# Fawn Nguyen’s incredible Euclidean Algorithm project Fawn Nguyen recently published an incredible blog post about a project related to the Euclidean Algorithm that she did with her students:

Fawn Nguyen’s “Euclid’s Algorithm

 The above link is broken, but Fawn’s post is here:

https://www.fawnnguyen.com/teach/euclids-algorithm

Fawn’s projects are usually very easy to do right out of the box, and this one is especially easy since you can just start with her pictures. So, we just dove in.

You’ll see from the comments my kids had that Fawn really has made using this blog post effortless:

Next I asked them to make their own shapes. They built the shapes off camera and then we talked about them.

At the end I asked them when they thought a shape would require 1x1x1 cubes.

After hearing their thoughts about relatively prime numbers at the end of the last video I asked them to make a shape that wouldn’t require 1x1x1 cubes to finish. Here’s what they made and why they thought it would work:

Such a fun project. Fawn’s work is so amazing. I love using her posts with my kids.

# Random walks with kids A week or so ago my older son did a short project on random walks out based on a page in Patters of the Universe:

Returning to Patterns of the Universe

By coincidence that week Kelsey Houston-Edwards’s new video was about random walks. So, we watched her video after that project:

Today my younger son is sick and wasn’t up to participating in a project. So, I thought it would be fun to revisit the random walk project and dive in a little deeper since my older son was a little more familiar with that topic.

I started by asking him what he remembered about random walks from the prior project and from the PBS Infinite Series video. One thing that he remembered is that 2d random walks do tend to return to where they started, but 3-d ones tend not to.

We started looking at specific random walks by studying a 1-dimensional random walk. We created a random walk by rolling dice and didn’t get quite what we were expecting, but that result led to a fun conversation:

In the last video we got more even numbers than we were expecting, so we decided to continue on to see if the walk would return to 0. Obviously we kept rolling even numbers . . . .

Next we moved on to studying a 3d random walk (and, of course, now rolled lots of odd numbers 🙂 )

We created the 3d random walk with snap cubes and it was pretty neat to see the shape that emerged from the dice rolls.

Despite the unexpected outcome with the even and odd rolls this was a fun project. I’d like to think a little more about how to make some random walk 3d prints. My guess is that those prints would be really fun to share with kids.

# Does (x + y)^2 = x^2 + y^2 In a few projects that we’ve done over the last couple of days my younger son has gotten a little confused on some basic algebra. Not something I’m worried about as ideas like does:

(i) $(x + y)^2 = x^2 + y^2$, or

(ii) $\sqrt(x^2 + y^2) = x + y$

(in case that latex isn’t displaying properly, the entire expression is supposed to be under the square root.)

are questions that confuse everyone when learning algebra.

Today we did a short project to talk about these equations. We ended up spending most of the time on (i) just because it was a little easier to talk about. First, though, was just a quick look at both equations:

Now we looked at $(x + y)^2 = x^2 + y^2$ more carefully. You can see my younger son’s confusion at the beginning. To help get past that confusion we looked at what $(x + y)^2$ actually means.

As we were talking during the last project I noticed a bunch of snap cubes near by (from one of last week’s projects). Rather than move on to the square root example I thought it would be better (and also fun) to view the square example from a geometric perspective.

This was a fun discussion and I especially enjoyed seeing the boys find a few different geometric approaches to the problem.

# Revisiting a counting project for kids that I learned from Jim Propp For our math project today we returned a tiling idea that is a really fun idea for kids to explore. Here are a few of our prior projects on the subject:

A fun counting exercise for kids suggested by Jim Propp

Counting 2xN domino tilings Today the plan was to look at 2xN tilings first and then move on to tilings of 3xN rectangles with 3×1 dominoes.

We stared by exploring some simple 2xN cases and looked for patterns:

In the first video we counted the number ways that we could tile 2×1, 2×2, 2×3, and 2×4 rectangles with dominoes. Now the boys noticed the connection with the Fibonacci numbers and we tried to find and explanation for why the Fibonacci numbers seemed to be showing up here. The nice thing is that the boys pretty much got the complete explanation all on their own.

Now we moved on to counting the tilings of a 3xN rectangle using 3×1 “dominoes” – what would be different? What would be the same?

One really interesting thing here is that my older son and younger son came up with different ideas for how to count the general arrangement.

So, in the last video my older son had a counting hypothesis that I couldn’t quite understand. In the beginning of this video I have him explain his process more carefully. The surprise was that for the 3×6 case we were looking at next both of their counting procedures predicted the same number of domino tilings.

In this part of the project we tried to follow both procedures to see how they worked.

Having sorted out the counting procedure in the last video, we now looked carefully at the 2xN and 3xN tiling procedures and saw that we could compute the number of tilings for the 2×100 and 3×100 cases if we wanted to.

