Puzzles Again

This is a short placeholder for my records. The synopsis for this week is I did the following logic grid puzzle:

I tried to pick one that would be finishable without using the entire period but as has often happened in the past most of the groups worked the entire time. On the positive side, these are always a hit.  For the one table that finished early I gave out a practice MOEMS test since that's coming up in a few weeks. I handed it out to everyone to look at on their own but I don't expect to get much participation without doing it together.

So I might wedge it in tomorrow. However, I've found an intersesting election based activity from NCTM which I plan to focus on given that its the real election. So we'll see how long everything takes.


Friday, October 28, 2016

1:2 triangles and their link to Pythagorean triples

I've mentioned before how instinctively it feels like the 1:2 triangle ought to have a more natural angle measure. In fact its in a 90 - 26.57 - 63.43 degree triangle. However when combined with a 1:3 triangle more integral relationships appear.

The 1:1 triangle ABD, 1:2 triangle AED and 1:3 triangle AGD all in a row.

Demonstration that the 1:1 + 1:2 + 1:3 triangles create a 90 degree angle. in the corner G.

  • First draw in a congruent 1:2 triangle GLK.  
  • The remaining angle between it and the 1:3 triangle forms a 45-45-90 triangle.
I just realized another extension today.  Multiple the angles in the corner G by 2 and you get

90 degrees + 2 1:2 triangles + 2 1:3 triangles = 180 degrees.

Stated another way if you take 2 of each of the base triangles they will form another right triangle.
And here's where it gets cool: out pops the 3-4-5 Pythagorean triple.  (Its also another demonstration why the in-circle has radius 1)

This meshes well with another 1:2 triangle exploration that led to a
 3-4-5 triangle:


Once you have medians, like F and E you create 1:2 triangles like ABF, AED, CDF. So its fairly natural that a 3-4-5 is lurking.  The normal approach would be to show AEG is similar to ABF and also a 2:1 and then find the length of EG, GD and then prove DFG is a 3-4-5 in proportion.

But we simplify by just noting FDG is 90 degrees - 2 arctan(1/2) angles and therefore is
2 arctan (1/3) i.e. one of the corners of the 3-4-5. We can either show FGD is 90 degrees using similar triangles or AFD is 180 - 2(90 -  arctan (1/2)) = 2 arctan(1/2) and the 2nd of the 3 needed angles. And we're done.

Second Example:

This doesn't immediately appear the same but once you add the radii and center and remove the circle you're left with:

The median at the bottom has created a 1:2 triangle CBH which its then easy to show similar to first AGH and then CGH.  You can then see angle DGC = 2 1:2 triangles angles from the parallel lines and it follows CDG is a 3-4-5.

What to do with all these connections?

It seems like there is a really fun project for kids exploring all these relationships. I just have to figure how to structure them since the conceptual leaps are fairly big.

Thursday, October 27, 2016

10/25 Gozen

It was an interesting week from a planning perspective.  I'm  almost finished emphasizing divisibility and trying to decide what area I'd like to turn to next. One possibility was to return to some math relays from Math Counts. We'd tried that last year and it was mentioned as being fun by several kids in the wrap ups for the quarter.   My friend Dan had mentioned the relays again and they would fit with my aim at finding delivery methods that excite the kids while working through problem sets. In particular, there are also some thematically unified sets which I prefer.

At the same time I saw an interesting set of topology projects that Mike Lawler did:

I tried them out myself and I think they would be hit based on my experience with fold and cut activities last year. Finally, there was a comment from Joshua Green that had me relooking at resources at after a fairly long absence.

In the end I decided to focus on one of the games from there:   (See: for the inspiration) I thought and this turned out to be true that the tower-defense structrure would appeal to my kids.  The one problem with this game is the fairly complex instructions. So going in, I planned to play a sample game as  a class to help everyone catch on.  This worked pretty well.   There was still some initial confusion and questions but after the group play was over, I only needed to walk around and answer about 1 question per group to have everyone on track. And once that was done, most of the kids really got into it.

As you can see I brought an assortment of colored pencils and the final boards are actually quite pretty. If I were to repeat I might tinker with the rules a bit. The defensive block shots don't really fit well with the permanent archers defending their squares. Perhaps you should be able to choose between two types of moves.

