Tuesday, January 31, 2017

1/31 Chessboard Problems or manipulatives on the cheap

This week's planning revolved around my desire to pivot away from the more conventional topics of last week.  I needed to give the kids more exposure to exponents but that being accomplished I wanted a lot more whimsy this week. I was casting around in some of my more Math Circle oriented resources but then I ended up watching a lecture by Maria Droujkova @ https://www.bigmarker.com/GlobalMathDept/Avoid-Hard-Work-Natural-Math-Adventures?show_register_box=true. Among the discussion, one particular problem caught my eye: the knight's tour which is done on a chessboard. I then independently found a different chessboard problem that I liked featured in a numberphile video. I also remembered a chess station I manned last year in the Julia Robinson Festival. All told, that was more than enough material and I thought it would make a fun themed day. The final problem was producing enough pieces for 12 kids to use.

Inspiration struck at the grocery store. For only a few dollars I purchased hundreds of dry lima beans. They worked perfectly on some printed out chessboards and the only issue was making sure they didn't end up all over the floor.

As you can see from above, I also bought some candy to reward the kids for reaching our problem of week point goal. The last few weeks, participation has been edging up again and I'm feeling good again about its function.

I also ended up borrowing a video projector so I could show the following video:

I played the first 5 minutes or so and then broke out the lima beans and had the kids work on solutions to the problem for the next 10 minutes. At the very end, I started to get questions about whether this was impossible. My response was can you come up with reasons for why that seems to be the case. We then reconvened for the back of the video. As usual media makes for very easy to manage Math Club sessions. I could very easily see running a permanent format where one did a 10 minute video every week.  I particularly like the focus on math practices and proofs embedded within this clip. Its almost perfect for the kids at this stage in their math careers.  Two immediately on point moments occurred first when the video asked whether it was possible to prove something impossible. I heard a lot of "yes' murmurs from the room.  Then later on when the video started talking about the infinite geometric series 1 + 1/2 + 1/4  ... I stopped to ask the kids what they thought that ended up summing to. Sure enough as the video would call out most answers were a fractional bit less than 2.

For the last 20 minutes or so we then turned to the Knight's Tour Problem. I explained the basic rules in a huddle, promised everyone this puzzle was solvable and then everyone was off.

All told, I was very satisfied with the engagement again this week. I have another Olympiad coming up in a few weeks but I hope to repeat another "pure" Math Circle session before then.

Bonus: http://www.msri.org/attachments/jrmf/activities/ChessCovers.pdf

a pythagorean puzzle from @solvemymaths.

Wednesday, January 25, 2017

1/24 Curve Ball

Sometimes random events complicate the best of planning. I was on my way to work when I received an email from my co-coach Kristie that  her plane was delayed and she was not going to make it back to town in time.  So I ended up taking both the fourth and fifth graders for Math club but I didn't have enough time to really modify what I had setup for the afternoon.  Off the bat, I knew there wouldn't be enough desk space for all the kids, the fourth graders hadn't done the problem of the week but I needed to review it since the fifth graders had and I also had picked a fairly formal main activity. Despite these concerns and fretting that it wouldn't be as fun for everyone, the day worked out generally well and the kids maintained their focus belying my worries.


See: http://cemc.uwaterloo.ca/resources/potw/2016-17/English/POTWC-16-NN-PA-14-P.pdf    Once again, about half the kids completed the sheet which is a success in my book. That allowed me to pre-choose one boy to demo that doesn't talk as much. (That's a persistent goal of mine: get everyone talking in front of their peers as much as possible.)   His solution was a good example of using a targeted guess and check algorithm to quickly solve a linear equation.  This is the kind of informal algebraic reasoning that most of the kids have already developed.  Next, I had one of those moments. After asking for any different strategies one of the girls came up and proceeded to write down a system of linear equations and very competently solve them via substitution.  This was both awesome and hard.  I was fairly sure most of the fourth graders didn't follow this let alone the rest of the fifth graders. But developing the groundwork for substitution was clearly not going to happen.  So I made a strategic choice. I asked if anyone had any followup questions about the algebra, gave a quick talk about multiple strategies and how over time everyone would gain more tools and then moved on.


Fortunately I had already decided to repeat the game of Median from last  week: http://mymathclub.blogspot.com/2017/01/117-3rd-olympiad.html  This required re describing the rules for everyone who was seeing it for the first time. We then did a communal set of rounds as a group with three volunteers.  Finally, I broke everyone up into trios and had them play with the guidance that they should look for strategies.    This time around, many of the kids noticed that ties were the most common outcome.  The general idea that if you were ahead then you should aim to lose rounds also was brought out. I ended with asking a take home question "Is Median like tic-tac-toe where three experienced will always end up in a draw?"


