Tag learning math

Matt Enlow and Suzanne von Oy’s geometry problem

Saw a tweet from Matt Enlow today that led to a fun discussion and also a fun project tonight with the kids:

The last tweet in the conversation was a new Desmos activity from Suzanne von Oy showing how the problem worked:

I couldn’t wait to try out this problem with the boys tonight. We aren’t (obviously!) going to go into a lot of depth – this isn’t really a problem for 5th graders! But, I thought the boys would have some fun talking about it.

Since the problem is a pretty challenging one for kids to even understand, I started the project by trying to explain the problem carefully.

Next we tried to pick some points at random and then draw some triangles. If there are infinitely many equilateral triangles passing through these 3 points, it ought to be easy to draw one of them, right?

My older son went first. The cool thing for me in both this and the next video was seeing kids experience the problem and struggle with both trying to understand it and trying to solve it. There really is a lot of great geometry for kids here:

Next my younger son gave it a try. His approach was absolutely terrific to watch – I never would have approached the problem the way he did.

Next we went to play with Suzanne von Oy’s Desmos program. We got interrupted by the new puppy in the house across the street between leaving the living room and heading upstairs to play with the program, so I took the first minute of this video to review the problem again.

My older son went first again. He quickly found a picture that didn’t satisfy the conditions of the problem and that threw him for a little loop. Once we got past that, though, he seemed to have a much better understanding of the problem.

My younger son went next and eventually found an arrangement of the points that didn’t work at all. That was actually a really cool surprise ( we’ll deal with that surprise in the next video).

So, we got a wonderful surprise in the last video when we stumbled on an arrangement of the three points that didn’t seem to have any equilateral triangles passing through them.

Talking about what went wrong was a fantastic little surprise and it really made this project for me.

This was a super fun project. Thanks so much to Matt Enlow and Suzanne von Oy for sharing both the problem and the Desmos activity. Math twitter is amazing!

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 🙂

Returning to Patterns of the Universe

As we searched for our old patty paper supply yesterday I ran across our copies of Patterns of the Universe:

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We’ve done a few projects with the book before:

Fibonacci, Zome, and Patterns of the Universe

Patterns of the Universe Part 2

I thought doing some mathematical coloring would make for a great project today.

My older son talked about his coloring project first – he chose a section of the book on a random walk dice game. It was fun to hear his thoughts on random walks (and walks constrained like the one in this activity). By coincidence Kelsey Houston-Edwards recently did a video on random walks, so there’s a really nice way to continue the discussion we had here.

My younger son chose a section on “snug bunnies” – a picture of bunnies tiling the plane. He picked this section because he likes tesslations. It was fun to hear him talk about his coloring pattern. At the end we took a look at some Penrose tiles just to extend the tiling idea.

I really enjoyed the project today – the seemingly simple act of coloring led to some really fun discussions and also gave me some ideas for fun follow up projects.

Returning to inclusion / exclusion

We’d taking a break from our inclusion / exclusion project but I wanted to return to it tonight. I picked a fairly challenging problem from one of my old math books:

How man 5 card poker hands have at least one card from each suit?

I didn’t have any idea how it would go . . .

We started by reviewing the ideas in inclusion / exclusion and the moved on to try to get our bearings in the problem:

Having formed a pretty good plan in the first video, we moved on to tackling the rest of the problem.

I’m really happy with how this went. It is fun to see the boys learning to break complicated problems down into problems that are slightly easier to deal with.

Revisiting Stephen Wolfram’s MoMath talk

Last week Stephen Wolfram posted an incredible summary of his talk at the Museum of Math:

We did a project using some of the code here:

Sharing Stephen Wolfram’s MoMath talk with kids

I think the ideas from the talk can provide kids with a really wonderful opportunity to explore math. We’ll hopefully revisit the ideas many times!

Today’s exploration follows the same line of ideas that we followed in the first project. The procedure we are looking at goes like this:

(1) Start with the number 1, and proceed to step 2.

(2) Whatever number you get here, cycle the digits to the left -> so, 123 becomes 231, 1045 becomes 0451 (so just 451 for computations), 110110 becomes 101101, and etc . . .

(3) Now multiply the number from step 2 by a fixed number N and add 1.

(4) Take the output from (3) and return to step (2).

We look at the sequence of outputs from this procedure in base 2, 3, 4, and 5 today. Quite amazingly, Stephen Wolfram showed that this entire procedure could be done with some very short code in Mathematica. Here’s a pic of the short code and also patterns we see in the digits when we multiply by 1, 2, 3, 4, 5, 6, and 7 at each step when we reun the procedure above in base 4.

Wolfram.jpg

If this seems way too complicated I’m not explaining the procedure well enough – go back to our first post on the subject or to Wolfram’s blog. I promise you’ll see that the explorations are totally accessible to kids.

 We started our project today by revisiting the results in base 2 and looking for strange or unusual or really anything that caught our eye in the digit patterns.

