What kids learning math can look like -> working through some ideas in one of Burkard Polster’s books

Yesterday I was given a copy of one of Burkaard Polster’s books (he is the face of the Mathologer series on youtube):

This morning I had the boys each choose a chapter that they found interesting and we walked through the proof that Polster in that chapter.

My younger son went first – the idea he found interesting was dividing up a square in different ways. Here’s the introduction and an explanation of two of the four ideas:

Here’s how we finished up the last two proofs:

Next my older son found a neat proof relating the volume of a sphere to the volume of a cylinder and a cone. He struggled a little bit to understand the proof, but the struggle that takes place in this and the next video is a great way to see how kids learn and think about math.

Just so there’s no confusion, the formula he derives for the area of the slice of the cylinder / cone we are looking at isn’t right. He’ll discover the mistake and correct it in the next video.

Here we find the second formula that we need to show how the volume of a sphere relates to the volume of the cylinder / cone combination.

Finally, we revisited an old 3d print that we had showing the relationship between the volume of a sphere, cylinder, and a cone. The print is designed by Steve Portz and is on Thingiverse here:

“Archimedes Proof” by Steve Portz on Thingiverse

Finding the coordinates of the vertices of a Tetrahedron and an Octahedron

My younger son is starting to learn about coordinates in 3 dimensions. I thought that spending a little time finding the coordinates of the corners of a tetrahedron and an octahedron would make for a nice project this morning.

We started with the tetrahedron and found the coordinates for the bottom face. Once nice thing about the discussion here was talking about the various choices we had for how to look at the tetrahedron:

Having found the coordinates for the bottom face, we now moved on to finding the coordinates for the top vertex:

Now we moved on to trying to find the coordinates for the corners of the octahedron. Here the choices for how to orient the object are a little more difficult:

Finally, we talked through how we would find the coordinates of the octahedron if we had it oriented in a different way. This was a good discussion, but was also something that confused the boys a bit more than I thought. We spent about 10 min after the project talking through how to find the height. Hopefully the discussion here helps show why this problem is a pretty difficult one for kids:

Sharing Laura Taalman’s slices with my younger son

I had an opportunity to visit Laura Taalman at ICERM today who made me a copy of her latest creation. I couldn’t wait to get home to share it with my younger son:

He had some interesting ideas about what the shape was in the last video. Now I shared where the slices came from and had him explain how Taalman’s creation worked:

This is such a fun way for kids to experience shapes from a different point of view. I’m really excited to see if we can create some similar objects to play with!

Labeling each vertex of a permutahedron is a terrific mathematical exercise for kids

Yesterday we did a fun project exploring a permutohedron:

A Morning with the permutohedron

Last night I thought it might be neat to have the kids try to label the vertices of a permutohedron with the permutations represented by each vertex. Fortunately, it was possible to build a truncated octahedron with the green Zometool struts:

We started out today’s project by talking about the rules for making a permutohedron in different dimensions. Here I used the labeling of the permutation of 3 objects as a base case to make sure the boys understood the directions properly.

Next I had the boys label each vertex of the permutahedron with the permutation of {1,2,3,4} that the vertex represented. Then, they talked about the process of figuring out the right labels.

I’m sorry that the video below runs 10 min, but if you listen to the whole discussion I think you’ll see that seemingly straightforward act of labeling these vertexes is a terrific mathematical exercise for kids.

A morning with the permutohedron

Today we are revisiting an old project on a really neat shape -> the permutohedron:

“A fun shape for kids to explore – the permutohedron

I learned about this shape thanks to Allen Knutson at Cornell – he included a fun pic of a large permutohedron in the comment of the blog post above:

permutohedron

He also pointed me to a 3d print on Thingiverse that we used in the last project and again today:

“Permutahedron” by PFF000 on Thingiverse

So, I started today by having the boys describe the 3d printed shape. We have two versions – a larger one that unfortunately broke a little and a smaller – but in one piece! – version. Here’s what the boys had to say about the shapes:

Next I had the boys read the Wikipedia page on the permutohedron for about 10 min and then we discussed some of the ideas that they thought were interesting:

Finally, we built the 2-D permutohedron and showed how it was embedded in a 3d grid:

Definitely a fun project and it is always great to be able to have kids hold interesting math ideas in their hands!

Sharing Vladimir Bulatov’s Tetrahedral Limit Set with kids

Last week I saw a really neat tweet shared by Alex Kontorovich:

I ended up buying Bulatov’s piece from Shapeways and it came today. Here’s a quick video look at it:

When the kids got home from school I asked them to take a look at it and share their thoughts.

Here’s what my younger son had to say about the shape:

Here’s what my older son had to say:

Sharing a new “picture” of the Milky Way with kids

I saw a new image of the Milky Way last week thanks to these two tweets:

After seeing the first tweet from Kayley Brauer I was hoping to find a way to talk about this new result with the boys, but didn’t really know what to do. Thanks to the tweet from Dr. Chanda Prescod-Weinstein, I learned that the LA Times had put together a terrific presentation that was accessible to kids.

First, I had my older son read the article on his own and then we talked through some of the ideas he had after reading it.

I thought that reading the article on his own would be a little too difficult for my younger son (he’s about to start 8th grade) so instead of having him read it on his own, we went through it together:

Obviously I’m not within 1 billion miles of being an expert on anything related to this new image of the Milky Way, but it was still really fun to talk about it with the boys. I’m very happy that advanced science projects like this one are being shared in ways that kids can see and experience.