I ended up reserving about 10 minutes at the end to demonstrate the divisibility rule for 7's since I had promised I would do so last week.

That left just enough time to hand out the problem of the week. This time I turned to a fun geometry exercise.  I didn't include any drawings so hopefully everyone interprets squares as geometric figures rather than square numbers (although its certainly approachable that way as well.) I'll probably stress that point in my weekly mail to the group.

"Find a way for any number n greater than 5 to divide a square into n parts each of which is also a square. Note: the subsquares do not have to all be the same size."

Looking forward, our first MOEMS contest is coming up as well AMC 8. I'm also feeling a strong urge to break out a grid logic puzzle.

Saturday, October 22, 2016

10/18 More divisibility

After a slow start last week, this time almost all of the kids in the Math Club finished the problem of the week.  (See the end of this post)  I had one of the boys demonstrate his solution on the board. He's one of the slower but more careful writers and almost always shows something interesting. So I tend to ask questions to the room to keep everyone involved while he gets his thoughts written down.

In point of fact, I don't really have any kids that are good at talking simultaneously while they write on the board. That appears to be a learned skill. So in the beginning of almost all student work on the whiteboard I usually choose between narrating what's being written or asking background questions to prep the room.  One of these days I'd love to see what other people do.

As an amusing side note I found a very similar variant on the puzzle in the Moscow Math Club diary book I was rereading this weekend.  (See below)

Besides the odd coincidence, two ideas stuck out for me in the book.
  • The importance of emphasizing games for younger students in a math circle. In the context of this book, younger meant around 5th grade as opposed to High School students. I continually find evidence that supports this idea in the various sessions I run.
  • The focus on working problem sets every week with relatively low student to  facilitator ratios. This was my vision of how a Math club would work going into the process. I don't really follow this model very often though.  So I decided to go with a more pure problem set based day to see where we're at.

To start up with, I decided to begin with the game of 100.  This is a fairly simple two player game with the rules being:
  • The score starts at 0.
  • The goal is to be the first person to reach 100.
  • Each player takes turn picking a number between one and ten and adding it to the score.

I was a little unclear in my first explanations so I ended up doing a few whole classs demos where I played against one of the kids and walking around the room to make sure everyone had understood the rules. After that startup activity, I let everyone play with the goal of finding a strategy to always win.

Gratifyingly, after about 5-10 minutes most groups had discovered part if not all of the structure of the game.  [If you reaches 89 you can always win, which means you need to reach 78, which means you need to reach 67 .... all the way back to 1]  I then had the group talk about their discoveries.

For the middle of the day, I returned to divisibility rules. I started with having everyone name all the rules for 2,4,5,6, etc. through 9. We then spent some time talking about adding multiples again and I brought up the question "How does this relate to adding odd and even numbers?" I then went over the logic behind the nine rules again on the board.  I had the kids pick random digits for our sample number which worked well.  Then I had them pick (mostly) random digits again so we could work on the rule for 11's.  My key question here was if "9 = 10 -1" led to the rule for nines what relationship would help with 11's i.e. 11 = 10  + 1?  We then did the follow exercises:
  1. Find a pattern for nearest multiple of 11 to a power of 10 and figure out why.
  2. Using that rule, and the distributive law breaking out the sample number to see how the divisibility rule worked.

This is all a bit trickier since for 11's the numbers alternate between positive and negative i.e.

10 =  11 - 1
100 = (11 - 1)^2   =  11^2 - 2 * 11 + 1
1000 = (11 - 1) ^ 3 = 11^3 ....  -1

Which was a pattern I wanted to emphasize without the benefit of knowing the binomial theorem.

We had about 18 minutes left at the end of this work and I gave the kids a choice between a worksheet with some extension problems about divisibility or a sample AMC 8 set of problems. Interestingly,  everyone seemed to prefer the random set. I spent this period walking around, checking on  progress and answering questions.

The most interesting moment was helping a student who had forgotten how to multiple decimals. I went through the idea of treating the decimal like this:

    1 2 3 4 . 5 6
x           7 . 8 9

=  $123456\times\frac{1}{100}\times789\times\frac{1}{100}=123456\times789\times\frac{1}{10000}$

Unfortunately, I didn't have enough time to see how well that explanation resonated. Overall everyone worked fairly diligently through this experience but I was still left not feeling sure about the structure. I'd like to figure out how to get more insight into what everyone's doing and to keep the interest level high.

