Exploring induction and the pentagonal numbers

Yesterday we did a fun project based on this tweet by James Tanton:

That project is here:

Exploring a neat problem from James Tanton

During the project yesterday we touched on mathematical induction and also on the pengatonal numbers. Today I wanted to revisit those ideas with slightly more depth.

We started with a quick review of yesterday’s project:

Now we looked at a mathematical induction proof. The example here is:

1 + 3 + 5 + \ldots + (2n - 1) = n^2

(the nearly camera ran out of batteries, that’s why this part is split into two videos)

Here’s the 2nd part of the induction proof after solving the battery problem:

To wrap up the project we went to the living room to build some shapes with our Zometool set. The Zome shapes really helped the boys make the connection between the numbers and geometry.

The boys really liked this project. In fact, my younger son spent the 30 min after we finished making the decagonal numbers ๐Ÿ™‚

Some simple proofs of the Pythagorean theorem

Yesterday we did a fun project on these two questions:

(1) Given a square with area one, find a way to make a square with area 2,

(2) Given a square with area one, find a way to make a square with area 3.

That project is here (where you can see that part 2 gave both kids a lot of trouble):

A neat and easy to state geometry problem

I decided to revisit a piece of that project today to show them that both of their solutions to part 2 were essentially proofs of the Pythagorean theorem.

We started by reviewing yesterday’s project:

Next we talked about how my younger son’s way of constructing the square with area three can be used to prove the Pythagorean theorem:

Finally, we looked at the slightly different way that my older son constructed the square with are 3. This approach proves the Pythagorean theorem in a different way:

This was a fun couple of projects that came from a really innocuous sounding question.

Kelsey Houston-Edwards’s ‘Proof’ video is incredible

Kelsey Houston-Edwards’s latest video is amazing:

It absolutely blew me away. The kids have a late karate class today so I had to wait a few extra hours to use the new video for a project – WORST WAIT EVER!!!

To start the project I had the boys look at three of the problems in the video to see what approach each kid would take to prove the mathematical statement. My younger son is in 5th grade and my older son is in 7th grade, so I’m not expecting perfect proofs (by any stretch of the imagination) but rather just looking for their ideas.

Here’s the first problem – can you cover an 8×8 checker board with 2×1 dominoes if you remove two opposite corners:

The next problem was to show that the sum:

1 + 3 + 5 + \ldots + (2n - 1) = n^2

Here’s what they had to say – both ideas here were really interesting and used arithmetic. I was excited to see their reaction to the geometric proof in the video:

The next problem was to show that ” n choose 2″ was equal to 1 + 2 + 3 + \ldots + (n-1).

My younger son had a nice idea to start small and work his way up. He got stuck so I helped him a little. As in the last video, my older son did the proof by calculating.

After working through these three problems we watched the new video together. The problem about the L-shaped tile covering the 2^n x 2^n grid caught my youngers son’s eye. That led to a short discussion of induction.

The problem about breaking the stick into 3 parts and forming a triangle caught my older son’s eye. He reconstructed the cool proof from the video. I’d like to show him some alternate proofs from geometric probability, too, since they are all so fun!

I’m really enjoying the math videos that Houston-Edwards is making. This one is especially amazing. How great would it be for every math class in the country to watch her video tomorrow! I think it would change the way that kids see math.

Sharing John Cook’s Fibonacci / Prime post with kids

Saw a neat post from John Cook about using a fun fact about the Fibonacci numbers to prove there are an infinite number of primes:

Infinite Primes via Fibonacci numbers by John Cook

Funny enough, we’ve played with the Fibonacci idea before thanks to Dave Radcliffe:

Dave Radcliffe’s Amazing Fibonacci GCD post

That project was way too long ago for the kids to remember, so today we started by just trying to understand what the Fibonacci identity means via a few examples:

Next we looked at the idea from Cook’s post that we need to understand to use the Fibonacci identity to prove that there are an infinite number of primes. The ideas are a little subtle, but I think the are accessible to kids with some short explanation:

We got hung up on one of the subtle points in the proof (that is pointed out in the first comment on Cook’s post). The idea is that we need to find a few extra prime numbers from the Fibonacci sequence since the 2nd Fibonacci number is 1. Again, this is a fairly subtle point, but I thought it was worth trying to work through it so that the boys understood the point.

