Tuesday, April 25, 2017

4/25 Platonic Solids

This week's inspiration started with a very late school bus. My son's bus driver has been on vacation and the substitute drivers have been really, really tardy. So much so that he missed most of Math the other morning when the class was going over the volume of a pyramid. I checked with his teacher and seeing he had missed the explanation for the formula decided to try out some activities at home to make up the gap.

To start, while I love much of Geometry the introduction of sundry area, and volume formulas in the middle school sequence seem pretty pointless to me. They don't connect with much before or afterwards and are often taught without sufficient explanations. Frankly you can go really far even in pure Mathematics without ever missing the pyramid formula. (Brainstorm topic: where would this fit more naturally? The calculus connection is fairly compelling ...)  The missed experiments in class compared prisms and pyramids and the volume of rice they held. As an experimental process this is not bad but as a mathematical foundation it doesn't totally satisfy me. Its neither universal "How do you know that if the pyramid dimensions shift the relationship stays constant?" nor does it speak to "Why is this happening?"  The question I want to provoke is "Why 1/3 and not 1/4?"

So at home, we started looking at the formulas and  I asked "What does the similarity between the volume of a cube/prism and pyramid suggest to you?" I was lucky that was enough of a prompt for him to suggest "Is there a way to cut a cube into 3 pyramids?" From there we printed out some templates and built some 3-D models to show the trisection.    This was enough fun that I thought I'd build a day out of it for the whole Math Club. To round things out I thought I'd bridge from there to an exploration of platonic solids.   At this point I worried a bit about timing and decided to have some Sudoku puzzles in reserve. But I stayed firm and left them for the end if needed which as it turned out was not the case.

The afternoon began for real with  me handing out spiced gum drops for reaching our problem of the week target. I also left out a sample tetrahedron I had built to see if I could garner any questions. (Nope)  Once we were upstairs I decided to have a short debate about last week's problem. Infinity Link.  I asked everyone to pick a corner of the room. One side for those in favor of Courtier A's offer, the other for Courtier B. A group of students actually remained off to the side and I asked what they supported. Their answer was they thought both offers were equal and since that seemed interesting I setup a 3rd corner for them.  We then went around the room with everyone offering positions on why their side was correct and rebutting the other side's idea. This went on for may 5-6 minutes which was fun . The disadvantage was this format really makes universal participation hard to achieve so I wouldn't rely on it a lot. (To be fair: repeated usage could make it more natural for more kids to speak.)   Secondly, a group of kids really wanted me to rule on the "correct" answer which  I demurred on.  Next time, I should also remember to close this with a final vote.

From there I did a version of my initial process with my son and  we bridged to building the pyramid templates. I used some cutout templates from here:
  http://www.korthalsaltes.com/model.php?name_en=three%20pyramids%20that%20form%20a%20cube and spread them among the various tables. We eventually cut out 3, folded and assembled them with scotch tape, and confirmed they were identical and formed a cube.

I was hoping that someone would complain that the pyramids weren't exactly the same as the regular ones we started with. That didn't happen so I prompted "Is there anything that doesn't seem quite right in this explanation?" That eventually brought out the idea and I gave a brief hand waving explanation of slicing the pyramids and rearranging them to have the same volume but centered rather than offset to the corner.

Next: I handed out a combination of further platonic solids from the site above (cube, tetrahedron, octahedron, dodecahedron,  icosahedron)  and from  http://www.senteacher.org/worksheet/12/Nets-Polyhedra.html.  Each table had a different one to assemble. At the same time I brought some pipe cleaners and straws to make companion wire models.

I had everyone work on the models and to tie things together chart the edges, faces and vertices per shape on a communal white board. My hope was to have the kids observe the the Euler Characteristics.patterns and I seeded things a bit by arranging the chart  V / E / F.

Some of the kid's handiwork.

This worked fairly well. Engagement was good among the modellers (which I had to rotate due to limited tape and scissors) but I had to work a bit to keep the other kids counting edges and faces and thinking about patterns. 

At the end I gathered everyone back at the board to discuss the data. There were a few fun observations.
  • All the numbers were even. (Followup for another time: could any characteristic ever be odd?)
  • You could more quickly and accurately calculate edges and vertices than counting by multiplying the number of faces time edges/vertices per shape and dividing by the number of faces that met at an edge/vertex.
I had to have everyone think about a numeric relationship for a few minutes but lucked out and one boy discovered   V - E + F = 2.  At that point my time was almost up so I left with some closing questions:
  • Is there an equivalent relation in 2-D?
  • Why do you think this is happening? 
  • Are there any other platonic solids you can discover?

In the end, this was a lot of fun. I actually had templates for stellated polyhedrons and Archimedean solids we never got to in my back pocket. We could easily do a followup day on the topic although my general style is to zig-zag around subjects.


This one from an old Purple Comet is interesting because its algebraic but linear not quadratic like I first suspected and you don't need to every find the exact dimensions to discover the perimeter.


