## Friday, June 23, 2017

### Ideas for Next Year

It was a good week for finding inspiration on the internet (and in some books from the library). This is a grab bag of ideas for next year based on what I happened to read recently.

#### Jim Prop's gam: Swine in a Line:

From a given starting position is there a move you can make so you'll always win? There is a followup video.

#### Brenford's law

(first digits are mostly ones (logarithmic) Easy to collect data and observe.  See: https://en.wikipedia.org/wiki/Benford%27s_law

#### Problem Stream using the Trivial Inequality

(Great for Whiteboard).
Prove the AM-GM inequality as a demo.

## Wednesday, June 14, 2017

### 2017 Year in Review

One of the questions I had going into this year was "how much would my practices change this time?"  Its never possible to figure everything out but after two years one can certainly establish patterns and routines and move towards smaller refinements.  Its fairly normal for improvements to slow down or become more subtle at the same time.

So I went back through my blog, especially through the last 2 year end reviews.

For the most part I still agree with my thinking from past years but on reflection I think this year has been different from the last two and I've made some important changes.

As always my key goal is to generate and sustain enthusiasm among the kids and to get them to engage with as many  challenging problems as I can manage without losing them along the way. This year in particular I think I made strides towards engagement.

#### VNPS (Vertical Non Permanent Surfaces)

This was probably my biggest discovery for the year. See: http://mymathclub.blogspot.com/2017/03/328-vnps.html  In the past while I used whiteboards during student demos or while I was explaining a concept, it never occurred to me that it would work so well for group problem solving.  I found I could keep kids going on a problem set much longer in this format than in paper even while both involved group work.  I think there was also good carry over week to week. A successful session set the stage for more focus the next time.   What I liked best was setting up 2-3 problems on all the boards and letting the kids switch between problems as they wished.  I then could stop by a cluster and watch and ask questions.

2 Things I'll keep working on here:
• Work on listening carefully and not just floating when this is going on. Its easy to watch the kids working on the boards and be excited about the energy rather than focusing on the work they are producing.
• Integrate gallery walks at the end, where the group goes through the solutions that were found. I started doing this more consistently at the end of the term.

#### Structure

As I've mentioned before, my thinking on warmups and how to layout a session has evolved a bit this year. My old structure was often a 10-15 minute "warmup" and then a main activity. Over time what I've come to realize is that Math is not baseball.  Warming up is the wrong metaphor for the process. In practice, after doing my first welcome speech for the day when I talk about what we're going to do and having kids go over their solutions to the problem of the week, I've usually provided enough routine for all the kids to transition to thinking about Math.  Instead, what I usually find is the kids benefit from breaks in the middle and the end of the problem solving process.   The need for  pauses is especially evident with problems that are challenging and not quickly solved. So nowadays I usually assume an average focus length of 30 minutes and I'll take something like a kenken puzzle and hold it in reserve for those moments. When kids flag, I'll have them switch gears for a little while or perhaps the remaining time.

This doesn't mean I don't sometimes do a 2 part activity.  This can still be valuable if the first part directly relates to the second. For example: coloring in Pascal's triangle for patterns was a great setup for looking at combinations.  But if I have a great main activity, my first inclination is just dive in and let it take the whole time if necessary.  As a result, I've been printing and saving puzzles over multiple weeks much more often.

#### Games

This was an early focus for me this year. I didn't end up using games every week but whenever I found one that I thought was mathematically interesting I consciously tried to build a session around it.  I think  my favorite 2 for the year were Rational Tangles  and Attack of the Clones.  When I repeat activities, The challenge here is to continue finding new ones when you've worked with the same kids for several years.  I usually find kids don't mind/don't remember the first time nearly as clearly as I do/fear and I do reuse several categories of puzzles like the skyscraper puzzles this spring.

I also am still working on connecting the games explicitly to underlying Mathematics and making the time to talk about the games after we play them.  (This dovetails with my structural shifts. When you just dive in, there is more time left for a post-discussions and you run out of time early less often.)

#### AMC

I was very happy with my experiment getting the kids to try out AMC8. These also provide a little structure in the beginning of the year.  Competition brings out focus that a normal sheet of problems would never elicit. Next year, I'm going to try to leverage that a bit and use practice tests as a way to motivate kids to try and discuss problems more explicitly.

#### Video Integration

This year I chopped videos up a lot more than I did previously. Wherever I saw opportunities to discuss or try something out I would just pause.  For instance, during the video on proofs every time a problem was introduced we turned on the lights and tried it out as a group first before hearing the solution. I'm also really happy with the serendipitous alignment of my random topic choices and the final "Infinite Series" video on slicing a n-dimensional cube.

