Tuesday, April 26, 2016

4/26 Pythagorean Theorem Day

If you've been following along, I've been preparing to do a day based on the Pythagorean Theorem. Last year I  tried out a fairly inquiry based approach where I had the kids try to find equivalent squares using cut out shapes on the way to one of the basic proofs. Overall, I wanted to improve on that approach this year especially since the fourth graders have much less exposure to the basics of the theorem. This time around, I decided to go at it more directly and outline some proofs as a group and work instead on extension problems.

To start Math Club off, I gave out jelly beans to reward the group for finishing all the take home assignments.  We then went over the solution to last week's problem:  problem of the week This was a fun factoring/basic number theorem exercise and  I was pleased that one of the girls outlined a really good proof why the equation was impossible on the whiteboard. In a nutshell:

let Y be the units digit and X be the other ones.

So the original number 10X  + Y and the transposed version is 100Y + X.

if 2(10X + Y )= (100Y + X) then  19X = 98Y and since 19 is not a factor of 98 and Y is a single digit. There is no way to satisfy the equation.

[Note from the future: I would flip this idea now - don't waste focus on the puzzle and jump right in. Save cool downs for when the kids lose focus and try to maximize time on the main task. I'd also have the kids  work on the whiteboard on the problem sets]

I also wanted to work about 30-35 minutes on the main topic to maintain focus. So I then had the kids do a warm up sudoku puzzle from http://www.websudoku.com/  for about 15-20 minutes.   Now that spring is really upon us, the kids often start out  highly energetic after the stress of a school day.  Its easy to get the kids excited about games and puzzles and this made the next transition  which required more focus a bit easier than jumping in directly. As a side note, I talked about my agenda for the day and it was interesting how several kids were already really interested in the topic.

At the midpoint I gathered everyone back to the whiteboard. For my presentation this year I started with some basics:

  • A review of what a square number is and how it relates to a geometric square. I had read on the web that students often don't realize this connection and sure enough when asking for ideas from the room it was not immediately obvious.  This made for good group brainstorming.
  • A quick review of what a right triangle is and which side is the hypotenuse and how to find the area of a right triangle.
  • Some background on the discovery of the theorem by Pythagoras, the Chinese et. al and then a discussion of the classic diagram from above emphasizing the idea that the theorem is about area.
  • Then I broke into 2 of the more accessible visual proofs:

The first one I chose uses the 2 equivalent squares below:

I had the kids tell me the area of each square after having everyone agree they were both the same. Its really easy to "cancel" the 4 triangles and the theorem falls out from there.

The second one involved rotation:

After drawing the two smaller squares side by side you construct the original triangle by adding in point B and rotate ACB to AA'G and then rotate BB'F to A'B'C'.  This one I did twice emphasizing the hinges of the rotation.  I'm glad I did two completely different methods since it showed there are multiple ways to a solution.

[2nd Note from the future: this proof using similar triangles is more awesome: http://loopspace.mathforge.org/CountingOnMyFingers/FavouriteProof/]

Next, we briefly discussed Pythagorean triples and I had the kids name the few they knew included the 3-4-5. Since the first few pages went further into that topic I stopped at that point.  With my presentation finished I then handed out some quick practice triangles to find the missing side to. Interestingly that sheet had some larger numbers and it showed that some of the kids don't know how to find a square root by narrowing down the range its in. I'll have to do a session on that and perhaps the old hand algorithm which while a novelty nowadays is fun to try out.

Finally we finished with the main  packet I found  online: http://agmath.com/media/DIR_12306/9$20Pythagorean.pdf  which had a lot of  good followup problems. By this point the kids only had about 10 minutes so they mostly worked on the first 2-3 pages. But from walking the room, it looked like the basics had gelled and they were able to make progress. I'm half tempted to continue on the packet again next week since the following pages looked fun.

I chose the following for this week's problem of the week:
This is a repeat from one of my pi day worksheets that I don't think most of the kids finished the first time around. I'm hoping to get good results from the repetition and the fact we've covered  more basic ideas by now.

Overall I was fairly pleased. The session had fairly good flow and the kids seemed excited along the way. I'll have to mull things over some more since I still feel like there is a way to get build up to the proofs more organically and this was the closest I've come to teaching a subject like an actual math class.

For next year:

Find the length AB.  (5-12-13 Pyth. Triple)

Also good:

Saturday, April 16, 2016

You don't always win

"I want to thank everyone for coming out over spring break. I hope your kids had fun.  I really enjoyed seeing everyone competing and talking with all the other coaches and parents. Overall  I think team did well and we were in the middle for our division (the most competitive one). I also wanted to stress my goals for the contest. The most important parts as I see it are to generate excitement, to have a good learning experience and to have fun working with your other team members. The math in these contests highly emphasizes speed. As you move forward and keep going this is not going to remain the case . I think its important as a parent to emphasize the positive parts of the competition and to focus on personal growth when talking about it with your kids. 
I'll handout the returned tests on Tuesday at Math club.
In the meantime I was sent the electronic results last night."

My least favorite part of math contests is telling everyone how they did.  We don't usually win and we certainly will always have kids scoring across a range of results. At the same time I can tell how excited the kids get at the prospect of a win during the contest and  I'm always worried that they'll get discouraged.  The text above is my latest attempt at handing out the news.  During the latest Math is Cool contest this Friday I brought 2 teams of fourth graders. From talking with the kids and parents during the process I think they all had fun. I tried to emphasize the important aspects (to me) along the way but its really hard to do.  I'm not immune to the siren song of comparing relative scores. At the every end of each of these events I usually have a few minutes of regret while I remind myself of the big picture.

  • Elementary Math Contests are hopefully just the start of a math career and bear little relationship to what comes later especially with their emphasis on speed.
  • My hope is to foster continued interest in Mathematics down the road.
  • And at the same time add exposure to interesting topics right now.

Tuesday, April 12, 2016

Geometry - Similar Triangles again

For spring break here's a geometry walk through I wrote up a while back but never got around to publishing.

Another walk through of a @five_triangles problem.  Given the following diagram find
the length of AB.

Total Time: 3 days (on bus, before bed)
In retrospect I circled around the solution and it seems much more obvious than it was while working on the problem.

1. First thought was that we'd definitely need a auxiliary line but my first idea was to extend the triangle containing AB.

2. Secondly I noticed the 30-60-90 triangles with the 2cm-4cm sides.

3. I calculated the other side lengths in the right triangles using the Pythagorean Theorem.

4. I wondered if AB was congruent with the 4cm side and  if I could prove it.

5. Then I noticed an isosceles 3 - 3 triangle but that didn't add any other angles.

6. I did some angle chasing to see if I could show AB was 4 and in the process realized I had another
big similar triangle. If I extended the other line.

7. This also created a 3rd smaller similar triangle.

8. Now I just needed one of the side lengths in the new triangle.

9. Because I knew a bunch of the original base lengths I started looking for ways to find the rest which would give a side of the similar triangles. But this became a dead end.

9. Then I realized the hypotenuse was 4 + the hypotenuse of the littler triangle and the littler triangle
had enough side lengths to figure the rest out.

Wednesday, April 6, 2016

4/5 Snap Cube Hotel

Before this week's Math Club meeting I had an extra math adventure. This weekend was the Julia Robinson Festival at the UW (http://jrmf.org/). I drove to the campus intending to wait on the side while about half of my kids participated. But the organizers ended up needing volunteers and so I manned tables for about 2 and half hours.  This was a ton of fun and I received a free volunteer T-Shirt. I was able to work one on one with about a dozen kids over the time. My favorite part was an interesting observation at the first table which was centered on chess pieces. If you attach the top and bottom of a square to each other and then the left and right you get a toroid! This is very easy to verify with a piece of paper folding it one set of sides at a time but also something I had never considered. Next year I intend to volunteer to help ahead of time.

Back on topic, while brainstorming for activities for the last session I came across the following interesting activity from @fawnpnguyen:  http://fawnnguyen.com/hotel-snap/.  I couldn't procur snap cubes quickly enough to use it last week but one of the third grade teachers got back to me and I arranged to borrow her supplies this week.

The basics are you break the group up into teams who design a "hotel" built out of snap cubes. These groups then earn income given one set of rules and are charged costs based on another (see rule sheets). I thought this would probably take the whole session but I printed an extra sheet of problems from the Julia Robinson Festival just in case. http://jrmf.org/problems/LittleBoxes.pdf  It turns out the process took the entire hour as expected (and everyone had a really good time playing around with building designs). I reserved the last 10 minutes for scoring and if I repeat I'd bump that up to 15-20 minutes.  The range in designs was really interesting and we didn't even probe all the optimization issues possible.

For the problem of the week I chose the current uwaterloo set which involves a number triangle from http://cemc.uwaterloo.ca/resources/potw/2015-16/POTWB-15-NN-24-P.pdf  This is not too difficult and hopefully will lead to lots of students trying it out.

Looking forward my backup idea would make a great session on its own. I will definitely try out a mini Julia Robinson problem set in future weeks.

Saturday, April 2, 2016

What's the best way to talk about the Pythagorean Theorem?

I've been reading Martin Gardner's "Gardener's Workout" recently and came across the following section at the end.

For example, teachers traditionally introduced the Pythagorean theorem by drawing a right triangle on the blackboard, adding squares on its sides, and then explaining, perhaps even proving, that the area of the largest square exactly equals the combined
areas of the two smaller squares. 

According to fuzzy math, this is a terrible way to teach the theorem. Students must be allowed to discover it for themselves. As Cheney describes it, they cut from graph paper squares with sides ranging from two to fifteen units. (Such pieces are known as "manipulatives.") Then they play the following "game." Using the edges of the squares, they form triangles of various shapes. The "winner" is the first to discover that if the area of one square exactly equals the combined areas of the other two squares, the triangle must have a right angle with the largest square on its hypotenuse. For example, a triangle of sides 3,4,5. Students who never discover the theorem are said to have "lost" the game. In this manner, with no help from teacher, the children are supposed to discover that with right triangles a^2+b^2=c^2. 

"Constructivism" is the term for this kind of learning. It may take a group several days to "construct" the Pythagorean theorem. Even worse, the paper game may bore a group of students more than hearing a good teacher explain the theorem on the blackboard

This is topical since I'm thinking about doing a Pythagorean Theorem day again this year. My attempt last year: http://mymathclub.blogspot.com/2015/01/16-winter-session-starts-pythagorean.html somewhat followed the design Gardner derides. Among other ways, I had the kids cut out triangles and try to find two squares with equivalent area that would then reduce into one of the easier proofs.

Generally, I don't think it worked as well I wanted it to either. Only a few kids found the necessary equivalent squares and even then it took some guiding to reach any insights.

So what I could do better this time around bearing in mind my primary goal is to engage with why the Pythagorean Theorem works?

After doing a cursory look on the web I've thought of the following ideas so far:

  1.  Remember to emphasize the geometric nature of square numbers i.e. that they are the area of squares before starting.
  2. Start with a measuring exercise and observe patterns? I could preprint various pythagorean triple triangles to make things easier to spot.
  3. Do a short brainstorm session and the introduce a few proofs directly.
  4. Concentrate on extension problems rather than the proofs.
  5. Focus on the Pythagorean triples and finding them and looking for patterns?
This feels like I haven't quite found the right answer yet....