Related Session: Euler Characteristic

Wednesday, April 19, 2017

4/18 the series "Infinite Series"

Spring break really flew by and yesterday to my surprise Math Club was already resuming. Things started with small snafu, the door to our room was locked. While we were waiting in the hall for the custodian I went over some administrative items. I'm still looking for a few kids to round out the group going to the upcoming WSMC Olympiad, I wanted to acknowledge the high participation in the problem of the week and that I'd bring candy in next week. Finally, I also started laying the groundwork for the talk next month and asked the kids to start thinking about questions to ask our guest mathematician. 

If only there was a whiteboard in the hall I would have gone over the previous problems of the week but sadly we waited a few extra minutes instead.

For this week I wanted to try out the Infinite Series youtube webcasts with the kids. I thought the above video on proofs was a good first choice since one of my priorities is to emphasize understanding why things work and how it will become increasingly important (and computation less) for the kids as they move forward. In fact, I'm trying as much as possible to add in comments about the math progression whenever appropriate. This is one of those areas I feel is not well understood in 5th grade.  Most of the kids know they're working towards algebra, geometry and probably Calculus. They don't necessarily know what Calculus is about even in the most broadest sense and they don't often think what happens after they finish that sequence. I also think they take it for granted that Math topics are all a roughly linear sequence which is not truly the case beyond school math.

What's also nice about the video is it structured around several problems and even has breaks where you're supposed to try them out first.

I took full advantage of that format and stopped 3 times:

1. The chessboard / domino coverage question was the easiest and one of the boys came up with the standard reasoning in a few minutes.
2. Probability of sticks forming a triangle. I wasn't sure if the kids had been exposed to the triangle inequality so I played that part before pausing. Interestingly everyone said "Oh yeah" even if they didn't recognize it by name.   No one came up with he answer but there was a lot of good discussion before I resumed.
3. Sum of odds formula:  Again no-one fully came up with an answer but I was satisfied with the thinking along the way.

In general this was a bit of a balancing act on how long to let the kids grapple with each problem, knowing they would probably not crack them. I wanted enough time so that the explanations really resonated afterwards but still allowed me to finish the video. In the end I had about 10 minutes of the session left. I thought the quality of discussion was particularly good even though everyone reasoned at their group of tables. Perhaps this was a residue of our work on the whiteboards the last few weeks.

Finally, to round things out I brought two sample Sudoku puzzles and an older purple comet problem set: http://purplecomet.org/welcome/practice.   I thought most kids would prefer the Sudoku but I was pleasantly surprised that many asked for both so they could try them out.  This represents a shift in my organizational thinking. I'm tactical about this but especially with new activities I'm not sure the length of, I'm jumping right in and saving my old warm-up ideas for the end instead.  I see more benefit from having a light weight activity for those whose focus is used up than a transitional one at the beginning and it means I'm shorting my main focus much less often. If the activity takes the whole time and everyone is engaged I'll just save the extra puzzle for another week.

I went with an infinite series conceptual riddle. My hope is to have a group debate next week.

You’re a venal king who’s considering bribes from two different courtiers.

  • Courtier A gives you an infinite number of envelopes. The first envelope contains 1 dollar, the second contains 2 dollars, the third contains 3, and so on: The nth envelope contains n dollars.
  • Courtier B also gives you an infinite number of envelopes. The first envelope contains 2 dollars, the second contains 4 dollars, the third contains 6, and so on: The nth envelope contains 2n dollars.

Now, who’s been more generous?

Courtier B argues that he’s given you twice as much as A — after all, for any n, B’s nth envelope contains twice as much money as A’s.

But Courtier A argues that he’s given you twice as much as B — A’s offerings include a gift of every integer size, but the odd dollar amounts are missing from B’s.

So who has given you more money?

Friday, April 7, 2017

Spring Break Geometry

[In exciting real news, I almost have a guest speaker from the UW Math department lined up for May. My hope is that this will be helpful in showing the kids that Math is a living field where research is still going on. My goal is to collect some questions ahead of time to prime the pump.]

In the meantime while we're on break, here is one of the latest problems  I've looked at from @go_geometry. This is a good example of the power of cyclic quadrilaterals and approaches to more difficult ratio problems. (original problem)

My first thought was that all segments in the ratio were on the same line. That's a problem because we only have a few tools to use that create ratios and they all need polygons.

1. Combinations of well known triangles.
2. Similar triangles.
3. Cyclic quadrilateral diagonals (which really are just similar triangles).
4. The angle bisector theorem (although I didn't initially think much of this one.)