One experiment I'm still evaluating in my mind, was showing the MathCounts final. Kids are still asking to see more clips from MathCounts. So it definitely was popular. I'm not sure if it sent the right message about speed and ability though.

#### Guest Speaker

Going through the work to arrange a guest speaker was definitely worth it. I will try to maintain my relationship with the UW Math dept. going forward. In my ideal world, I could have a roster of speakers through the whole year.  Since that's not possible video clips act as a surrogate for this experience letting the kids see and hear more mathematicians. Also I'm toying with the idea of having the kids expand on my: http://mymathclub.blogspot.com/2017/05/questions-for-mathematicians.html. I'm hoping we could get a lot more responses.

#### Challenges

The biggest one for me is trying to ensure that the club transitions beyond my departure. I'm currently working hard to recruit folks for next year.  At the same time, I'm also starting to brainstorm about the differences in running a M.S. vs an E.S. club. I  expect to have to make many changes next year with older kids.

## Tuesday, June 13, 2017

### 6/13 And its a wrap

I was really heartened after sending an email out last week that all of the kids made it to Math Club today despite the transportation difficulties posed by a class field trip. Since this was the last session for the year I planned a mostly celebratory day.

First up there was a Math Club cake I ordered from the supermarket.  One unexpected hitch, on examination there were almonds in the cake ingredients. Fortunately I had brought some candy to give out for anyone with a dairy allergy that I also used for those who were allergic to nuts.  I'm still not convinced where the almonds were hidden but I'll be more careful before ordering next time.

We then went upstairs for the rest of the afternoon. I gave a quick speech thanking everyone for their hard work and then went around to survey what the kids liked this time. There was no overwhelming favorite unlike last year. Beside food which is always popular, the kids mentioned the Rational Tangles from last week, Pascal's triangle, some of the videos, Attack of the Clones, etc.  Overall, I think I executed pretty well on my goal of trying lots of different things to draw the different kids in.

Today I brought the projector in so I could show the following video:

Its incredibly cool and  I've been waiting to show this since by happy coincidence this tied into 3 of the topics we explored this quarter:

After that was done we finished with a game day as is traditional. I brought in my normal assortment of board games as well as one printed grid logic puzzle since these have been a great hit.

I was struck at the end at how smoothly the day went. (Granted I completely stacked the deck) Kids went excitedly from games to the puzzle and organically interacted with each other. I'm also really happy with classroom culture that had developed.

## Wednesday, June 7, 2017

### 6/6 Rational Tangles

This Math Club  was a growth exercise for me. I had decided a few week's ago that I wanted to do John Conway's rational tangle game: http://www.geometer.org/mathcircles/tangle.pdf in a future session.  It seemed great for a couple of reasons.

• The problem was posed in a game format that didn't require a lot of supplies.
• The game was physical (Good ones in this class are always hard to find.)
• The connection between the game and rational numbers had a lot of depth.

I also really wanted to stay hands off and maximize the kids own thinking as much as possible. So my challenge to myself was to allow the time for experimentation but keep the kids going all the while sticking to asking questions rather than telling answers.

This is the structure I chose.  First I outlined the rules and had a demo set of kids try out twists and rotates just to make sure everyone understood what we were doing.

For the next 10 minutes or so I had kids in each group create their own tangles and unknot them through experimentation. I mostly observed through this point.  The one exception I made was that its fairly easy to do a twist / rotate / twist combo that gets you back to the starting point. If the kids fell into this path, I'd ask them to add more twists at once i.e. twice 2 or 3 times.

At the end of this phase, I had members from each group make the tangle for the other ones and I asked them to try to make them as challenging as possible.   By this point the kids had developed a reasonable set of strategies that revolved around studying the loops and intuiting the sequence of steps to untangle bit by bit. What was particularly noticeable was they would often rotate through all 4 configurations to find 1 that would improve the tangle if twisted.

Next: I introduced the idea that we were going to map the moves to arithmetic operations. Everyone quickly came up with the idea that twists were  a  +1 operation. Rotations  remained mysterious.  After playing around a few more minutes I added the suggestion that they should try simple configurations and record all the moves they made.

Several ideas developed over the next phase: including are rotate/twists -2? I asked them to try doing 2 twists and seeing if the rotate/twist combo reversed it. (No)  One boy also jumped to the idea of infinity so I was able to ask questions about if we had any states that behaved like infinity i.e. if you twisted them they stayed the same.

Finally, the kids were starting to flag so I intervened  more directly by asking the kids to come up with ideas for what the rotate could mean and had them conduct experiments on simple tangles (usually double twists) to see if they would work. We did this as a group with one of the kids recording the results and tried out 3 or 4 options like multiple by -1.

