Tuesday, December 19, 2017

2017 AMC 8 Questions

We finally received our scores. Overall I always have to remind myself that "comparison is the death of joy".  But really I think the kids did very well and when the final stats come out most will be at or above average. I also hope that I get a chance to see some of the same students take it next year and that I'll have evidence how much everyone has grown.

Art of Problem Solving has posted the 2017 AMC 8 problems and solutions at: https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_8_Problems

To get a feel for this year's set, I tried them out and timed myself. It took me about 50 minutes to complete them carefully.  Since I don't find it as interesting, I didn't use guess and check to speed things up or look at the multiple choice answers unless the question required it. The kids however only had 40 minutes which made it fairly difficult in my mind.  You had to work really quickly and do some intelligent strategic guess work from time to time to finish everything.  I definitely will mention my own timing when we talk about it.  From what I can gather the test was a bit harder this year than 2016 with overall cut scores about 2 points lower for the top 1 and 5 percent overall.

Favorite Parts:

22) In the right triangle $ABC$$AC=12$$BC=5$, and angle $C$ is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?

[asy] draw((0,0)--(12,0)--(12,5)--(0,0)); draw(arc((8.67,0),(12,0),(5.33,0))); label("$A$", (0,0), W); label("$C$", (12,0), E); label("$B$", (12,5), NE); label("$12$", (6, 0), S); label("$5$", (12, 2.5), E);[/asy]
[My personal preference is always for the geometry ones plus this had a 5-12-13 in it.]   The 3 different ways to solve listed on the site are part of why I like this. Note: its always simpler to look for similar triangles rather than jumping straight to the Pythagorean theorem.

24) Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchildren?

A stealthy number theory problem.  Its actually fun to make out a chart for the first 60 days after which it repeats anyways.   And as often is the case its easier to find the inverse than the positive case i.e. 2/3 * 3/4 * 4/5 = 1 - (1/3 + 1/4 + 1/5 - (1/12 + 1/15 + 1/20) + 1/60)

11) A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 37, how many tiles cover the floor?

I just  enjoyed visualizing this one.

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 at 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. 


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. Of course, 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 of kids are already worn out. Moreover,  a teacher in a classroom has a whole set 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.

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.

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. 


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.

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.


  • 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


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.

Tuesday, October 31, 2017

10/31 Put a bird on it.

My surprisingly well read retweet from today.

I joke (or recycle Ed's joke) above but that's essentially what I did today in Math Club.  It's about 2 weeks until AMC8  and I really felt the need to have the kids do one sample test before hand so they are familiar with the format. At the same time, I also know from experience handing kids a multi-page set of problems doesn't work that well and the group tends to get unfocused.  So I went through several options beforehand:

  • Select 3-5 representative questions. 
  • Do a few problems over several weeks.
  • Create some kind of competitive relay.
  • Slog through a larger selection and just work really hard at keeping everyone focused.
There are drawbacks to all of these ideas. What I finally went with was a 30 minute practice. I printed out the entire 2011 test from https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_8 
but stripped away the multiple choice answers.  I also slapped a Halloween pumpkin on the sheet. 

I then had all the kids put their names on the whiteboard and had them work through every third problem individually.  After each problem was done I told everyone to put the problem number on the board under their name and find someone else who had finished the same problem to compare answers with.

This structure worked out surprisingly well. A lot of the kids got into running up to the board to write down what they finished but it didn't feel super competitive.  At the same time, this farmed out the answer checking so I didn't need to act like a living answer key while there was still feedback for everyone. I was also able to keep the process going with a bit of nudging over multiple problems and circulate to help out with questions. Further by skip counting questions the kids were able to try  the range of problems from the easier ones in the beginning to the more difficult ones at the end.  I didn't really need it but by not condensing the questions there were enough to not worry about running out.

To keep things interesting, I interrupted several times during this process to tell some corny Halloween Jokes:

"Why do mathematicians mix up Halloween and Christmas. Because Oct. 31 = Dec. 25" and to take a field trip to my brand new Math Club Poster Board:

After the 30 minute mark I did a cool down / party. First I handed out candy to celebrate Halloween. This was strategically delayed until after the serious practice was done and then I found a lovely set of Halloween themed logic puzzles from here http://geekfamilies.co/halloween-math-and-logic-puzzles-for-kids/   Today confirms these are still irresistible even to older students.

