February 2016
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Month February 2016

A neat idea about primes I saw from Dave Richeson

Saw a neat tweet from Dave Richeson earlier this week:

His post is about a “new to me” proof that there are an infinite number of primes. Same warning as in Dave’s blog post – don’t click the link unless you want to know the punchline!

“Prime Number Generator” on Futility Closet’s website

Here’s the problem without giving away the punchline:

Take the set consisting of the first n prime numbers – { 2, 3, 5, \ldots, p_n }. Divide the set into two (non-intersecting) sets, and let A and B be the product of the numbers in each of those two sets. What can you say about A + B, and A – B?

I thought it would be fun to talk through the idea with the boys today – it really did seem like a fun way to talk through some introductory ideas about mathematical proofs.

By coincidence, about a year ago we also talked about a “new to me” proof that there are an infinite number of primes. That project was also inspired by a twitter post (!) and is here:

“A “new to me” proof that there are infinitely many primes

I began today’s project by first asking the kids to talk a little about prime numbers and then I introduced the idea from Futility Closet’s post.

The first thing they noticed was that exactly one of A and B would be even, and both A + B and A – B would be odd.

 

Next we looked at the example where the original set of primes was {2, 3, 5, 7}. Playing around with this set let to a few more conjectures – some of which we were able to show were not true, but (despite a little clumsiness by me) one of the ideas about finding new primes looked interesting.

 

For the next part of the project we looked more carefully at why we seemed to be getting new primes when we looked at the two numbers A + B and A – B. There are a couple of neat to talk through here. The one we focus on here is about divisibility or modular arithmetic. This idea helps us understand why we are seeing new primes:

 

A second idea that we need to understand is more subtle. This idea is about the difference between “an integer is prime” and “an integer is not divisible by any of the first n primes”. The kids began to understand this idea and then I got a big surprise – working together the boys came to see that the ideas here would lead us to conclude that there were infinitely many primes!

So, when that idea appeared on the table I ran with it 🙂

 

So, a great introductory proof project, and a fun problem to talk through with kids in general. There are lots of math ideas that are so close to the ideas here, and so many different directions that the talk could go that I think you’d find new ideas with every group of kids working on this problem.

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Zometool and minimal surfaces

Two weeks ago my older son was in a program over at Merrimack college and I happened to chat with Dana Rowland, who is the co-chair of the math department. She showed me a neat minimal surface project you can do with soap bubbles.

A few days later Paula Beardell Krieg sent me this link pointing to a minimal surface project with Zometool pieces:

Minimal surfaces were in the air! Today we tried out a little project:

I started by having each of the boys build two shapes from the Zometool set. Here’s their description of the shapes that they built – it was funny to me that the each built a prism but described it in totally different ways:

Next we dipped the shapes one at a time. The octahedron was first – “it looked like something from the 4th dimension”:

The second shape was the prism that my younger son built – “and there’s an archway . . . interesting”:

Now the smushed and stretched cube – “whoa – it is like a smushed hypercube . . . . except that it is a hyper square.”:

Now came the last of the original shapes – the tetrahedron – “. . . who knew that bubbles could find the center of a tetrahedron?” and then an amazing suprise . . . :

Finally, the boys wanted to try a cube. The boys expected to see a shape similar to a hyper cube, but instead we got a shape that was similar to what we saw with the “smushed” cube. Eventually, though, we did get the shape they were expecting to see. That led to the conjecture that for platonic solids if “you catch a bubble” you’ll get the original shape on the inside.

So, a fun (and pretty easy) project. They boys played with it for about 15 minutes after we turned off the camera. Definitely a neat little project for kids!

A really nice thing that happened this week

Earlier in the week my phone “pinged” because of this tweet:

Clicking through I learned that Joel David Hampkins hard turned the Fold and Punch and Fold and Cut projects into an amazing activity at his daughter’s school:

Math for nine year olds: fold, punch and cut for symmetry!