Also, I’m sorry that the zoomed in shots are so fuzzy (so, the first minute here and basically all of the 4th video). I didn’t realize how bad the footage was until it was published. Even with the fuzziness, though, you can still hear how engaged this kids are and how interesting it was for them to explore all of the strange patterns:

For the 2nd part of the project we looked at the patters of the digits in base 3:

Then we looked at base 4 and immediately saw something that we’d not seen before:

So, having explored bases 2, 3, and 4 we went back to some of the patterns we’d seen and got a nice surprise – we were able to find structure in some of those patterns. This video is the exploration that led to us finding the pattern in base 2.

Again, I’m sorry this video is so fuzzy – wish I would have caught that when we were filming 😦

Now we moved on to exploring some of the patterns that we’d seen in base 3 and base 4 – that exploration allowed us to predict a pattern in base 5 even though we’d not yet looked at any of the digit patterns in base 5!

I can’t wait to play with Wolfram’s ideas a bit more. The ideas are such a great way to expose kids to exploration in math!

 

Sharing a shape from calculus with kids

Finding the volume of the intersection of two cylinders is a common calculus problem. The shape also plays a role in this old (for the internet!) video from Brooklyn tech that inspired me to get a 3d printer:

Today for a fun project to start the week I decided to share the shape with the boys and see what they thought about it. My younger son went first:

After playing on the computer I had him explore the printed version of the shape – make sure to stay to about 1:25 to hear where he thinks this shape might occur in “real life” 🙂

Next my older son played with the shape on the computer. He remembered seeing it before in a project from a month ago on the intersection of 3 cylinders:

Exploring 3 intersecting cylinders with 3d printing

Next he played with printed shape. I asked him to describe how he thought you’d be able to figure out that the shape was made out of squares – I thought his answer was pretty interesting. This question gets to the math ideas behind the calculus problem.

It is sort of fun for kids to see and play with shapes like this – no need to wait for calculus anymore to explore interesting shapes!

Extending our coordinate change project with Desmos

Last weekend we did a fun 3d printing project involving changing coordinates:

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Here’s the link for that project:

Exploring some fun 3d transformations

A youtube commentator – mxlexrd – made the following Desmos version of one part of the activity after seeing the project:

mxlexrd’s Desmos version of our coordinate change project

Last night I had the boys play with it since I’d be traveling for work today. My younger son went first:

Here’s my older son:

The Desmos version of our activity is really fantastic. Even if the concepts are a little bit too complicated for my kids to understand in detail, I love how easy the program made it to explore the mathematical ideas.

Playing with the Cubeoctahedron

Last night I was flipping through the book I bought to understand a bit more about folding – Geometric Folding Algorithms by Erik Demaine and Joseph O’Rourke:

folding-book

and I ran across a short note on the cuboctahedron. The boys were taking a short trip today (school vacation week!) and I was looking for a short project to do before they left – folding up the cuboctahedron seemed perfect.

Making my life much easier was a template on Wolfram’s website:

Wolfram’s folding template for a cuboctahedron

Here’s what the boys had to say after creating the shape:

After the short discussion about the shape we went upstairs to look at the shape using the F3 program. My idea for the ~10 min discussion here was inspired by a talk by Keith Devlin I saw over the weekend:

I thought that an approach similar to a game with our F3 program would help the boys create the shape.

Here’s how we got started. The F3 program allows us to create a cube and an octahedron. It also allows you to add and subtract shapes. How can we use these 4 ideas to create the cuboctahderon?

I think the video here really shows what Devlin calls “mathematical thinking.” The conversation here was really fun (for me at least!) since trying to discuss the ideas through equations would be impossible. However, the geometric ideas are accessible to the boys via the F3 program, just as the number theory ideas are accessible to kids through Devlin’s “Wuzzit Trouble” program.

I broke the discussion into two pieces – at the start of the 2nd half of the discussion we are trying to figure out how to – essentially – flip the shape inside out. My son comes up with an idea that was very different than what I was expecting, and it worked 🙂

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 🙂

Writing 1/5 in binary

I’ve spent the last couple of days talking about binary with my younger son. We were inspired a bit by Kelsey Houston-Edwards’s latest PBS Infinite Series video on binary. It has been a fun little review.

Tonight we talked about how to write 1/5 in binary. I didn’t really know how the conversation would go, but it ended up being a nice little arithmetic review.

We started talking about the problem and he settled on the idea that we needed to find a number that would equal to 1 when we multiplied by 5. That got us going on the arithmetic review since that idea works in any base.

Now we had to figure out now to divide 1.000000000…. by 101 in binary. This long division problem gave us an opportunity to talk about subtraction (and borrowing) in binary:

The last step was multiplying the number we thought was 1/5 by 101. Once again this was a great opportunity to review some basic ideas about arithmetic and multiplication.

So, an unexpectedly fun project! We learned what 1/5 was in binary and had a nice review of subtraction, division, and multiplication along the way 🙂