For the main task for the club I chose some work on exponents which I structured around a whiteboard discussion, small group investigation and problem set.  First I wrote some sample numeric exponents like 2^3 on the board and asked for definitions of what an exponent means. Fortunately, one girl almost immediately put out the idea it was a shorthand for multiplication. That let me expand the sample exponents on the whiteboard a few times. I also demo'ed with variables like x^4 to show they were no different. My main message was that exponents are just repeated multiplication and that you can usually expand them out if you're unsure of the semantics. We then went over some common cases which I used the expansions to show how they worked.

1. What happens when you multiply two exponents.
2. What happens when you divide two exponents.

In each case I asked for hypotheses first and then had the kids give me the answer once I expanded on the board.

Next:  I asked what they thought the 0th power would equal i.e. 2^0.  Again,  I received the correct answer. But this time, I asked for reasons why this was true which was a little harder. After waiting a while, one of the kids came up with idea that it fit the pattern which I emphasized on the whiteboard. I then introduced the formal argument using the rules for exponent division.

Next up was negative exponents. Again I asked for ideas from the room. This proved more confusing. Many kids believed they would probably produce a negative number. So I went back to the pattern chart and asked if negative exponents followed the pattern what should they be using the example of 2^-1.    I then demonstrated the formal argument using division again.

For the last portion I asked if we had tried all the integers was their anything else we could use as the power?  There were a few jokes but no ideas so I threw out what's \(9^\frac{1}{2} \)?  For this one I decided we would do an extended brainstorming session in groups. So I wrote some more rational exponent examples on the board and asked the kids to work in a group and use what they knew about exponent rules so far to come up with ideas.  When they came back to share, I got a lot of interesting but not quite correct ideas. Many found patterns that worked for the sample exponents but were not generally true. So to close this section off I guided everyone through this type logic:

\(2^\frac{1}{2} \cdot  2^\frac{1}{2} = 2^1 \) using the general exponent multiplication laws.  This implies if \(x = 2^\frac{1}{2} \) that \(x^2 = 2\) and therefore x is \(\sqrt{2}\).

Problem set:

Finally for the last 15 minutes of this session I had photocopied the review problems from the exponent chapter in the AoPS pre-algebra book. I had everyone work on these and floated around the room helping out and correcting any misconceptions I saw. As usual I'm never quite satisfied with this format. I assume that since the kids like to work together they will mostly catch each other's errors and raise their hand if they need help. But I still worry about errors creeping through.  However, I don't want to bring an answer sheet because that quickly degenerates into a line of kids asking me to check their work which is not scalable.  So this is still one area for me to think about improving.


Looking forward

After this week I want to switch tacks again and work on something more free-form. I'm leaning towards trying out the knight's tour problem after watching a program from Natural Math.

Wednesday, January 18, 2017

1/17 3rd Olympiad

We started this week with the pdf from further maths that I gave out as a problem of the week: http://furthermaths.org.uk/docs/FMSP%20Problem%20Poster%201.pdf. To my satisfaction half the kids worked the problem so I had a lot of choices on whom to choose to show their work on the whiteboard. Thus I had a kid demoing who usually doesn't volunteer.  This problem is a clever riff on the Pythagorean theorem. Along the way I interrupted several times to draw out a few key ideas from the group  via questions i.e. how the Pythagorean theorem worked, the formula for a triangle's area, and the formula for the area of a half circle.  My only idea for improvement would be to draw out the area arithmetic at the end on top of the student explaining it to make sure the logic was clear.


Despite it being only the second Math club meeting for the quarter MOEMS released the third Olympiad for us to take. This was a bit too early for some of the kids' tastes and I elicited a few groans when I told everyone what we would be doing. I would also have preferred at least one more week before taking this on.  I have several topics I'd like to broach including exponents and I also want to throw in some more recreational math activities. But once we started, everyone worked very diligently on the contest and it appeared on a  quick glance that many of the kids found solutions to  most of the questions. So the experiment with the middle school level after a rocky start seems to be going well.

Some general notes:

  • The first problem was rather clumsy and included the expansion for (a + b)^2 and then asked the kids to evaluate it for 2 specific values. I thought this was a failure on two scores. It was most likely to result in blind plugging in of numbers and the phrasing actually ended up confusing some of the kids. Interestingly some of them skipped using the formula entirely and just tried grinding through the calculations in the expanded form. In general, I'd save this one for Algebra when everyone has more background context.
  • The last problem involved some combinatorics which even I missed in my quick try out. Basically there was some normal combinations to sum but then you had to recognize one case was double-counted.  As expected almost everyone missed the hitch,
  • Embarrassingly this was the first time I could properly have the group go over the solutions together on the whiteboard at the end.  As usual, the kids were enthusiastic about showing off their work and finding out if they had the correct solutions. (Never wait or delay talking about problems as a group if you have the time).

