Thursday, May 18, 2017

5/16 Expected Values

My planning process this week went something like this: after last week's talk I either wanted to do some group white-boarding or find a new game to explore. I also was thinking more about combinatorics. I've never done anything on combinations (n choose m) and I mulled choosing that as a theme. Then in the middle of the week the Math Counts finals occurred. Watching the live stream was fun for me and I thought the kids would like that too. So initially, I thought about showing pieces of the video and then pausing and have everyone do the problems on the whiteboard. But after some more thought, I worried that it would emphasize the speed of the competitors too much and I also wanted to dig into the Chicken problem more deeply.

Finally this was the structure I ended up with:

We warmed up with some individual skyscraper puzzles from I really like doing these and they went over well engaging everyone. I cut this short after everyone had at least finished the first one of the set I provided in the interest of time.

After watching the videos I transcribed a few of the problems I liked and thought would be good to try on the board:

Whiteboard Problems from the video:

  • Caroline is going to flip 10 fair coins. If she flips  heads, she will be paid $. What is the expected value of her payout?

  • Sammy is lost and starts to wander aimlessly. Each minute, he walks one meter forward with probability 1/2 , stays where he is with probability 1/3 , and walks one meter backward with probability 1/6. After one hour, what is the expected value for the forward distance (in meters) that Sammy has traveled?

  • A finite geometric sequence of real numbers with more than 5 terms has 1 as both its first and last terms. If the common multiplier is not 1 what is the value of the 4th term?

  • The length of a 45 degree arc on circle p, has the same length as a 60 degree arc on circle q. What is the ratio of the areas of circle p to circle q?

  • The novel Cat Lawyer is 300 pages long and averages 240 wd/pg. The sequel Probably Clause is 60 pages longer and 30 more words per page. Probable Clause has what % more words?

  • Ian is going up a flight of stairs. Each time he takes 1,2 or 3 steps. What is the probability that he steps foot on step 4?

  • How many 6 digit integers are divisible by 1000 but not 400?

But I decided to focus on the final question:

"In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately to its left or right. What is the expected number of un-pecked chicks?"

To build up to it, I started with discussing expected value and used dice questions as starters in a group.

1 What's the expected value of a single roll of a six sided die?
2. What's the expected value of 2 rolls?

After we went over the concepts as a group, out came the blue markers and  I had everyone work on the followup problems up on the various whiteboards.

3. What's the expected value of the product of 2 rolls?
4. What the expected value of the product plus the sum of the rolls?
5. What do you notice about this? 

The kids all worked on these followup questions in groups over about 20-30 minutes and then we gathered together to discuss what we found.  I had to bring out the linearity of expectation relationship and did not go into the proof at this point since its a bit too complex. [This is always a struggle to resolve whether to dig into every observed pattern and find the reason or accept its probably true in order to reach a target for the day.]

Then I posed the chicken problem expecting everyone would work again for a while. I was surprised but several kids almost immediately shouted out the answer. It turns out the scaffolding may have been a bit too much after all. If repeating I might lead off with the final question, work on it a few minutes and then go into the build up.

At any rate for the last 10 minutes we did watch the video communally.  Before showing it I prefaced it with a short talk about speed and emphasizing both the competitors had really trained to build this up and that it wasn't really important outside contest while the problems on the other hand were fairly interesting.  So hopefully, I didn't damage any of the kids self-conceptions. As expected, the room was fairly rapt watching the competitors even many of the parents on pickup stopped and watched it until the end.

Continuing the MathCounts theme I chose this weeks problem from their site: although I'm not completely keen on the questions.

Tuesday, May 16, 2017

MathCounts Final

Since it was fun Last Year to think about the Math Counts final question, here is the 2017 version:

Oddly enough when I saw this year's final question, I almost immediately said out loud 25% of the total number or 25. Somewhere very recently I'd seen this problem (I can't remember exactly where), I didn't recall the reasoning offhand but the answer came to mind instantly.  Math Counts at the national level works a bit like that. Seeing lots of problems and being able to quickly either recall the entire answer or the efficient means to solve it is critical to win where kids are answering questions like above in a few seconds. In fact, I couldn't even "borrow" some of the questions since they were answered before being fully read out or printed on the screen.

