A short project inspired by a Holly Krieger tweet

Saw this tweet from Holly Krieger this morning:

and, oh bother, the embedded tweet isn’t coming through. Here it is:

I thought it would be fun to talk about some sort of similar shape and stumbled on this neat design by eduardoviruena on Thingiverse:

eduardoviruena’s “Sierpinski cubes” on Thingiverse

sierpinski-copy

I sent the print to the printer as I was running out the door. Unfortunately in my haste I made the print much smaller than I intended. Oh well . . . we still got to have a fun conversation.

Here’s what he had to say about the eduardoviruena’s “Sierpinski cubes” – I was really interested in his description of how to make this shape:

Because the print was so small I wanted to see if he had any other thoughts when he saw a larger version on the screen:

I’m really excited to do more 3d printing projects like this one – hopefully giving kids shapes like this to play with will help them see a fun and exciting side of math 🙂

Extending our project on Kelsey Houston-Edwards’s Infinity video

Yesterday we did a project inspired by Kelsey Houston-Edwards’s latest math video:

Here’s a link to our project:

Sharing Kelsey Houston-Edward’s Infinity video with kids

Last night my younger son and I were talking a little bit more about the project and he asked me why Cantor’s diagonal argument for why the set of real numbers is larger than the set of Natural numbers doesn’t work for rational numbers!! Yes!!

We explored that question today. First we did a quick review of the diagonal argument (which was the last part of yesterday’s project) and then we began talking about the rational numbers:

Next we looked at what would happen if you applied the diagonal argument to the rational numbers:

After getting our arms around the diagonal argument when applied to rational numbers, we backed up and looked at the argument why rational numbers are countable.

Unfortunately I made an easy concept hard in this part of the project. I was trying to explain the “easy for me” idea that if a set that is larger than the rationals was the same size as the natural numbers, that meant the rationals must also be the same size as the natural numbers. My explanation started off terribly and went down hill . . . .

Finally we looked at one of the strangest consequences of all of this infinity stuff. In math language – the rational numbers have measure zero.

The idea here always blows my mind and is a really fun idea about infinity to share with kids.

Sharing Kelsey Houston-Edwards’s Infinity video with kids

The latest PBS Infinite Series video came out this week:

This is the 4th video in an incredible series from Kelsey Houston-Edwards. Our projects on the first 3 are here:

Sharing Kelsey Houston-Edward’s [higher dimensional spheres] video with kids

Sharing Kelsey Houston-Edward’s Philosophy of Math video with kids

Sharing Kelsey Houston-Edward’s Pigeonhole Principle with kids

I had the boys watch the new video together and started today’s project by asking them what they thought was interesting.

After hearing what the kids found interesting, we dove into the idea of bijections. We talked a bit about how a bijection has to work both ways using the bus idea from the video.

After the bus example we moved on to the example of the bijection between the points in an interval and points on the real line.

We finished up by talking about the bijection between the national numbers and the positive even integers.

Since we’ve done several prior projects where infinity played some role, the next thing I asked the kids was for some thing that they already knew about infinity – both things that they thought made sense and things that they thought didn’t make sense. The discussion and examples here were amazing – “no one knows what infinity divided by infinity is” 🙂

Finally, we wrapped up the project talking about why the infinity associated with the real numbers is larger than the infinity associated with the natural numbers.

I thought this would be a fun way to end the project since it was one of the key ideas in Houston-Edwards’s video:

So, another really fun project from the new set of math videos from PBS Infinite Series. I love this new series – can’t wait for the next one!

Playing with some 3d printed knots

Today we looked at some 3d printed knots designed by Laura Taalman and Henry Segerman.

Two are versions of Taalman’s “rocking knot” which we found here:

Laura Taalman’s Makerhome blog: Day 110 – the Rocking Knot

The second is the Torus knot from Segerman’s new book Visualizing Mathematics with 3D Printing.

We started the project today by just talking about the knots. Comparing the two knots that are actually identical was useful in refining the language they used to talk about knots.

Next they wanted to try to compare the two identical knots by looking at their crossings. My older son had the idea of assigning a +1 to every “over” crossing and a -1 to every “under” crossing. My younger son noticed that this counting method should always produce a net 0 because we counted the over and under crossing for each crossing exactly once.

New we tried to compare Segerman’s torus knot to Taalman’s rolling knot. Here we used the “tangle” from Colin Adams’s book Why Knot?

One fun thing that came up by accident in this video is an amazing shadow cast by Taalman’s knot – that was a really fun surprise.

Unfortunately, it proved to be a bit difficult to get the tangle back together so we had to pause the video at the re-connect the tangle off camera. It is really neat, though, to watch kids try to make a copy of a knot.

Once we got the tangle connected we started the next video. Since the tangle can move around, it isn’t that hard to manipulate the tangle from the form Segerman’s knot to the form of Taalman’s knots. In fact, it happened more or less by accident!

As I mentioned above, it is actually a pretty difficult task for the kids to describe the features of the knots when they compare them – even with a knot as simple as the trefoil knot. I think one of the neat parts of this particular project is working on using more precise mathematical language.

So, a fun project. We have a new 3d printer and I’m really excited about using many more 3d printing ideas from Taalman and Segerman to explore math with the boys.

Talking about Henry Segerman’s 5-cell with my 5th grader

Last night we printed a shape from Henry Segerman’s new 3d printing book Visualizing Mathematics with 3D Printing:. We’ve done many project based on Segerman’s work and even were lucky enough to be able to attend his talk at MIT earlier this fall:

segerman

The shape we printed last night is Henry’s 3d representation of the 5-cell – a 4 dimensional “platonic solid” ( You can read more about the shape here: The Wikipedia page for the 5-cell)

If you search “Segerman” in the blog you’ll find more than 10 projects we’ve done based on his work!

I started off the project today by asking my younger son for some thoughts on the 5-cell:

One interesting thing that he remembered is that he’d seen the shape previously in some of our bubble projects, so we brought out the bubble solution to make the shape out of bubbles. It was really interesting to hear how he viewed the two shapes differently.

Sorry for the absolutely awful camera work in this video – you’d think I’d have gotten the hang of this after 4,000 videos . . . . .

A challenge relating to a few problems giving my son trouble

I’ve seen some interesting ideas from Tracy Johnston Zager over the last week about the relationship between learning math and intuition. For example:

Although I’ve been traveling a bit for work this week the relationship between learning math and intuition has stayed in my head. Sometimes my thoughts have drifted to and old blog post about a problem from the European Girls’ Math Olympaid:

A Challenge / Plea to math folks

That post, in turn, was inspired by an old post by Tim Gowers where he “live blogged” his work while he solved a problem from the International Mathematics Olympiad.

It can be really hard for anyone to know what math intuition looks like because everyone sees polished solutions way more often than they see the actual process of doing math.

That’s part of the reason I make the “what a kid learning math can look like” posts – so everyone can see that the path kids (or anyone!) actually takes to the solution of a problem is hardly ever a straight line:

Our “what a kid learning math can look like” series

The other thing on my mind this week has been some old AMC 10 problems that have really given my older son some trouble. These are pretty challenging problems and require quite a bit of mathematical intuition to solve.

So, I’d like to make the same challenge with these problems that I made with the problem from the European Girls’ Math Olympiad – “live blog” yourself solving one of these problems. Post the though process rather than a perfect solution. Let people see *where* your mathematical intuition came into play.

problem-19

 

problem19

problem

Playing with Dan Anderson’s rotation program

I’m not sure of the history, so apologies for surely leaving out credit for everyone involved, but Dan Anderson published an really neat math program last night:

Dan Anderson’s rotating polygon’s program

This morning I had my younger son play with the program – it is just amazing to watch and hear what kids have to say. It is also amazing to play with this program yourself!

Part 1:

Part 2:

A nice AMC 10 angle bisector problem

I love how the AMC 10 problems combine multiple ideas. A problem from the 2010 AMC 10 a gave my son some trouble today because he didn’t understand one of the words and also because he needed a little review with the angle bisector theorem. It made for a good morning math talk. Here’s the problem:

 

Problem16.jpg

We started off talking about the problem to understand what “nondegenerate” meant also ended up getting to a point where the Angle Bisector Theorem was the stumbling block:

Since the Angle Bisector Theorem was the stumbling block – we took a short detour to review that theorem:

Following that short review (and unfortunately it had to be short since he needed to get out the door to go to school) we revisited the problem:

So, another great AMC 10 problem. I love how these problems combine different concepts and highlight areas to review. I’d never be able to make up problems like this one on my own.

A nice Simpson’s Paradox example for kids

Last night I was looking for a project for today and grabbed this book off the shelf:

In the middle of the book I found a really nice Simpson’s Paradox example and tried it out today with the kids. For more on Simpson’s Paradox, the Wikipedia page is actually great:

Simpson’s Paradox on Wikipedia

So, here’s the first part of today’s project – we have 4 boxes that have fixed amounts of red and blue cubes inside of them. First we divide them into two groups of 2 and ask which one in each group gives you a better chance of selecting a red block. It turns out that this is also a good introductory fraction exercise for kids, too!

Next we see the “paradox.” We combine the two winning boxes into one box and combine the two losing boxes into one box. Now which of the two remaining boxes gives you a better chance of selecting a red block?

So a fun and strange example for kids to see. Again, the Wikipedia page linked above gives a few more fun (and famous) examples. Really happy to have found this example in Moscovich’s book last night!

Sharing Kelsey Houston-Edwards’s Pigeonhole Principle video with kids

The 3rd video in Kelsey Houston-Edwards’s amazing new series was published last week. I’ve already used the first two videos for projects with the boys – I love this series so much!

Sharing Kelsey Houston-Edward’s [higher dimensional spheres] video with kids

Sharing Kelsey Houston-Edward’s Philosophy of Math video with kids

the latest video is about the Pigeonhole Principle and begins with the question – Do any two human beings have exactly the same number of body hairs:

Before diving into the video I asked the boys what they thought about the hair question – fortunately I got two different answers!

Next we watched Houston-Edwards’s new video:

Here’s how the boys reacted to the video:

(1) They were excited about the hair result and were also able to understand and explain it.

(2) They gave a nice summary of the Pigeonhole Principle.

(3) They really liked the example about 5 points on a sphere, so we took a really close look at that example. One of the tricky parts of that problem is understanding *why* you can draw an equator through any two points – both kids gave nice explanations of that idea.

Now I moved on to a couple of fun Pigeonhole Principle examples that weren’t covered in the video. I wanted to show the boys that the idea comes up in lots of different situations, including some that are not at all obvious Pigeonhole Principle situations!

The first example comes from my college combinatorics textbook – Applied Combinatorics with Problem Solving by Jackson and Thoro:

Screen Shot 2016-12-03 at 8.47.57 AM.png

Small twitter math world fun fact – the professor for this class (~25 years ago!) was Jim Propp!

Here’s the problem (which is example 5 on page 35 of the book):

Suppose that we are given a set X of 10 positive integers, now of which is greater than 100. Show that there are two disjoint nonempty subsets of this set whose elements have the same sum.

I had to do a little bit of work on the fly to translate the problem into something that the boys could understand (and also explain quickly why there are 1024 subsets), but it seemed like they enjoyed this example:

The last problem is one I remembered when reading through some of the other examples in Jackson and Thoro’s book and is one that I talked about with the boys last year:

A challenging arithmetic / number theory problem

Here’s the problem:

Show that every positive integer has a multiple whose base 10 representation consists of only 1’s and 0’s.

It certainly isn’t obvious at all at the start why this is a Pigeonhole Principle problem!

As I said at the beginning – I love this new series from Kelsey Houston-Edwards. I’m so happy to be able to use these videos to explore fun mathematical ideas with my kids!