Idea: I saw another teacher post about doing problem sets and stipulating all work must be done on the whiteboard to facilitate discussion. This seems interesting and I think I have enough space to make it practical.


I went with a scaffolded version of the divisibility problem I found online:
10.18 Problem of the Week

I also at the very end asked if the kids would like a demo of the rule for divisibility by sevens. About half were interested so I may do so next week even though I don't expect them to remember it. This is more aimed at impressing that complex divisibility rules exists and there are patterns that extend up through the integers.

Thursday, October 20, 2016

More Geometry in a Box (Trisection)

Continuing an occasional topic, I saw another great simple box construction.

[Prev post in this series:]

As always what  I like about these is both the complexity hidden in relatively simple constructions and the interesting, often surprising relationships that fall out.

This one started:

Given M and N are medians, find the ratio of the area of triangle DKL to the area of rectangle ABCD.

Initially I was looking for similar triangles and decided to add a parallel line in the middle DKL. That splits it into two triangles that are similar to larger ones CLN and BDN. Interestingly not only are the two sets similar but they are similar in the same proportion since both sets have the same length sides.

CN = NB since its a median and the new line is the base for both new triangles 1 and 2.

This approach works, you can use the similarity to setup a equation and derive the proportion but it didn't feel elegant at the time.

As it turns out there is a better way and it reveals cool fundamentals that were obscured above.

My second approach was to try to create a single similar triangle rather than two. So I added in the parallel line AN.  This creates DNO which is similar to to DKL.

But just adding this line now starts to reveal some more interesting facts. To start from the symmetry its very easy to show triangle DKM is congruent to its reflection BNO.    That implies BO = DK,
But even more interesting DKM is similar to ADO. That implies that K is the middle of DO since  M is the middle of DA. In other words, this construction has trisected the line DB.

(Note given the symmetry you could also show O is the middle of KB using the triangle BCK)

You can now use triangle proportions to quickly solve the problem. Triangle BDN is 1/4 of the rectangle. Triangle DNO is 2/3 of triangle BDN since its base DO is 2/3 of DB. And triangle DKL is 1/4 of DNO since its a similar triangle scaled 1:2.  

So Triangle DKL is 1/24 of the entire rectangle.   But more importantly its not just a coincidence that we started with only medians/dissections and ended with ratio that contains a multiple of 3!

Wednesday, October 12, 2016

10/11 Don't Fence Me In

My initial thinking for how to structure the start of Math Club was influenced by last week's  problem of the week:

This sheet revolves around a graph paper and dice based game where you roll the dice and fill in the boxes until someone can't go using either a rectangle with the same perimeter or area as the product of the dice roll. I suspected the kids would really like playing the game before we started talking about the questions. So I brought in my graph paper and dice and we spent the first half trying it out.

Sadly I didn't take any pictures of the finished games. But the kids really enjoyed the experience and several asked if we could keep going.  This definitely helped with our white board discussion afterwards. Interestingly for the final question, the best the kids could think of was a 7 square solution that would cover the whole board. I know of at least a 6 square one with an upper bound of 5 so I may return to this problem. For the end of this portion I asked everyone to think about if they could find a better solution.

We then transitioned to a talk about divisibility rules. I made several improvements over last time I tried this. First we spent some time group brainstorming about what happens when you add a multiple of n to another multiple of n, a multiple of n to a non-multiple of n and two non-multiples of n together.

After several minutes reflecting I had the kids report their ideas and why they thought that they worked.  Interestingly, I had to introduce the notion of the distributive law here. I skipped focusing on it this year since we did so last year but perhaps I should circle back.

For next time: On reflection I think a really good followup question would be how does this relate to the rules for adding odd and even numbers. The hope would be to build the connection that this is the same as multiples of 2 and that its a special case where 2 non-multiples add to a multiple.

With that foundation we talked about the simpler rules for multiples of 2 and 5 and why they must work. This flowed fairly well from the previous ideas. I had kids volunteer solid reasoning for both rules.  I ran out of time as we just started to work through the more complicated nines case.