To make up for jumping into the contest, I picked some really fun activities for everyone to try out while they waited finishing. First up: Median https://gilkalai.wordpress.com/2017/01/14/the-median-game/ was awesome.  This game needs no more than a pencil and paper to keep score and yet has some really interesting game theory embedded within it. It was a bit tricky accumulating groups of 3 as the kids finished the contest. But beyond that the rules were simple enough for them to get going and soon you started hearing a steady 1,2,3 countdown coming from the clusters.  A few kids didn't initially realize the scores were cumulative and asked why you'd ever want to choose an 8 or 1. I replied that sometimes you want to lose in order to keep your overall score in the middle which highlighted that point. So I think I'm going to reuse the game at the start of the next session and do a group play once so we can have a formal discussion about what strategies everyone came up with. This one is highly recommended.

I also finally got around to trying out tiny polka dot from Math4Love: https://www.kickstarter.com/projects/343941773/tiny-polka-dot-the-colorful-math-game-for-young-ki.  This is really multiple games in one. Many of them are leveled for slightly younger children so I wasn't sure how it would go over. While the memory style variants and simple arithmetic weren't very interesting, the kids reported the pyramid variation of tiny polka dots was difficult and fun to try.

In this version you need to form a pyramid of 4 - 3 - 2- 1 cards where each layer of 2 cards when subracted  is the next one above. Note: you can try this out without any cards.  The goal is to use some of the blue and orange  numbers cards (each  between  0-10)  to produce this arrangement.

(Solution completed at home by the beta tester who found this interesting enough to keep working on his own.)

This all made me think of a tweet I read reflecting how the teacher didn't regret not using "competitive games" anymore. In my experience, games including competitive ones are always popular so I wondered  "Why the lack of love?"  It turns out some some games are just not very game like. What was being described here was a timed relay that pitted teams of students against each other. These type activities are really still just math exercises where the only way to win is to go faster.  They succeed or not based on the strength of the problems chosen and suffer from the serious drawback that often most of the kids are just waiting their turn to go. Generally, I try to never let kids wait around because mine at least will always find some other way to entertain themselves. (It generally involves crumpling up paper and throwing it at each other.)  Math Club or a regular class for that matter is too short to intentionally miss using ever minute anyway.  For me a successful Math game involves strategy or logic of its own and must always focus on play.  The Mathematics is embedded in the rules and not ancillary Preferably everyone is involved as much as possible of the time. You win by figuring out the game works and developing better strategies. These type games can be competitive or cooperative and still usually everyone has fun.


Tuesday, January 10, 2017

1/10 New Year's Celebration

Its gratifying when you plan a Math Club session where your time estimates work out and you achieve really good engagement.  I had a few goals for this time since we're starting up.

1.  Redo introductions and talk about club principles for the new kid.
2. To celebrate the new year with some work on factoring.
3. Try out a cool puzzle I'd seen online.

Having worked out a sequence in my mind, I started to worry that I'd need a little more filler activities.  I have a new game from Math4Love:  Tiny polka dots that I brought with me just in case. But in the end my instincts were correct. This worked to out to a good hour long session.

First, I had everyone assemble on the carpet. I'm trying to use that more often since being closer seems to sharpen focus for everyone. We went around and everyone introduced themselves, named their homeroom teacher and their favorite activity from last quarter or why they joined this time.  As usual candy for Problem of the Week celebrations was often mentioned. I chuckled when someone asked if they could use Pi day from last year. This seemed like a good time to promise we'd have pie again this year.

I also had the kids debrief about the last Math Competition. Overall, everyone seemed to be fairly upbeat about it.

With that out of the way I went over my core principles for this quarter.
1. Respect the room and leave it as we found it.
2. Listen to me and each other.
3. Work on perseverance or "What to do when you're stuck" . For this one I asked several kids for real examples of their thinking during the last MOEMS test and gave a bit of talk about how real math problems aren't always solved in a  few seconds. The one strategy I talked up the most was moving on and coming back to something you couldn't get.  My example was some geometry problems I've worked on.  For the kids, I mentioned that they should try this out with a problem of the week.

This was a natural bridge to talk about one of the problems from last contest which was given 2016 = 2^a * 3^b * 7^c find a,b, and c.  First I asked about the exponential notation and we quickly went over what it means. (This is a subgoal of mine for the quarter, work a bit on exponents) Then I asked what kind of problem was this? After a few false starts one of the girls came up with the idea it was factoring.

Next I asked for ideas from the room on how to factor. Most kids know about factor trees but have a weak idea of the entire process. For example, there was a lot of "start by dividing 2". When I asked what if you can't see any obvious factors there was a long pause. So I decided to switch things up and ask everyone how can we factor 2017? (which is prime and doesn't have any easy factors.)

Eventually I started them off with the statement "I can try dividing 2017  by every integer between 2 and 2017 and I'll find the factors but gosh that will take a while, are there any ideas you can come up with to speed this up?"

