A terrific prime number question from Matt Enlow

A great question from Matt Enlow inspired a super fun conversation with the boys last night:

Before diving in to the project, I’d really recommend thinking about the question – even just for a few seconds – just to see what your intuition tells you.

We started the project by looking at the tweet and trying to make sure that the boys understood the question. The question itself was harder for them to understand than I expected. One reason was that they weren’t used to thinking about ages in terms of days.

Next we went to Mathematica and wrote a little program using the “PrimePi” function which tells you the number of primes less than or equal to a number.

We played around a little bit. Their initial instinct was to zoom in on a specific number like 30 years old. There were some fun surprises since the number of primes between two numbers bounces around a bit. They also had some really interesting ideas about prime numbers.

Eventually they decided to check a range of ages.

At the end of the last video we decided to check a range of ages, and we did that with a “For” loop. Once we did that we found a couple of really fun surprises 🙂

Running the program over night, the largest age that I found was 179,676 years old! I doubt that’s the highest number, though, and I love that the boys thought that there might be infinitely many solutions to this problem.

Thanks to Matt Enlow for posing this problem!

A neat geometry problem I saw from David Butler

I saw this problem today when it was re-tweeted by Matt Enlow:

It is a little advanced for my younger son, but I still thought it would be fun to turn into a mini project tonight with the boys.

We started by talking through the problem and taking a guess at what we thought the answer was -> Is there enough information to determine the side length of the square?

Although we didn’t really make any progress towards a solution in this initial discussion, I really like the ideas that we talked about. Specifically, I liked how much thought my older son put into how to label the diagram.

In this part of the project we began to discuss how to solve the problem. We found two equations, but had 3 variables. My older son began to think that we weren’t going to find a solution.

In trying to simplify one of our equations my younger son made a common algebra mistake. I spent most of the video slowly showing him how to tell that the algebra he thought was right was actually off.

At the end of the last problem we found an equation that seemed to be a step in the right direction of finding a solution to the problem. In this part of the project we explored that equation.

At the beginning my older son was really confused. I think he’s used to seeing problems where there is always a solution – the open endedness of this problem seemed to leave him puzzled.

We did get our sea legs back, though, exploring a few specific cases. The happy accident was that the two solutions we found to the problem gave us the same perimeter for the square – was a unique solution hiding here?

To wrap up the project we went up to the computer to look at our equation using Mathematica. We’d covered the important mathematical ideas already, but finding some of the exact solutions was going to be a chore and certainly finding the maximum perimeter wasn’t going to be in reach.

Nonetheless, there were a few fun surprises to be found 🙂

Sharing Stephen Wolfram’s MoMath talk with kids

I saw an amazing tweet from Stephen Wolfram today:

Based on the blog post, his talk at MoMath must have been incredible!

I decided to try out one of his explorations with the boys tonight. We did the first few parts by hand and the last part using Mathematica and the code from Wolfram’s blog post.

The process we studied works as follows:

(1) Pick an integer to start with and pick a number $n$ to multiply by in step (3),

(2) Cycle the digits of the number to the left. A few examples will make the process clear:

123 goes to 231
402 goes to 024, or simply 24
111 would stay 111

(3) Multiply the new number by $n$ and then add 1.

The video below shows how our exploration began. Our initial integer was 12 and we multiplied by 1 at each step (so, starting easy, though I picked 12 at random so I really didn’t know what was going to happen):

Now we moved to a slightly more complicated example -> the same process as in the first part but we’ll be working in binary rather than in base 10.

We started with the number 6 (110 in binary) and multiplied by 2 at each step. Once again we found a fun surprise:

To get one more round of practice in before moving upstairs to the computer we looked at the same situation as in part 2, but this time starting with 1 and looking at several cases – multiplying by 1, by 2, and by 3:

Finally, we went to the computer to explore the process in many different situations. We used code from Wolfram’s blog post to recreate the work from the MoMath talk:

What I *love* about this project is that the exploration works really well with kids on the whiteboard and on the computer. The whiteboard exploration gave us a great opportunity for a little practice with arithmetic, with binary, and with algorithms. We also saw some really fun surprises!

The computer exploration is obviously fantastic, too. I’m so grateful that Stephen Wolfram shared the ideas from his talk!

A morning with the icosidodecahedron thanks to F3

A few weeks a go I saw this shape in a display case at the MIT math department:

The shape is mislabled, unfortunately, it is an icosidodecahedron. We’ve already done a few projects based on the shape. Last week’s project is here:

A zometool follow up to our Cuboctohedron Project

And there’s our fun Zometool Snowman, too, where the icosidodecahedron is the head:

The name suggests that it is made from a combination of an icosahedron and a dodecahedron – but how?

(i) the icosidodecahedron
(ii) a dodecahedron with an icosahedron removed
(iii) an icosahedron with a dodecahedron removed

Here’s what the boys thought of those shapes:

Next we went upstairs to play with some code in the F3 program. Looking at the video now I see that I forgot to publish it hi def – sorry about that. I hope our explanation of the code is good enough if the code is too fuzzy to read:

Definitely a fun little project – it is so fun to be able to play with these shapes on the computer and then hold them in your hand!

Sharing advanced ideas in math with kids via 3d printing

Yesterday (after a few false starts!) I printed several different versions of the torus in different L^p metrics. Here they are next to spheres in the corresponding metric

The idea was inspired by an old project that was inspired by a Kelsey Houston-Edwards video

Sharing Kelsey Houston-Edwards’s Pi video with kids

Prior to the prints finishing I talked through some of the shapes as they appeared on the computer with my younger son:

Exploring different L^p versions of the torus

When the various torus prints were done I asked each of the boys to tell me what they thought about the shapes. I love how 3d printing allows you to share advanced ideas about math with kids so easily!