Finally, we went upstairs to the computer to explore some of the results a bit more using Mathematica. Luckily Mathematica has both a Fibonacci[] function and a Prime[] function, so the computer exploration was fairly easy.

One thing that was nice here was that my older son was pretty focused on the idea that we might see different prime numbers in the Fibonacci list than we saw in the list of the first n primes. We saw quickly that his idea was, indeed, correct.

This project made me really happy ๐Ÿ™‚ If you are willing to take the Fibonacci GCD property for granted, Cook’s blog post is a great way to introduce kids to some of the basic ideas you need in mathematical proofs.

Steven Strogatz’s circle area exercise

Saw this really neat tweet from Steven Strogatz yesterday:

I asked him if he had downloadable versions of the circle and he was nice enough to share the templates with me (yay!)

So, with a little enlarging and a little cutting we had the props ready to go through the exercise.

We began with a short conversation about circles. My older son knows lots of formulas about circles from his school’s math team practices, but my younger son doesn’t really know all of the formulas. The quick review here seemed like a good way to motivate Strogatz’s project:

Now we moved into Strogatz’s project – how do we show that the area of a circle is \pi * r^2? We cut the circle into the 16 sectors and rearranged them into a shape that was more familiar to us:

Next was the big challenge and the really neat idea in Strogatz’s first tweet – there is a different shape we can use to find the area. The boys were able to find this triangle fairly quickly, but then we had a really fun discussion about what the triangle would look like if we used more (smaller) sectors. So, the surprising triangle from Strogatz’s tweet led to a really fun and totally unexpected discussion! It is so fun to hear kids think through / wonder about math questions like the one they asked about the new triangles.

The last part of the project today was inspired by a tweet from our friend Alexander Bogomolny that was part of the thread Strogatz’s tweet started on Twitter yesterday:

I love it when Twitter writes our math projects for us ๐Ÿ™‚

I had the kids look at the picture and describe what they saw. At the end I asked them why they thought the slanted lines in the triangle were lines and not curves – they had interesting thoughts about this little puzzle:

The amount of great math shared out twitter never ceases to amaze me. Thanks (as always!) to Steven Strogatz and to Alexander Bogomolny for inspiring this project about circles. Can’t wait to try out this project with other kids.

A great question from Dave Radcliffe

During a twitter conversation earlier in the week, Dave Radcliffe presented this question:

The question is a really deep and really challenging one for kids. Truthfully it is probably a little over the head of my kids, but I thought I’d give it a try anyway. I’ll revisit this one (hopefully!) several times over the course of this school year – although the question confused my kids a little bit, I really like it.

Here’s my older son’s (started 7th grade today!) thoughts on Dave’s question:

Here’s my younger son’s thoughts – he’s in 5th grade. I took a little extra time at the beginning with him to work through some examples with numbers so that the abstract symbols wouldn’t be so confusing:

Does 1 – 1 + 1 – 1 + 1 . . . . . = 1/2

This morning, for a little first day of school fun, we played with Grandi’s Series.

I’ve seen the series pop up in a few places in the last few days – first in part of a little note I wrote up inspired by a Gary Rubenstein talk:

A Talk I’d live to give to calculus students

and then a day or two later in this tweet from (the twitter account formerly known as) Five Triangles:

So, what’s going on with this series? What would the boys think?

Here’s their initial reaction:

And here’s their reaction when I showed them what happens when we assume that the series does sum to some value x:

We have touched a little bit on this series (and my favorite math term “Algebraic Intimidation”) previously:

Jordan Ellenberg’s “Algebraic Intimidation”

It is fun to hear the boys struggle to try to explain / reconcile the strange ideas in Grandi’s series. I’m also glad that they are learning to think through what’s going on rather than just believing the algebra.

3 proofs that the square root of 2 is irrational

My younger son has been learning a little bit about square roots over the last couple of weeks and I thought it would be fun to show him some proofs that the square root of 2 is irrational. Because this conversation was going to explore some ideas in math that are both important and pretty neat, I asked my older son to join it.

I wasn’t super happy with how this little project went – it felt a bit rushed while we were going through it. Hopefully a few of the ideas stuck.

We started by talking about the square root of 2 and what basic properties the boys already knew about it:

After that short introduction we moved on to the first proof that the square root of 2 is irrational – I think this is probably the most well-known proof. The proof is by contradiction and starts by assuming that \sqrt{2} = A / B where A and B are integers with no common factors.

