Tag learning math

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.

Β 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!

Β 

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” πŸ™‚

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:

screen-shot-2017-02-18-at-10-22-50-am

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 πŸ™‚

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 πŸ™‚

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 πŸ™‚

Extending our project on Kelsey Houston-Edwards’s Infinity video

Yesterday we did a project inspired by Kelsey Houston-Edwards’s latest math video:

Here’s a link to our project:

Sharing Kelsey Houston-Edward’s Infinity video with kids

Last night my younger son and I were talking a little bit more about the project and he asked me why Cantor’s diagonal argument for why the set of real numbers is larger than the set of Natural numbers doesn’t work for rational numbers!! Yes!!

We explored that question today. First we did a quick review of the diagonal argument (which was the last part of yesterday’s project) and then we began talking about the rational numbers:

Next we looked at what would happen if you applied the diagonal argument to the rational numbers:

After getting our arms around the diagonal argument when applied to rational numbers, we backed up and looked at the argument why rational numbers are countable.

Unfortunately I made an easy concept hard in this part of the project. I was trying to explain the “easy for me” idea that if a set that is larger than the rationals was the same size as the natural numbers, that meant the rationals must also be the same size as the natural numbers. My explanation started off terribly and went down hill . . . .

Finally we looked at one of the strangest consequences of all of this infinity stuff. In math language – the rational numbers have measure zero.

The idea here always blows my mind and is a really fun idea about infinity to share with kids.

Sharing Kelsey Houston-Edwards’s Infinity video with kids

The latest PBS Infinite Series video came out this week:

This is the 4th video in an incredible series from Kelsey Houston-Edwards. Our projects on the first 3 are here:

Sharing Kelsey Houston-Edward’s [higher dimensional spheres] video with kids

Sharing Kelsey Houston-Edward’s Philosophy of Math video with kids

Sharing Kelsey Houston-Edward’s Pigeonhole Principle with kids

I had the boys watch the new video together and started today’s project by asking them what they thought was interesting.

After hearing what the kids found interesting, we dove into the idea of bijections. We talked a bit about how a bijection has to work both ways using the bus idea from the video.

After the bus example we moved on to the example of the bijection between the points in an interval and points on the real line.

We finished up by talking about the bijection between the national numbers and the positive even integers.

Since we’ve done several prior projects where infinity played some role, the next thing I asked the kids was for some thing that they already knew about infinity – both things that they thought made sense and things that they thought didn’t make sense. The discussion and examples here were amazing – “no one knows what infinity divided by infinity is” πŸ™‚

Finally, we wrapped up the project talking about why the infinity associated with the real numbers is larger than the infinity associated with the natural numbers.

I thought this would be a fun way to end the project since it was one of the key ideas in Houston-Edwards’s video:

So, another really fun project from the new set of math videos from PBS Infinite Series. I love this new series – can’t wait for the next one!

A challenge relating to a few problems giving my son trouble

I’ve seen some interesting ideas from Tracy Johnston Zager over the last week about the relationship between learning math and intuition. For example:

Although I’ve been traveling a bit for work this week the relationship between learning math and intuition has stayed in my head. Sometimes my thoughts have drifted to and old blog post about a problem from the European Girls’ Math Olympaid:

A Challenge / Plea to math folks

That post, in turn, was inspired by an old post by Tim Gowers where he “live blogged” his work while he solved a problem from the International Mathematics Olympiad.

It can be really hard for anyone to know what math intuition looks like because everyone sees polished solutions way more often than they see the actual process of doing math.

That’s part of the reason I make the “what a kid learning math can look like” posts – so everyone can see that the path kids (or anyone!) actually takes to the solution of a problem is hardly ever a straight line:

Our “what a kid learning math can look like” series

The other thing on my mind this week has been some old AMC 10 problems that have really given my older son some trouble. These are pretty challenging problems and require quite a bit of mathematical intuition to solve.

So, I’d like to make the same challenge with these problems that I made with the problem from the European Girls’ Math Olympiad – “live blog” yourself solving one of these problems. Post the though process rather than a perfect solution. Let people see *where* your mathematical intuition came into play.

problem-19

 

problem19

problem

Dave Richeson’s Knotted bubbles project

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

here’s the video in cast the twitter link doesn’t work:

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 πŸ™‚

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!