  • My second thought was that BC is congruent to every other side of the square so that could at least give sides to one triangle CD  and CG for instance but FG still looked hard.
  • Triangle EFG is similar to ADE  which does generate some ratios involving FG and AD but I wasn't sure I could do much with them. The algebra looked fairly complex when playing with such ratios.
  • It looked clear from everything so far that it would be a combination of ratios to produce the result.
  • I then noticed ABEC was a cyclic quadrilateral since angle ABC = angle AEC = 90 degrees. That's useful for angle chasing and produces a set of similar triangle including ABF and CEF.
From those triangles one gets:

\(\frac{BF}{AB} = \frac{EF}{EC}\)  Since AB = BC that converts to \(\frac{BF}{BC} = \frac{EF}{EC}\)

That's about half way to the desired ratio \(\frac{BF}{FG} = \frac{BC}{CG}\) so I rearranged the goal  to the same form on the left side:

\(\frac{BF}{BC} = \frac{FG}{CG}\)   which meant  I still had to show  \(\frac{EF}{EC} = \frac{FG}{CG}\) 

  • My next observation was that angle DEB sure looked like a right angle also.  I then stopped to measure and check in geogebra. That appeared correct so I looked around some more for reasons why this was the case. I started angle chasing and found BECD was also a cyclic quadrilateral since angle DBC = DEC = 45 degrees. This could be used to show that the original intuition DEB was in fact a right angle.

At this point I stopped and had a "duh" moment. If you add in the diagonals of the square and the circle that circumscribes it ABECD are all on it.  The diagonals of the square are the diameters of the circle and meet at its origin and its obvious why DEB had to be a right angle since its a triangle made of the diameter and a point on the circle.

This gives a lot of underlying structure for angle chasing. I could find all the angles at the top in my triangle of interest CEF including that CEG = FEG = 45 degrees.  (FEG inscribes the same arc as ABD which is a 45 degree angle in the square, then its simple angle subtraction)

I then stared at \(\frac{EF}{EC} = \frac{FG}{CG}\)  and realized the form looked familiar. This is a slightly rearranged version of the angle bisector theorem and EG does bisect angle FEC!  So
\(\frac{EF}{FG} = \frac{EC}{CG}\) and when everything's combined you're done. Looking back this flowed fairly quickly from intuitions and observed patterns. The whole process was actually a bit chunky and done during various points in the morning when I had a moment.

Tuesday, April 4, 2017

4/4 Spring Quarter Begins

This quarter began with a seamless transition the week after the old one ended  However, I had a little bit of turnover with 2 kids leaving and 2 new boys and 1 girl joining.  I always want a math club session to be compelling but knowing its the first time for some of the audience adds a bit of pressure to get the balance right. So this week, I spent a lot of my planning time work deciding on what to do as an icebreaker and where to focus our main activity. I actually made several adjustments along the way until I settled on what occurred and still hope that I tuned the difficulty level correctly.


To start off, I had all the kids gather on the rug in the front of row and introduce themselves. As usual I had everyone state their name, homeroom teacher and either their favorite activity from last quarter if they were returning or why they decided to join if they were new.  Interestingly, there was a strong consensus that Pi Day was the favorite. I'm hoping that it wasn't just the literal pie I served that influenced everyone.

Human Knot

I really wanted to do something physical at the start and I had used up most of my ideas already in previous quarters. After looking around I didn't find anything new that was really satisfactory. There's a lot of ideas that revolve around Simon Says or Duck Duck Goose that just don't feel very authentic to me. So I went with a short team building exercise I used in cub scouts. http://www.group-games.com/ice-breakers/human-knot-icebreaker.html  Basically, you have the kids stand in circle grasp hands and then cooperate to untangle the resulting knot., If you're being generous you could say this relates to topology or knot theory but really its about having the kids interact together and practice cooperating. I found that my initial knot was  too difficult  so I split the group in half (6-7 kids per knot) which worked better.  [I'd actually like to come back to knots from a mathematical perspective at some future point in time.]


Afterwards I went over the the serious part of the day, the basic rules for the club. This time I boiled it down to the 3 core values:

  1. Respect  - As guests in the classroom, towards each other etc.
  2. Listening  - To me and to each other when they are sharing, I like to stress this is both hard and really important.
  3. Perseverance -The only section where I solicited opinions this time. I went around and had the kids talk about how they handled getting stuck. As I remember I went off on a short tangent about how long it took to solve Fermat's Last Theorem for my real life example.
Math Carnival

For the main activity, I decided to explore using the whiteboard more this week. I went back and forth on leveling and finally settled on the following 3 problems which I wrote on three different sections of the board. After explaining each problem, I  handed out markers and told the kids to pick which problems they wanted to work on.

Cue Ball
http://mathforlove.com/lesson/billiard-ball-problem/   This one flowed really well so I spent most of my time asking questions like "I see you have a pattern for even numbers, what about the odds" or "What happens when you grow or shrink this row by 1?" I also worked a little on emphasizing charting results to look for patterns. Kids in the group tended to stay put the entire time in contrast to the other 2 problems which were a bit quicker to crack.

Letter Magnets. A store sells letter magnets. The same letters cost the same and different letters might not cost the same. The word ONE costs 1 dollar, the word TWO costs 2 dollars, and the word ELEVEN costs 11 dollars. What is the cost of TWELVE?

Interestingly, most kids found the solution to this through a combination of guess and check rather than equations. This was actually easier to do than I realized. So where algebraic approaches sprang up I tried to encourage the kids to go down that avenue.

