Tuesday, February 21, 2017

My own Geometry Puzzle

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

Monday, February 20, 2017

Mid-Winter break Geometry

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

- Courtesy of @solvemymaths

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

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

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

Before going any farther I noted some expressions:

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

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

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

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

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

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

5 Squares

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

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

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

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

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

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

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

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

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

Wednesday, February 15, 2017

2/14 Valentine's Day Math Olympiad #4

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

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

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

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

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

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

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


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

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


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

Tuesday, February 7, 2017

Making Explorations Successful

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

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

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

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

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

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

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

Tuesday, January 31, 2017

1/31 Chessboard Problems or manipulatives on the cheap

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

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

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

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

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

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

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

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

a pythagorean puzzle from @solvemymaths.

Wednesday, January 25, 2017

1/24 Curve Ball

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


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


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


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

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

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

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

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

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

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

Problem set:

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


Looking forward

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

Wednesday, January 18, 2017

1/17 3rd Olympiad

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


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

Some general notes:

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

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

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

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

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

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