James Tanton’s rectangle problem

Saw this neat question from James Tanton today and thought it would make for a nice last project of the year. We are still stuck in Omaha due to flight issues, so no videos . . . .

Their first thought was to try out a few rectangles. After a few tries they found one that worked.


Next they tried some larger numbers and found that large rectangles didn’t seem to be close to having the property Tanton was looking for. They speculated the 2×3 rectangle was the only one.


My younger son thought that the reason the 2×3 rectangle was the only one was that “one side stays the same and the other side doubles” when you add one to both sides.

That observation led to my older son writing down an algebraic equation describing the situation. We checked that the 2×3 rectangle satisfied the equation.


Now, how is this equation going to help us? It took a while for them to figure out that solving for one of the variables would be helpful. Part of the problem is that the result doesn’t look that helpful – but both x and y have to be integers, so it is helpful.


Next we tried a few small values of y to see when (y+1) / (y – 1) would be integers. We found 2, but those two values described the same rectangle.

My younger son also noticed that the value of (y+1)/(y-1) was decreasing as y increased.


Finally, I showed them a “simple” way to see that the expression (y + 1) / (y – 1) decreased as y increased. Unfortunately I wrote the inequality backwards, but the proof at least shows how the right proof would go πŸ™‚


So, a fun little project to end the year. Can’t wait to see what math 2016 brings!

Megan Hayes-Golding’s Clue tweet

A tweet from Megan Hayes-Golding led to some fun:

The fun was that the very next day my sister had a first guess win playing Clue with the kids:

On our way out to eat last night the kids were wondering what the probability of a first guess win was!

They told me that she had three cards in her hand, but didn’t know what they were. Today we worked out the probability – turns out to be roughly 1 in 200.


This is a great problem for kids and a really nice way to reinforce some basic counting ideas.

Here’s a sketch of how we worked it out:

(1) Step 1. How many different three card hands are there? In Clue there are 6 suspect cards, 6 weapon cards, and 9 room cards, for a total of 21 cards. All of the cards are distinct, so we computed that there were 1,330 different 3 card hands ( 21 choose 3).

(2) Step 2. Compute the different kinds of three card hands, and how many of each type of hand there is. For example, we found that there were 324 3-card hands with 1 suspect, 1 weapon, and 1 room card.

After we found all of the different types of hands we double checked that this way of counting the hands also led to 1,330 different 3 card hands. It was nice to have a way to check that we’d not made any mistakes (yet!).

(3) Next we found the probability of a first guess win with each type of hand. For example, the probability of guessing right on your first guess if you have 1 suspect card, 1 weapon card, and 1 room chard is (1/5)*(1/5)*(1/8).

In case you aren’t familiar with the game, you are trying to determine who committed a crime, in what room, and with what weapon. The cards revealing this information are in an envelope at the center of the board. Each card you hold tells you a particular person / item / room that was not involved in the crime. So, if you hold exactly 1 suspect card at the beginning of the game, you have 5 suspects remaining who may have committed the crime.

(4) Finally, we found the average probability from all 1,330 cases and found that it was:

3,047,827 / 603,288,000 which is about 1 in 200 πŸ™‚

You can see on the paper that we made one arithmetic mistake computing this average. We accidentally entered 81/10 instead of 81/50 in the calculation. That error led to us computing a probability of about 1 in 100, but that answer didn’t make sense because all of the individual probabilities were less than 1 in 100, so the 1/100 couldn’t have been the average.

So, a fun project and an amazing coincidence with the earlier tweet. I think this would be a great activity for a stats class. It is also a nice basic counting example, too.

Reacting to James Tanton’s Math Wars essay

Saw this nice essay from James Tanton today:

What especially caught my eye was the 2nd question – What is the difference between familiarity and understanding?

A version of that question has been in my mind for a few days because of this comment on my blog from Paula Beardell Kreig:

A comment from Paula Beardell Kreig that’s been on my mind for a while

Also, Amy Hogan made a nice point when I showed this comment to Tanton today:

I wish I had a good answer to Tanton’s question. I’m don’t, and I’m not even sure where I fall on the mythical “traditionalist” to “reformer” line sometimes used as a measuring stick in the math wars.

One piece I go back to again and again that is sort of an indirect answer to the question, though, is the interview that Wild About Math did with Julie Rehmeyer. What’s always stuck with me from this interview is the story that begins around 31:30 and in particular the part beginning around 34:40 about proving that 0 + 0 = 0.