That's not super interesting in the long run, but I think its balanced out by what happens when the larger set of kids are preparing for the contest and even when bystanders read an article in the nytimes and spend some time thinking about a problem.

This one is fairly fun to model. Many people eventually came to the reasoning that for an individual chick there is a 25% chance it won't be pecked. But this is not independent of what happens to all the other chicks. Because this chick wasn't pecked 2 other neighbors were.  I like thinking about this as a chain of  Ls and Rs where your counting the number of transitions between letters.

Behind all of this is the somewhat counter intuitive: Linearity of Expectation.  Even though the individual outcomes are dependent the expected value can be had by simply adding them up anyway.

I'm planning to do a session  around this today divorced from time pressures. Building up to the Math Counts question through a series of exercises and observations about expected value should make an excellent white board #vnps activity.

Followup interesting tweet stream on the probability distribution:

Tuesday, May 9, 2017

5/9 Dating for Elementary Students

Today was a special Math Club session. Annie Raymond from the UW Math dept. came and gave a talk to the combined fourth and fifth graders on the topic of combinatorics. So I had the unusual opportunity to act as a photographer more than a facilitator. We started with me asking the kids if they had any initial questions. There were few basic ones like "How do you spend your days?"  Answer: teaching a lot of the time and thinking about research in between.

Annie went a bit bold and chose to talk about the Stable Marriage Problem which is right up my alley as a computer developer. I was a little worried we'd end up with a lot of nervous laughter and asymmetric interest from the boys and girls but the kids exceeded my expectations and were very attentive and engaged.

The kids did a great job calculating the total number of combinations was 3! at this point.

In the middle she brought an interactive version of the algorithm to test out. There were ~10 boy and girl preference sheets handed out. She then had the kids work through the algorithm in rounds with the boys going to their next choices and the girls picking the top selection. Much amusement and chatter soon followed.  This would be fun to do as an ordinary combinatorics exercise on its own.

What's very nice is there is a not too complicated proof by contradiction that the algorithm works. That fit really well with our recent session on proofs.

Then we observed that the algorithm is asymmetric (with the kids volunteering if they got a good "match") its much better to ask than to choose partners.

Variants you could build more extensions on. She didn't bring it up but I thought a bit about complexity as a tangent. I think the algorithm is O(n^2) for instance in the worst case where everyone has the same preferences. Could this be optimized?

 Overall I think this was a grand success with the kids and I'm hoping I can continue this relationship with the UW Math folks next year.

Monday, May 8, 2017

Questions for Mathematicians

I've been prepping for our guest talk from the UW Math Dept. One of the tasks I've done is survey the kids to generate questions for the talk. Jayadev Athreya  emailed me back with some answers  which I really like:

1. What's your normal day like i.e. what does a mathematician actually do?

We teach, we think, and we write- but mostly we play with patterns- exploring ones we think we understand to see if there is a deeper pattern hiding behind it. Like you might notice that all prime numbers bigger than 2 are odd, then you notice that all prime numbers bigger than 3 aren't divisible by 3, and so on... that's a series of patterns that all come from the definition of a prime number! We do spend quite a bit of time using computers to find patterns too!

2. What did you have to do to become a mathematician and when did you decide to go down that path. What motivated the choice?
I was very lucky in that my mom is a physicist and my dad taught math. So I had great role models and I saw how exciting math and physics could be!

3. Were you really good at math when you were our age?

I worked hard at it, and I liked learning it and exploring it. I didn't always do well on tests. My dad, who is also a mathematician, was not very good at all as a kid but enjoyed playing with problems, and became a really good mathematician.

4. What do you do when you get stuck on a problem?

I follow the advice of a famous mathematician, Polya, who said that for every problem you can't solve there is a simpler problem that you can't solve! So I look for the simpler problem, try and work out a bunch of examples, and try and and play with patterns to see if I can unlock the problem. Sometimes this takes months, or even years- so patience and hard work are key!

Annie Raymond also sent back some answers:

1. What's your normal day like i.e. what does a mathematician actually do?

It depends on the day!