I’d love to come up with more counting projects for kids. These projects are accessible to young kids and I think shows of some really fun ideas from advanced math that kids probably don’t usually see in school.

# Connecting arithmetic and geometry I’ve been kicking around a few ideas about connecting arithmetic and geometry. My first thoughts were revisiting an idea we’ve played with a few times before:

So, today I decided to look at these two sums with the boys: $1 + 2 + 3 + 4 + \ldots + n$, and $1^2 + 2^2 + 3^3 + \ldots + n^2$.

We got off to a slower than expected start because my younger son didn’t remember the formula for the first sum correctly. I’m trying hard to break through the idea of relying on remembering formulas, so I was actually happy to review where the formula came from, though.

After the introduction we moved on to studying the first series using snap cubes. What is the geometry hiding behind the formula.

This part of the project to a totally unexpected turn, though:

I decided to keep going with the new sum that added up to $n^2$ to see if we could make another connection. The boys did remember that the sum of odd integers connects to perfect squares, so I challenged them to find the connection between that formula and the new one they just stumbled on.

Finally we moved on to the sum of squares formula. Lots of fun questions from the boys here, including if the idea extended to 4 dimensions!

The shape here is more difficult to build that it initially seems, but they got through it and now hopefully have a better idea of where the formula comes from.

We wrapped up by looking very briefly at pyramids.

I’d like to do more projects like this one and develop a bunch of different ways to share connections between arithmetic and geometry with kids.

# Grant Sanderson’s “Fair Division” video shows a great math project for kids [sorry for a hasty write up – had to be out the door by 8:15 this morning . . . ]

Yesterday I saw the latest video from Grant Sanderson, and it is incredible!

I couldn’t wait to share the “fair division” idea with the boys. I introduced the concept with a set of 8 yellow and 8 orange snap cubes. To start, we looked at simple arrangements and just talked about ways to divide them evenly:

Next we looked at the specific fair division problem. We made a random arrangement of the blocks and tried to find a way to divide the cubes evenly with 2 cuts:

To finish up we looked at a few more random arrangements. Some were a little trick, but we always found a way to divide the cubes with two cuts! We also found an arrangement where the “greedy” algorithm from the 2nd video didn’t work.

After we finished the project I had the boys watch Sanderson’s video and they loved it. So many people are making so many great math videos these days – how are you supposed to keep up 🙂

# John Golden’s visual pattern problem We seem to always start our year off with a Fawn Nguyen-like problem. Today it happened by accident when I saw this visual pattern problem from John Golden:

We tried out the problem for a little after dinner math challenge tonight. Here’s what the boys thought initially – I was happy to see that they noticed that they could look at the pattern going backwards as well as forwards:

At the end of the last movie the boys wanted to make the base for the next tower. We did that with the camera off and then started looking at the pattern again.

I was a little surprised that they wanted to make this next piece rather than just talk about it, but making it did seem to help them see what the pattern was. In fact, their initial guess at the pattern was totally different from what I saw 🙂

So, although we didn’t get all the way to the formula for the nth step, we did find a way to determine (in theory) the number of blocks on any of the steps. I remember playing around with these difference tables in high school and being absolutely amazed – it is fun to be able to play around with them with the boys now.

# Connecting arithmetic and geometry Last week we did a short project on approximations (based on a section in Mazur and Stein’s new book in primes):

Looking at Mazur and Stein’s new book about primes, part 2

I thought it might be fun to revisit approximations this weekend. My idea was to look at the two sums $1 + 2 + 3 + 4 + . . . + n$ , and $1^2 + 2^2 + 3^2 + . . . + n^2$. The idea would be to use some ideas from geometry to approximate these sums.

We started by looking at $1 + 2 + 3 + 4 + 5$ and thinking of ways to approximate this sum. The boys were a little confused at the beginning because they already new the value of the sum, but eventually they landed on the idea that we could approximate the value of the sum as the area of a triangle.

Next we looked at a few more sums of consecutive integers from 1 to n to see if we always had a “good” approximation according to the definition we’d seen last week in Mazur and Stein’s book.

Now we moved on to adding up perfect squares from 1 to $n^2$. We started with $1^2 + 2^2 + 3^2 + 4^2 + 5^2$. Initially the 5 snap cube squares were arranged sort of in the shape of a right triangle. The kids had a hard time seeing how to use this triangle, though – that surprised me a little. Eventually they discovered that the squares could be arranged in a pyramid-like shape, and we used that pyramid as the basis of our approximation.

One difference between the sum of squares and the sum of integers is that the approximation we used for the sum of squares was not “good”.

