Wednesday, December 13, 2017

12/12 End of Quarter

The end of this quarter really snuck up on me.  In my planning process I decided to finally use the video below on  Schumann's enumerative Geometry. What I particularly liked was the triangle puzzles discussed in the video and the fact they linked to a modern discovery.  But on looking through the video for a last time I became a little worried. The end part gets quite complex and I wasn't sure if the kids would be able to follow it. So I started to look back old Julia Robinson Festival questions and assemble a min-festival we could do in Math Club. Then I remembered that it was the last session and I wanted to do game day to celebrate. So in the end I decided to go with video, take the problems with me in case it looked like a dud as well as the games.



What's especially nice is that there are several natural breaks in the video for pausing and trying the math out yourself. I took advantage of the ones in the beginning and had the kids try out assembling triangles and looking for patterns:


Base shapes:


I brought lots of colored pencils and had the kids draw versions on their own paper. If I had more time I might have precut out base triangles out instead. 


Examples:



Overall this went pretty well after all. The kids wanted to see the end of the video when I offered them the chance to move on so we did watch the whole thing.  The only other mistake  I made was pausing a hair early the first time and having to explain the rules more than I expected.

In the back half of the day I brought in my usual assortment of board and card games:

  • pente
  • set
  • prime climb
  • terzetto
  • rush hour
These are still popular with the middle schoolers although I really need to pick up a new one before next time. My favorite moment here was one student pulled out last week's skyscraper puzzle to finish working on it today. I really like this display of persistence.

Finally, we also had a club discussion about recruiting. The kids decided to talk to their friends and in front of their math classes as well as one is going to make a PA announcement. We'll see how this effort works. I like that I'm offloading some of this to the kids and hopefully I'll find some 7th graders next quarter.



Thursday, December 7, 2017

Motivating Kids

I recently saw this tweet



The rather interesting gist of the research was how much better US students performed on PISA when given a monetary incentive.  That made me immediately think of my recent success and failures in getting everyone in the Math Club engaged.
 
If you haven't run an after school club, you might be excused thinking its nothing like a Math class. Everyone is there voluntarily and excited to work on Math problems.  The truth is a bit more complicated.  For one kids show up for a variety reasons including the dreaded "my parents made me do this."   Secondly, a student's temperament varies day to day.  The club meets after six long hours of school has already taken place. Some days even the best kids are already worn out.  Moreover,  a teacher in a classroom has a bunch of tools to leverage to make kids participate such as grades.  The club is a purely voluntary affair, buy in on everything from talking in front of the rest of the kids, to doing a problem of the week at home is a hard fought battle. Each day I need to find a way to create flow and draw kids into the topic I want us to explore.

There is no perfect answer to the problem and I continue to evolve in how I think about this issue. That by itself, is the first and foremost principle.   After each session I try to be critically honest with myself about how well it went and what I could do to improve.  In practice, I  almost always find I do better presenting a topic the second time.   Since I'm continually searching for new material this is something I have to keep in mind. For every really new activity, leverage whatever connection it has to previous ones to inform how it will be done and fall back to more tried and true formats/topics after experimenting.  I don't want to always be on the bleeding edge.

The culture of the club builds on itself.  First that means I always try to emphasize and reinforce when I see notable participation. I'm also ambitious in the sense I want the kids to engage with complex Math that requires a lot of focus.  In my  ideal vision we would just do a challenging problem set that I'd print out each week.  That would in reality be a recipe for disaster.  Instead I'm very mindful of the need to thread in puzzles/games/activities that are particularly playful. This is especially true when starting up with new kids I haven't worked with.


There are several general strategies I'm currently following that are working reasonably well



  • Games and Puzzles are always great as long as they are mathematically relevant.  Often they can be repeated multiple times and kids will develop more insightful strategies.
  • Leverage media. I'm super careful not to show a video most days. But sometimes after working really hard one week, a numberphile video is just the right change in tempo to keep everyone going. 
  • Have the kids use the whiteboard as much as possible. I've written about VNPS before: http://mymathclub.blogspot.com/2017/03/328-vnps.html  This remains an excellent strategy.    
  • I utilize a very minimal  common  routine to get everyone into a Math frame of mind. Mostly this consists of an introduction and talk about what we're planning to do for the day and a group review of the problem of the week.
  • If things don't go as well as I want one week - move on and change things up next time. 
  • Use competition from time to time. I'm also super cagey about this but official contests bring out a lot of energy and focus in most kids. 
  • Shamelessly bribe them with treats. I'm still giving out candy for homework participation. I only give one problem a week and the goal is to have time to think about something interesting over more than a few minutes. When enough work is handed back as a group I bring in treats. The ends seem to justify the means.
  • Talk candidly about where I think things are with kids. If I see a problem or direction I want the kids to go, I'll usually mention it up front. For example, last week I knew we were going to walk through student solutions to the the MOEMS contest. So at the start I told everyone that was coming up and I wanted to focus on listening to each other.
Overall lest this paint a picture of perfection, I still work on motivation from week to week. I'm always looking for other people's ideas on what works and what I might adapt.   Engagement is very near the heart of mathematics teaching, its complex and its not easy.

Looking forward:
Now that I've experienced 4th-8th graders I can definitely see the growth in maturity as kids get older.  Right now I only have 3 eighth graders. If I can recruit more of them, I'm hoping to leverage their leadership potential more. 



Tuesday, December 5, 2017

12/5 Olympiad #1

Today was the very delayed first MOEMS middle school contest day.  As I mentioned before this contest was supposed to be on the same day as AMC 8 so we had to push it out and then I needed some buffer. Fortunately, MOEMS is a low key organization and as long as you get all the contests into the system at the very end in March you can move individual dates around.  I was really curious going in how the kids would do and react to the contest. In looking at the questions before hand I thought this was slightly easier than any of the ones from last year.  My great worry was actually that it would be too easy for everyone.

That turned out to not be the case. When I polled at the end although the kids thought it was easier than AMC 8, they also generally all enjoyed it. That's great since I think its a good format: 5 questions over a half hour gives enough time for most kids to solve what they are capable of solving. And the split over 5 different weeks allows you to parcel the questions out and discuss them in manageable chunks as a group.

Which brings me to the other win for the day. This was probably the best whiteboarding session I've done yet this year. Almost everyone volunteered and there were multiple solutions presented for each of the 5 problems. There was just a ton of enthusiasm. Sadly, I'm not allowed to discuss any of the details of the problems but the kids came up with a lot of good problem solving solutions and really listened to each other.  I'm hoping to extend this streak to next week's whiteboarding and have some more interesting details to record here.

As usual to occupy everyone who finished early I brought a low-key puzzle. In this case I went back to the skyscraper puzzles from https://www.brainbashers.com/skyscrapers.asp  and printed  an easy and hard 6x6 one.

P.O.T.W
This is my absolute favorite linear systems problem:

Assume that are real numbers such that





Find the value of

Wednesday, November 29, 2017

11/28 Egyptian Fractions




Brainstorming this week I became interested in Egyptian Fractions because they dovetail nicely with the math history from last time.  Here's a topic that is both historical and mathematically interesting. I was going to originally title this week Funny Fractions and do a unit on both Egyptian Fractions and Farey Sequences but on consideration I decided there was enough to deal with just focusing on the first idea.  That was right decision to make based on actual time management.  As I discovered also over the hour, these provide a great platform for practicing other more basic skills,

To start off I had everyone guess when fractions were first documented as being used. I mentioned the late entry of decimals as a starting point. I was pleased someone remembered the Babylonian base 60 fractions from last week.  I then did a quick read of the background of the  Rhind papyrus with some information and a printout of the scroll from: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html

I then used a modified version of the series of questions and activities from here:
https://nzmaths.co.nz/resource/egyptian-fractions   I particularly focused on finding ways to break Egyptian fractions apart into sums of other Egyptian fractions and discovering algorithms to find an Egyptian fraction sum for a regular fraction.   Once kids started to brainstorm on the whiteboard I started feeding further problems as different groups progressed:

Further Problems:

1. The Mullah's horse: The former Grand Wizier, Mullah Nasrudin was approached by three men with 19 horses. The men asked him to adjudicate the will of their recently dead father which required that his horses be divided among his three sons so that the oldest son receives 1/2, the middle son gets 1/3, and the youngest son would get 1/7. With little hesitation Nasrudin added his own horse to the herd and said, "What is half of 20, 1/4 of 20, and 1/5 of 20" After some time the men replied, "10, 5, and 4". The eldest son then took 10 of the horses, the middle son took 5 of the horses, and the youngest son took 4 of the horses. The Mullah Nasrudin, then took the remaining horse and rode home. Can you explain what occured?


2. Find all the solutions (there are less than 10) to the problem (n-1)/n = 1/a + 1/b + 1/c, where a < b,
b < c, a, b, and c are positive integers with least common multiple n. Note. a = 2, b = 4, c = 6, and n = 12 gives one solution.


3. How many different egyptian fractions can be used to describe 2/3? Two of them are 1/2 + 1/3 + 1/6 and 1/3 + 1/10 + 1/15.


4. Want to solve an unsolved problem? One of the most famous problems on Egyptian Fractions asks, "Can every proper fraction of the form 4/q be expressed with an egyptian fraction with less than 4 terms?" Can every proper fraction of the form 5/q be expressed with an egyptian fraction with less than 4 terms?


5. The sailor, coconut, and monkey problem: Five sailors were abandoned on an island. To provide food, they collected all the coconuts they could find. During the night one of the sailors awoke and decided to take his share of the coconuts. He divided the nuts into five equal piles and discovered that one was left over, so he threw the extra coconut to the monkies. He then hid his share and went back to sleep. A little later a second sailor awoke and had the same idea as the first. He divided the remainder of the nuts into five equal piles, discovered also that one was left over, and through it to the monkies before hiding his share. In turn each of the other three sailors did the same - dividing the observable amount into five equal piles, hiding one, throwing one left over to the monkies. The next morning the sailors, looking innocent, divided the remaining nuts into five piles with none left over. Find the smallest number of nuts in the original pile.

6. find 1/a + 1/b + 1/c + 1/d + 1/e  = 1

7. 355/113 approximates to 6 places. (355/113) - 3 = 16/113. Find an egyptian fraction whose sum is 16/113
I also printed out a fun geometric puzzle for tired students to relax with when they needed a break.
https://blogs.adelaide.edu.au/maths-learning/2016/10/19/panda-squares/

Overall, this went well but I'd improve several things if repeating:

  • This time too many kids went over to the puzzle a little too quickly. I have to think of way to keep everyone on task longer. 
  • I also wanted to do some notice/wonder activities around patterns in the puzzle but that was not possible while focusing on the main activity.  
  • I needed one or two more problems in the set to fully round things out. Several were of the type that kids could become stuck on. So a few more easier warm ups would work well. I'd have kids work out a variety of easy equivalent fractions next time i.e. find 3/4, 2/7 etc.
  • I had one student who out of character just wanted to read and not work on math today. Given the other needs of the kids I let her do that but I want to make sure next week she's engaged.

Also during the time I noticed a lot of fluency issues while the kids worked on the math. 
  • Adding fractions like 1/4 + 1/5.
  • Long division.
  • Mental math for fairly easy computations like 84 divided by 4.
In each of these cases I ended up doing mini walk-throughs  and I think the session acted as a way to review rusty skills.  But overall, I'm toying with the idea of  finding other activities that also stress these again. 


P.O.T.W





Wednesday, November 22, 2017

11/21 The intersection of History and Mathematics

My goal for this week in Math Club was to do something low key after AMC 8. Originally, I had been thinking about some tangram or panda block puzzles. I also had seen a recent Infinite Series video with a interesting triangle puzzle embedded within it that I thought looked promising.

But as often occurs, I ended up going in a different direction. Earlier last week I was thinking about what educators mean by "humanizing math".  The main claim is that mathematics is cold and sterile which I don't find totally convincing especially in the context of math circles. But some of the of the ideas bundled in with this subject are really interesting. In particular I liked this paper https://www.jstor.org/stable/27968440?seq=1#page_scan_tab_contents  about using Math History to humanize a classroom. This dovetails with my worry that kids don't really understand the trajectory of their Mathematics education in the same way as other subjects and tend to view Mathematics as a complete set of knowledge rather than a developing field that people still work within.  Even I as a student, couldn't imagine what Mathematics research really looked like.

So I started researching Mathematics History videos that might be a good fit for a session. I found several candidates.  My initial pick was "The Story of Maths"  a BBC documentary that looked promising.  But after picking out the clips when I went to prepare I found that they had all be taken down from youtube due to copyright issues.   It turns out I can borrow the DVD version of this from the library which I will remember for the future.  So instead I went with the following lecture given by Dr. John Dersch:



What I like about this talk is that it covers a lot of ground and gives a good historical framework. Conversely, it lacks flashy visuals and does assume a college level background. So I prepped by starting with a talk with the kids where I had them guess when various mathematical discoveries occurred ranging from addition, to algebra to geometry to calculus.  That set the stage for video. I also liberally stopped the video and talked about various topics. This was especially true when I thought the subject was new i.e. logarithms or derivatives.  This led to several tangents that might be fun to do a whole session on:

  • How did 17th century mathematics calculate square roots or logarithms?
  • Why can't you solve a 5th degree polynomial in a general fashion?
  • Fermat's Last Theorem.
  • Egyptian Fractions

Overall, I thought this went really well. If I repeat this topic, I do hope to find a better video resource or perhaps develop a slide deck of my own.  I also wonder if I could thread various historical discoveries in during a year i.e. a talk on Babylonian tables, or Napier's bones.

Image result for clock image

To round things out before this started I actually went to back to clock problems. This was strategic since I had noticed some of the kids were really interested in when the clock hands coincided already and had been looking up tables of the values online.  (If I repeat and this wasn't already the case I would ask kid to observe during school beforehand.)  The day really started with me drawing clocks on the whiteboard and having a few kids talk about what they already knew.  I was hoping they had noticed a regular pattern but since that wasn't the case we worked on that in club.  One focus I asked a few questions to point out was that here is a point of coincidence at every hour except 11. We then developed the basic equation to discover the actual values.   

m  = h * 5 + m / 12 

I was surprised that this seemed fairly new to everyone and the basic process of solving was not as smooth as expected especially developing the minute to hash mark ratio. I plan to return to ratios at some point.    On the bright side kids quickly found the method for find when the hand form a straight line by the times we were done.

P.O.T.W
This time I gave out a sample MOEMS test to prep for the first one of the series.  I'm probably going to do it in two weeks which means I'll have to continue to slide the other ones around in order to balance activities out. I'm actually very curious to see how the kids do on it.





Wednesday, November 15, 2017

11/14 2017 AMC 8 and a digression


Math Club was super easy for me today. I paced outside the classroom while everyone took AMC8. 



I was happy that the kids all were very focused. Hopefully we'll get good results and it was a positive experience for everyone. The problems are released in a few weeks. I'll come back to them if anything interesting appears.

So to fill the week here's the problem I looked at last night before bed. Its interesting to see the vast difference in approaches between mine and another online. Once again this is why I love geometry.



Thought Process:

(Unusually this was a fairly linear process where each observation led farther forward.)
  • I immediately noticed the right triangle and thought about the Pythagorean Theorem.
  • Then it occurred to me that D was the incenter and it would be interesting to draw in all the altitudes from it and to connect it to C.
  • That also meant CD would bisect angle C into 2 45 degree angles. 
  • At about this point I noticed the square that formed.
  • I then started to think about the line AE and how it bisected the triangle and could be used with the angle bisector theorem.
  • I thought this was almost enough and I actually used 3 variables at this point to see how much I could combine. That didn't quite work so I actually plugged a sample number in just to watch how it played out.
  • At that point I went back to the picture and angle chased to find the similar triangles. That gave me a way to only use 2 variables and I was sure I was almost there.
  • I did some algebraic simplification and at this point I wasn't sure if I needed another equation/invariant. 
  • But I lucked out since I was looking for the sum of the 2 variables, everything was in place.






Setup:  Note O is the incenter since its the intersection of the angle bisectors. So drop another one from point C.  This forms the 45-45-90 triangle CHO,   let r = CH  = HO = GO = the inradius, let x = DH .  We want to find r + x.

1. After angle tracing triangle AGO is similar to DHO. so \(\frac{AG}{GO}=\frac{HO}{DH}\)
   \(AG=\frac{HO \cdot GO}{DH}  =  \frac{r^2}{x}\)

2. From the angle bisector theorem:   \(\frac{AC}{CD}=\frac{AB}{BD}\)
 \( \frac{\frac{r^2 }{x} + r}{r +x}=\frac{\frac{r^2}{x} + x + 3}{3} \)   which simplifies to: \(3r = r^2 + x^2 + 3x\)

3. We also know from the Pythagorean theorem on triangle BHO that  \(4^2 = r^2 + (x+3)^2\) which simplifies to \(r^2 + x^2 = 7 - 6x\)

4. Substitute r^2+x^2 from the 2nd into the 1st equation: \(3r = (7 - 6x) + 3x  = 7 - 3x\)
   \(3(r+x)= 7\)  or  \(r +x = \frac{7}{3}\) and we're done.


The Trig Approach:

Another user @mathforpyp put this soln up. Notice how completely differently this works. I like to think of trig as a bulldozer for these type problems but applying it is actually a bit tricky. The key observation here which I didn't use above was the relationship between the angles  A and B.




Thursday, November 9, 2017

11/7 Decoding

I decided to do a second smaller sampler of AMC 8 problems for Math Club this week.  Unlike last week (see: http://mymathclub.blogspot.com/2017/10/1031-put-bird-on-it.html)  this time I wanted to approach them as a group and only do 5-6 max but concentrate on the hard ones and have the group demo solutions.

So I picked the last 6 problems from AMC 2014: link to partial set  and had the kids divide up and work them in groups. My goal was to only spend 15 minutes but because the work looked productive we ended using about half the time again.

Overall:


  • focus was less good today especially during the demos. I'm going to need to work on improving the classroom norms here or be more mindful to limit this to fewer problems.
  • There was one really interesting argument about the solution to the 2nd problem in one group. One girl had a general solution and her partner didn't understand how it worked. I intervened to try to get the two students to slow down and listen to each more carefully.  
  • I noticed a general hole in modular arithmetic that would make a good topic for one of the upcoming sessions.
  • One other student has a weakness for linear systems. I really like his thinking but he almost always tries to setup a system regardless of the problem. My personal goal here is to work to get him to expand his tool set.


At this point we switched to my main focus, encoding problems which ran better than the first half. I've done these in the past but this time I switched my sequence of problems up a bit which I think worked really well.

Encoding Problems:

First  I started with a general open-middle type problem: Using the digits 1-9 form  a valid addition
equation with the form below:

   _  _  _
+ _  _  _
=======
   _  _  _


This was great for a low barrier to entry and due to the many/many possible solutions. After the group had found 3 or so we move on to a class decoding problem.

Each letter stands for a distinct digit

  DOG
+GOD
======
 DEEP

Interestingly, my best solver in the first part also cracked this one first.

Finally we finished with this multiplication problem which was not solved before time ran out:


A B C D E F               A B C
x                6   and    + D E F
-----------------            ---------
  D E F A B C                9  9 9


Spare problem we didn't reach:

                         _ 5 3 
            -----------------
_  _ 9  |  6 _  8 _ _  _
             _  _  _ _

                ======
                 _ 9 _  _
                 _ _ 4  _
                    =======
                     _ _  4 _
                     _ _  _  _


Overall, I would have preferred to have only done my main activities but I think again for AMC 8 it was worth one more session of prep. I also had a few issue with a few students rough housing today that I'm working hard to nip in the bud.  I'm going to go over behavioral standards and pull one student aside before we start next time.   Looking forward, I'm super excited to see the kids take the test next week. I have a small scheduling issue with MOEMS which is on the same date. I really don't want 2 contest in a row so my plan is to to slide the MOEMS tests around fairly aggressively to free up time for more focused math circle sessions.