Wednesday, May 31, 2017

5/29 Combinations and Pascal's Triangle

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

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

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

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

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

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

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

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

           \( 2 \choose{0}\)  \( 2 \choose{1}\)  \( 2 \choose{2}\)

      \( 3 \choose{0}\)  \( 3 \choose{1}\)  \(3 \choose{2}\)  \(3 \choose{3}\)

 \( 4 \choose{0}\)  \(4 \choose{1}\)  \(4 \choose{2}\)  \(4 \choose{3}\)   \(4 \choose{4}\)

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

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

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

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

Problem of the Week
An algebra one from the mathforum that doesn't really need formal Algebra to work it out:

I just saw a recent video on infinite series using pascal's triangle to look at hypercubes. This may make a cool followup for next week.

Tuesday, May 23, 2017

5/23 Triangle Conundrum

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

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

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

There was mention of the following problem:

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

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

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

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


Some probability work form Waterloo:

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

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!

This set is from Kevin Knudson

1. What’s your normal day like? 

Busy! On Mondays, Wednesdays, and Fridays I teach a class. I’m also the associate chair of my department, which means I have a lot of administrative responsibilities to manage. I have two Ph.D. students, so i spend some time working with them on their projects, and I also have a couple of undergraduate students who are working on their honors theses with me. And, I have my own research problems that I’m trying to solve, which involves some quiet time thinking, reading, and writing. Oh, and I like to drink coffee, cook, walk the dog, and play my guitar.

2. When did I decide to be a mathematician? What motivated the choice?

I always loved math in school. My earliest mathematical memory is of my first grade classmates cheering me on after I solved a hard problem at the overhead projector (this was in 1975). The problem? Write the number 8 using tally marks (I know, that’s not *that* difficult). But I definitely decided I wanted to be a mathematician when I was in high school, learning algebra, precalculus, and calculus. I was hooked.

3. Was I really good at math in middle school? 

 I was ok. I was on my school’s math competition team but I didn’t always make an A in class. It was definitely my favorite subject, though, and so I was motivated to be good at it. But sometimes new concepts are tough and I still struggle sometimes when I’m trying to learn something new. The way to tell that you really like math is that you don’t get completely discouraged when you’re stuck. If you love it in spite of the struggle then you know you’re in the right place.

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

I take a walk. I try to pay attention to my surroundings while doing so, looking at the leaves on the trees, the flowers, the waves on the surface of a lake, the birds and other animals. It’s very important not to be distracted by other things while doing this, so no headphones. What I don’t do is think about the problem. It will still be there after I clear my mind. However, sometimes I don’t have time to take a walk, or maybe it’s raining, so then I get out my yo-yo. Or I do some busy work that doesn’t require a lot of thought (answering emails, for example). The main thing is to not get stressed about it. When your mind is open and relaxed the solution will come.

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."