Hardly any of the projects we do involve any planning – in fact, they’d probably require two extra levels of planning to get to “fly by the seat of our pants” status. So, it was cool to see a long and incredibly well though out write up of this project, and especially cool to see where a professional mathematician took the project 🙂

I hope that people are able to take advantage of the wonderful write up by Hamkins – this is a tremendously fun project for kids!

A nice series problem for kids from Five Triangles

Back in 2013 we did a neat problem on Numberphile’s “Pebbling the Chessboard” video:

That video also reminded me of a neat “proof without words” that Patrick Honner had written about:

Our project is here:

Numberphile’s Pebbling the Chessboard game and Mr. Honner’s Square

and Patrick Honner’s blog post is here:

Proof Without Words: Two Dimensional Geometric Series

Tonight I saw a neat tweet from Five Triangles that reminded me of the prior project:

I thought it would be a fun one to try out with my older son, though I didn’t quite know how to introduce the problem. I started with a slightly easier series as a trial: 1/2 + 2/4 + 3/8 + 4 / 16 + . . .

Since things seemed to go pretty well with the first problem I decided to go ahead and try out the series posted by Five Triangles:

So, a neat problem for kids building off of a the “simple” infinite series 1 + 1/2 + 1/4 + . . . . As our project from 2013 shows, the more complicated versions can have interesting geometric interpretations, but I’ll leave those for another time. Tonight it was just fun to see some neat arithmetic with infinite series.

Matt Enlow’s Fibonacci problem

Saw this tweet from Matt Enlow yesterday:

I liked this problem both as an illustration of mathematical thinking and as a problem you can share with kids.

For mathematical thinking, the point is made really well in this interview with Julie Rehmeyer. The ~5 min part beginning around 31:30 about proving that 0 + 0 = 0 is what I’m thinking of specifically:

Julie Rehmeyer’s “Inspired by Math” interview

I had a similar feeling on seeing Matt’s problem – it wasn’t obvious to me why the Fibonacci numbers should have this property, but I had some ideas about how you’d prove that they did:

  

For sharing with kids, I like this problem because it is (i) accessible, but (ii) probably not as obvious how to solve it. I shared it with my son this morning and although we didn’t solve the problem it was very interesting to hear the ideas he had about how you might go about solving it:

 

I’m excited to finish up this problem with my son later this week and also excited to try out this problem with a larger group of kids sometime.

Honeycombs

I was reading a bit more on tilings of 3 space and found these two neat pages on Wikipedia that were related to our tilings with truncated octahedrons:

The “Bitruncated Cubic Honeycomb” article on Wikipedia

The “Cubic Honeycomb” article on Wikipedia

I was a little more interested in the topic than usual since my younger son had made a new “smushed” truncated octahedron after we finished yesterday’s project:

yesterday’s project is here:

Tiling 3 dimensional space with our Zometool set

For today’s project we built some more of the shapes that my younger son built yesterday and explored if these shapes could also tile space. Here is their description of the shape and how it is similar to and different from yesterday’s “smushed” truncated octahedron:

Next we looked at how the “smushed” truncated octahedron and the new shape tiled space:

Finally, I showed the boys the two Wikipedia pages from above so that they could see all of the incredible honeycomb patterns:

So, a fun project inspired by my son just playing around. 3D tiling / honeycomb patterns are really neat!

Tiling 3 dimensional Space with our Zometool set

A while back we did a fun project from Zome Geometry about tiling the plane with different shapes you can make out of Zome struts – these pentagons, for example:

IMG_0555

Zome Tilings

 

Yesterday we did another project out of Zome Geometry looking at Archimedean Solids:

A Quick Zometool Project

Today we sort of combined the two projects and looked to see if the truncated octahedron (and our smushed truncated octahedron ) could tile 3 space. We have looked at 3d tilings previously, too, so the 3d tiling idea isn’t totally new to the boys:

Revisiting the Rhombic Dodecahedron

The main difference between today’s project and the one using the rhombic dodecahedron is that building the truncated octahedron is a tiny bit more difficult because of the connections with green struts are slightly more complicated.

