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.

A day in the life: building and extending number sense

I’m not entirely sure why but I’ve been spending a lot of time recently thinking about different ways to build up number sense.  About a week ago I started a chapter on similar triangles with my older son, and the problems in that chapter have helped me gain a better understanding of the importance of building “algebra sense” (for lack of a better phrase) too.    I’m surprised how many opportunities there are to focus on both of these topics now that I’m actively paying attention to them.   An odd coincidence today made me want to write up the conversations I had with my kids this morning.

But first I want to back up to a coincidence from yesterday.

As I mentioned above I’ve been studying similar triangles with my older son for a week or so.  The bit of math that seems to be giving him the most difficulty isn’t the geometry, it is working with the ratios that arise in the problems about similar trirangles.   Here’s one of the problems we worked through yesterday just to give an example of the ratios that come up in these problems:

I felt that it would be good to review some of the algebra behind these ratio equations before finishing the similar triangle section and found several sets of practice problems on Khan Academy that provided more or less exactly the review I was looking for.  Here’s one set for example:

Khan Academy Ratio problems

Although people have widely differing views about Khan Academy, I think one nice advantage that it has is that the problem sections are great for this type of review.

Interestingly last night Steven Strogatz posted this picture on twitter:

The similarity between the homework I wanted my son to do and the homework assignment in Strogatz’s tweet got me thinking about context.  The motivation to learn more about geometry was enough for my son to understand the purpose of the Khan Academy problems.  Actually, he even asked to do more.   I don’t know the context of the other homework assignment, but do think that without proper context that assignment could seem quite dull.  This coincidence from yesterday reminds me to be careful to be clear about why I’m asking the boys to do the homework I give them.

Now on to today . . . .

This morning my younger son and I were talking about palindromes (section 6.5 in Art of Problem Solving’s Introduction to Number Theory book).  We began with several simple examples – numbers like 11, 454, 34543 – and then he stopped me:

kid:  “I know a long list of palindromes.”
me:  “what is it?”
kid: [ writes the first 4 rows of Pascal’s triangle on the board ]

This example is definitely a fun one for looking at palindromes, but it also turns out to be a great one for building on number sense.  The connection I wanted to focus on was how the rows related to powers of 11, and how that connection seems to break down in the row:  1 5 10 10 5 1.

My first question to him was whether or not this specific row was a palindrome.  He surprised me by saying that although the number you get by putting all of the terms together, namely 15101051, was not a palindrome, you could get a palindrome you looked only at the last digits, so 150051.  Interesting observation.  We’ll have to return to this topic later when we talk about modular arithmetic!

My next question for him was about the powers of 11.  Starting at 11^0, the powers of 11 are 1, 11, 121, 1331, 14641, and 161,051.  Why did we lose the connection to Pascal’s triangle when we computed 11^5?  This led to a wonderful conversation about place value and eventually to showing why we did not actually lose the connection to Pascal’s triangle at all.  Really fun, and I think a neat way to talk through place value while getting in a little arithmetic practice, too.

Later in the morning my older son got tripped up on this problem from the 2006 AMC 8:

2006 AMC 8 problem 24

The problem has a really lucky connection to palindromes since an important observation in solving it is that one number is equal to another number multiplied by 101.  Talking through this problem also led to a good conversation about place value.  Luckily the notes from the conversation about Pascal’s triangle and place value happened to still be up on the board when this second conversation took place.

Seeing some of the earlier work that was on the board my older son said that he thought you could make the row 1 5 10 10 5 1 into a palindrome by working in base 11.  Ha – another unexpected response, but also now a wide open door to talk a little about what I’m calling “algebra sense.”

We quickly reviewed the place value conversation I had with my younger son about how the rows connect to powers of 11, but then looked at what happens in base 11.  Surprise –  powers of 12!!  Don’t think he saw that coming 🙂    Now maybe 5 to 10 minutes of conversation about what the polynomials (x + 1)^n and (x + y)^n look like and we’ve quite unexpectedly done some neat work that helps build up familiarity with algebra and algebraic expressions.

So a fun morning.  As I have the goal of working on number sense in the back of my mind, I’m excited to see all of opportunities that come up to work on it.  Algebra sense, too, but Strogatz’s post from yesterday reminds me to be extra careful about context.  It is fun to take advantage of the lucky times like this morning when that context appears almost by magic!

Mixing Problems

One more trip up to Boston today for the last Brute Squad practice weekend before nationals.   Follow along next weekend and maybe watch some great ultimate frisbee on ESPN:

Before heading out the door we went through a quick Family Math project about mixing.  I illustrated the problem using a glass of red liquid and a glass of green liquid.  The problem is this:

(1) Take an amount of liquid from the red glass and put it in the green glass.

(2) Now take the same amount of liquid from the green / red mixture and put it back in the red glass.

(3) The question is – does the green glass have more red in it or does the red glass have more green in it?

It is a pretty cool problem for kids to talk through:

After talking about the problem with liquid, we now try a hopefully easier version of the problem using snap cubes.   One of the lessons I’m hoping for here is learning to find versions of a complicated problem that are a bit easier:

So, a fun problem for kids that we can do with food coloring and with blocks.   Hopefully these two different approaches hope clarify what is a somewhat non-intuitive result.

A little fun and a little math wtih Rubik’s cubes

Sometime last year my kids became fascinated by Rubik’s cubes.  Not sure why or how it happened, but once they started playing around a little they were hooked.  So much so that learning more about how to solve the cubes seemed like a fun topic to include as part of the school year, so we’ve been studying some of the 2×2 and 3×3 speed solving techniques for fun since September.   Even though I’m not practicing the speed solving with the boys, I’ve gotten a little hooked, too 🙂


Following a few folks on twitter also led to some interesting Rubik’s cube related reading in the last year.  Christopher D. Long (@octonion) tweeted about the book “Adventures in Group Theory:  Rubik’s Cube, Merlin’s Machine & Other Mathematical Toys.”  Definitely a fun read if you are into math, though it’ll be a while before I can pull much from it for the boys.

I also ran across Cathy O’Neil’s (@mathbabedotorg) old post about math contests :

Math contests kind of suck

This comment really struck me  – “I have never been particularly fast at working out the details of something from the conceptual understanding (for example, it takes me a long time to solve a 7x7x7 Rubik’s cube) but it turns out the Rubik’s cube doesn’t mind. And in fact mathematics in real life isn’t a timed tests- the idea that you need to be original and creative really quickly is just a silly, arbitrary way to select for talent.”  (as an aside, you should definitely follow her blog and follow her on twitter – you’ll not find a more interesting blog.)  I agree with Cathy O’Neil’s point that there’s not anything special about solving the cubes fast.  The kids seem to like it and enjoy learning the techniques, but it is mostly just a matter of practice.  That said, the world record solve times (~5.5 seconds for solving the 3×3, for example) really are  mind blowing.

So, what can kids learn from these cubes?

There is quite a bit of interesting math related to the cube solving algorithms (see the book mentioned above).  A simple introduction to these algorithms probably has some benefit, but I’m aiming a little lower right now.

One interesting advanced topic is parity.  This position on the 3x3x3 is impossible to solve:

Cube Reverse

You can not create a position with just one middle reversed with legal moves.  In order to make this postion, you have to take the cube apart.

However, this position on the 4x4x4 cube, which seems pretty similar to the picture above, is solvable:

4x4 Reverse

At least for my (very, very very slow) solving technique, figuring out how to solve the 4x4x4 from this position was the final obstacle to overcome in learning how to solve the larger cubes.

For my younger son, it turned out that the cubes were also fun tools for learning about topics like fractions:


and exponents:


** Update **  Imaginary and non-commutative numbers!!

It isn’t hard to believe that kids will be more excited about learning when they are having fun, but it great to see that excitement in practice.    Of course, it has also been really fun for me to use the Rubik’s cubes to help teach a bunch of different math topics.  Maybe one day we’ll even be able to replicate something like this 🙂