At this point I was just about out of time so for the last 5 minutes we switched and I  let them do sequences of moves and I would call out the actual state values. Then they were finally able to discover that rotations were a negative reciprocal and that let me do a quick wrap up for the day.

All in all, we worked on the problem for most of the hour and while there were points when the kids were ready to give up, I was able to draw them back into the problem and re-establish flow. Hopefully, I'll have chance to try this again and see if my facilitation can improve further.

#### Further Highlights

In going over the problem of the week: MathForumProblem  a mostly standard linear system story problem with a small twist I had expected a blend of informal and formal solutions.   In past years, I'd get different strategies from bar charts or guess and check all the way to fully symbolic answers. This time around, I had 3 different kids demoing on the whiteboard all using substitution.  I'm impressed how many kids have already made the algebraic transition prior to middle school.

Finally, the most touching moment for me had to do with next week.  I realized after a parent question that a large group of the kids would be absent next Tuesday on a field trip. The excursion is downtown and finishes without transportation back to the school.  This will be our last session for the year and I assumed most of these kids wouldn't be able to attend. I've sent out an email and so far it looks like the parents will arrange carpools to bring the kids back especially to come to Math Club. The fact they were willing to do this makes me feel really happy.

## Sunday, June 4, 2017

### 15 - 75 Triangle Redux

I came up with this problem after looking at the original one from @five_triangles (Find the area of the trapezoid ABCD) That's a lot of fun but along the way while modelling the solution in geogebra I noticed AF is also on the diagonal of the trapezoid.

Note the 15-75-90 triangles at the bottom. Nowadays when I see them I also think of the following construction:  http://mymathclub.blogspot.com/2015/07/sometimes-one-direction-is-lot-tricker.html
which allows one to find the ratio of the sides without trigonometry:  1 : $2 - \sqrt{3}$ for the legs.

Bonus: Another problem with one of these in it: The $2 - \sqrt{3}$ in the expression is a dead giveaway.

## Wednesday, May 31, 2017

### 5/29 Combinations and Pascal's Triangle

This week I decided to hit a bit of combinatorics before the year ends. I know most of the kids understand permutations fairly well but not combinations and that seemed fairly accessible.  I like the connection between combinations and Pascal's triangle and that led me to the idea of warming up with a repeat from "This is not a Math Book".

In probably my favorite page of the book, Anna has a coloring exercise with Pascal's triangle where you search for patterns after coloring all square that are multiples of 2 different numbers, 2 different colors. I've used this before and it still drew the kids in (even the ones who were here last year).

After maybe 10 minutes of coloring, we gathered together to discuss what we had noticed. There were a lot of mentions of symmetry and triangles. I pointed out several examples of Sierpinski's triangle since these keep recurring this year. [Maybe a whole session on fractals is in order at some point.]  I told all the kids that we'd come back to the triangle but we now were moving to our main task.

From there we talked first about permutations with some group questions on the order of how many ways are there to pick a pair of socks to wear  for a week out of a sock drawer with 7 pairs.  Factorials are pretty well known by now so I just reviewed them on the board after one boy mentioned them.

Next we moved to combinations (my main target).  I started with a group question about picking teams emphasizing the order in which you pick was not important. I chose $4 \choose{2}$ as a starting point. We took predictions on the result and then the kids worked in tables to enumerate and figure out the answer. Predictably some thought at first it would just be 4 x 3 but after a few minutes the truth was discovered and the kids were able to give some informal reasoning about removing duplicates by dividing by 2.  I then expanded and asked what they thought $10 \choose{4}$  would be.  Some kids again predicted that it would be like the permutation but divided by 2.  Again I let everyone work on finding the enumerated answer.

At this point I wrote out the formula $n \choose{m}$ = $\frac{n * (n-1) * (n-2)... (n-m+1)}{m!}$
and asked if anyone could figure out why this was happening?

Unfortunately, no one had a good idea about the denominator. I tried some leading questions how would you calculate the number of duplicates given a concrete set of say 4 items. But in the end, I related this back to combinations by saying the top was the total permutations but included duplicates and for each individual class of duplicates the denominator showed the number of permutations i.e. for pairs: duplicates come in 2 but for trios they come in 6 etc.  [If repeating I think I would linger here and try some more examples as group work to see if more intuitions would develop.]

Finally, for my favorite part I had the kids calculate and write all the combinations for 2, 3 and 4 on the board in a pyramid:

$2 \choose{0}$  $2 \choose{1}$  $2 \choose{2}$

$3 \choose{0}$  $3 \choose{1}$  $3 \choose{2}$  $3 \choose{3}$

$4 \choose{0}$  $4 \choose{1}$  $4 \choose{2}$  $4 \choose{3}$   $4 \choose{4}$

What do notice now?  This elicited some wows when Pascal's triangle re-emerged. So again because this is a bit mysterious I went into an informal explanation centered on the there being 2 cases:

• The new element n is in the set you pick and then there are combinations of the n - 1 elements for the rest of the set i.e. the left parent.
• The new element n is not in the set and there n combinations with the rest of the set i.e. the right parent.
This also works best with concrete examples.

Finally, I found a decent problem set that I based  my own problem set off of: My sheet. Going in though I was a bit worried. This is the end of the year, and I wasn't sure how much focus I could count on. So I hedged my bets a bit and brought another set of skyscraper puzzles with the idea that I would offer them to the kids if they started to flag at the end.  This turned out to be prescient. What I hadn't counted on was today was also a standardized testing day and therefore the kids were more drained than usual after several hours of SBAC testing.  If I had known that ahead of time, I think I would have compromised and picked maybe 3 problems to do on the whiteboards instead of at the table.

For the long run, I still have the aspirational goal of being able to have a group of kids spend 20-30 minutes working through a short problem set (10 or less) of interesting problems. I'm not completely certain that's realistic (ok I'm fairly certain that if it is, its not easy) and I've pivoted more towards group white-boarding or providing choices in these scenarios which allow the kids to work on the problem set or a lighter puzzle and switch between tasks.  What I'd really like is some kind of reward/competition that was motivating but not discouraging for the room in these scenarios. Based on the odd fact the kids really liked the Lima beans we used as counters several times, I'm tempted to try spray painting a bag different colors and handing them out as prizes to see what happens.

Problem of the Week
An algebra one from the mathforum that doesn't really need formal Algebra to work it out:
http://mathforum.org/pow/teacher/samples/MathForumSampleAlgebraProblem.pdf

Todo:
I just saw a recent video  https://youtu.be/KYaCtHPCARc on infinite series using pascal's triangle to look at hypercubes. This may make a cool followup for next week.

## Tuesday, May 23, 2017

### 5/23 Triangle Conundrum

In the middle of last week, the MOEMS awards for the year arrived. So I started handing out patches and medals. I'm fairly happy with our overall performance at the Middle School level. Almost all the fifth graders who were present for all 5 tests was in at least the top 50% of 6-8th graders and we had one boy crack the top 10%,  This was not as high a level of achievement as last year when we used the Elementary level but confirmed that this was providing a good level of challenge and was not too difficult. Week to week, almost everyone could access at least 1-2 problems (often more) and we had good discussions about the entire set. As I said before, the MOEMS format has grown on me over the last 3 years. I think I will bring this with me to the middle school level.

On another note, I thought the MathCounts problem of the week was not super interesting so I took a poll of the kids after we discussed it today. Interestingly, the kids seemed to generally like it. I mean to think about this some more. Was it because these type questions are more straightforward? Its definitely a caution for me to remember to vary activities. My taste in Math is my own (and perhaps a bit quirky) and I want to make sure to try to appeal to everyone over time.

Today's main Math Club activity was inspired by the following tweets:

There was mention of the following problem:

That made me think of the classic Martin Gardiner missing square puzzle:

These problems seemed like a good progression of fishy triangle issues and all seem well suited to group problem solving on the whiteboard. So I had everyone getup and circulate among them during the main part of the hour. I liked the general activity.  The most difficult one turned out to be the Tanya Khovanova "triangle". This was the only one the kids didn't fully solve although it brought out some great questions about the Pythagorean Theorem and experimentation with various triangle configurations. As kids cracked the other ones, there were occasional excited shouts "This isn't really a triangle!" I was particularly happy they also connected the problems back to slopes to prove what they discovered.

To close the day out I wanted another game. This time I turned to one I found on Sara Vandewerf's site: 5x5.

I pretty much followed Sara's format. (I always appreciate time estimates for a game in a writeup) We did 5 founds and the kids were just as engaged as promised.  Beforehand, I had wondered if all the scores would bunch around a few values. Even with 14 players that didn't generally happen except when going for low scores.  As a thought experiment: since all the kids loved the lima beans we used as tokens a few months ago it occurred to me afterwards I could spray paint them gold and give them out as "prizes" in the future.  [Would older kids find this corny or fun?]

P.O.T.W:

Some probability work form Waterloo:

http://www.cemc.uwaterloo.ca/resources/potw/2016-17/English/POTWC-16-DP-30-P.pdf

Unused: I actually had some more skyscraper brain teasers and  a little bit of combinatorics in my back pocket.