A final holiday themed geometry problem, I found from @five_triangles:

Tuesday, October 24, 2017

10/25 Strange Bases

This was a funny week. All the eighth graders were out on the class trip so Math Club skewed younger. That played a part in my planning. I aimed a bit less complex this time and hoped to draw all the sixth graders in.

My main inspiration was an article by @RobJLow on  balanced ternary number systems. This got me thinking about the Global Math Project and the exploding dots work James Tanton has been doing. I've seen a lot of folks participating and it looked like it might be a fit.

Then last Friday, our official MathCounts team packet arrived with the lovely poster from my last post.    My own son found the poster intriguing and I was fairly sure everyone else would too. So I started with a group whiteboard attempt at the problem.

With closer to 9 kids,  I could have everyone work in to two groups up at the board. I brought in some magnets and tacked the poster in the middle.  We worked on the problem for about 10 minutes and  I circulated making sure to keep stragglers at the board working.   What was nice was the first group to solve was fairly collaborative so I could have the kids take turns explaining their thinking to the other half of the group. (This was a theme for today: making lots of opportunities for kids to demo to each other.)

This week I remembered to tackle the P.O.T.W early on. After 3 or 4 kids presented it was clear we didn't have a solution yet to the problem:

So I made the executive decision to spend some more time engaging with it on the whiteboard. Once again, I had everyone come up. This time I seeded the groups with a few ideas after a few minutes like: try adding the origin of the circle and radii from there to all the points on the circumference.

One of the topics I worked on a bit with several students was applying the Pythagorean Theorem to find hypotenuse side lengths. At the end, I lucked out and had two different solutions from two students that again I had each explain to the larger group.

As a consequence, we were now half way into the time and just about to start working on my main focus: alternate bases. It  was  clear, we'd only have time for one of the two examples. So I  told the group "We can work on base 3/2 or a balanced ternary base system, which would you like to explore?"  The room voted for base 1.5, probably because it sounded less formal.

In looking through the source material: I found Tanton's videos not quite to my taste and I wasn't sure if I'd have a projector. So I decided to focus on topic 9: source link but work the material directly. I started by asking who had played with alternate base system before. Most of the room had some experience. I also asked if anyone had seen exploding dots (nope).  So I spent a few minutes explaining how the model worked on base 2 and then jumped right into base 3/2.

First we started by counting up and recording the numbers. Here I enlisted one of the students as a scribe which freed me up to talk more. We then worked on what was the meaning of the place values, why were some values missing, and was there any pattern to the numbers. Just like the source material, I had the kids verify that the number worked and converted correctly back into the familiar decimal system. This all went fairly well. The back half of the exploration, we built multiplication and addition tables and figured out how to manipulate numbers in this system  With a minute to go we started to look for divisibility rules: like the one for multiples of 3.

All told this material worked really well. Building the tables up was particularly satisfying. If I had another half an hour, I would have loved to have broken into the second system but I think not rushing and digesting the topics we already had was necessary today.

I took a page out of the MathCounts manual so the kids would have an initial exposure to some of these style problems

Monday, October 23, 2017

This year's Math Count Poster

The poster arrived in the mail with the rest of the contest materials. I think its charming and hopefully I can get permission to post somewhere in the school.

Friday, October 20, 2017

Heron's Formula

The MathCounts guide for the year arrived today and I was looking over the problems. The following one caught my eye.

Given a triangle of side lengths 13,14, and 15. What is the radius of the largest circle inside it?

This is a slightly convoluted way of saying what is the radius of  the incircle. I then thought about it for a moment and came up with the following approach.

  • Find the area with Heron's Formula  = \(\sqrt{s(s-a)(s-b)(s-c)}\)  $$\sqrt{21\cdot6\cdot7\cdot8} = 84$$
  • The area is also \(\frac{1}{2} r \cdot 2s\)  So in this case that means r = 4.