Julie Rehmeyer’s “Inspired by Math” interview

When I think about familiarity and understanding my thoughts always drift to this interview. Though I show my kids lots and lots of different ideas with the idea of trying to get them familiar with all sorts of different mathematical ideas, the part about mathematical understanding that comes in the “0 + 0 = 0” story is a good description of my long term goal in terms of building their understanding.

Relations from i to geometry

Saw this video from Numberphile a few weeks ago:


The boys and I had some fun talking through a few of the examples. For instance, we saw that 5 is not a prime in the Gaussian integers because 5 = (2 + i)(2 – i).

We also saw that the set of integers combined with integer multiples of \sqrt{2} has some unexpected factorization properties, for example, like 2 = (2 – \sqrt{2})*(2 + \sqrt{2}), and 14 = 2 * 7 = (4 – \sqrt{2})(4 + \sqrt{2}). We also played with the \sqrt{-5} example in the video.

I spent the last few days kicking around a few ideas about how to explore these ideas in other ways with the boys. Part of the struggle was figuring out how to translate advanced math ideas like the polynomial ring Z(x] / (x^2 + 1) into something that the boys could understand.

I couldn’t crack the code for that, but thinking about this idea led to a fun coincidence when I saw this problem from Matt Enlow today:

I won’t give away the precise solution to the problem, but my approach used the idea of a polynomial relation. A simple relation like x^5 = 2x^3 + x (but not that exact relation) arose from the conditions in the problem and that relation allowed me to solve Enlow’s problem without actually having to solve for x.

It is neat to see similar algebraic ideas arising in totally different contests.

Moving beyond the quadratic formula

It has been interesting watching my son learn a bit more algebra this year. He seems to have made quite a bit of progress studying linear equations, but he’s still at a point where the quadratic formula is the first thing he thinks about with any sort of non-linear equations (which, given that he’s just learning algebra are mostly quadratic equations).

Today he ran across a problem that was probably designed quite specifically to help kids see beyond the quadratic formula. The problem is #20 from the 2007 AMC 10 a

Here’s the problem:

Suppose that the number $a$ satisfies the equation 4 = a + a^{ - 1}. What is the value of a^{4} + a^{ - 4}?

This problem gave him some difficulty and I asked him to explain his original approach using the quadratic formula first:


Next we talked about how to approach this problem without solving for a first. We had briefly talked through this approach in the morning but this was his first time trying to explain it.


Next we went to Wolfram Alpha to see that the solution he’d found with the quadratic formula actually produced the answer of 194. After that we talked about the graph of y = x + 1/x. It was a little hard for him to see that the minimum value on the graph occurred at x = 1, but zooming in a little helped him see it.


Finally, I wrapped up by showing him one way that we could use the quadratic formula to help us see where that minimum occurred. I took this approach to help him see that even though the quadratic formula wasn’t so helpful in solving the original problem, it still could be helpful as a way to learn a little bit about x + 1/x.


So, a nice little problem that provides a good example of a situation where the quadratic formula isn’t so helpful. Hopefully examples like this one will help him see that there are lots of to approach non-linear equations, and the quadratic formula is just one of them.

Continuing our talk about primes

Yesterday we had a fun talk about prime numbers starting with a problem I saw on twitter:

A neat problem from 5 Triangles and Dave Radcliffe

Today I wanted to spend a little time with my younger son revisiting some of the ideas from yesterday. I started by asking him to find 10 consecutive integers that weren’t prime. He remembered most of the ideas from yesterday, but there were one or two things that caused a little difficulty. This is a challenging argument for kids to follow since you are talking about numbers without really knowing what those numbers are.


In the first part of today’s discussion my son ran into the number 10! + 1. I thought it would be fun to use this number as a starting point to talk about (i) a simple proof of why there are infinitely many primes, and the related topic (ii) how to find numbers that we know contain “new” primes. We used Mathematica to make help out in this part of the discussion.

Oh, and one point I didn’t explain at all but definitely should have. Mathematica’s function FactorInteger[n] returns a list of primes that divide n as well as the highest power of that prime that divides into n. So, when we do the example FactorInteger[4! + 1] = FactorInteger[25] = {5,2}, the output “{5,2}” is saying that 25 = 5^2 not that 5 and 2 divide into 25.


So, a fun follow up to yesterday’s project. Hopefully today’s talk helped a few of the ideas from yesterday sink in for my younger son.