On Monday, Wednesday and Friday, I teach two different classes, one on multivariable calculus, mostly to engineering students, and one on how to prove things to math students. Outside of the two hours when I actually teach, I meet individually with students who need extra help, grade some of their work, come up with new material for them. On Wednesdays, I often go to a talk about combinatorics, one of the fields that I work in, to hear about new work done by colleagues from all over the world. On two of those days, I also meet with some collaborators to discuss our progress on a common project that we are working on. If I'm lucky, I'll have a couple of hours to do research or work on papers as well, but that's not always the case.

Tuesday and Thursday are the days when I actually do my research and write papers. Doing research for me means sitting down and thinking about some problem I'm hoping to solve. The nice thing about problems in math is, when you solve one, it usually opens up ten new ones, so you never run out of problems to solve. Going to talks also helps with finding out about new problems too. It is very hard to explain how you get the good idea that allows you to solve a problem. Usually, it just clicks all of a sudden after you've spent hours and sometimes weeks or even years playing with it.

Finally, on Tuesday night, I teach college-level math to inmates at a prison. I believe making education more accessible to everybody is the best way to create a strong and fair society.

I do need to mention that traveling to go to conferences and give talks and meet with other mathematicians from all over the world is a pretty regular thing too. Of course, those days are completely different!

2. What did you have to do to become a mathematician and when did you decide to go down that path. What motivated the choice?

I had to study a really long time: I first got a bachelor's degree in mathematics (and music!), and then I went to grad school to get a phd. The nice thing is that, in science, you usually get paid while you do your phd, so you're not a starving student. I'm now finishing up a postdoc which is something you do after you get your phd to prove that you're ready to be a professor. You do more or less the same thing as a professor, but your job is temporary. Next year, after 4+5+3 years of being at a university, I'll finally be a professor.

I decided to go down that path right before going to college. I went to math camp the two summers prior to college, and I really loved it. Up until then, I knew I liked math, but I didn't have a good idea what more advanced math looked like, and I thought---wrongly---that math was a pretty useless field, and I wanted to do something useful. Math camp opened my eyes on how amazing math can be.

3. Were you really good at math when you were our age?

I was pretty good at school overall---it came easily to me. I did find college very hard however. We all find things very hard at some point. How we deal with that and how we persevere are both more important than how long we found things easy.

4. What do you do when you get stuck on a problem?

Being stuck on a problem is my normal state, and the normal state of most mathematicians. I have spent a few years working on a few problems. But that's normal: there are many problems in mathematics that have been open for 10 years or 100 years! I've learned not to be frustrated if I don't know immediately what to do and I try to enjoy the phase where you play blindly with the problem, where you try to look at it from every possible side. If I ever get too frustrated or don't know what to try next, I move on to a different problem or different task I need to accomplish: often, new ideas come when I am doing something else. Discussing the problem with friends and colleagues also help: it helps make my ideas clearer and combining our ideas together often leads to a winning strategy!

I think this is fairly interesting. Maybe next year I'll have the kids write letters and see if we can get more responses.

5/2 Assessment

This is a short placeholder entry.  I decided to administer the AoPS algebra assessment last week since the 5th graders are currently choosing next year's math course. In our district 6th graders have the option to opt-up to Algebra I. This is determined via a set of opaque measures which often leave parents uncertain about the best choice.

Some notes for future reference.
  1. While this is useful for parents. I think its best done out of club after all. I would just offer links in the future even knowing only  a small portion of parents would take advantage of them.
  2. There is no time limit for the 12 questions. I thought that would be sufficient but most kids only finished about 7.  So it probably takes closer 1:30-2 hours.   (This could be threaded through several weeks.)
  3.  To make up for that, I'm going to grade what I have and give the parents links to the full test so they can finish it they they desire.
  4. It takes a lot of work to keep kids focused on something so "class-like". Another reason why I don't think I will do this again. I did decently since I know all the kids well but it required constantly moving between tables and encouraging them to keep going / checking in.  I was a little surprised how difficult some of the kids found the questions "This makes my brain hurt."

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: 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  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:   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.  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   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: 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: 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: 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 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: but I found a tweet from Sarah Carter that looked interesting about slant puzzles:  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:   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

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:

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.  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 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.


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