Here’s what my younger son had to say:

Here’s what my younger son had to say:

These are the kinds of math conversations that I’d like to have with kids.

Exploring different L^p versions of the torus

A few weeks ago we did a fun project on L^p spheres after watching Kelsey Houston-Edwards’s video on different ways of measuring distance:

Sharing Kelsey Houston-Edwards’s Pi video with kids

Playing around a little with our 3d printing software last night made me want to try a similar project with a torus in various L^p metrics. I made 5 different ones and set the printer to print them overnight . . . and the print failed. Boo 😦

So, I’m re-printing them to use for a project this afternoon, but for now the project with my younger son just used the shapes on the computer.

Here’s what he thought about the usual torus and the torus in L^1

Next we moved on to looking at the torus in the L^3 and L^5 metrics:

Finally, we looked at some of the shapes when p was not an integer. We looked at p = 0.75, 1.5, and 1.05.

Using the computer program was a nice way to save the project after the print failed. I’m really hoping that the 2nd time is a charm with the print and we can explore the 3d printed shapes this afternoon!

Using 3d printing to share 4-dimensional spheres with kids

A few weeks back we did a project on 4-dimensional spheres intersecting a different sorts of 3d worlds:

What if Flatland wasn’t a plane!

Last night I got around to 3d printing some of the shapes from that project:

Today we talked through the idea of how objects from higher dimensions “look” as they pass through lower dimensional shapes. We started by talking about the idea from Flatland – a 3d sphere passing through a 2d plane. After that we moved on to talking about what the intersections would look like if the sphere was passing through a plane that was creased in to a “V” shape:

Next we moved on to talking about a 4d sphere intersecting the same sorts of objects – a flat 3d space and a “V” shaped one. To create the “V” shape, I just assumed that the 4th dimension – call it w – had a value equal to the absolute value of the x-coordinate.

Next we looked at the 3d printed shapes I made last night. These shapes show a few different stages of a 4-d sphere passing through the “V” shaped 3 dimensional space:

Finally, rather than looking at 4d sphere passing through a “V” shaped 3d space, we went and looked at the shapes made when a 4d sphere passes through a 3d space that is bent like a parabola. So, using my language from above, the 4th coordinate in the space, w, is set equal to x^2.

The shapes here are really cool and also pretty surprising.

Exploring 3 intersecting cylinders with 3d printing

Calculating the volume of 3 intersecting cylinders is a classic calculus problem.  The 3 cylinder problem caught my attention a few years ago when Patrick Honner shared this video about the 3d printing lab at his high school:

I wrote about my reaction to the video here:

Learning from 3D Printing

Today we used our 3d printer and the F3 program to explore the intersection of three cylinders. Here’s what they boys had to say when they saw the setup on the F3 program – my older son went first:

Here’s what my younger son had to say:

After the shape finished printing I had the boys talk about their thoughts when they had the shape in front of them. Here’s my older son’s thoughts:

And next my younger son:

Even though this is probably a better Calculus example, I loved being able to share the shape with the boys. It is fun to hear kids talk and wonder about fun shapes like this one.

Sharing the amazing F3 program with kids

Yesterday I stumbled on an incredible blog post from Roy Wiggins:

I desperately wanted (and still do!) to figure out how to 3d print the last creation in his blog post. Wiggins pointed me to a program by Reza Ali (and sorry for saying “Renza” in the first video) that he thought would do the trick:

I’m grateful to both of their help in pointing me to and helping me understand how this program works. I still don’t quite know how to make the shape from Wiggins’s blog, but part of the reason is that the F3 program is so cool that I’ve been having a ton of fun just playing around with it. (The other part is that I don’t know what I’m doing at all . . . . 🙂 )

Tonight I showed the program to the boys and played with some “sums” and “differences” of 3d shapes.

Here’s the introduction and what the boys thought a few shapes would look like:

I let the boys pick some shapes that they thought would be fun to see. Here’s what my 5th grader thought would be fun:

We wrapped up with my older son picking a few shapes. He wanted to move some shapes left and right which I’d not done before, but luckily the program was intuitive enough for me to guess how to do it. Small victories . . . . The next thing I’d like to learn how to do is angle the shapes differently so that I can make the “Prince Rupert Cube” shape, which was one of our first ever 3d printing projects:

The last thing that we looked at was a shape created by removing a cube from an octahedron:

So, I love this program. Can’t wait to learn more about it and maybe even get to the point where I’m using 1% of its capabilities!

Images of a 4d sphere intersecting a 3d cone

For our Family Math project today we played with the idea of a spheres intersecting objects other than a plane. The idea was to explore how a 2d being on a cone, say, would “see” and describe a sphere passing through the cone. That project is here:

What if Flatland wasn’t a plane

Tonight I revisited the project just to see the different images of a 4d sphere intersecting a 3d cone. I wanted to see the analogous and presumably not really spherical shapes.

Here are some pictures.

(1) A 3d sphere intersecting a 2d cone as it descends down the middle of the cone:

(2a) The analogous picture of a 4d sphere intersecting a 3d cone:

(2b) Note the intersection does have a hole in the middle:

(3) Now I shift the sphere so that it is descending down the cone slightly off center. For the 3d sphere intersecting the 2d cone you get this picture:

(4) And here’s the same situation with a 4d sphere intersecting a 3d cone:

It is fun to see and try to imagine these images. The usual way of thinking about constructing a sphere (either 3d or 4d) makes some sense when you think about stacking up the images of the intersection with a 2d plane or with 3d Euclidean space. Trying to “stack” the images if the sphere intersecting the cone to make either a 3d or 4d sphere is really not intuitive for me at all!