The geometry here was a bit harder than I expected for everyone. I ended up scaffolding a bit and ran into some issues with knowledge about calculating the area of obtuse triangles. I was pleased that one group came up with the idea to split the shaded shape in half on its own. On the downside this one in particular was a bit susceptible to encouraging answer seeking. Next time, I need to remember to tell the kids to check their answer with another group when they think they have a solution.

Once again I was pretty happy with overall group engagement and thinking during this process. The whiteboards proved superior to paper in keeping the group fully involved. Also I noticed that they make it a bit easier for me to drop in as I walk between groups and absorb where they're at. The larger format is easier to access.


I couldn't decide between the following 2 problems so I gave them both out. We have a week of Spring break before the next meeting so that seemed reasonable.

From AOPS:
Buses from Dallas to Houston leave every hour on the hour. Buses from Houston to Dallas leave every hour on the half hour. Te trip from one city to the other takes 5 hours. Assuming the buses travel on the same highway, how many Dallas-bound buses does a Houston-Bound bus pass on the highway (not in the station).

From Blaine:
Suppose that N is an integer such that when it is divided by 3, it leaves a remainder of 2, and when it is divided by 7, it leaves a remainder of 5. How many such possible values of N are there such that 0 < N ≤ 2017?

Friday, March 31, 2017

Not so Innocuous Quartic

\(x^2 - 16\sqrt{x}  = 12\)

What is \(x - 2\sqrt{x}\)?

The above problem showed up on my feed and my first thought was that doesn't look too hard it's either a factoring problem or you need to complete the square.  That's the same reaction my son had too when I showed it to him.

But a little substitution (\(z = x^2)\) shows that its actual a quartic equation in disguise:

$$z^4 - 16z - 12 = 0$$

The wording strongly suggests that \(z^2 - 2z\) or some variant is a factor which is a useful shortcut but that led me down the following path on how to generally factor a quartic.  The good news here is that the equation is already in depressed format with no cubic terms.

Some links for the procedure:


A little easier to read:


How it works:
1. First we need to find the resolvent cubic polynomial for \(z^4 - 16z - 12 = 0\).
That works out to \(R(y) = y^3 + 48y - 256\).

2. Using the rational roots test we only have to look at \(\pm2^0\) ... \(2^8\) for possible roots but since we only can use roots that are square we only have to test \(\pm2^0, \pm2^2, \pm2^4, \pm2^6\) and \(\pm2^8\).   Plugging them in we find \(2^2=4\) is indeed a root. So there is a rational coefficient factorization for our original quartic.

3.  Now we can use the square root of the resolvent root i.e. 2 and its inverse  to get the following factorization (they are the coefficients of the z term): $$(z^2  - 2z - 2)(z^2 + 2z + 6) = z^4 - 16z - 12 = 0$$

4. At this point we could factor the 2 quadratics and plug the solutions  back in to find  \(x - 2\sqrt{x}\)  which in terms of z is \(z^2 - 2z\).   But we can shortcut slightly for one of the solutions since the  if the first factor is the root then \(z^2 - 2z - 2 = 0\) which implies \(z^2 - 2z = 2\)

5. Interestingly for \(z^2 + 2z  + 6 = 0\) we have the two roots \(1 \pm i\sqrt{5}\)  Plugging either
one into \(z^2 - 2z\) and you get -6 anyway!

Tuesday, March 28, 2017

3/28 #VNPS

Today was a fascinating learning experiment for me. I recently watched the following lecture:

https://www.bigmarker.com/GlobalMathDept/Building-Thinking-Classrooms by Peter Liljedahl.

Several of the ideas seemed relevant but I was particularly interested in his talk about the value of whiteboards  or VNPS (Vertical Non-Permanent Surfaces in his parlance) for working problems. I've talked previously about how I've been learning to more effectively use the double whiteboards in the room this year. Like previous years, I always have the kids demonstrate the solutions to problems on them like the Problem of the Week and after Olympiads I've taken to writing the problems across all the boards and doing a review  by moving among them rather than erasing and I'm more mindful of switching orientation and moving between the front and back ones for various transitions. But for the most part most group work I give out is done at the desk pods in groups with paper and pencil. Liljedahl's research suggests you can get much more effective engagement having kids work standing up on the boards. This is something I hadn't considered although I have always noticed the kids are irresistibly drawn to try and write with the markers.

So I decided to dive right in and try out an experiment. I looked through some of the suggested problems on his website: http://www.peterliljedahl.com/teachers/good-problem and noticed the four 4's one.  I use the game of 24 cards from time to time and actually had tried this exact exercise 2 years ago: http://mymathclub.blogspot.com/2015/06/62-pentagrams-and-some-inspirational.html. The problem involves using four fours and any operations you'd like to derive the numbers 1 .. 30. For example:  (4 / 4) + (4 - 4) = 1 and  ( 4 / 4 ) +  ( 4 / 4 ) = 2.  Last time, I wasn't entirely happy with how things went. That gave me a baseline to compare today with.  So after a quick review of the problem of the week I decided to dive in.  First I gave out a blue marker to everyone and told them to form into group on the board and then I talked through the challenge.


