# Struggling through a challenging AMC 10 problem

This problem gave my son some difficulty yesterday – it is problem #19 from the 2011 AMC 10a

Last night we talked through the problem. The talk took a while, but I was happy to have him slowly see the path to the solution. Here’s his initial look at the problem:

Next we looked at the equation $y^2 - x^2 = 141$. Solving this equation in integers is a nice lesson in factoring. Unfortunately by working a bit too quickly he goes down a wrong fork for a little bit.

In the last video we found that the original population of the town might have been 484, and it might have changed to 634 and then once more to 784. We had to check if 784 was a perfect square.

Finally, we needed to compute the approximate value (as a percent) of 300 / 484. The final step in this problem is a great exercise in estimating.

So, a really challenging problem, but also a great problem to learn from. We went through it one more time this morning just to make sure that some of the lessons had sunk in.

# Sharing Kelsey Houston-Edwards’s philosophy of math video with kids

Kelsey Houston-Edwards is making a series of math videos and the first two are outstanding. We looked at the first one last week:

Sharing Kelsey Houston-Edward’s video with kids

This week’s video is about philosophy and math. A deep subject, for sure, but one which the kids thought was interesting. Here’s the video (and the twitter link so you know when the new videos appear!):

Here’s my older son’s reaction and a few things he thought were interesting:

and here’s what caught my younger son’s eye:

It is so great to see someone doing such an incredible math outreach program. I’m so excited about this video series!

# Revisiting our Zometool Snowman

When we first moved into our house we did a couple of fun and large Zometool projects because we didn’t have any furniture ðŸ™‚

This week I saw a fun tweet from Eli Lubroff that reminded me of one of those projects:

Here’s a part of that old project ðŸ™‚

Today we revisited that old snowman and had the boys talk about each of the Archimedean solids in the shape. This is a fun project – not just because the shapes themselves are cool – but you get a nice opportunity to talk about counting and symmetry. You’ll see in the videos that my older son is a bit more comfortable with the idea, but my younger son seems to catch on by the 3rd video.

Here’s a link to all of the Archimedean solids on Wikipedia:

The Archimedean Solid page on Wikipedia

And here’s our project:

First the bottom of the snowman – the Truncated Icosidodecahedron

Next was the Rhombicosidodecahedron

Next was the Icosidodecahedron

Finally the Archimedean Solid Snowman ðŸ™‚ Two years later and he still fits!

Definitely one of my all time favorites and a really fun way to discuss counting and symmetry!

# Fun with rotopo

Saw this neat tweet from Jim Propp yesterday:

After playing the game for a just a few minutes I knew that my kids would love it.

Here’s each of their reaction to seeing and playing the game.

My younger son first:

My older son next:

So, definitely a fun little game for kids. They need to be fairly fluent with the arrow keys on the keyboard, but that’s really all that’s required. Definitely some fun puzzles to solve!

I happened to see the Raspberry Pi set below at a store earlier in the week:

Â Today I showed it to the kids and we played with it for a bit. Here’s their initial reaction:

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There wasn’t much to assemble which was nice. We did have some trouble connecting it to my laptop, so we switched to connecting it to our TV. Here’s what the boys had to say after we got it running:

Finally, the kids discovered a few of the games that came loaded on the computer. This squirrel one made them laugh. From start to finish was about an hour – and at least 20 minutes of that was trying and failing to get connected to the laptop.

We’ve only scratched the surface of what was in the kit. I’m excited to have the boys play around with the computer a bit more. It has Mathematica (yay!) and some software for introductory programming like Scratch. Hopefully there will be many more projects to come.

# Why I love watching my kids learn math

Had a great night with the boys tonight. My older son was working on some old AMC 10 problems and we talked through one that stumped him for his movie:

It was #15 from the 2013 AMC 10a:

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Next I spent some time with my younger son. He’s been studying the basics of lines using Art of Problem Solving’s Introduction to Algebra book plus a little bit of Khan Academy (when I’ve been traveling for work). I asked him what he’d learned so far and loved his response. It was a great reminder of the joy of learning new math ideas for the first time:

# A terrific problem from Matt Enlow

Saw a fascinating problem and request from Matt Enlow when I was on the road on Thursday:

Everyone always talks about thinking through problem carefully – here’s a great problem and a great opportunity to give some examples of that kind of deep thinking.

Give him some feedback!

# Dave Richeson’s Knotted bubbles project

Saw this tweet from Dave Richeson last week which basically “had me at hello”:

We’ve done a few bubble projects in the past, so the boys were already familiar with the basic concept:

Zometool and Minimal Surfaces

Trying out 4 dimensional bubbles

More Zome Bubbles

Anyway, I ran out to home depot and got some wire and we made some knots. I had each of the boys make a trefoil knot and then make a random knot of their own choosing. In retrospect I wish I’d spent maybe just 5 minutes explaining some of the ideas in Richeson’s blog post – oh well, the excitement got the better of me ðŸ™‚

Here’s my older son playing with his trefoil knot and making a Mobius strip bubble. I love the “hey, I actually think I got it” moment:

Here’s him playing with the knot me made – in retrospect I’d argue for a knot that was slightly less complicated:

Next up was my younger son. First up was the trefoil knot and we got another great moment “I think this might be a Mobius strip” !!

Finally we made his own knot and explored. Again, I’d probably ask for a less complicated knot if I was doing this again:

So, that so much to Dave Richeson for posting his old project – this is an incredible project, and an especially great one for kids. The appearance of the Mobius strip is really quite an amazing little math miracle!

# Sharing Kelsey Houston-Edwards’s video with kids

I saw this amazing video from Kelsey Houston-Edwards yesterday:

I wrote a bit about it last night because I was so excited about it:

An amazing new set of math videos from Kelsey Houston-Edwards

Here’s what my kids thought was neat about the video:

My younger son thought the “central sphere” problem was fun, and my older son thought the shape of the n-dimensional spheres was fun. I originally intended to talk about both this morning, but our talk about the shape of the spheres took enough time for one project.

So, below is our initial look at the shape of the spheres. There’s a lot of nice introductory geometry (and fractions!) in the discussion. Also, I made the choice to talk about 1/2 the length of the long diagonal – that choice sort of confused the kids, so I’d focus on the full length if I was doing this again.

Finally, we talked about how the diagonal changes as you go up in dimension. This is a fairly straightforward application of the Pythagorean theorem, so it isn’t that hard to talk about. The boys saw the pattern fairly quickly.

Then I introduced the volume formula for even-dimensional spheres and we calculated the ratio of the volume of a 30-dimensional sphere to the 30-dimensional box it is inscribed in.

I’m super excited for this new series of videos from Kelsey Houston-Edwards, and I can’t wait to share the next one with my kids!

# An amazing new set of math videos from Kelsey Houston-Edwards

I happened to see this old tweet from Steven Strogatz today:

Here’s a direct link to the video:

I was super excited to see this new work from Kelsey Houston-Edwards since, for one(!), I was really hoping that mathematicians would publicize the sphere packing result and find ways to make it accessible to the general public:

A challenge for professional mathematicians

I was fortunate to be able to attend Maryna Viazovska’s talk at Harvard and Henry Cohn’s talk about BU about the new sphere packing results. Although I don’t have nearly the mathematical sophistication to be able to write about the result in any detail, both talks were great. There’s also a link in the post below to a nice talk from Henry Cohn about the history of the sphere packing problem, but I think that’s as close as you can get to the problem without diving into very heavy math:

Maryna Viazovska’s Sphere Packing talk at Harvard

Because of the difficulty of the problem I haven’t been able to figure out to much to do with kids – but I did try two projects in 2 dimensions:

The 2-dimensional version of the sphere packing problem is a fun problem to explore with kids:

Sphere packing – well . . . Circle packing with kids

Using a Natalie Wolchover article to talk about hyperuniform distributions kids

The two higher dimensional sphere problems that Houston-Edwards discusses in the 2nd part of the video are ideas that are accessible to kids.

An old tweet from Steven Strogatz had inspired me to try to talk to kids about the area and volume of circles in different dimensions:

Showing the kids about the area of a circle

We’ll try a new project tomorrow to understand some of the volume properties mention in Houston-Edwards’s video.

I learned the problem about the central spheres from Bjorn Poonen earlier this year and wrote about it here:

A Strange Problem I overheard Bjorn Poonen discussing

Bjorn Poonen’s n-dimensional sphere problem with kids

A fun surprise in Bjorn Poonen’s n-dimensional sphere problem

I called the problem “Bjorn Poonen’s n-dimensional sphere problem” because I learned about it from him. So, to be 100% clear, he wasn’t taking credit for the problem. I learned later from Alexander Bogomolny that at least the two parts of the problem discussed in the video were attributed to Leo Moser

I don’t know the origin of the 3rd part of Poonen’s problem.

The “fun surprise” in the last post discusses an unexpected relationship between $\pi$ and $\latex e$ that makes the 3rd part of Poonen’s problem work.

Anyway, I’m really excited for this new video series – can’t wait to see what comes next!