An interest rate question extending Kate Nowak’s rate post

Earlier in the week Kate Nowak wrote a neat post about rates. The perspective in the post (in my words) is coming from writing curriculum materials for 6th grade math:

Here’s an alternate perspective on the same (or at least similar) issue that I encountered at work this week.

Suppose I ask you to play the following game:

(1) You pay me $2 today.
(2) I’ll then select an integer from 1 to 10 at random (uniformly)
(3) At the end of year 1 you pay me $1, and if my random number was 1 I’ll pay you $10 and the game stops. If my number wasn’t 1 we’ll meet again next year.
(4) In general, at the end of year n, you’ll pay me $1 and if the random number I picked was n, the game stops.

The interest rate question relating to this games is this: What is your expected rate of return for playing my little game?

Here are two different ways to think about it:

(1) Internal rate of return

You’ll see an expected set of cash flows that look something like this:

Screen Shot 2016-01-23 at 11.09.40 AM

The “internal rate of return” on those cash flows is about 12%, so you might say (and I think that many people would be quite comfortable saying) that your expected rate of return playing my game is about 12%.

(2) Accounting for the costs and the investment returns differently

One possible objection to the internal rate of return calculation is that your cash outflows are really part of your investment in the game and so are quite different than the investment return. In fact, to play the game all the way through, in addition to the $2, you need to be sure that you have access to $10 over time to play.

So, you might prefer to discount your cash outflows at a less risky rate – I’ve picked 4% just for example purposes – and discount the inflows (the investment returns) at a risky rate to measure your return. That calculation looks something like this:

Screen Shot 2016-01-23 at 11.14.43 AM

Using this method the expected investment return you’ll get for paying $2 to play my games is more like 8% per annum.

So, what is the correct way to think about the rate of return for playing my game?

I think the rate of return question here is pretty interesting to think about and gives a real life example of the things that Nowak is thinking about writing 6th grade curriculum.


Revisiting the Surreal Numbers

I bought a copy of Donald Knuth’s Surreal Numbers recently:

Reading it yesterday gave me the idea to revisit the surreal numbers with the boys today. Our previous look at the surreal numbers was inspired by Jim Propp’s off-the-charts-excellent post:

Life of Games

Although I hope today’s project stands on its own, our previous projects inspired by that post are here if you are interested:

Walking down the path to the surreal numbers Part 1

Walking down the path to the surreal numbers Part 2

Today I started by reminding the kids about the game “checker stacks” that Propp describes in his post and then playing through a few simple examples:

Next up was our first challenge – finding the value of a red + blue checker stack. The boys determine that the value is negative fairly quickly. After a bit of investigating they determine that the value is -1/2. I think this is a wonderful example of what kids doing math looks like:

Now we moved to a problem that is a bit more challenging – a blue – red – red stack. This is a more difficult investigation, but the boys spend about 10 minutes exploring and experimenting. Over the course of the next two videos they determine that the value is 1/4.

Again, I think this is a really nice example of what kids doing math looks like. I’d love to try out this investigation with a larger group of kids:

There are lots of directions to go with this investigation – tomorrow I’d like to explore the “deep blue,” “deep red,” and “deep purple” checkers discussed at the end of Propp’s post. I’m excited to see if kids can understand the ideas that come up with these special checkers.

Henry Segerman, John Edmark, and . . . biology??

We’ve done lots of 3d printing projects including a few inspired by Henry Segerman’s work:

Henry Segerman’s Flat Torus


as well as the work of John Edmark (which we saw via a Math Munch post):

If you like great math for kids, check out Math Munch


As I was walking through the lobby of the Koch Cancer Research center at MIT this morning I noticed this picture hanging on the wall as part of a public art exhibit:


with this caption (sorry for the glare, my camera skills were not able to avoid it)


Despite having nothing whatsoever to do with math or 3d printing, the picture reminded me of Segerman’s and Edmark’s work. Maybe it is just a coincidence, but I wonder if there’s any structure in that picture that math could help uncover.

