Pyramids and Count Like and Egyptian

Last night I was looking through Count Like and Egyptian by David Reimer looking for a new project and stumbled on an idea about square pyramids that we’ve touched on before. Seemed like a nice review project.

Prior similar projects to this one are:

A neat geometry project inspired by a James Tanton / James Key tweet

Revisiting an old James Tanton / James Key Pyramid project

and

Octahedrons, Tetrahedrons, and the Pyramid Puzzle

Some of our old projects from Count Like and Egyptian are here:

Going through Count Like an Egyptian with the Boys

Count Like an Egyptian Part 2

Count Like an Egyptian Part 3

I started today’s project introducing the pyramid topic from the book and having the boys play with some small pyramid pieces we’d 3D printed for previous projects.

They boys were able to use the pieces to see that the volume of the square pyramid we had was 1/6th of the volume of a cube whose square sides were the same size as the base of our pyramid.

For the next part of the project I wanted the boys to build a cube made out of these 6 square pyramids. We started by building the small pyramid shapes and talking about how you could use them to form a cube:

After building the smaller pieces the boys worked for about 10 minutes to build the cube with the square pyramids inside. We talked about the various properties of that cube here:

Finally, after the brief discussion of volume formulas in the last video, we returned to the whiteboard to have a (slightly) longer discussion. The point here wasn’t to derive the formula but rather to show that at least the two sets of pyramids we studied in this project had volumes that we could find using the formula.

So, a nice little review project which hopefully helped some of the ideas from previous projects sink in a little more. I like working through these 3d geometry project with the boys.

AND, after we were done our little Zometool pyramid captured BB-8 ๐

Revisting a challenging AMC 10 problem with Patty Paper

Earlier this week my older son and I looked at problem #11 from the 2011 AMC 10a:

That earlier project is here:

and the entire 2011 AMC 10 a can be found here:

The 2011 AMC 10a hosted at Art of Problem Solving

Today I wanted to revisit the problem from the perspective of a project rather than as a contest problem. This approach came to mind after watching Po-Shen Loh’s use of a problem from the 2010 International Mathematics Olympiad as fun project during a public lecture at the Museum of Mathematics:

So, inspired by Loh’s approach, I started with a much easier version of the problem – how do we make perpendicular bisectors if we are given a line segment? My younger son (4th grade) makes the perpendicular bisector with a ruler and protractor, and my older son (6th grade) uses some folding ideas:

At the end of the last video I was asking the boys about some special properties of perpendicular bisectors. Upon reflection I don’t know why I assumed they’d notice that the line was equidistant from the two endpoints of the line segment – it isn’t as if this is an obvious property for kids to see. Although I was surprised when we sort of hit a wall at the end of the last video, I really shouldn’t have been.

Instead of just telling them what the property was, I let them play around a little more and eventually my older son noticed the it. We then talked about why the perpendicular bisectors had this property. This 5 minute detour wasn’t really planned, but I didn’t want to just tell them about the property – I wanted them to find it. I really liked how Loh had success with a similar approach in his MoMath lecture. There he showed a simpler version of the IMO problem first – and allowed lots of exploration from the audience – which prepped everyone for thinking about the harder problem.

Now, with a little bit of background on perpendicular bisectors, we looked at the old AMC 10 problem. The background work on perpendicular bisectors helped my younger son approach the problem – it was actually pretty cool to see him take the lead in drawing the region R. After the boys had drawn the region, they had lots of great questions about the picture.

I didn’t go all the way through to finding the area of the region because I didn’t think that would be a great use of time with my 4th grader – after you’ve got the region drawn, finding the area is mostly just calculation.

Finally, we wrapped up the project by approaching the problem using folding techniques. It was nice that this approach revealed additional properties of the pentagon / region R that were not as clearly visible in the last video.

So, I was happy to see the success that Loh had using an IMO problem in a public lecture, and it was fun to mimic his idea on a smaller scale with a contest problem that was also accessible to kids. Contest problems might lend themselves to this approach a little more easily that research problems since, essentially by design, they have short solutions. Still, though I’m excited to think a little bit more about how to use an approach like Loh’s share non-contest math ideas like Larry Guth’s “no rectangles” problem and John Conway’s Surreal Numbers with kids.

A nice probability problem to talk thorugh with kids

Saw this problem from the 2012 AMC 10 b tonight:

you can find the problem (and the entire test) here:

The 2012 AMC 10b on Art of Problem Solving’s site

It looked like a neat problem to talk through with both of the boys. I started with my older son first:

Next up was my younger son who had some fun with it, too:

Some 3D geometry for Pamela Rawson

I got a nice comment on a blog post form Pamela Rawson and I ended up reading her neat post talking about her new 3D geometry class:

https://rawsonmath.wordpress.com/2016/01/26/how-do-we-know-that/

The point of the Week 3 MTBoS Blogging initiative isn’t the course specifics, I know, but still the idea of a 12 week course focusing on 3D geometry captured my imagination.ย  I never studied 3d geometry in any formal way, but exploring that subject with my kids has been incredibly fun.

For example:

and more recently this tweet from Simon Gregg (and the animated gif inside of his blog post) inspired a super fun project:

My work prepping to explore folding a dodecahedron into a cube with the boys is here:

A neat post from Simon Gregg

and the project with them is here:

Can you believe that a dodecahedron folds into a cube!

Moving away from our Zometool-related projects, we’ve done a few fun 3d geometry projects using just snap cubes. For example, this tweet from Five Triangles made for a neat project:

A neat counting problem shared by Five Triangles

This project explored the connection between the sum of squares – $1^2 + 2^2 + 3^2 + . . .$ and geometry:

Summing up Squares

and it was realted to another 3D geometry project inspired by a tweet from James Key:

A neat geometry project inspired by a James Tanton / James Key tweet

I could go on and on and on and on, but I’ll just stop at one more fun one – the volume of a sphere:

The Volume of a Sphere via Archimedes

All of our 3D geometry projects have the “3D Geometry” tag on the blog, that list of projects is here:

Our 3D Geometry projects

It is such a fun subject!!

I went through the project below with my older son on Sunday night. The question we were talking through is here:

This is problem 20 from the 2011 AMC 10 A. My son really struggled with this problem, and I also struggled finding ways to help him.

It was one of those times where I think the overall struggle was productive, but I was left feeling that I could have done a lot better. After the project I saw that the Week 3 MTBoS blogging challenge was about good questioning. Since I wasn’t satisfied with how this one went, I thought I’d use it as an example of where I’d like to improve.

In the first part of the project we discussed the problem and why it was giving him trouble. I was actually pretty happy with how this part went – we identified an initial misconception and then made some good progress toward drawing a good picture.

Having walked oh so close toward perpendicular bisectors in the first part of our talk, I thought that the next part would go pretty smoothly. We got off track, though, and struggled to get back on track. I really did nothing productive to help him here. Part of my problem is that I was overly focused on the two equilateral triangles in the picture and he was having a hard time seeing those triangles.

In the next part my struggles continued. My son was focused on one of the 30-60-90 triangles and I was hoping beyond hope that he’d see the equilateral triangles. I really should have found a good question about the 30-60-90 triangles he was seeing.

So, at the end of the last video my son had made some great progress and had reduced the problem down to finding the area of a pentagon. In the last part of the project he finishes that calculation.

So, a challenging problem for sure and a nice opportunity for me to look back and wonder what I could have done differently. It is funny that from my perspective during this conversation my son was stuck because he was so focused on some 30-60-90 triangles, but what prevented me from helping him get unstuck is that I was so focused on some equilateral triangles. Hopefully I can learn from this and come up with some better questions next time.

Amazing math from mathematicians to share with kids

About two years ago I saw this Numberphile interview with Ed Frenkel:

One of the ideas that Frenkel mentions in the interview is that professional mathematicians haven’t done a good job sharing math with the general public. Although I’m not really the kind of professional mathematician Frenkel was talking about, I took his words to heart and have been on the lookout for math to share – especially with kids.

It turns out that there are some fantastic ideas that are out there for kids to see. Some surprising fun I had sharing Larry Guth’s “no rectangles” problem with kids earlier this week (see below) made me want to share some of the ideas I’ve found in the last couple of years, so here are a few examples:

(1) One of the most incredible lectures that you’ll ever see is Terry Tao’s “Cosmic Distance Ladder” lecture at the Museum of Mathematics in New York City:

I used Tao’s video for three projects with my kids – but there are probably 20 math projects for kids you could get out of it.

Part 1 of using Terry Tao’s MoMath lecture to talk about math with kids – the Moon and the Earth

Part 2 of using Terry Tao’s MoMath lecture to talk about math with kids – Clocks and Mars

Terry Tao’s MoMath lecture part 3 – the speed of light and paralax

(2) The Museum of Math’s public lectures are a great source beyond Tao’s lecture.

Here’s a project based on Bryna Kra’s lecture:

Angry Birds and Snap Cubes – Using Bryna Kra’s MoMath public lecture to talk math with kids

Eric Demaine’s lecture was part of our Fold and Cut theorem project:

Fold and Cut part 3

and I can’t say enough good thinks about Laura Taalman’s work – she’s inspired dozens of our projects.ย  Just search for her name on the blog:

(3) and Speaking of Fold and Cut . . .

Katie Steckles and Numberphile put together an incredible video about the Fold and Cut theorem. I used the video this week for project with 2nd and 3rd graders at my younger son’s school earlier this week.ย  Steckles’s presentation is so incredible – this is the kind of math that really inspires kids:

We used it for three projects (including the Eric Demaine one above):

Our One Cut Project

The Fold and Cut Theorem is Awesome!

In prepping for the grades 2 and 3 projects I also totally coincidentally ran across a “fold and punch” exercise that is a great activity to try with kids before trying out fold and cut:

(4) Another great success with the 2nd and 3rd graders was Larry Guth’s “no rectangles” problem. I had a great time playing around with this problem with my kids, but nothing prepared me for how enthusiastic the kids in the two programs were about this problem.

Larry Guth’s “No Rectangles” problem

After the 3rd grade night, Patrick Honner sent me this picture that I used to wrap things up with the 2nd graders.

(5) The Surreal Numbers

I’d seen John Conway’s surreal numbers previously via an amazing Jim Propp blog post:

and I wanted to revisit them after finally reading Donald Knuth’s book:

Revisiting the Surreal Numbers

Infinity + 1 and other Surreal Numbers

Playing with the surreal numbers via checker stacks is an incredibly engaging way for kids to learn about mathematical thinking.

(6) Speaking of John Conway –

In the 2014 edition of the Best Writing in Mathematics Conway had an article about variations on the Collatz conjecture. It was a fascinating article that even gave us the idea to translate some of the math into music.

The Collatz Conjecture and John Conway’s “Amusical” variation

I’ve also talked with the boys about the standard version of the Collatz conjecture:

It is a great way to introduce kids to an unsolved problem in math while also sneaking in a little bit of arithmetic practice!

(7) Occasional contest math problems

I happened to run across another MoMath lecture yesterday – this one by Po-Shen Loh. He was talking about “Massive Numbers.” I thought maybe he’d be talking about the book “Really Big Numbers” by Richard Evan Schwartz:

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

or maybe Graham’s Number:

An attempt to explain Graham’s number to kids

The last 4 digits of Graham’s number

His presentation is fascinating and I even talked through the first version of the problem with my younger son:

Show that any positive integer n has a (positive) multiple which has only the digits 1 and 0 when represented in base 10.

A challenging arithmetic / number theory problem

(8) Building off of popular books by mathematicians as well as public lectures

I was surprised at how much great math writing and speaking there has been for the general public in the last couple of years.

Jordan Ellenberg’s “How not to be Wrong” inspired several projects – probably my favorite was using his idea of “algebraic intimidation” to talk about the famous 1 + 2 + 3 + . . . = -1/12 video by Numberphile. :

Jordan Ellenberg’s Algebraic Intimidation

Jacob Lurie’s Breakthrough Prize public lecture inspired two projects about a year apart from each other:

Using Jacob Lurie’s Breakthrough Prize Lecture to Inspire Kids

Using Jacob Lurie’s Breakthrough Prize talk with kids

And, Ed Frenkel, who got me thinking about sharing advanced math with kids in the first place has inspired a few projects, too:

Fine Ed Frenkel – you convinced me

Ed Frenkel, the square root of 2, and i

and one of my all time favorites:

A list Ed Frenkel will love

(9) Finally, it would be impossible to write a post like this one without mentioning the work that Evelyn Lamb is doing writing math articles for the general public. I’ve lost count of how many projects she’s inspired, but it is probably well over 20. I’m especially grateful for her talk about topology which have generated really fun conversations with the boys. For example:

Using Evelyn Lamb’s Infinite Earring with kids

Evelyn Lamb’s fun torus tweet

and

Henry Segerman’s Flat Torus

which arose after Lamb pointed out this video:

So, I’m really happy that mathematicians are sharing so many amazing ideas. I think this is the sort of math promotion that Frenkel had in mind. Hopefully it continues for many years to come ๐

Exploring infinity and other Surreal Numbers

[post publication update on March 14th, 2016. Lazily doing a google search to get a link for this blog post I learned of this article in Discover Magazine from 1995:

http://discovermagazine.com/1995/dec/infinityplusonea599

I’d not seen this article before – though I should have as it is linked in Jim Propp’s “The Life of Games” blog post which was the seed of my interest in the Surreal numbers. The original title of this post was completely by accident the same as the title of the Discover Magazine article. After learning about the prior article I have revised the title of this post.]

Yesterday we revisited the surreal numbers by looking at the game “checker stacks”

Revisiting the Surreal Numbers

We explored the values of some of the positions in the game and found some simple stacks that had values of 1/2 and 1/4.ย  Today we studied some of the more unusual ideas in the game, looking at positions that seem to have infinite and infinitesimal values.

Just to be super clear from the start -I’m not trying to be even remotely formal about the surreal numbers in this project.ย  Rather, I’m stating a few rules and ideas and we are exploring some simple consequences for fun.ย  The ideas here are something that I think that many many kids will find fascinating.

So, on to the game following the ideas and terminology in Jim Propp’s Life of Games.

The first thing we looked at was the “deep blue” checker.

The boys seemed to catch on to the idea that the deep blue checker had a value of infinity fairly quickly, though the idea that it was strange that a piece could have an infinite value didn’t become clear until the next part of the talk.ย  I was pretty happy to see that they wanted to explore what happened when a deep blue played against a deep red – that game seems shows that infinity minus infinity equals 0!

After the discussion about infinity in the last part of the talk, we explored some of the strange properties of these new numbers. First we looked at infinity + 1, which the boys assumed would be the same as infinity. Surprise ๐

For the last part of the project this morning we looked at a new stack – a blue with a deep red on top of it. I had to do a little bit of review of the game for my younger son at the beginning of this part of the talk, but once we got the rules straight he understood the strange property of the blue + deep red stack – it has a positive value, but that value seems to be less than any positive number we can think of.

So, a fun project showing the boys some neat, though odd, ideas from math. I love how easy it is to lay out some basic properties of the game “checker stacks” and have kids explore the implications of these properties. To me, this is what learning math looks like.

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:

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:

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.