For the last part of the project I had the boys build two more snap cube pyramids. They new from some prior project that pyramids could form a cube, so they tried to see if our approximate pyramid could also form a cube. A few samples of those prior project are here:

Summing up Squares

Pyramids and Count Like and Egyptian

It turned out that our three of our approximate pyramids do not form a cube, but the shape they do form provides some additional insight into the actual sum. In fact, it gives the exact formula 🙂

So, a fun way for kids to see a connection between arithmetic and geometry. It was also a nice way to see some examples where the “good approximation” idea from Mazur and Stein worked and didn’t work. Nice little Saturday morning project 🙂

# Henri Picciotto’s factoring activity

[sorry for the tired write up – I’m a little sick today]

Saw this neat tweet from Henri Picciotto last night:

At the end of the blog post is this activity:

Geometry and Graphing Connection by Henri Picciotto

and the last page of this activity really caught my attention. I’d been doing a bit of work on factoring and completing the square with my older son anyway, so I thought I’d give it a try with him tonight.

One warning is that I’m pretty sick and out of gas today. I wanted to see how he’d react to using factoring ideas with geometry, but I didn’t have the energy to pursue all of the ideas. That said, I’m actually very happy with how he reacted to the ideas – especially with little help from me since I was fully in the useless zone tonight. We’ll definitely be revisiting this activity.

I started off with a quick introduction to how the blocks could be used to represent the various pieces of a quadratic polynomial. He seemed to catch on fairly quickly using the example $x^2 + 3x + 2$:

At the end of the last video he’d picked a 2nd example quadratic with negative coefficients. I wanted to do an example with negative coefficients, but I wanted to wait to do that example later in the project.

Here he pics the example $x^2 + 5x + 6$ which makes another rectangle. He’s able to see how the blocks show that this quadratic factors into $(x + 2)(x + 3)$.

The next two examples showed what this method would look like for a perfect square, and how it could be used for completing the square.

Now we returned to a polynomial with negative coefficients. The geometric ideas here were a little harder for him to see. I need to find a better way to communicate the ideas for this part so that he sees the -2 as -3 + 1:

My only idea for helping him see the geometry in the case with negative coefficients was to use an example that was a little easier. That example was $x^2 - 2x + 1$. He did seem to see the geometry here, which made me happy. I was especially happy that he saw how a block was taken away twice.

So, I think this is a really great project from Henri. I’m really excited to try some more with it this weekend.

# Summing up Squares

Earlier in the week my son’s math team had a practice problem with this sum:

2 + 5 + 10 + 17 + . . . .

The question was how many terms would it take in order to be above 650?

I though that this problem would make a good starting point for reviewing the formula for the sum of squares: $1^2 + 2^2 + 3^2 + \ldots + n^2 = (n)(n + 1)(2n + 1) / 6$

This project didn’t go as well as I’d hoped, but maybe there’s a way to come back to it again later to reinforce some of the ideas. The difficulty the kids had – arranging 6 stacks of $1^2 + 2^2 + 3^2$ cubes into a box – comes in the 3rd and 4th videos below.

We started by talking about sums of integers. Of course the infinite sum came up at the end!

With the sums of integers out of the way, we introduced sums of squares. After we wrote down the formula, we started exploring that formula with snap cubes:

Now we studied the slightly more difficult geometric cases of $1^2 + 2^2$ and $1^2 + 2^2 + 3^3$. The second case gave the kids a lot of trouble.

In the last video we weren’t able to construct the box relating to the sum of the first 3 squares. We kept trying here. I didn’t expect this piece to give them as much trouble as it did, so, unfortunately, I didn’t have any good ideas to help them see what to do.

One thing that was interesting to me is that the pattern with the colors seemed to hinder their progress as much as help them:

As prep for this project yeterday, I 3d printed 6 of the block sets for $1^2 + 2^2 + 3^2$. This was a fun 3d printing project all by itself, and came in handy in this project to see if they were able to build the shape without the colors.

Although, as I mentioned above, the color patters seemed to give them as many difficulties as help, my older son thought it was actually harder to build the shape without the colors:

To wrap up the project we took a look at the three pyramids that fold together to make a cube. This idea was a fun project all by itself last year:

A neat geometry project inspired by a James Tanton / James Key tweet

We also looked to see if the sum of squares was approximated well by $n^3 / 3$ for a few values of n.

So, although this one didn’t go exactly as planned, it was still pretty fun. After we finished I had they try to build the box made out of 6 $1^2 + 2^2 + 3^3 + 4^2$ pyramids, and they built that shape pretty quickly.

I like the connection with the sum and the 3 dimensional box and am excited to come up with ways to revisit that connection. I’m also trying to think through ways to show a connection with the sum of cubes and a 4D box – I bet that will be fun!