My favorite – watching problem solving ideas develop

This is sort of an accidental “my favorite” but I spent 30 minutes with my older son today and found myself thinking about how much I love watching problem solving ideas develop in kids.

“Problem Solving” is notoriously hard to define – and since I’m in a happy mood, I’m not going to try to define it 🙂  It is, at least in my mind, though, a skill that you can watch develop over time.

Today my son worked through four old AMC 10 problems that had given him difficulty the first time through them.  We had not looked at or reviewed these problems since he worked through these tests, so, although he had seen the problems before, he’s not previously been able to solve them.  This afternoon with just a few nudges here and there he was able to work through all of them.  Along the way are some pretty nice examples of what a kid looks like doing math.

All of the problems can be found on Art of Problem Solving’s site here:

The 2009 AMC 10 A hosted on Art of Problem Solving’s website

and here:

The 2008 AMC 10 b hosted on Art of Problem Solving’s website

The first problem was #14 from the 2009 AMC 10 A – it is a problem about absolute values:

Screen Shot 2016-01-18 at 5.29.06 PM

There are a lot of plus and minus signs to keep track of in this problem, and he does a nice job of approaching the problem in a pretty systematic way to help keep track of all of those signs.


The second problem is #17 from the 2009 AMC 10 a – it is a problem about 3-4-5 right triangles

Screen Shot 2016-01-18 at 5.29.24 PM

There are a lot triangles similar to a 3-4-5 triangle in this problem and keeping track of the sides is made a little extra difficult by how the various triangles scale. In one of the triangles the longer leg has length 3 and in another triangle the shorter leg has length 4. That trickiness does trip him up, but luckily he does catch the mistake (because his original answer wasn’t one of the 5 choices).


The next problem was #14 from the 2008 AMC 10 b – it is a problem about rotations:

Screen Shot 2016-01-18 at 5.48.28 PM

This problem gave him a lot of difficulty. There’s a little bit of geometry to keep track of and also you have to keep track of a few plus and minus signs at the end. His solution here is a good example of working through a few initial misconceptions to arrive at the correct solution:


The last problem was #19 from the 2008 AMC 10b.


Screen Shot 2016-01-18 at 6.08.54 PM

This problem looks like a super challenging 3d geometry problem – it is really easy to have a reaction similar to – “well, I really don’t know anything about water in cylinders.”

What I loved about his work in this problem is that he figures out what he can do and then figures out how to use that information to solve the problem. At one point he says something like “wouldn’t it be great if these were 30-60-90 triangles?” Loved that!


So, the work today really did make me happy. I love having the opportunity to work on math with my kids, and I love watching their problem solving skills slowly develop over time.

Some book suggestions for a 14 year old who loves math

I saw this tweet from Aperiodical retweeted by Steven Strogatz this morning:

Thinking about book recommendations for a 14 year old interested in math was sort of fun, so I pulled a few books that we’ve used for projects (plus one or two more) off of our bookshelf.  Here are some that I think a kid interested in math would probably find interesting:

A wonderful mathematical coloring book from Alex Bellos and Edmund Harriss, we’ve done three projects from it already!

Fibonacci, Zome, and Patterns of the Universe

Patterns of the Universe Part 2

I found out about this book through one of Evelyn Lamb’s blog posts. It gives you a fascinating look at the various aspects of Egyptian mathematics – we did three projects from it that are all linked here:

Count Like an Egyptian Part 3

I assume this is the Matt Parker book that was referenced in the Aperiodical tweet. I just found it last week at the same time I found Patterns in the Universe:

It looks like a great way to introduce a kid to some fun ideas in math.

This book is one that you’d have to use more selectively with a younger kid because lots of the examples assume knowledge of Calculus. However not all do and there are some fantastic ideas that really show the power of mathematical thinking.

Zome Geometry combined with a Zometool building set opens up the world of 3d geometry to a kid in ways that are almost impossible to describe. I wish there was a cost-effective way to get the Zometool sets in the hands of every kid.

Here’s our latest Zometool project and there are probably 40 more on the blog.

Can you believe that a dodecahderon folds into a cube?