After we finished the builds this morning we talked about the shapes. First up was the truncated octahedron:

Our incorrectly built truncated octahedron also could tile space – that was sort of a miracle! It turned out to have a second use, too, as it was easier to see some of the symmetry in the shape because it had struts with different colors:

After seeing some of the symmetry in the “smushed” truncated octahedron, we went back to the first collection of real ones to see those same symmetries:

Definitely a fun (and challenging) project today. I love how the Zometool set opens up the world of 3D geometry for kids.

A quick Zome project

The boys were running out the door early today to go skiing and we were looking for a short project this morning. I told them to go grab “Zome Geometry” and find something that they’d like to build.

 

img_0726

Turned out to be a nice little activity for them since they are getting to the point where they can complete most of the project on their own.

They ended up choosing the section on Archimedean Solids and built two – here are their thoughts:

(1) The Truncated Octahedron:

(2) The Icosidodecahedron

So, a fun project, and, as usual with Zome constructions, it is neat to hear kids talk about 3D geometry. It was also neat (though an accident) to see the kids build the truncated octahedron incorrectly the first time and then figure out the right way to build it.

I wish I could have seen more 3D geometry when I was a kid – it is so neat 🙂

A writing assignment courtesy of Clifford Pickover

I’m on the road for work today and wasn’t able to do a face to face project with the boys before I left.  Instead, their assignment was to pick a page out of the one of two books we have from Clifford Pickover and write a short pieces about the page. The two books can be found here:

The Math Book

The Physics Book

My 6th grade son picked a page from the Math Book about ants:

Ants

and wrote this summary:

Ant Writeup.jpg

My 4th grader picked a page from the Physics Book about the greenhouse effect:

Greenhouse

 

and wrote this summary:

Greenhouse Writeup

We actually used to do something similar every day back when we were home schooling. I need to do more project like this 🙂

Pension Math

Last year I saw the graph below detailing the funding level for the pension plan for the teachers in Illinois:

Pension

What surprised me was that the funding level hadn’t improved that much since 2009 even though the stock market had more than doubled during that time.  Earlier this week I saw a few articles about pensions in Illinois and wondered if the 2015 report had come out yet. It had:

The Financial report as of June 30, 2015 for the Teachers’ Retirement System in Illinois

The report has lots of information about the pension plan. This summary chart below caught my eye.  The total contributions in the last 10 years growing from roughly $600 million to $3.5 billion is surprising.   I don’t see how growth like that can continue.

Screen Shot 2016-02-16 at 11.09.56 AM

Even with that growth in how the pensions are funded, though, the shortfall in the pension has grown to just over $65 billion.  That means the pension plan holds about $46 billion of assets and calculates their total liability to be $112 billion as of June 30th, 2015.

Screen Shot 2016-02-16 at 11.01.11 AM

However, the information in the presentation that jumped off the page when I read it was in the chart below.   The total pension liability of approximately $112 billion (and shortfall of $65.5 billion) was computed using a discount rate of 7.47%.

For simplicity, that means all  of the estimated future liabilities have been discounted to June 30th, 2015 using a rate of 7.47%.   So, for example, a payment of $100 due in 10 years would have a value of $100 / (1 + 7.47%)^10 = $48.66.

If, instead, the liabilities are discounted at a rate of 6.47% the shortfall grows from $65.5 billion to $81 billion.  Just glancing at a chart of the 10 year treasury interest rate, at June 30, 2015, it looks like that rate was around 2.3% and the rate on the 30 year treasury bond was about 3%.

As the sensitivity chart below shows, the higher the discount rate, the smaller the liability.  There are ways to use the information provided in the report to estimate the value of the liability for lower discount rates, but I don’t want to go into too much detail and guessing here.  Suffice it to say that if a 1% change from 7.47% to 6.47% increases the liability by $15 billion, changing the discount rate closer to the 3% 30 year treasury rate at June 30, 2015 will increase the liability by more than $15 billion.

 

Screen Shot 2016-02-16 at 11.16.33 AM

So, I’m glad that Illinois publishes such a comprehensive report.  For a few other State pension plans I looked at the reports were not nearly as easy to read.  I’m pretty concerned about the health of these plans and by what could happen to them (and to the people who rely on them for retirement) if their health doesn’t improve.