Friday, October 28, 2016

1:2 triangles and their link to Pythagorean triples

I've mentioned before how instinctively it feels like the 1:2 triangle ought to have a more natural angle measure. In fact its in a 90 - 26.57 - 63.43 degree triangle. However when combined with a 1:3 triangle more integral relationships appear.

The 1:1 triangle ABD, 1:2 triangle AED and 1:3 triangle AGD all in a row.

Demonstration that the 1:1 + 1:2 + 1:3 triangles create a 90 degree angle. in the corner G.

  • First draw in a congruent 1:2 triangle GLK.  
  • The remaining angle between it and the 1:3 triangle forms a 45-45-90 triangle.
I just realized another extension today.  Multiple the angles in the corner G by 2 and you get

90 degrees + 2 1:2 triangles + 2 1:3 triangles = 180 degrees.

Stated another way if you take 2 of each of the base triangles they will form another right triangle.
And here's where it gets cool: out pops the 3-4-5 Pythagorean triple.  (Its also another demonstration why the in-circle has radius 1)

This meshes well with another 1:2 triangle exploration that led to a
 3-4-5 triangle:

Field Guide to spotting 3:4:5's (examples)


Once you have medians, like F and E you create 1:2 triangles like ABF, AED, CDF. So its fairly natural that a 3-4-5 is lurking.  The normal approach would be to show AEG is similar to ABF and also a 2:1 and then find the length of EG, GD and then prove DFG is a 3-4-5 in proportion.

But we simplify by just noting FDG is 90 degrees - 2 arctan(1/2) angles and therefore is
2 arctan (1/3) i.e. one of the corners of the 3-4-5. We can either show FGD is 90 degrees using similar triangles or AFD is 180 - 2(90 -  arctan (1/2)) = 2 arctan(1/2) and the 2nd of the 3 needed angles. And we're done.

Second Example:

This doesn't immediately appear the same but once you add the radii and center and remove the circle you're left with:

The median at the bottom has created a 1:2 triangle CBH which its then easy to show similar to first AGH and then CGH.  You can then see angle DGC = 2 1:2 triangles angles from the parallel lines and it follows CDG is a 3-4-5.

Third Example:
ABCD is a square containing a half and quarter circle and their intersection. What is the ration of CF:EF:CE?

After angle chasing one could apply the Pythagorean theorem (or similar triangle / ratios) and eventually find BF is  2/5 of the side BC and EF is 4/5. But there is again a quicker way

What to do with all these connections?

It seems like there is a really fun project for kids exploring all these relationships. I just have to figure how to structure them since the conceptual leaps are fairly big.

Thursday, October 27, 2016

10/25 Gozen

It was an interesting week from a planning perspective.  I'm  almost finished emphasizing divisibility and trying to decide what area I'd like to turn to next. One possibility was to return to some math relays from Math Counts. We'd tried that last year and it was mentioned as being fun by several kids in the wrap ups for the quarter.   My friend Dan had mentioned the relays again and they would fit with my aim at finding delivery methods that excite the kids while working through problem sets. In particular, there are also some thematically unified sets which I prefer.

At the same time I saw an interesting set of topology projects that Mike Lawler did:

I tried them out myself and I think they would be hit based on my experience with fold and cut activities last year. Finally, there was a comment from Joshua Green that had me relooking at resources at after a fairly long absence.

In the end I decided to focus on one of the games from there:   (See: for the inspiration) I thought and this turned out to be true that the tower-defense structrure would appeal to my kids.  The one problem with this game is the fairly complex instructions. So going in, I planned to play a sample game as  a class to help everyone catch on.  This worked pretty well.   There was still some initial confusion and questions but after the group play was over, I only needed to walk around and answer about 1 question per group to have everyone on track. And once that was done, most of the kids really got into it.

As you can see I brought an assortment of colored pencils and the final boards are actually quite pretty. If I were to repeat I might tinker with the rules a bit. The defensive block shots don't really fit well with the permanent archers defending their squares. Perhaps you should be able to choose between two types of moves.

I ended up reserving about 10 minutes at the end to demonstrate the divisibility rule for 7's since I had promised I would do so last week.

That left just enough time to hand out the problem of the week. This time I turned to a fun geometry exercise.  I didn't include any drawings so hopefully everyone interprets squares as geometric figures rather than square numbers (although its certainly approachable that way as well.) I'll probably stress that point in my weekly mail to the group.

"Find a way for any number n greater than 5 to divide a square into n parts each of which is also a square. Note: the subsquares do not have to all be the same size."

Looking forward, our first MOEMS contest is coming up as well AMC 8. I'm also feeling a strong urge to break out a grid logic puzzle.

Saturday, October 22, 2016

10/18 More divisibility

After a slow start last week, this time almost all of the kids in the Math Club finished the problem of the week.  (See the end of this post)  I had one of the boys demonstrate his solution on the board. He's one of the slower but more careful writers and almost always shows something interesting. So I tend to ask questions to the room to keep everyone involved while he gets his thoughts written down.

In point of fact, I don't really have any kids that are good at talking simultaneously while they write on the board. That appears to be a learned skill. So in the beginning of almost all student work on the whiteboard I usually choose between narrating what's being written or asking background questions to prep the room.  One of these days I'd love to see what other people do.

As an amusing side note I found a very similar variant on the puzzle in the Moscow Math Club diary book I was rereading this weekend.  (See below)

Besides the odd coincidence, two ideas stuck out for me in the book.
  • The importance of emphasizing games for younger students in a math circle. In the context of this book, younger meant around 5th grade as opposed to High School students. I continually find evidence that supports this idea in the various sessions I run.
  • The focus on working problem sets every week with relatively low student to  facilitator ratios. This was my vision of how a Math club would work going into the process. I don't really follow this model very often though.  So I decided to go with a more pure problem set based day to see where we're at.

To start up with, I decided to begin with the game of 100.  This is a fairly simple two player game with the rules being:
  • The score starts at 0.
  • The goal is to be the first person to reach 100.
  • Each player takes turn picking a number between one and ten and adding it to the score.

I was a little unclear in my first explanations so I ended up doing a few whole classs demos where I played against one of the kids and walking around the room to make sure everyone had understood the rules. After that startup activity, I let everyone play with the goal of finding a strategy to always win.

Gratifyingly, after about 5-10 minutes most groups had discovered part if not all of the structure of the game.  [If you reaches 89 you can always win, which means you need to reach 78, which means you need to reach 67 .... all the way back to 1]  I then had the group talk about their discoveries.

For the middle of the day, I returned to divisibility rules. I started with having everyone name all the rules for 2,4,5,6, etc. through 9. We then spent some time talking about adding multiples again and I brought up the question "How does this relate to adding odd and even numbers?" I then went over the logic behind the nine rules again on the board.  I had the kids pick random digits for our sample number which worked well.  Then I had them pick (mostly) random digits again so we could work on the rule for 11's.  My key question here was if "9 = 10 -1" led to the rule for nines what relationship would help with 11's i.e. 11 = 10  + 1?  We then did the follow exercises:
  1. Find a pattern for nearest multiple of 11 to a power of 10 and figure out why.
  2. Using that rule, and the distributive law breaking out the sample number to see how the divisibility rule worked.

This is all a bit trickier since for 11's the numbers alternate between positive and negative i.e.

10 =  11 - 1
100 = (11 - 1)^2   =  11^2 - 2 * 11 + 1
1000 = (11 - 1) ^ 3 = 11^3 ....  -1

Which was a pattern I wanted to emphasize without the benefit of knowing the binomial theorem.

We had about 18 minutes left at the end of this work and I gave the kids a choice between a worksheet with some extension problems about divisibility or a sample AMC 8 set of problems. Interestingly,  everyone seemed to prefer the random set. I spent this period walking around, checking on  progress and answering questions.

The most interesting moment was helping a student who had forgotten how to multiple decimals. I went through the idea of treating the decimal like this:

    1 2 3 4 . 5 6
x           7 . 8 9

=  $123456\times\frac{1}{100}\times789\times\frac{1}{100}=123456\times789\times\frac{1}{10000}$

Unfortunately, I didn't have enough time to see how well that explanation resonated. Overall everyone worked fairly diligently through this experience but I was still left not feeling sure about the structure. I'd like to figure out how to get more insight into what everyone's doing and to keep the interest level high.

Idea: I saw another teacher post about doing problem sets and stipulating all work must be done on the whiteboard to facilitate discussion. This seems interesting and I think I have enough space to make it practical.


I went with a scaffolded version of the divisibility problem I found online:
10.18 Problem of the Week

I also at the very end asked if the kids would like a demo of the rule for divisibility by sevens. About half were interested so I may do so next week even though I don't expect them to remember it. This is more aimed at impressing that complex divisibility rules exists and there are patterns that extend up through the integers.

Thursday, October 20, 2016

More Geometry in a Box (Trisection)

Continuing an occasional topic, I saw another great simple box construction.

[Prev post in this series:]

As always what  I like about these is both the complexity hidden in relatively simple constructions and the interesting, often surprising relationships that fall out.

This one started:

Given M and N are medians, find the ratio of the area of triangle DKL to the area of rectangle ABCD.

Initially I was looking for similar triangles and decided to add a parallel line in the middle DKL. That splits it into two triangles that are similar to larger ones CLN and BDN. Interestingly not only are the two sets similar but they are similar in the same proportion since both sets have the same length sides.

CN = NB since its a median and the new line is the base for both new triangles 1 and 2.

This approach works, you can use the similarity to setup a equation and derive the proportion but it didn't feel elegant at the time.

As it turns out there is a better way and it reveals cool fundamentals that were obscured above.

My second approach was to try to create a single similar triangle rather than two. So I added in the parallel line AN.  This creates DNO which is similar to to DKL.

But just adding this line now starts to reveal some more interesting facts. To start from the symmetry its very easy to show triangle DKM is congruent to its reflection BNO.    That implies BO = DK,
But even more interesting DKM is similar to ADO. That implies that K is the middle of DO since  M is the middle of DA. In other words, this construction has trisected the line DB.

(Note given the symmetry you could also show O is the middle of KB using the triangle BCK)

You can now use triangle proportions to quickly solve the problem. Triangle BDN is 1/4 of the rectangle. Triangle DNO is 2/3 of triangle BDN since its base DO is 2/3 of DB. And triangle DKL is 1/4 of DNO since its a similar triangle scaled 1:2.  

So Triangle DKL is 1/24 of the entire rectangle.   But more importantly its not just a coincidence that we started with only medians/dissections and ended with ratio that contains a multiple of 3!

Wednesday, October 12, 2016

10/11 Don't Fence Me In

My initial thinking for how to structure the start of Math Club was influenced by last week's  problem of the week:

This sheet revolves around a graph paper and dice based game where you roll the dice and fill in the boxes until someone can't go using either a rectangle with the same perimeter or area as the product of the dice roll. I suspected the kids would really like playing the game before we started talking about the questions. So I brought in my graph paper and dice and we spent the first half trying it out.

Sadly I didn't take any pictures of the finished games. But the kids really enjoyed the experience and several asked if we could keep going.  This definitely helped with our white board discussion afterwards. Interestingly for the final question, the best the kids could think of was a 7 square solution that would cover the whole board. I know of at least a 6 square one with an upper bound of 5 so I may return to this problem. For the end of this portion I asked everyone to think about if they could find a better solution.

We then transitioned to a talk about divisibility rules. I made several improvements over last time I tried this. First we spent some time group brainstorming about what happens when you add a multiple of n to another multiple of n, a multiple of n to a non-multiple of n and two non-multiples of n together.

After several minutes reflecting I had the kids report their ideas and why they thought that they worked.  Interestingly, I had to introduce the notion of the distributive law here. I skipped focusing on it this year since we did so last year but perhaps I should circle back.

For next time: On reflection I think a really good followup question would be how does this relate to the rules for adding odd and even numbers. The hope would be to build the connection that this is the same as multiples of 2 and that its a special case where 2 non-multiples add to a multiple.

With that foundation we talked about the simpler rules for multiples of 2 and 5 and why they must work. This flowed fairly well from the previous ideas. I had kids volunteer solid reasoning for both rules.  I ran out of time as we just started to work through the more complicated nines case.

My idea for followup is to walk through nines again and then let groups work on 11's with the starter idea that 10 = 11 - 1, 100  = 11* 9 + 1, 1000 = 11*91 - 1 etc and see if they can follow the logic.
Two years ago I showed the rule for 7's just for fun. I may pull that out again although it serves no practical use because of its inherent interest.

I also had a sheet of problems from AoPS that I planned to use that we didn't get to that I might bring back.  Buy coincidence yesterday I saw the following problem:

This would dovetail really well with the divisibility work.  I'm thinking about having groups brainstorm about the left hand side to see if they can see if must be a multiple of 6^2 by examining concrete values.


Todo: About 7-8 kids were volunteering answers out of the 12 who were here today. I need to draw out the quieter ones in future weeks,.

Tuesday, October 4, 2016

10/3: Year 3 (Bigger and Better)

I always get an odd mixture of impatience and trepidation right before after school activities start up. This time most of my concern is around getting two rooms running smoothly.  I really want a strong start to the year to ultimately prepare for passing the Math Club onto the next generation of parents.

Final Demographics:  26 kids
14 5th grade 1 girl
12 4th grade 9 girls

As you can see above its a weird lopsided year. So I have 10  girls but almost all are in 4th grade. In theory I'd like a few more 4th grade boys and lot more 5th grade girls.  This wouldn't actually be an issue except I'm splitting the club with another parent by grade. I have one lead for a future participant and I'll have to think what more I can do to recruit next quarter. One idea is to ask the 5th grade girl's parents if she has any friends who might want to join her.  I'm also tempted to randomly mix the kids up to achieve a better gender balance but that cuts against reusing material with the 4th graders and the opportunities to more closely fit the skill levels that we get from a grade split.

Update: I've found one more female fifth grader. So its not quite so bad. 

One other short term goal: Most of the fifth grade boys are coming back from last year and very comfortable (probably a bit too much) with each other. I really want to shake things up. So I'm thinking about either focusing on individual activities for a little while or forming temporary assigned groups to force kids to partner with new people. I'm actually pretty excited about this in the long run. You can often see really interesting development over the span of two years.

The afternoon started with me unable to get into either assigned room. The after school coordinator assured me that both rooms would be unlocked while the kids came down. Ultimately that did happen but I wasn't sure what state either room was in (I ended up with another room that didn't have much in the way of desks). For the first session I talked with my new co-coach and we agreed to do a joint starting session that I would captain. She brought name tags which were  useful except when kids lost them or hid the names under their pony tails. On the bright side, knowing half the kids before hand meant a lot less names to memorize. Also this was a new record for me handling a real full classroom of students.

I began with my normal introductory activities. I always have each kid introduce themselves, say their grade and teacher and why they joined. I had a sprinkling (thankfully not too many) of "my parents made me" and a few "my older brother was in math club" but for the most part the kids were there out of desire.

We then followed my normal outline about behavioral norms and goals. See: Club Charter Discussion Points  I like to emphasize the big ideas of "What do you when you're stuck?", "Mistakes are part of the process", and "We're working on listening to each other" and ask for lots of examples from the kids.  I actually thought a bit about whether to change this time after I saw a bunch of posts online about skipping discussing rules until the second day of class and jumping straight into math. I'm just not brave enough yet to try that out. My fear is that if we don't establish some of the basic norms right off the bad precedents will be set. That being said I think all my returning students work really well and the new ones seemed very amiable so I'm not sure if I'm being overly cautious.


I went with  Sara VanDerWerf's 100 game.   This was a hit and interesting to watch. We did 2 rounds with a 3 minute reflection on strategies in between and then a 6 minutes wrap up reflection.  Overall, I would definitely repeat this. The kids were captivated and they did not finish the whole sheet like I feared. You could easily bump the time per round from 2 to 3 minutes and run an entire 3rd round but I had only printed enough sheets to go twice.  On the final reflection it didn't look like the groups were going to notice the pattern so I asked everyone to very slowly replay a few steps and look for patterns again. That did the trick and almost every group found the number spiral based on that strategy.  One other fine tuning I might try if repeating.  I only had yellow highlighters due to supply constraints. If I could have handed out multi color high lighters in the second round, that probably would have more naturally led to the key insight.

We then started a dot pattern exercise. ala:  I wrote four questions on the board:

  • What is the next step for each of the patterns?
  • What is the 7th step for each of the patterns?
  • What would the step prior to beginning of the patterns look like?
  • Can you find an equation to model how many squares or sticks  are in each step?
We only had about 10-15 minutes to work on these. I went around the room and most of the kids were just getting the basic idea coming up with the next step for one or two of the patterns. So I definitely want to come back to them to at least have a group discussion and some white-boarding. One possibility is to warmup this way for a few sessions.

P.O.T.W Amusingly the first thing most of the returning fifth graders asked about were homework points and treats. That apparently had a very strong effect on everyone. So I'm hoping for strong participation again this year,