My idea for followup is to walk through nines again and then let groups work on 11's with the starter idea that 10 = 11 - 1, 100  = 11* 9 + 1, 1000 = 11*91 - 1 etc and see if they can follow the logic.
Two years ago I showed the rule for 7's just for fun. I may pull that out again although it serves no practical use because of its inherent interest.

I also had a sheet of problems from AoPS that I planned to use that we didn't get to that I might bring back.  Buy coincidence yesterday I saw the following problem:

This would dovetail really well with the divisibility work.  I'm thinking about having groups brainstorm about the left hand side to see if they can see if must be a multiple of 6^2 by examining concrete values.


Todo: About 7-8 kids were volunteering answers out of the 12 who were here today. I need to draw out the quieter ones in future weeks,.

Tuesday, October 4, 2016

10/3: Year 3 (Bigger and Better)

I always get an odd mixture of impatience and trepidation right before after school activities start up. This time around most of my concern is around getting two rooms running smoothly.  I really want a strong start to the year to ultimately prepare for passing the Math Club onto the next generation of parents.

Final Demographics:  26 kids
14 5th grade 1 girl
12 4th grade 9 girls

As you can see above its a weird lopsided year. So I have 10  girls but almost all are in 4th grade. In theory I'd like a few more 4th grade boys and lot more 5th grade girls.  This wouldn't actually be an issue except I'm splitting the club with another parent by grade. I have one lead for a future participant and I'll have to think what more I can do to recruit next quarter. One idea is to ask the 5th grade girl's parents if she has any friends who might want to join her.  I'm also tempted to randomly mix the kids up to achieve a better gender balance but that cuts against reusing material with the 4th graders and the opportunities to more closely fit the skill levels that we get from a grade split.

Update: I've found one more female fifth grader. So its not quite so bad. 

One other short term goal: Most of the fifth grade boys are coming back from last year and very comfortable (probably a bit too much) with each other. I really want to shake things up. So I'm thinking about either focusing on individual activities for a little while or forming temporary assigned groups to force kids to partner with new people. I'm actually pretty excited about this in the long run. You can often see really interesting development over the span of two years.

The afternoon started with me unable to get into either assigned room. The after school coordinator assured me that both rooms would be unlocked while the kids came down. Ultimately that did happen but I wasn't sure what state either room was in (I ended up with another room that didn't have much in the way of desks). For the first session I talked with my new co-coach and we agreed to do a joint starting session that I would captain. She brought name tags which were  useful except when kids lost them or hid the names under their pony tails. On the bright side, knowing half the kids before hand meant a lot less names to memorize. Also this was a new record for me handling a real full classroom of students.

I began with my normal introductory activities. I always have each kid introduce themselves, say their grade and teacher and why they joined. I had a sprinkling (thankfully not too many) of "my parents made me" and a few "my older brother was in math club" but for the most part the kids were there out of desire.

We then followed my normal outline about behavioral norms and goals. See: Club Charter Discussion Points  I like to emphasize the big ideas of "What do you when you're stuck?", "Mistakes are part of the process", and "We're working on listening to each other" and ask for lots of examples from the kids.  I actually thought a bit about whether to change this time after I saw a bunch of posts online about skipping discussing rules until the second day of class and jumping straight into math. I'm just not brave enough yet to try that out. My fear is that if we don't establish some of the basic norms right off the bad precedents will be set. That being said I think all my returning students work really well and the new ones seemed very amiable so I'm not sure if I'm being overly cautious.


I went with  Sara VanDerWerf's 100 game.   This was a hit and interesting to watch. We did 2 rounds with a 3 minute reflection on strategies in between and then a 6 minutes wrap up reflection.  Overall, I would definitely repeat this. The kids were captivated and they did not finish the whole sheet like I feared. You could easily bump the time per round from 2 to 3 minutes and run an entire 3rd round but I had only printed enough sheets to go twice.  On the final reflection it didn't look like the groups were going to notice the pattern so I asked everyone to very slowly replay a few steps and look for patterns again. That did the trick and almost every group found the number spiral based on that strategy.  One other fine tuning I might try if repeating.  I only had yellow highlighters due to supply constraints. If I could have handed out multi color high lighters in the second round, that probably would have more naturally led to the key insight.

We then started a dot pattern exercise. ala:  I wrote four questions on the board:

  • What is the next step for each of the patterns?
  • What is the 7th step for each of the patterns?
  • What would the step prior to beginning of the patterns look like?
  • Can you find an equation to model how many squares or sticks  are in each step?
We only had about 10-15 minutes to work on these. I went around the room and most of the kids were just getting the basic idea coming up with the next step for one or two of the patterns. So I definitely want to come back to them to at least have a group discussion and some white-boarding. One possibility is to warmup this way for a few sessions.

P.O.T.W Amusingly the first thing most of the returning fifth graders asked about were homework points and treats. That apparently had a very strong effect on everyone. So I'm hoping for strong participation again this year,

Tuesday, September 13, 2016

In Which I start to Organize

We're about 3 weeks out from the start of Math Club and there's a fair amount of organization to do. One big initiative this year has been growing to two instructors and two rooms.
  • The school only assigned one room so I have to fix that up.
  • Although we still only charge $45 per head, I need a more formal structure to deal with finances. Hopefully its not too hard to form a non-profit org.
  • Recruiting I've sent out my email to the teachers to try to bring the number of girls participating up again, It would be great to reach 50/50 again.
  • Confirm whether fingerprinting is needed or not.
  • I still need a good rhythm for planning with the expansion.

On the positive side: 18 students have already signed up after 1 day. I was a bit worried that I was going to lose many to soccer practice etc.

Update: I'll be at 25 students for the first quarter. I'm pleased that a lot of last year's fourth graders are coming back. I also lucked out and procured another classroom so we can run two proper sessions. This has probably used up all of my building karma for the year.

Ideas for the first day.

  • Play Buzz again.  Cons: if a lot of kids return that may be less fun.
  • Sara VanDerWerf's 100 game.
  • Math Biographies. Cons: writing intensive. This depends on the students who sign up. With lots of repeats I feel like I mathematically understand the kids already.
  • Dot pattern / equations. ala:  I like the dot patterns where you predict what comes next and then model an equation.
  • Untangle the human knot. Need to get some specific starting points or investigate outcomes.

Friday, August 26, 2016

Another Fundamental Square Construction

Its amazing how much for want of a better word beauty is lurking in very simple constructions. I've talked about some square variants before:

I've been thinking about another one a lot this week: Take a square, draw the quarter circle arc from its two corners and then draw a line from its median M to the corner D. The original problem from @five_triangles was to find the length of MN but I'm more interested in the sub problem embedded within it and all the things I noticed and wondered  Sadly this is all a bit too advanced for the kids this year, although I could guide them through part of it through a combination of guess and check, exploration of samples etc. Its the spirit of what I want to happen though and how I interpret what mathematical questions we should be asking. 

So instead lets draw the internal right triangle this creates BNE and see what kind of triangle it is.

If the side of the square is length x, then so is the hypotenuse BN.  Likewise since BM is the median its length is x/2. Triangle MNE is similar to MDC which has its sides in ratio 2:1. So lets let NE be y and ME be y/2. We can then apply the pythagorean theorem:

$$x^2=  (x/2 + y/2)^2 + y^2$$ and simplify a bit to

$$x^2 = \frac{x^2}{4} + \frac{xy}{2} + \frac{y^2}{4} + y^2$$
$$4x^2 = x^2 + 2xy + y^2 + 4y^2$$
$$0 = -3x^2 + 2yx + 5y^2$$

Apply the quadratic equation we can solve for x and arrive at $$x = \frac{5}{3}y$$.
If this doesn't look familiar at first consider that if x was 5, y would 3 or in other words BNE is a pythagorean triple in the 3-4-5 family! That seemed like a really cool relationship: Take a square, run a line from its median to one vertex and you have constructed a 3-4-5 triangle.

Thinking about this some more the wonder was a bit tempered. By picking points on the arc and drawing the right triangle from corner B to that point one can construct every right triangle from 0 to 90 degrees.  So of course all of the Pythagorean triples are included somewhere in that set.  But it still seemed cool that the median produced one rather than some random fraction or irrational number point.

But then I thought some more. What if you start with a Pythagorean triple? All the sides are integers in length.  The point M is going to be in exactly the same ratio to the squares side as the difference of point N from the sides of the square since they are in similar triangles. Those differences are going to be the differences of two integers so of course they will also be integers.

For example: Lets take the 5-12-13.  Its point N will be 13-5 or 8 from the top and 13-12 or 1 from the side.  So if we take a square and take the point 1/8 of the way from the side, draw a line from there to vertex D and create the corresponding triangle we will reconstruct the 5-12-13.

Thus every Pythagorean triple will be produced from a line through a point at some fairly simple ratio of the bottom of the square. The 3-4-5 happens to occur at 1:2.

Ok so what happens at some of the other ratios? What if you put point M at 1/3 of the length of the side for instance?  We can go back to our original quadratic equation and instead of using the ratio 1:2 for the sides ME and EN  let it be a variable k.

Solving you now get:

$$ x^2 = ((x/2 + (ky))^2 + y^2$$  resulting in:

$$x = \frac{2ky  \pm 2y\sqrt{4k^2 +3}}{3}$$

In the Pythagorean triple case k was 1/2 and 4k^2 + 3 was 2. In the case where k is 1/3 liked I originally wondered  the square root of 4k^2 + 3 is irrational.  So all the other ratios produce less interesting non-Pythagorean triple triangles.

Then I started to wonder what's the pattern to the ratios that produce Pythagorean triples? My first thought was lets make a table but then I thought some more about how all the Pythagorean triples can be generated from two integers a,b by computing the following:
$$a^2 + b^2, a^2 - b^2 \text{ and } 2ab$$

That means the computing the ratio in terms of a and b we will end up with

$$R = \frac{(a^2 + b^2) - 2ab}{(a^2 + b^2 )- (a^2 - b^2)}$$

(or its reciprocal) and that simplifies down to:

$$R = \frac{(a-b)^2}{2b^2}$$

When (a,b) = (1,2) the 3-4-5 generator we get 1/2 or the median as expected and its much easier to play with the ratios from here. One easy observation that falls out is that its not a coincidence that the first two ratios we looked at were powers of 2. All ratios of the form:

$$ \frac{1}{2^n}$$ where n is odd produce triples.

You can construct them by choosing b = 2^n and a = to b + 1.

Finally in one last riff, I was left thinking about the original 1:2 triangle MDC. It somehow seems inelegant that its angle is the rather messy 63.43 rather than some pretty fraction of 90 degrees.  Although as I was reminded there is a nifty relationship between \(\frac{\pi}{4}\) and the 1:2 and 1:3 triangles.

Thursday, August 11, 2016

Book Review: The math book : from Pythagoras to the 57th dimension, 250 milestones in the history

Last weekend I finally had enough down time to read through The Math Book by Clifford Pickover.
Overall I thought it was a very good read. The book is structured as a chronological series of one page discussions of mathematical breakthroughs throughout history. Each article also includes a related illustration which I found both visually arresting and useful.

The articles range from pre-history (30000 year old bones engraved with tally marks) through modern day discoveries like the Lie Group E8. This format is good for getting an appreciation of the progress of mathematics through history without necessarily delving into details. In many ways this sense of scope and of Mathematics as a living field of research where discoveries are being made is what is left out of normal school textbooks. So any resource that breaks the static/linear impression students might have is useful.

Also because the articles don't necessarily relate to each other (although Pickover is very good about cross-linking related topics) you can jump around easily.  The first night I picked the book up from the library, I played a game with my kids where they picked a year in history and we looked at what had happened then.

A small sampling of topics:

  • Magic Squares
  • Zeno's Paradox
  • Archimedes Spiral
  • Al-Khwarizmi's Algebra
  • Fibonacci's Liber Abaci
  • Golden Ratio
  • Kepler Conjectur
  • Projective Geometry
  • Euluer-Mascheroni Constant
  • Goldbach Conjecture
  • Buffon's Needle
  • Bessel Function
  • Quarternions
  • The Mobius Strip
  • Pick's Theorem
  • Principia Mathematica
  • Hilbert's Grand Hotel
  • Cellular Automata
  • Ulam Spiral
  • Surreal Numbes
  • The ABC Conjecture
  • Perfect Magic Tesseract
  • Checks is Solved
Overall, for Math Club purposes this serves more as a starting point for thinking about what I might plan a day around. There's not enough  here to make anything complete but its a great survey of potential ideas. If you wanted to have kids do book reports or presentations (which is outside my thinking of  the scope for the club) this would also be a good resource. One other idea I came up with was reading or adapting a single individual page to the group from time to time. 

Author's page: