A challenging counting problem from the 2011 AMC 8

My older son has been preparing for the AMC8 and this problem from 2011 gave him a little trouble:



We talked through it this morning and he was still a little confused about why his original answer isn’t correct. The error is pretty subtle – especially for a test that gives you roughly 2 min per problem:

So, we talked for a bit more and he was able to find some numbers that he counted that did not fit the requirements of the problem:

Finally, when he got home from school tonight we revisited the problem and counted the number of solutions directly:

I like this problem a lot – it is a great one for helping you learn how to count carefully!

Steven Strogatz’s circle area project – part 2

Yesterday we did a really fun project inspired by a tweet from Steven Strogatz:

Here’s tweet:

Here’s the project:

Steven Strogatz’s circle-area exercise

During the 3rd part of our project yesterday the boys wondered how the triangle from Strogatz’s tweet would change if you had more pieces. They had a few ideas, but couldn’t really land on a final answer.

While we punted on the question yesterday, as I sort of daydreamed about it today I realized that it made a great project all by itself. Unlike the case of the pieces converging to the same rectangle, the triangle shape appears to converge to a “line” with an area of \pi r^2, and a lot of the math that describes what’s going on is really neat. Also, since my kids always want to make Fawn Nguyen happy – some visual patterns make a surprise appearance 🙂

So, we started with a quick review of yesterday’s project:

The first thing we did was explore how we could arrange the pieces if we cut the circle into 4 pieces.

After that we looked for patterns. We found a few and my younger son found one (around 4:09) that I totally was not expecting – his pattern completely changed the direction of today’s project:

In this section of the project we explored the pattern that my son found as we move from step to step in our triangles. After understanding that pattern a bit more we found an answer to the question from yesterday about how the shape of the triangle changes as we add more pieces.

Both kids thought it was strange that the shape became very much like a line with a finite area.

The last thing that we did was investigate why the odd integers from 1 to N add up to be $late N^2$. My older son found an algebraic solution (which, just for time purposes I worked through for him) and then we talked about the usual geometric interpretation.

So, a great two day project with lots of fun twists and turns. So glad I saw Strogatz’s tweet on Friday!

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 🙂

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.

What a kid learning math can look like – fraction confusion

My younger son is currently in the review section on percents in his algebra book. Last night he chose a fairly standard problem on percents for our movie. The arithmetic with fractions tripped him up a little, though. The first video shows his struggle:

After we finished the problem I decided to propose an alternate solution just to get him a second round of fraction practice. His work here was really good, but the fraction arithmetic at the end still also gave him a tiny bit of trouble:

You never know what’s going to give a kid difficulty and it is interesting to watch them try to work through these unexpected struggles.

An absolutely mind blowing project from James Tanton

I was flipping through James Tanton’s Solve This last night and found a project I thought would be fun. I had no idea!

We’ve done a few paper cutting projects before. For example:

Cutting a double Mobius strip

Tadashi Tokieda’s “World from a Seeht of Paper” lecture

Our one cut project

Fold and cut project #2

Fold and cut part 3

Fold and Punch

and this list also need this incredible project from Joel David Hamkins:

Math for nine year olds: fold, punch and cut for symmetry

So, having see a few folding / cutting project for kids previously the projects about Möbius strip in Chapter 8 of Tanton’s book caught my eye. As I said above, though, I had no idea how cool this project would turnout to to be!

We started with the standard project of cutting a Möbius strip “in half”. What happens here??

Oh, and before getting to the move – most of the movies below have a lot of footage of us cutting out the shapes. I was originally going to fast forward through that, but changed my mind. The cutting part isn’t that interesting at all, but I left it in to make sure that anyone who wants to repeat this project knows that the cutting part (especially with kids) is a tiny bit tricky. You have to be careful!

Now, once we’d done the cut the boys were still a little confused about whether or not the result shape had one side or two. I thought it would be both important and fun to make sure we’d resolved that question before moving on:

With the Möbius strip cutting out of the way we moved on to what Tanton describes as “a diabolical Möbius construction”. All three shapes start as a thin cylinder with a long ellipse cut out. You then cut and twist the strips outside of the ellipse making a Möbius strip-like component of the new shape. Hopefully the starting shapes will be clear from the video.

Try to guess what the resulting cut out shape(s) will look like prior to them getting cut out 🙂

The first shape involves putting one half twist into one of the strips left over after cutting the ellipse out of the cylinder.

The second shape also starts as a cylinder with a long ellipse cut out. This time, though, we make put half twists (in the same direction) in both of the long strips that are left over after cutting out the ellipse.

Sorry about the camera being blocked by my son’s head a few times – oops!

The final shape for today’s project is similar to the second shape, but instead of two half twists going in the same direction, they go in opposite directions.

All I can say is wow – what an incredible project for kids. Thanks James Tanton!!

Taking kids through John Baez’s post about the Gyroid

[sorry for no editing on this one – had some computer problems that ate up way too much time. I finished typing with 2 minutes to spare before rushing out the door.]

I saw this neat tweet from John Baez earlier in the week:

You should be able to click through to Baez’s blog post from the tweet, but just in case that isn’t working, here’s the link:

The Butterfly, the Gyroid and the Neutrino by John Baez

I spent the rest of the week sort of day dreaming about how to share some of the ideas in the post with kids. Last night the day dreaming ended and I printed a gyroid that I found on Thingiverse:

The specific gyroid that I printed is here:

Alan Schoen’s Gyroid on Thingiverse by jamesosaurus

This project connects with several of our prior projects on 3d printing (particularly the recent ones inspired by Henry Segerman’s new book) as well as projects on minimal surfaces. Though the list below is hardly complete, here are a few of those projects:

Zometool and Minimal Surfaces

Trying out 4 dimensional bubbles

More Zome Bubbles

Playing with shadows Inspired by Henry Segerman

Playing with more of Henry Segerman’s 3d Prints

Henry Segerman’s flat torus

Using Hypernom to get kids talking about math

So, with that introduction – here’s what we did today.

First we revisited the zome bubbles to remind the kids about minimial surfaces – it is always fun to hear kids describe these complicated shapes:

Next we looked at the Gyroid that I printed last night. This shape is much more complicated than the zome bubbles and the kids sort of had a hard time finding the words to describe it – but we had a similar shape (and I don’t remember why or where it came from) that helped the kids get their bearings:

So, after playing with the blue shape for a bit and seeing some of the symmetry that this shape had (yay!) we returned to the Gyroid. The boys still struggled to see the symmetry in the gyroid (which is really hard to see!) but we made some progress in seeing that not all of the holes were the same:

Finally, we turned to Baez’s article to see the incredibly surprising connection with butterflies and physics. There’s also a fun connection with some of the work we’ve done with Bathsheba Grossman’s work and Henry Segerman’s 3D printing book:

So, a fun project. I love how 3d printing helps open up advanced ideas in math to kids. After we finished the boys kept reading Baez’s article to find the connection with neutrinos – it is really gratifying to see how engaged they were by today’s project!

Playing with Bathsheba Grossman’s “Hypercube B”

Last week we went to Henry Segerman’s talk at MIT:


During that talk one of the 3d-printed objects he passed around “Hypercube B” by Bathsheba Grossman. I was excited to have the boys play around with this shape a bit more so we ordered a copy from Shapeways:

Hypercube B by Bathsheba Grossman on Shapeways

It turns out that Grossman is also the creator of “Double Zarf” which we ordered previously and my older son discussed here:

Exploring some shapes from Henry Segerman’s new book

Hypercube B arrived in the mail today and I was really interested to hear what the boys had to say about the shape. My older son gets home from school first, so he was the first one to play with it:

My older son recognized the shape from the talk (which was fun). My younger son didn’t right away, but he came to believe that it was a hypercube because of some of the projections that it made (when viewed in the camera lens!). I loved hearing what he had to say about the shape!

In the next couple of days we’ll also play around and see what sorts of (2d) shadows this shape makes. Can’t wait to do that!

Martin Hotham’s spirals

Saw a great tweet from Martin Hotham last night:

I thought it would be fun to let the boys play around with this activity. We’ve actually talked about spirals before a little. First with our Zome set:

Fibonacci Spirals and pentagons with our Zometool set

Then with Anna Weltman’s “loop-de-loops”

Anna Weltman’s Loop-de-loops

and Weltman’s project spiraled us into some of Frank Farris’s work:

Extending Anna Weltman’s loop-de-loops with Frank Farris’s “Creating Symmetry”

The difficulty with the math behind the spirals is that it’ll be a few years before either of the boys encounters trigonometry. However, Hotham’s Desmos programs are so stunning and so easy to use that I’m fine just letting the boys play with them.

My younger son played around for 5 minutes – here’s what he had to say:

Next I let my older son play around – here’s what he had to say when he was done:

We’ve played around with one of Hotham’s creations before – when he created a program based on Ann-Marie Ison’s art:

Extending our project with Ann-Marie Ison’s art”

I love his work and per the law “any sufficiently advanced technology is indistinguishable from magic” I can only assume that he’s some wort of wizard. Can’t wait to see what he comes up with next!

Playing with Shadows – inspired by Henry Segerman

Last night the whole family attended Henry Segerman’s talk at MIT:


Here’s Henry in action:

He’s already inspired a ton of projects, for example:

Playing with more of Henry Segerman’s 3d Prints

Henry Segerman’s flat torus

And this joint work with Vi Hart, Andrewa Hawksley, Henry Segerman, Emily Eifler and Marc ten Bosch got a big reaction from the crowd last night:

Using Hypernom to get kids talking about math

As a follow up to Henry’s talk last night we played with shadows of a cube and a tetrahedron this morning. First up was the tetrahedron – it is really fun to hear what the kids have to say about the shadows.

Also, here’s the “hypercube B” shape that my older son mentions:

Hypercube B on Shapeways

After talking about the tetrahedron shadows we moved on to the cube. Again, it is really fun to hear what the kids have to say about these shapes. Several of the questions I asked about the cube shadows came from Henry’s talk:

Finally, we looked at one of Henry’s prints. We’ve studied this print before (and my younger son even brought it to the talk to show Henry!). I wanted to use this shape to explore the question:

“If two lines come together in the shape, do they have tocome together in the shadow?”

So, a fun talk last night and a fun project this morning. I want to live in a world where all kids have the opportunity see Henry’s work!

Examples of kids and problem solving

Both kids gave nice examples of the problem solving process in the two videos we did last night, so I wanted to highlight those videos with a short blog post.

First up was my younger son. He’s learning algebra this year and has a really nice way of thinking and talking through problems. I love how deliberate he is and how he discovers his own mistakes. The problem that he’s working on here is to find 3 solutions to the equation 3A – 5B = 9.

Next up was my older son. The problem he’s working on is an old Mathcounts problem, and it is pretty challenging:

What fraction of the first 100 triangular numbers are divisible by 7?

His work is a nice example of, for lack of a better phrase, the discovery process. Initially he does not see how to solve the problem, but I love his path to the solution.

After he finished I showed him two other approaches to solving the problem, just to help him see how a few other ideas in math can connect to this problem:

I wanted to share these examples to show that problem solving in math isn’t all about speed. A slow, deliberate process is a great way to get to the solution of a problem.