In the end, I thought this was a total success. All the kids worked excitedly at the boards this time versus two years ago. There was a fair amount of cross communication between the sides of the room as answers were discovered, A few times. I thought a kid was sitting down in a char to disengage, but in each case they were only thinking and then got up and went back to the board to write down a new idea. Afterwards even though I had brought boards games for an end of the quarter celebration some of them  even continued to work on the problem looking for solutions to 31, 32 etc.   I'm definitely going to keep playing with this format. Perhaps this is also part of the answer for middle school next year.

I actually had my end of quarter / game day activities planned as well for the day. Since the kids had seen all the materials (pente, prime climb, terzetto, rush hour,tiny polka dots) and were excited to play with them the previous experiment was even more impressive. There was very little attempts to break out during the 20 minutes or so. In addition to the above mentioned games I also had https://en.wikipedia.org/wiki/Sprouts_(game) in hand to try out on the board.  This game was new to the group I thought this would dove-tail well with the previous activity.

We were a bit short on time due to being temporarily locked out of the room in the beginning so rather than having the entire group play, I strategically pulled pairs of kids out showed them the rules and had them try it out. In the end I probably drew about half of the Math Club in. We will be looking at Sprouts more in the future to look for patterns and strategy.

Wednesday, March 22, 2017

3/21 Graph Pebbling

This week I went back to a pure math circle format with my favorite activity from the recent Julia Robinson Festival: Graph Pebbling. Based on my experiences at the festival I thought it would occupy 30-40 minutes so I decided to do a warm up puzzle as well. Initially I had considering doing a battleship puzzle (see: https://www.brainbashers.com/battleships.asp) but I found a tweet from Sarah Carter that looked interesting about slant puzzles: https://mathequalslove.blogspot.com/2017/03/slants-puzzles-from-brainbasherscom.html.  These have a fairly simple set of rules: put a line through every cross and make sure to have the requested number of lines connecting to each square with a number. Unmarked square are free and can have any number of connections.

Simple is often good though. All the kids really liked them:

We then transitioned to graph pebbling: The full rules are here: https://drive.google.com/open?id=0B6oYedIeLTUKc1hWSWtHMi1vbHM   A series of graphs are included as well as 5 variations. For Math Club I used lima beans again as "knights"

My only issue was I have one table of boys that are harder to keep on task. I tried separating them a bit this time which didn't quite work but I may do it again next week but from the start. They're not disruptive per. se but they are distracting each other and only stay on task when I come over and work with them.


A fun factoring / number theory problem for this week:


Wednesday, March 15, 2017

3/14 Pi Day

Every 7 years or so accounting for leap years, Pi day actually occurs on a Tuesday. Yesterday was the first time that occurred while I've  been running the Math Club. Because most of the kids were here last year I did not go over my usual conceptual question "Why is the circumference of a circle in a constant ratio with its radius, and why such a funny value?"


I fall into the camp that its fun to celebrate as long as something mathematically meaningful occurs during the party. I also try to de-emphasize anything to do with memorizing digits. So due to all the apple pies being taken this year I picked up a strawberry rhubarb pie at the local super market which I served as everyone arrived in the cafeteria. This kept the mess to a containable minimum and as expected the kids were all very excited by the treat.

Like last year I decided to also do a pi day themed video after the following one showed up in one of my feeds:

After we were done I had another NASA packet to try out:

I tried this type material once before (space map session).  Since some of the kids liked it before, I thought 20-25 minutes would be about the right amount of time to try a similar activity again. I'm not completely keen on the formula plugging involved but in watching the kids, its actually useful every once in a while to use real, messy physical values and reason a bit how to apply basic geometry.

Overall everything went smoothly including setting up the video (cabling + wifi). The setup time did mean the kids fooled around for the 2 minutes before I could start but that just took a little extra talk to get the room's attention and settle in.

A not too hard but perhaps counter-intuitive circle property from brilliant.org


Wednesday, March 8, 2017

3/7 Olympiad #5

Today started with a small mix-up. A boy I recruited at the Julia Robinson Festival to join Math Club showed up. But the next quarter doesn't start for 3 weeks. I offered to let him join us anyway but I think he was too embarrassed. Hopefully, he'll still come on the real first day. The whole incident is a reminder that even though I assume I know most of the "mathy" kids in the grade, hidden depths are out there.

After that, the rest of the day went  more smoothly and had several small rewarding moments. We started by running down the  Problem of the week as a group.  I only had one student demonstrate how to divide the boards (its a stair like cut) and unfortunately this didn't generate as much problem solving discussion as I prefer.  From there, we completed the last MOEMs Olympiad for the year. Looking this one over, I thought it was among the trickiest of the series. We'll see how the scores go but several of the problems had fairly complex instructions to deduce the answers and I think the general trend will be a bit lower than the last one. I did have a good group problem solving session afterwards and had the kids show solutions for all the problems.  One small tweak I've implemented is to write the problems on all the whiteboards while the kids are working so we're set to go for the group discussion.  Kids were well focused through the entire time with the only extra chatting being about how to solve the problems differently. The one future topic I noticed among the problems was to work a bit on explaining how choosing unordered sets work i..e  \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)  This pairs well with a dive into Pascal's triangle. I'm going to take a look at Arthur Benjamin's book to see if he has an approach that is adaptable for a group.