A neat problem from 5 Triangles and Dave Radcliffe

Saw a neat exchange on twitter last night:

Though my kids are far too young (4th and 6th grade) to find this solution on their own, I thought that going through this solution with them would be a useful and fun exercise. Each step in the solution is something that they can understand and Dave’s approach is also a great lesson in problem solving.

So, instead of our typical morning projects this morning we talked through this problem.

Here’s the introduction to the problem and a few initial thoughts. Right off the bat the kids have some nice thoughts about prime numbers:

The boys had some good thoughts about simplifying the problem in the last section. We looked at a few other simple examples – 3 consecutive integers in which two are prime (this led to a nice discussion about twin primes).

Next we moved to the computer to take a look at Dave Radcliffe’s idea. Luckily Mathematica has a function – PrimePi[n] – that counts primes less than or equal to n. We wrote a little program to count the number of primes occurring in a list of 2015 consecutive integers.

In this part of the project we began to use this program to explore the number of primes in various lists of 2015 consecutive integers.

At the end of the last section my younger son noticed the point that Dave Radcliffe had made last night -> as you go down the list the number of primes changes by +1, 0, or -1, but no other number.

In this part of the project we discussed why the changes were never greater than 1 and also how this property might help us solve the original question.

Finally we discussed how we could find a long list of consecutive integers with no primes. Unfortunately we were running a little longer than I expected so this part was a little more rushed than I would have liked. Still, though, they seemed to understand the idea.

So, I think walking through this problem with kids is a fantastic exercise. There are lots of interesting mathematical ideas from arithmetic and from number theory kids might find fascinating. Also, the idea that the proof shows a list of 2015 integers with exactly 15 primes exists without actually finding it is also an amazing idea (and likely one that is brand new to kids).

Finally, Dave Radcliffe’s idea to look at lists of 2015 consecutive integers to see how many primes they have is a fantastic problem solving idea. Seeing how a simple idea like Dave’s changes an almost unapproachable problem into a one that is now much easier to understand is an important example for kids to see.

Definitely a fun project.

Henry Segerman’s Flat Torus

[Sorry for writing this one up at one million miles per hour. We are seeing Star Wars at 9:30 am today and I wanted to get this one published before we left. Hopefully the fun of this amazing shape outweighs the quick and unedited writing.]

Saw another amazing post from Evelyn Lamb a few weeks ago:

In that post she showed this video by Henry Segerman:


I decided to buy this shape from Shapeways and do a project with the boys. Here’s where you can order it:

Henry Segerman’s Flat Torus on Shapeways

Before starting the actual project this morning I had the kids explore the shape on their own. Although these individual explorations were a little more difficult than I was expecting I’m glad that we did them because it gave the kids a bit more familiarity with the shape. The videos from that part of the project are here:

Background for our project on Henry Segerman’s Flat Torus

So, with that background out of the way, we began today’s project by talking about how to make MΓΆbius strips. The discussion to a little detour when I asked them to describe how to make the shape, but it was a cool and important detour. It is neat to hear kids talk about complicated shapes!


Next we talked about how to make a torus from a single sheet of paper. The kids had a hard time imagining exactly how a square would fold up into a torus, but that’s sort of the point of the project! At the end we looked at how the shape was sort of like a bagel, but sort of not like a bagel!


For the 3rd part of our project we looked more carefully at Henry’s shape an how a flat torus was different than the bagel.


The last part of the project was a quick discussion of the Klein Bottle and the projective plane, which we can also understand from a single square piece of paper. The boys were really interested in the ideas here, and especially how difficult it was to make the Klein bottle in three dimensions! They did make a connection between the Klein bottle and the MΓΆbius strip, which was really cool to hear.


So, a super fun project exploring some really complicated shapes. Infinite thanks to Evelyn Lamb and Henry Segerman for inspiring this project!

Background for our project on Henry Segerman’s flat torus

This videos here are just background for a project on Henry Segerman’s flat torus. I’ll link that project after I write it up.

Before diving into that project, though, I wanted the kids to experience the shape without knowing what it was. It is both an engineering and math challenge.

My older son had a tough time, but made the folded up shape eventually. That it took nearly 15 minutes to make sense of this shape gives a little peek at how difficult this shape is to understand:




My younger son was also puzzled about what to do with this shape. He even wondered if it was hard to understand because it was 4 dimensional πŸ™‚ At the end he thought it was a torus, which was cool: