Wednesday, February 24, 2016

2/23 Prepping for Pi Day

Riff on the what I want the group to focus on

I was reading a post on the natural math site:  around designing a math circle. This has made me reconsider my informal focus for the group. Under the taxonomy used there, we definitely fall into the Mathematical Olympiad category. I like working on more complex problem sets (with multiple strategies) as well as throwing in a fair helping of puzzles and games. Recognizing this is my comfort zone and I'm unlikely to design free form activities involving dance for instance I do think I want to rotate in some occasional more open ended explorations.  I've been calling these notice and wonder exercises when I've tried them out. The natural math suggested tasks seem like a fertile area to look into this further. Likewise, I also have flirted with some more art related activities this year. For instance, I've used some pages from "This is not a Maths Book" several times. I think I also want to consciously balance these in the mix going forward as well. I suspect these would also work really well as warm ups.


The motivation for the activities for today was my informal map of what  I want to do in the next few weeks with the math club.   Among other things I plan to:
  • Celebrate Pi Day
  • Run another game day around checker stacks.
  • Fit in the last Olympiad
  • Prep better for Pi Day
The last item falls out what I learned last year: 2015 pi day. Most kids even in fifth grade while conversant with pi related formulas had very little conceptual knowledge about how they are derived. In a single day I was able to run a light-hearted relay using pi related problems but not cover all the background material as much as I wanted. This year I'm compensating by spending several days prior to Pi day working through some basic  models and more interesting problems around at least the concepts of pi and circumference and area. 

To get ready I threw together a progressive circumference related worksheet: Worksheet link. I'm planning to keep this in mind and add to it as I see other related problems. One issue I  immediately found is that its hard to keep the Pythagorean Theorem out of any such sets. I broke the 2 problems I found that need it into a challenge section at the back.  However, next year it would make sense to squeeze in a Pythagorean Theorem day as well. 

I already expected this topic would take the whole hour so after the kids finished  showing each other the problem of the week solutions I dived right in.  To start I had everyone brainstorm first for a few minutes what the meaning of pi is beyond its numeric value. There were some observations about it being irrational and how it was used in various formulas but no one was very crisp about it so I suggested "lets think of pi as the ratio between the circumference of a circle and its diameter or radius."   That lead naturally to a longer follow up brainstorming activity to think of reasons why this should even be true i.e. why is there a constant ratio?  After a few minutes of heated discussion I surveyed the small groups everyone had broken into. The best answers used some informal reasoning about how the radius needed to get bigger to reach the sides as the circumference. As I pointed then however its not immediately clear that the two quantities need to grow at the same rate. 

Moving forward, I gave a small demonstration of approximating the circumference using regular polygons on the whiteboard. After a small nod to the ancient Greeks I drew the regular hexagon which breaks into 6 equilateral triangles. Each triangle has a side length equal to the radius of the circle. Putting this all together this suggests that pi is approximately 3. I then glossed over how using the Pythagorean theorem we could keep subdividing our polygon and get closer and closer to the the true value. At this point one boy raised his hand to mention the Chinese had done this with a 200-gon. 

Note:  on reflection I had only about a third of the kids actively contributing during this portion. I want to focus on cold calling on different kids  more in these cases.

Hopefully, if nothing else sticks this portion and the basic idea of approximating does. From here I handed out the worksheet and the kids spent the rest of the hour working on problems. Focus was reasonable with some prodding here and there. Most kids were able to do 2-3 problems at the beginning which indicates to me that I want to add a few problems at about that difficulty level or a bit easier in the future.   I also ran into an issue where a few groups immediately went to the challenge problems at the back. I ended up asking them to try the beginning problems first when I thought they looked stuck. Perhaps on repeat, I should only give out the second sheet after the first one is handed in to enforce a rough ordering. 

Finally for the problem of the week I took a cool tessellation problem from a a UK contest :

Friday, February 12, 2016

President's Day Break Geometry

There's no school on Tuesday for President's day and therefore no Math Club. If I had realized this more fully I would have perhaps picked an extra problem of the week last time. In the meantime I saw this really interesting article on The Atlantic about extracurricular Math:  Leading an after school Math Club, I think I sit squarely in the middle of this movement such as it is. It's probably telling that I also recognize every organization mentioned.   The article ends with the hope that Gifted and Talent programs will bring deeper Mathematics into the school. I'm a bit skeptical of this premise. I definitely want to see improvements in the curriculum but I'm unsure how they can be achieved. Even our relatively shallow sequences are a struggle to cram into the year right now if you want to bring entire classrooms along (even if they are full of gifted students). For some more interesting discussion check out:  I'd settle for just knowing how to make sure the Math Club continues beyond me let alone spreading to other schools. However, some of the efforts mentioned in the comments are very inspiring.

With less prep work to do, I focused on another interesting geometry puzzle from @five_triangles. Again my aim with these posts is to explore the problem solving process and how one can experiment towards the proper approach.

My first thought on glancing at this one was the 1:2 ratios of interior line DE make me think of a Centroid. This is probably due to the recent numberphile video on the Euler Line that I watched with my son:  But it clearly wasn't quite a centroid yet and I thought maybe if I extended a parallel around there something my fall out.  That made me think of adding a parallel line on the bottom to DE so that there were similar triangles.  That also seemed interesting and I decided to explore it first.

Some ratios fallout fairly quickly. The new segments must be 2:3:1 to satisfy the original ratios and the 1:2 ratio needed in the similar triangle. Likewise you can show based on the other parallel lines that the smaller similar triangle has a single in ratio 1/3:1  to the large one. So this creates a lower triangle with edges 3x larger than the top one. But that seemed to be about all that was to be had here. I needed a breakdown of the triangle along AG.

[Note: I was a little sloppy and inconsistently labelled a few points between diagrams.]

So when I came home from work last night I picked up my original idea. DE is not the centroid of the ABC but if you divide up AB into 3 equal segments it is the centroid of the resulting triangle AHE.

You can then start by assigning an area A to ADF and find the areas of all the other pieces of this triangle relative to A via the ration of the interior  segments. For example since DF:FE is 1:2 triangle AFE has an area of 2A. In the lowest diagram I include the fact that since its a centroid there are intersections with the medians on all 3 sides and each segment is in a 1:2 ratio.  So our entire triangle AHE has an area of 6A.

From this point you can use various scaling ratios in the interior triangles to find more areas. My first two breakdowns were dead ends. Example:

I finally decided I really wanted to find the area in the reverse triangle on the bottom DEG instead. That's convenient since its base is a median so the area of DHG = HEG. And at this point I also realized that breaking up  the segment AB was useful so I also did the same for AC.

I ended up assigning another variable X  for the area of these two triangles. At first I thought I might need to solve for X. That's doable by breaking up the big triangle differently to find its total area in terms of  A and then assembling the pieces. But I suspected that the X's might cancel out so I decided to directly check the ratio.  You can find ABG and AGC by scaling ADG and AEG.  Sure enough the ratios are pretty and both A and X cancel leaving the fraction 3/5!  What's also interesting to note is how the inner core ADG and AGE of the triangle have equal areas its only the scaled outer ones that produce the final ratio.

A cleaned up version without even the centroid.

On reflection the construction works even cutting out a few steps.

Todo: add Mass Point Geometry solution which is a cool extension.

Could I give this out?

So this was originally advertised as Y6 which is within range but our curriculum doesn't cover triangle scaling as far as I can tell at this point. Its not a particularly hard point to teach beforehand although again the basic triangle formula can be a little weak given the de-emphasis of geometry. 
So let's assume I did that much prep, I think I would only provide it with the auxiliary lines filled in. That more than anything is what separates High School from Middle School problems for me. Even then there is the algebraic  expression in 2 terms to simplify and the need to scale several different triangles.  All told I would probably need to provide sub steps to do and it would still be a hard lift. So I think I'll leave this one for HS where it makes a very interesting exercise especially if you covered the mass point method after solving it conventionally.

Thursday, February 11, 2016

2/9 Olympiad #4

This week started with the kids getting agonizingly close (19/20 problem of the week points) to a reward. I'm hoping this tension plus the fact I printed this weeks problem up will lead to high participation rates next time. However, despite this I had a really satisfying answer to last week's problem which was to think more about the maximum number of tokens in the 4x4 no-rectangles problem.  One of my fourth graders came up with the following pigeon-hole proof that it could not be higher than 9.

1. You can have 8 tokens at most without having a column with  3 tokens in it.
2. To get to ten you have to columns with 3 tokens in it. [Not strictly true there is a subcase here with 4 tokens in a column that is also provable]
3. If you have 2 columns with 3 tokens in it they must overlap on two rows forming a rectangle.

This was probably my favorite moment so far this year.

For our main activity of the day, the club took the 4th MOEMS Olympiad. Overall this problem set was the best yet of the year despite an unintentional ambiguity in one of the questions that required me to accept two different answers.  There were two interesting observations that came out of it.

1. A surprisingly large number of the kids did not know the definition of adjacency. I would not have predicted this going in.
2. As usual many did not know what a counting number is. This I chalk up to our school curriculum which does not use that term. I believe it uses natural numbers instead.

Because the problems were a bit hard/more interesting fewer kids finished early. I brought some follow-up kenken problems for them to do anyway. We're now up to 5x5 medium versions.

Finally, everyone was generally enthusiastic about discussing their answers after we were done with the contest.  I did reasonably well choosing different people to answer each problem but there were maybe 4-6 kids who did not raise their hand. I'm going to try to track this and get them up more in the following weeks.

Problem of the Week:

Decode the following equation. Each letter stands for a unique digit.
Hint: we’ve talked about a key idea that you’ll need to use to figure this out in an earlier week.

YE * ME = TTT.

[I unfortunately didn't catch the fact I used * for the multiplication symbol which can be confusing if you aren't a professional programmer.]

Tuesday, February 2, 2016

2/2 Space Math

This week started with lower problem of the week participation than in the past probably due to the fact I didn't provide a handout and just orally gave the problem. (Take home lesson: its always worth making a hard copy of anything.) So after white-boarding I told the kids that I'd give the no rectangle 4x4 problem out again for another week and to see if anyone could beat our best solution so far of 9. Of course this was again given orally, so I'm going to take care to send an extra copy of the problem home in the weekly email send to the parents.

We then moved onto a Kenken warm up. As I learned from last year the first time you give these out its best to choose the easiest versions. So I printed a simple 4x4 and 5x5 version from  These went over well and I'm hoping to ramp up to more complex versions over the next few weeks.

For the main activity I then chose a pre-canned NASA packet that I learned about from a
another Math facebook group a week or two ago that I'm trying out: 1001 Math Circles.  So far the page looks promising for connecting with other math circles. I've also read about some of their books like "Mobeius Noodles" in the past but haven't yet had a chance to try them out myself.

Space Math Packet:

After trying out the packet myself, I thought of two problems. First, it involved a lot of photocopying so it would be a bit of a preparation splurge and I'd need to have the kids share packets to make this practical. Secondly, I thought the Astronomy was interesting myself but I wasn't sure how the problems would go or how many of the sets the kids would finish. I did like the contextual material and I thought the ratio problems were a good curriculum fit. This was partly why I chose a warmup activity. So if things went South we wouldn't spend the whole hour on it.

Overall, I'm still not sure how well received it was. Most kids finished the first two problem sets but it took a bit of hand holding to keep the room moving. My one thought was that if repeated it might make sense in these situation to have the kids read aloud some of the intro and to do the first quiz together as room. I asked a few kids what their impression was and received mostly positive reviews but my own son was not a fan. I think I'll open up next week with a post-mortem with the whole group to get some more data.  With this much photocopied material, I usually hope for a lot of bang for the buck.