Patty Paper Geometry, like Zome Geometry, is an eye opener. I’ve never seen an approach to geometry (2d geometry in this case) like the one outlined in Serra’s book. Essentially an approach that simply involves tracing and folding figures allows idea after idea from geometry to fall right into your lap.

Really Big Numbers is a book for kids with some “really big” ideas hiding in the background. We used it for a neat project here:

A few project for kids from Richard Evan Schwartz’s “Really Big Numbers”

and Jim Propp did a nice review of the book here:

Jim Propp on “Really Big Numbers”

Keith Devlin’s book isn’t aimed at kids, but I think a kid interested in math will find it fascinating. It walks you through some of the most challenging unsolved (at least at the time of publication) problems in math today and is a great introduction to the ideas that mathematicians think are important.

Like “Street Fighting Mathematics”, not all of this book is going to be accessible to a 14 year old. Parts of it are, though, and those parts plus the incredible pictures might be incredibly inspirational to a kid who is interested in math. Here’s one project we based on Farriss’s ideas:

Frank Farriss’s Patterns

Tanton’s book is really hard to find, but if you do stumble on it you’ll find tons of clever math ideas, questions, and projects that should delight a kid interested in math. If you can’t find it – don’t worry too much, Tanton is an incredibly active writer and sharer of math. Just follow him on twitter – he’s inspired tons of our projects!

Our projects inspired by James Tanton

“Bridges to Infinity” was a gift from my high school math teacher when I was 15. It was my first introduction to math that was outside of traditional school math / math contest math. It was an amazing thing to read back in 1987 – I had no idea that the world this book describes even existed.

Pickover’s book is full of amazing ideas from 100’s of different areas of math. Each comes with a picture and a short, one-page explanation. Great fun to just flip through and if something catches your eye just hop on the internet to find out more. We’ve done many projects based on my kids asking questions about something they saw in this book. For example:

Banach Tarski, Hilbert curves, and Infinite sets


Counting Geometric Properties in 4 and 6 dimensions

These last three pics come from some fun books by Ivan Moscovich and Theoni Pappas. The books by these two authors should be on the shelf of any kids who are interested in math – they are absolutely wonderful.

Christopher Danielson’s square problem

Saw an interesting problem on twitter last night:

with this one extra clarification on the shape:

Seems like a good problem for a kid learning geometry, so I tried it out with my older son tonight. First we looked at the two twitter posts to make sure that he understood the problem and I asked him to come up with a plan for how to solve it:


Next we went to the white board to work out the solution. He did a good job following through on this plan, which was nice. One thing that I thought might give him a little difficult was that the 4th root of 3 appears in his solution. However, he did not simplify the numbers in the problem the way I expected him to, so the 4th root of 3 didn’t appear 🙂


Definitely a fun little problem. One alternate problem that I skipped because I thought it would be too difficult was finding the area of a different square hiding in Danielson’s picture. The “center” of the 4 pointed star is also a square – what area does it have if the kites have area 1?

Fibonacci, Zome, and Patterns of the Universe

The kids wanted to do an other project using Patterns of the Universe today:



Flipping through the book last night I found a neat page featuring golden rectangles, so I turned that into a little half coloring / half zometool project for this morning.

We started by talking about the picture in the book. The kids were interested in the spiral pattern and my older son remembered a few of the properties of “golden” rectangles:


Next we built a golden spiral on the living room floor using our zometool set. We actually did a similar project about a year ago, so the ideas weren’t totally new:

Fibonacci Spirals and Pentagons with our Zometool Set

I liked hearing their description of how the Fibonacci sequence appear in the shape – it was a nice way to get talking and thinking about math ideas:


While I cleaned up the Zometool pieces the kids colored in Patterns of the Universe. Here are their descriptions of their colorings. Here’s the one from my younger son:


and here’s my older son:


So, a fun little project for kids connecting the Fibonacci numbers to some geometric patterns. It is neat to hear how the kids think about these connections. I really love how the coloring ideas in Patterns of the Universe lead so naturally to fun projects!