From there there were a lot of suggestions about individual divisibility tests for various numbers. I was able to tease out what made the numbers interesting was that they were prime after several rounds of questions and observations from the class. So finally, I was able to get to the key question do I have to try all the primes less than 2017?  This was a bit less satisfying. One kid finally volunteered we need only go to the square root but didn't have a reason why. So for the final part, I gave an informal discussion of why this is true on the whiteboard. If  I repeat this again I might have the kids try out a sample and look for patterns first. I was then easily able to get some estimates from the room that the square root of 2017 lay between 40 and 50. I finally had everyone go back to there tables and try the process out on 2017. My main suggestion was to be orderly and assign the different primes to different table members to avoid repeated work.

While everyone was computing I went around the whiteboards and put up some more 2017 number trivia I'd been collecting off  various sites. Sara VanDerWerf  has a great summary here: https://saravanderwerf.com/2017/01/02/geekin-out-on-2017/. After all 4 tables had confirmed 2017 was prime I pointed each one out. I didn't stress these much but left them up for inspiration for the rest of the afternoon.

Finally from there we transitioned to an awesome geometry puzzle I found from Sarah Carter:

The Zukei puzzles involve finding a given shape on a coordinate plane from among a set of potential vertices. These were highly engaging and kept the entire room's focus for the rest of the day.  Interestingly, I was able to work a bit on geometry definitions along the way as questions about what is an isosceles triangle or rhombus came up. I love that this flowed naturally from the problems rather than just being a giant exercise in taxonomy.

Finally I chose another one of the "favourite problem" posters for the P.O.T.W:

Bonus: I'm trying my hand at T-shirt design for the club. Here's my version for this year.

Sunday, January 8, 2017

Knight's of Pi 2017

High School full of mathy kids assembling for a Math competition.

This weekend was my third time back at the Knight's of Pi math competition and I brought 2 teams of fifth graders this year. I've written about my complex relationship with this one before: http://mymathclub.blogspot.com/2014/12/knights-of-pi-math-competion.html.

How I handle this now:

1. I'm upfront with the parents about expectations and leveling.
2. I stress focusing on the problems with the kids. This year I gave everyone the task to remember their favorite one so we could talk about it over the dinner pizza.
3. I try to encourage everyone to bring games and books so the waiting periods are more fun. I think this part was really successful this year. The kids always like pizza and a minecraft design book was the surprise hit. We also had some good rounds of pente and smashup.
4. I also tend to encourage everyone to skip the awards ceremony. This year it ran especially late and we endured 40 minutes of kids reciting digits of pi before the awards were handed out.
5. I'm making a more concerted effort to scan the questions and answers so I can email them out. I'm hoping some of the kids will be motivated to go over problems they missed and dig into them at home.

The ceremony at the end.

Overall looking through the questions, the quality varied quite a bit.

The good:
Some classic algebraic age problems. "In 3 years John's father will be 3 times as aold as John but 2 years earlier John's father was 4 times as old. How old is John now?"  While solvable with algebra this can also be attacked via various intelligent guess and check strategies, bar charts etc. Given as a group problem with (barely) enough time I think this worked pretty well.

"What is 2017 base 8 expressed in base 5?" Several kids thought this was their favorite one from the individual section.

The bad:
Calculate the probability of drawing 4 of kind from a deck of cards.   Too far out of scope for this age level. A rather large hand calculated fraction anyway.

The ugly:
Find all the zeroes of y = x^2 - 7x + 10.  Definitely out of scope for 5th grade (maybe even the vocabulary) and solving it via a guess and check strategy seems too expensive given the time limits per question.   Full disclosure: at least one of my teams figured this out anyway.

Final Irony:
The first winter session of Math Club is coming up this week. One of the fun topics I was planning to talk about were number facts about 2017. I was toying with having the kids factorize it to discover that its prime (and practice factoring)  Of course, this was one of the questions. So I may modify my plans a bit now. Fortunately there are 3 or 4 fun observations that I still have in my back pocket. I love that its part of a Pythagorean triple.

OK so what would you do to improve things?
I've thought about this a lot in the last few days. For a start there a few practical changes that would be helpful.

  • Pair schools and seat them together in the beginning to encourage interactions between students. Keep them in the same room for the entire contest.
  • Build in a social math oriented activity in the middle ala a Julia Robinson Festival type question.
  • Use extra time to talk about solutions rather than pi. Kids are never as excited to talk about how they solved problems than right after doing a contest. If this was formalized I think you could do a lot of learning and shift the focus back towards the problems.
  • Standards based awards. Define a threshold for "honors" and give everyone who reaches it a recognition.
  • Less problems that take more time.  A typical MOEMS contest will have 5 complex problems in 30 minutes vs. 40 problems in 45 minutes here.