My second thought was I don't think any of the kids in Math Club know Heron's formula. How would they approach this?  And then I realized they could drop an altitude and compute it via the Pythagorean theorem.  That would get them to being able to find the area from the standard formula \(\frac{1}{2}\)base x height.

Note: if you were lucky you might pick the base to be 14 and then find its made up of 2 Pythagorean triples the 5-12-13 and 9-12-15 but I'm going to ignore that possibility in favor of some more investigation.

From here   \(14^2 - x^2 = 15^2  - (13-x)^2\)   All the x^2 terms cancel out and you're left with a simple linear equation that reduces to \(x = \frac{70}{13}\) .

From there you reapply the Pythagorean theorem to find:
$$h=\sqrt{14^2 -\frac{70^2}{13^2}} = 7 \sqrt{\frac{4\cdot 13^2 - 100}{13^2}}$$
$$  =  7 \frac{\sqrt{576}}{13} = \frac{168}{13} $$

Now you can compute the area =  \(\frac{1}{2} \cdot 13 \cdot  \frac{168}{13} = 84 \) again.

And then I had an epiphany (which I'm sure has occurred in many textbooks.)  This would be a great intro to actually derive Heron's formula.  After finishing the concrete problem redo it with side lengths of a, b, and c.    

Repeating our steps. 

  \(b^2 - x^2 = c^2  - (a-x)^2\)     The x^2 terms cancel out again leaving:

$$ x = \frac{a^2 + b^2 - c^2}{2a} $$

You again apply the Pythagorean theorem to find the altitude:

$$ h = \frac{\sqrt{4a^2b^2 - (a^2 + b^2 - c^2)^2}}{2a}$$

This looks complex at first but if stare at it long enough there are a lot of differences of square here that we can take advantage of.  This would be a good time to review 

$$a^2 - b^2 = (a-b)(a+b)$$

So first  \(4a^2b^2 - (a^2 + b^2 - c^2)^2 = (2ab + (a^2 + b^2  - c^2))(2ab  - (a^2 + b^2 -c^2))\)
You then can combine the a and b terms to get:

$$ ((a+b)^2  - c^2)(c^2 - (a-b)^2)$$

That's nifty because we can reapply the difference of squares formula again:

$$((a+b)+c)((a+b)-c)(c + (a -b))(c - (a - b))$$

Putting this back into the height formula and with a little rearranging we get

$$ Area  = \frac{1}{2} a \cdot \frac{ \sqrt{(a+b+c)(a+b -c)(a+c-b)(b+c-a)}}{2a} $$

We're now ready to substitute the semiperimeter s  = \(\frac{(a+b+c)}{2}\) in which results in:

$$ Area  = \frac{1}{2} a \cdot \frac{ \sqrt{(2s)(2s -2c)(2s-2b)(2s - 2a)}}{2a} $$
$$           =  \frac{1}{2} a \cdot \frac{ \sqrt{ 16 \cdot  s(s -c)(s-b)(s - a)}}{2a} $$
$$           =  \sqrt{s(s -c)(s-b)(s - a)} $$

I'm guessing this would take almost the full hour especially if I let the kids experiment on their own first.

As a followup: something nifty happens when you investigate the Pythagorean triple triangles. The numbers in the formula are always the side lengths of the 2 triangles and the square/incircle radius that make them up.

This is not a coincidence and its easy to prove. Another good extension for a right triangle's in circle radius:

$$r = \frac{ab}{a+b+c} = s - c $$  

Wednesday, October 18, 2017

10/17 Pythagorean Triples

http://mymathclub.blogspot.com/2016/10/12-triangles-and-their-link-to.html The inspiration for this week was a puzzle from the recent Pythagorize Seattle event thrown by MoMath that my friend Dan recommended.

Link: https://drive.google.com/open?id=0B6oYedIeLTUKOFpYR1VZek80Y1RYeEJqLUN4ZmtVWHVqRTVN

Each of the triangles above is a Pythagorean triple with one edge given. To solve the puzzle, you need to find the length of the side marked with a question mark.  Then translate each number into its position in the alphabet. For example if you find a side of length 5 that becomes the letter "E". Altogether this forms a 8 letter word scramble.