The next proof is a geometric proof that I learned a few years ago from Alexander Bogomolny’s wonderful site Cut The Knot. It is proof 8”’ here:

Proof 8”’ that the square root of 2 is irrational on Cut the Knot’s site

If you like this proof, we have also explored some geometric infinite descent proofs in a slightly different setting previously inspired by a really neat post from Jim Propp:

An infinite descent problem with pentagons

Finally, we looked at a proof that uses continued fractions. It has been a while since I talked about continued fractions with the boys, and will probably actually revisit the topic soon. It is one of my favorite topics and always reminds me of how lucky I was to have Mr. Waterman for my math teacher in high school. He loved exploring fun and non-standard topics like continued fractions.

So, although I don’t go deeply into all of the continued fraction ideas here – hopefully there’s enough here to show you that the continued fraction for the \sqrt{2} goes on forever.

So, although this one didn’t go quite as well as I was hoping, I still loved showing the boys these ideas. We’ll explore them more deeply as we study some basic ideas in proof over the next year.

Math school year in review part 1 – our tiling projects

Roughly a year ago the boys started their first year in school after 5 years of home schooling. Both kid’s packets for the next school year arrived last week. That got me daydreaming about the math we’ve done in the last year. Turns out that we’ve done a lot and I wanted to write about some of it before it all slips out of my mind.

One subject that we’ve spent a fair amount of time on this year is tilings. Not by design but I just happened to see a lot of neat tiling ideas in the last year.ย  Here’s a review of the tiling projects we did in the last 12 months:

(1) The project from Zome Geometry that got us going

This project was one of the few ones that I didn’t film.ย  The reason was that we had several kids from the neighborhood over working on it and I don’t feel comfortable filming kids that aren’t mine!

Anyway, this was a really fun project from Zome Geometry by George Hart and Henri Picciotto

Zome Tilings

IMG_0555

(2) That project led to two 3d versions:

The natural thing to do after this project was to look at ways that you could cover 3 dimensional space with shapes – again we made use of our Zometool set:

Tiling 3-dimensional space with our Zometool set

Honeycombs

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(3) A problem from a UK math exam led to a fun tiling project

I saw this neat problem from a UK math test circulating on Twitter back in February.

The UK Intermediate mathematics challenge part 2

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(4) A domino counting exercise form Jim Propp

We’d done a couple of projects based on Jim Propp’s blog, he thought that we might enjoy studying how 2×1 dominoes tile a 2xN square.ย  The project was so fun that we actually did it twice!

A fun counting exercise for kids suggested by Jim Propp

Counting 2xN domino tilings

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(5) Propp’s suggestion above came after we did these two projects on the Arctic Circle Theorem

I learned about the Arctic Circle theorem from a graduate student at MIT who thought it might be possible to share this fairly advanced mathematical idea with kids:

The Arctic Circle Theorem

A second example from tiling the Aztec diamond

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(6) Dan Anderson’s Gosper Curves

Maybe stretching a little to call this tilining, but we had fun exploring how the Gosper Island’s that Dan Anderson sent us fit together:

Dan Anderson’s Gosper curves

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(7) Inspiration from Eugenia Cheng’s Shapes video

I saw this neat video from Eugenia Cheng over the summer:

Thinking about how to use it with my kids inspired these two projects:

Tiling pentagon cookies

Learning about tiling pentagons from Laura Taalman and Evelyn Lamb

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(8) Richard Stanly’s Tiling presentation

Finally, just last week I stumbled on a presentation that Richard Stanley – a math professor at MIT who specializes in combinatorics – had put together about tilings.ย  There were a couple of ideas that were accessible to kids:

Talking through some examples from Richard Stanley’s tiling presentation

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The beginnings of abstraction

I’ve been working on basic techniques of proof with the kids for the last couple of weeks and I’m hoping to incorporate that work in with what they are studying in their books.

My older son is working through Art of Problem Solving’s Introduction to Geometry. Today he picked a problem for our movie which leads pretty naturally to a discussion about proof and abstraction.

The problem is:

A regular polygon has the same number of diagonals as sides. What is the sum of the interior angles of this polygon?

Here is his solution to the problem:

After he finished I tried to get him to think about the problem more abstractly, and that led to a nice discussion. His approach to counting the number of diagonals was really interesting.

This was a fun way to start the 7th grade school year – can’t wait to see where the next 12 months take us ๐Ÿ™‚