For the light activity I had all the kids who finished early working on an Euler path exercise from "This is not a Maths Book"

The kids found this very interesting and it again could be a topic for a whole session.

Other Ideas from around the web I'm thinking about for future meetings:

Problem of the Week

Wednesday, March 1, 2017

2/28 Infinite Series

For this session of Math Club I wanted to revisit one of the ideas from the "free the clones" games: (See: http://mymathclub.blogspot.com/2017/01/131-chessboard-problems-or.html)

What is the sum of the infinite series  1 + 1/2 + 1/4 + 1/8 ...

On reflection, I decided this would make a nice connection with converting repeating decimals back to fractions. I had actually tried this 2 years ago and it went okay. Most kids can convert fractions to decimals but can only handle non repeating decimals in the other direction. But in the intervening time I had lost the worksheet I used back then.  This time, I wanted to risk it and just work on the whiteboard, have the kids go off and experiment and come back and discuss what they found.

Planned Questions

1. What is .999999... equal to and why?

2. How can we represent .99999.... as a series of fractions.

3. Warm up with some easier ones.

S = 1 + 1/2  +1/4 ....
S = 2/1

S = 1 + 1/3 + 1/9 ....
S = 3/2

S = 1 + 1/4 + 1/16
S = 4/3

4. Find the pattern and then come up with the general case:

S = 1 + 1/n + 1/n^2 .....
S = n/n-1

5. Ok let's go back to decimals

S = 1/10 + 1/100 + 1/1000 just like above. Can you use the same technique?

How about if the digits differ

S = 12/100 + 12/10000 + 12/100000

Final Conundrum

1 = 2 / 3 - 1 vs  2 = 2 / 3 -2 as continued fractions.


I actually started by having everyone talk about the Julia Robinson festival. A couple kids mentioned the final flatland talk and this was of sufficient interest that I ended up spontaneously repeating a huge section of it for those who weren't there.  My retelling was accurate except I didn't have any klein bottle pictures on hand other than one on my phone. This ended up taking at least 10 minutes and I would repeat and make a day of it based on how it well it was received.

Basically you have a town in a 2 dimensional world  and the inhabitants assume they live in an infinite plane but have never explored it. Then finally one tries it out and discovers if he goes north and leaves a trail he arrives back in the town from the south side etc. Given the behavior when the inhabitants go N, E and then NE you conjecture the existence of a sphere, torus and then klein bottle. 

As I result I ended up skipping my planned kenken warm up. We made it through about question 5 from above but by this time I had exhausted the focus of the group, it was getting harder to keep everyone on task. So I made the executive decision to pull out the kenken puzzles after all and "cool off". Fortunately, that pulled everything together again.   My take away from this is:

  • Kids were aware that .9999 =  1 but the explanation was a bit fuzzy (no numbers between 9 and one) but I didn't have enough time to circle back at the end and show why this must be the case.
  • This was still too much material, I need to break it up with something "lighter" if I try again. I think I want either a visual interlude (color in one of these infinite series?) or to gameify the middle somehow.


I went with this puzzle from The Guardian:

Julia Robinson Festival

(The flatland talk at the end of the afternoon.)

For the second year in a row, I volunteered at the Julia Robinson Math Festival over the weekend. This is among my favorite mathematical activities to do for the whole year. This time around  I went to the training session before hand. That was useful, since I had a chance to look at the problems I would be facilitating prior to actually jumping in.

My first one was a bit daunting from the perspective of maintaining interest. The first part was to figure out the brain teaser: What comes next in this sequence?

      1 1
      2 1
   1 1 12
   3 1 1 2
2 1 1 2 1 3
3 1 1 2 1 3

This took me almost 25 minutes to see by myself and I worked through a bunch of different ideas. My goal was document all my wrong approaches so I could anticipate what students my do. I also knew it involved some lateral thinking. As I remember my main thought was "Gosh I hope this isn't something silly like number of curves and lines in the numbers."

At any rate, I was pleasantly surprised during the actual Festival.  Based on the prep work I managed to keep multiple students occupied for 30+ minutes in the productively stuck state. The main thing I did was to have folks work together, keep close tabs on everyone and ask about what they were trying. I also tried to emphasize regrouping the pyramid as a triangle and looking for patterns.

My second table was a really cool graph theory game. https://drive.google.com/open?id=0B6oYedIeLTUKc1hWSWtHMi1vbHM.  I'm going to use this in Math Club and I will talk about it more then.

Tuesday, February 21, 2017

My own Geometry Puzzle

(This is based on my previous explorations of the @solvemymaths problems. As far as I know its a new so I'm very happy with it. Usually I just collate problems.)

Monday, February 20, 2017

Mid-Winter break Geometry

By tradition, I'm going off on some problem solving walk-throughs:

- Courtesy of @solvemymaths

This problem is a good example of the power of working backwards.

To start off with like all of these type problems, I draw the center of the circles in and connect all the tangent points to find the inner structure and look for triangles.

One immediate simplification is to only find the ratio of BI to BK since its the same as the larger rectangle (1:2 scaling).  Secondly the inner right triangle EGJ is ripe for the Pythagorean theorem.

Before going any farther I noted some expressions:

  • BI = 2R + T 
  • BK = 2S + T
The required ratio to prove is \(BI= \sqrt{5}BK\) so squaring each side to get rid of the radical you get  \(BI^2= 5BK^2\) or \(4R^2 + 4RT +  T^2 = 5(4S^2 + 4ST + T^2) \)  This simplifies to \(R^2 + RT = 5S^2 + 5ST + T^2\)

For the rest of the exercise I kept this in mind as the target (although as you'll see I adjusted as I noticed more).

The second thing to immediately try was what fell out of the Pythagorean relationship in the triangle EGJ. Using \((R + S)^2 = (S+T)^2 + (R+T)^2\)   That simplifies to: \(RS = ST + RT + 2T^2\).  Which unfortunately doesn't look much like the target.  For one there is no R^2 or S^2 term and there is an extra RS and none of the coefficients are near yet.

I then munged around a bit and tried algebraically manipulating this expression to get it closer with no luck. So I looked back the drawing and noticed something I had missed initially  BK = 2S + T but it also is the radius of the large circle in other words 2S + T = R.  This immediately simplifies the target of   \(BI^2= 5BK^2\) to  \((2R + T)^2  = 5R^2\) or \(R^2 = 4RT + T^2\) which already looks closer to the Pythagorean expansion. But what's nice is you can also rewrite that as well with the segment  GJ = R - S rather than S + T.  

So I redid the Pythagorean relationship and found \((R + S)^2 = (R-S)^2 + (R+T)^2\) which simplifies to \(4RS = (R+T)^2\) Again this looks more regular than our starting point but still not exactly the same. Then since our target is only in terms of R and T we need to substitute out the S which we can do given 2S + T = R so 2S = R - T and applying that you now have \(2R(R-T) = (R+T)^2 \) or \(2R^2 -2RT = R^2 + 2RT + T^2\).    Combining like terms you get \(R^2 = 4RT  + T^2\) which is what we needed to show!

However what i actually did for the last step was the exact opposite of that explanation. Instead I took the target and put it into a form closer to what we had to see what was missing i.e. 
$$R^2 = 4RT + T^2$$
$$R^2 = (R + T)^2 + 2RT$$  (Completing the square)
$$R^2 - 2RT = (R+T)^2$$
It was this final form that reminded me to substitute back in for S since it was so close. And note how it was much easier to match the two expression after simplifying both of them rather than just going with the Pythagorean relation and trying to end at the initial goal.

5 Squares

Also from @solvemymaths.  Prove the area of the square is equal to the triangle.

This one was is closely related to http://mymathclub.blogspot.com/2015/05/cool-geometry-1problem.html and both rely on the  fact that the triangles formed between touching squares have equal areas.   See the previous link for the proof. 
The 4 key observations here are the

1) bottom two triangles around the square are congruent. This is the start of a Pythagorean Theorem proof in fact.  (See below if KH = a and JL =  b then each of the triangles is an a x b and FI = c where \(a^2 + b^2 = c^2\).

2) Each of the lower and middle triangles pairs have the same area because they are formed between squares. (i.e. CDF and FHI)

3) So all the lower and middle triangles have the same area (1/2 ab)!

4) You can create a new triangle with the same area as ABC that's easier to work out.

That's pretty nifty but I noticed something interesting when modelling a bit in Geogebra. If you let the 3 generator squares be a Pythagorean triple i.e. a = 3, b = 4, c = 5 all of the points in the model and all the areas are also integral.  That didn't look like a coincidence.   In fact I could roughly see the upper 2 squares had areas \(2(a^2 + c^2) - b^2\) and \(2(b^2 + c^2) - a^2\). But why was this happening?

The key idea I first came up with was squaring or boxing off the figure and finding the new triangles.

1. First I found the base of the new triangle and then the height.

[More variations on  square boxing problems]

These all revolve around boxing or squaring off a square with 4 congruent right triangles.


Most elegant solution comes through boxing the large square.

2. http://www.sineofthetimes.org/a-geometry-challenge-from-japan/

All the triangles are isosceles and all the quadrilaterals are rhombi. Find the  area of the square at the top. 

Wednesday, February 15, 2017

2/14 Valentine's Day Math Olympiad #4

By the luck of the draw (well really modular arithmetic), this year Valentine's day fell on a Math Club Tuesday. I don't really go in for holiday themed activities much but I was in the drugstore and in a fit of whimsy bought a bag of heart shaped gummies. So I ended up handing them out to the kids as they arrived yesterday which always makes the start of the session more exciting.  As I was going around the table, the thought crossed my mind "Gosh I hope they didn't eat a ton of candy already from their various class parties. If so some of the kids are going to bounce off the walls." That was fortunately not the case.

Thematically, I was in a bind again this week. We lost last week to the snow, next week is Winter Break and  I had to give another MOEMS Olympiad to stay on schedule. This made for a little too few free form sessions since the last one.  Looking forward, I'm going to try to fit in some kind of circle geometry oriented activity to build up to Pi Day. I have also been excited by some reading on function machines and am thinking if there is a fun game or activity inplicit in them.  That said, I appreciate the structure the MOEMS contest enforces. Done properly, this results in a lot of intense focus on the part of the kids on 5 problems over a half hour.   Providing this exposure to more challenging material is part of my over arching goals.

To start off the day, we went over the P.O.T.W (see: http://mymathclub.blogspot.com/2017/01/131-chessboard-problems-or.html)  The kids came up with two different approaches. The first leveraged guess and check and the fact that the overall perimeter was supplied to narrow down on the boxes dimensions. I wouldn't have thought to go this way, in fact I had considered removing the given perimeter since its not needed, but with it in hand this strategy works fairly efficiently.  The second was a more traditional completely Pythagorean Theorem based approach.

Moving on,  I proctored the MOEMS contest. Exponents reared there head again which seems to be a recurring theme for this year.  From what  I can tell so far, there was less conceptual issues with what does the notation mean.  But my work is not done.  Most kids given something like:

$$\sqrt{4^6}$$ will compute  \(4^6\) first and then search for a root manually rather than notice that this is the same as \(\sqrt{(4^3)^2}\) and thus the same as \(4^3\). I'm hoping calling these problems out on the whiteboard afterwards will lead to growth over time.

On the positive side, I had one student who usually has not talked much this year raising his hand frequently and volunteering to demonstrate solutions during our followup whiteboard session. Noticing that trend was my favorite part of the day.

I went with 2 KenKen puzzles of differing degrees of difficulty for the kids to work on if they finished early.  These worked well, but I'll bring 3 next time since 1 student actually managed to finish them both before I was ready to move on.


(This is a slightly modified version of a twitter problem I found from @five_triangles)

2/3 of the kids in one classroom exchanged cards with 3/5 of the kids in a second classroom. What fraction of the total kids didn’t participate?


I'm still brainstorming about next year.  I'm not sure if its going to be easier or harder to keep 6th-8th graders on task. One of my thought experiments, is whether I could present circle activities at different levels on different weeks and have the kids who found it either too hard or too easy due to the age gap work on practice MathCounts based activities.  Its also quite possible to use the pre-canned MathCounts curriculum which I'll definitely experiment with and see how I and the kids find it.

Tuesday, February 7, 2017

Making Explorations Successful

In a fit of perhaps excessive caution, the district cancelled all after school activities today despite the snow being almost completely melted.  So I'm tabling my plans for Math club for this week.  I really look forward to working with the kids so I'll have to work some of that energy out with my own children instead. I'm particularly fond of watching Numberphile videos as a family.

In the meantime, I saw a quote that I wanted to bounce around:

"If you group kids by "ability", those who are struggling may never see the pattern. Groups need to be mixed"

My first reaction to this idea was contrarian. For one,  if you don't really believe in ability i.e. quote the word to signal skepticism, then why does it make a difference if you mix the kids or not?  If ability doesn't matter groups are basically random already.  Kids should succeed anyway based on their own potential regardless of which peers they are with.  If it does matter, then what kind of learning is happening exactly in these situations?  My fear would be that basically you end up with a set of kids forging ahead and a second set copying what the others have learned. For me this is a poor man's version of direct instruction.  Rather than having an adult who has specialized in instruction showing the way, you devolve to peer to peer tutoring.  And having a reasonable amount of experience, I can safely say even kids who really get a concept are usually not nearly as good at communicating it.

So how does this relate to my Math Club?  First, I do have semi-random groups since I let the kids self select who they work with. The clusters tend to be gendered as a result and split along lines of friendship not necessarily skill. The kids obviously have volunteered to join the club which correlates mostly to some passion for math but in practice there are differences among them that are still probably comparable to a classroom.  When we do non-trivial explorations or tasks which is most of the time, kids discover concepts at vastly different rates.   This is one of the great weaknesses of this structure. In a one on one setup, I could slow down and scaffold just the right amount to let each individual "get it".  In a group, I'm always balancing the needs of the many against each other.

I try to compensate for this by having group discussions where everyone shares and by working individually with clusters during any activity.  I also work really hard to focus on having everyone participate. Those mitigate to some extent, but I still don't achieve a truly even amount of learning. Some kids still regularly have more breakthroughs than others. In a way, I think this shows the need for individual tasks. There needs to be a space, where everyone can struggle with a problem without having  it short-circuited by a peer finding the answer.  I'm sensitive when giving advice to not do all the work. Friends on the other hand jump right to the answer.

But in the end of the day, group inquiry based learning works best for me the more level the playing field to start off with and I'm not sure I've found an entirely satisfying way to resolve the issues that arise when it really isn't. And in thinking about this more, to me this is the crux of why teaching is non-trivial in general.

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.