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:

I then used a modified version of the series of questions and activities from here:   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  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:  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.