With 12 kids, I printed and cut out 3 sets of these triangles so we could have 3 groups working at the same time. To start off I quickly reviewed the basic Pythagorean Theorem on the whiteboard. Because I knew we would talk about it more in the middle I deferred any review of why it works.

Then I let each group work for about 10 minutes. What I found was that a lot of kids took out calculators and started plugging numbers. Walking around, if kids looked stuck I suggested estimating the missing sides edges and just trying out numbers near the estimate.

At this point I paused everyone to watch the following video:

Some of the parts here are a bit advanced so I stopped in the middle a few times to go over ideas like the Complex plane. And at the end I emphasized the generator functions:

  • 2uw
  • u^2 + w^2
  • u^2 - w^2
I then sent everyone back to finish working on the puzzle. From here the groups fairly quickly finished although not at the same time. I gave out a followup Pythagorean Triple geometry problem for the last few minutes:

See the first proof here:

No one had fully solved this yet before we left for the day so I'm tempted to come back here. I'd also like to have the kids find some of the patterns in the triples i..e one of them is always a multiple of 3, 4, and 5. This might make a good bridge with a day on modular arithmetic as well.

Continuing with the Pythagorean theme:

When handing this out I told everyone to be on the lookout for another hidden triple. I think a few kids found it already before they left.

I'm getting to know the kids a little better. This week I was hoping the initial puzzle would have enough entries for everyone. That was mostly the case but I ended up working with one student a bit to get him started.  Now that I've heard what class everyone is in, I realize I have a larger gap than was indicated in the entry forms from Math7 to Algebra II.  If I'm going to throw algebra into the mix I'm going to need either to run different activities at the same time or plan how everyone will be able to approach and work on the problem at the same time.

Tuesday, October 10, 2017

10/10 Middle School

In a careless move I deleted my original post on this one. Here's a skeleton version until I have time to rewrite it.

  • Bylaws and ASB officer election
  • Demographics
  • 4 color map problem game.
Divide kids into 2 teams. I had group of 2-3 per team. Give each one a different colored marker. Then
take turns drawing dots on the whiteboard and then connecting the dots to form a "map". It works best if there aren't any overly small shapes.  After that the real game starts. Take turns now coloring in the map segments. The only rule is you can't touch a segment of your own color. The first team that can force the other side to not have a move wins.
  • 3 squares: pick 3 numbers such that any 2 add to a square.
  • Leveling/Behavior
  • Outlook so far

Wednesday, September 20, 2017

Teasers for this year

Things are moving along. I have a provisional process to get going in 3 weeks and I've sent out the initial signup forms.

Some of the challenges so far:

  • Pickup policy.  Without any backup I'm having parents give permission for everyone to self-release.
  • Fees.  Its harder to collect money under the new structure. I still have to check on the reimbursement process as well.
  • Gender Balance - this is looking promising so far this year. I'll see when the actual signups start coming in. 
  • Age balance - Initially I'm seeing most of the interest come from 6th and 8th graders.  If that turns out to be the final split its going a bit more difficult planning-wise. I'm already starting to think about how to directly deal with getting kids to work with / respect younger/older peers.

Here are my teaser problems for this year.

Inline image 1

There are 100 people in line to board a plane with 100 seats. The first person has lost their boarding pass, so they take a random seat. Everyone that follows takes their assigned seat if it's available, but otherwise takes a random unoccupied seat. What is the probability the last passenger ends up in their assigned seat?

Inline image 3

Friday, September 1, 2017

Status Updates


School is about to start next week and I am still trying to get all the logistics in place for this year. Hopefully by next week I'll have made contact with the ASB coordinator and have an idea when things will start, how to advertise etc.  I'm definitely getting excited/antsy and am ready to interact with kids again.

[Update] - I put out a FB post on the school page and have already generated 11 inquiries of interest including several girls. So far so good.

To Do:

  • payment process
  • pickup procedure
  • Remember to offer to email students directly now.


My new revised repository of interesting internet problems is here: http://mymathclub.blogspot.com/p/collected-problems-2.html.  The new version has my custom note button, some rough levelling  and proper attributions.

Of Note:

Found on FB, original source unknown. I like this one as a starter. [With the numbers switched up a bit]

Tuesday, August 15, 2017

Elegant Solutions vs. Hacking

This is a study in contrasts around a fun problem by @eylem:

An elegant solution to this would be as follows:

  • After angle chasing to find angle DEA is 45 degrees extending DE to make a full right isosceles triangle with side lengths of 15 sqrt(2)/2. 
  • Use the Pythagorean Theorem to find the missing side length of AGE. This gives you the base of the triangle DE also,
  • Then note AGE is congruent to the halves of CDEs so you also know the altitude.
  • Apply the triangle area formula.

But there are other shadier options for solving the problem:

Instead, we can take advantage of patterns in the values of the lengths and the then use knowledge about 3-4-5 triangles (see: http://mymathclub.blogspot.com/2016/10/12-triangles-and-their-link-to.html) to crack the problem.

This actually exposes a bit of structure that was not seen in the first solution.  If you didn't think to try the educated initial guess, you could more directly find it by setting up a simple quadratic equation based on the Pythagorean Theorem:

$$ (5\sqrt{5})^2 + (15 - x)^2 = x^2 $$

Tuesday, August 8, 2017

How I use Twitter

This post started with some musing about the meta conversations occurring online right now in twitter over hashtags.  There was enough activity that I was distracted for a bit and then I came to the realization: This is not why I'm on social media.

I started logging on to Twitter in January of 2015 after seeing some math posts on google+ that seemed to indicate there was a lot of interesting activity going on.  After a few weeks I was convinced:  twitter was a great place to find other people actively discussing mathematics and working with kids.

I started by monitoring #mathchat which has a mixture of spam/ads and real people.  From there I found #mtbos which is much more focused on teaching, has less spam and a more communal feel. Over time, I've been slowly growing my follow list of people who post interesting ideas.  Building this network has tended to focus the tweets and made the experience more useful.  Although compared to average I have both less followers and follow less people which probably reflects my own style for engaging with social media.

What I'm finding over time is there are several different types of posters that I enjoy the most.

1. Puzzle and problem producers. These are folks like @gogeometry, @eylem, @cuttheknot, @five_triangles, @sansu_seijin, @solvemymaths.   They regularly post interesting problems that I like to try out and reuse.

2. Animated Gif Makers. There are a ton of really interesting animated math gifs being produced by folks like @gohio, @dynamic_math, @beesandbombs.

3. Individual Bloggers. These are folks who's blog or channel I regularly read anyway like @mikeandallie, @hpicciotti, @fawnpnguyen, @mrhonnor, @math8_teacher, @standupmaths
I've found a lot of sources for ideas for ideas about activities or pedagogy this way.

4. Conversationalists: These folks are usually less for blogs and more for the conversations they produce in twitter itself. @mpershan, @trianglemancsd

The flip side is that twitter is not all goodness. Its very hard to express complex or nuanced ideas in the limits of a tweet. And then there are viewpoints out there with which I don't agree. I'm definitely susceptible to the "But someone's wrong on the Internet!" phenomena, especially if a person posts a lot.  So I try very hard to filter these out rather than responding most of the time. Arguments in 140 character tweets are not terribly exciting. In fact, there are even a few folks I stopped following even though I agreed with them because most of their time was spent in debates that were not useful for me. In the end, my mantra is try to produce the content that I would also like to consume.

Monday, July 31, 2017

Angle bisector

I've officially reached the point of the Summer where I'm missing interacting with kids besides my own.  In the meantime, this is another geometry walk-through of a problem from Cut The Knot this time. It's a  good example of trying different ideas to utilize angle bisectors.  Plus, if you haven't checked out this site, its well worth the time.

Easy Observations

To start off as always I took a look at the drawing and went with my instincts: that really looks like AZ is bisecting BAD and is 30 degrees.  I was so sure, I didn't even try modelling to confirm that impression before moving on.

The second observation I made was that BI was also the angle bisector of angle B since it intersected the other 2 bisectors at point I (the incenter).

Next I went through the obvious angle chasing. Initially I assigned a to the 2 bisectors of angle B and b to the 2 bisectors of angle C. So 2a + 2b = 60.  After walking through this for a while, I realized that it was a lot more convenient to let b = 30 - a and only use one variable.  The one interesting thing that popped out was EIB was 30 degrees. What I really wanted next was angle CDE or BZD and those were not deducible yet.

Initial Targets

From there I noticed if DAZ was 30 that AEZI would be a cyclic quadrilateral. So I needed to somehow show that was the case. Nothing immediately came to though for finding more angles or side lengths to do that.

I like to walk backwards from the angles sometimes to find more patterns. This showed CED was also 30 degrees and triangle EIZ was isosceles. If I could get to that point I would also be closer. But yet again I didn't see a connection to make.

Some Experiments

I picked this up again a day later, and tried a couple of experiments with the angle bisector. First, I thought it might be easier to manipulate if I extended the triangle to make it isosceles AD would extend to be a perpendicular bisector.

This produced some more semi-interesting angle in the new portion but no similar triangles or such and didn't seem helpful.

Then I decided to instead increase symmetry by mirroring on the right with an explicit angle bisector:

My hope was that I could prove the right and left similar and thus show the left was also a bisector.  It was easy starting with the bisector to show the right hand side was  a cyclic quad and it was noticeable that the combined isosceles triangle would form a larger cyclic quad but again I didn't quite see how to connect all of this. It also needed to be shown that the bisector on the right really intersected the other lines at the same point.

I then thought about creating multiple equilateral triangles through extension. This seemed interesting since there would then be a giant parallelogram. Again, it was interesting but there was not much I could do with this. I added the diagonals in which intersect at right angles but even that didn't give me new ways to solve the problem.


I stopped experimenting but then while trying to go to sleep I thought of a new angle.  Point Z was also the incenter of the smaller triangle ABD if I could prove that DE was an angle bisector I would have a way forward. That was attractive because I could use the angle bisector theorem. I just needed to show  BD and AD were in the same ratio as AC and BC.

Since those two segments weren't adjacent I trying picturing ways to move them closer. By overlaying them or adding reflections. Eventually I settled on reflecting AD over to intersect AC and immediately I got excited. When that happened I formed a new equilateral triangle and a parallel line (DF) and similar triangles (ABC and CDF).  From there I could pretty much see the way forward:

Final workup as transcribed by @CutTheKnotMath (Which is a lot prettier than my sketch)
Draw DFAB. By the Exterior Angle TheoremAFD=FDC+FCD=ABC+FCD=60, making ΔADF equilateral. In particular, AF=AD.
A Problem from the 1985 Balkan Mathematical Olympiad (Shortlist), solution 2
By the Internal Angle Bisector theorem, AEBE=ACBC.
By the Thales' theoremBDBC=AFAC=ADAC, i.e., ADBD=ACBC, implying ADBD=AEBE.
By the converse of the Internal Angle Theorem, DE is an angle bisector in ΔABD and so is AZ. Thus DAZ=30.

So this took in all 3 separate 20 minute sessions over 2-3 days with several dead ends as is often the case for me with trickier problems.  What also seems to be the true is that experiments tend to deepen my understanding of the construction. So while they aren't always fruitful they lead the way towards the eventual productive ideas.

Trigonometric Technique

What's also fascinating is looking on the site at the other solution. First it would not have occurred to me to use Stewart's  theorem.  So an interesting technique to remember for future use. But also new to me was the variant here:

$$AD = \frac{2bc \cos(\frac{A}{2})}{b +c}$$

I found a good explanation here: https://annoyingpi.wordpress.com/2010/04/14/angle-bisectors/. Basically if you use the law of cosines to find the area of the subtriangles ABD and ACD as well as ABC and simplify this form pops out. If you think to go this way, it avoids the need for auxiliary lines (as often happens when trig is introduced).

I'm particularly proud of this effort because my solution was added to the site. Now I just need to start working so that I can tackle some of the hairier inequalities problems discussed there.