Writing fractions in binary

I saw an interesting tweet last wee that got me thinking about fractions:

Today as sort of a unusual way to play around with fractions I thought it would be fun to try to write some fractions in binary. It has been a while since we talked about binary, though, so I had my son tell me what he knew about binary first:

Next we moved on to writing fractions in binary – we started with some simple cases:

Finally, we tried to write 1/3 in binary. This video shows what a kid thinking through a math problem can look like, and also shows why I thought this exercise would be a nice fraction review:

Sharing a new result about the Cantor set with kids

Earlier in the week I saw a tweet announcing a new (and really cool!) result about the Cantor set:

The new result is that any number in the intrrval [0,1] can be written as the product x^2 * y where x and y are members of the Cantor set.

After reading the paper, I thought that it would be really fun to try to share some of the ideas with kids. The two ideas I wanted to highlight in the project today were (i) the geometric ideas in the construction of the Cantor set, and (ii) the interpretation of the Cantor set in base 3.

I started with a question about base 3 -> how do you write 1/2 in base 3?

Now we looked at constructing the Cantor set by removing intervals. The boys had lots of interesting ideas about what was going on

Next we looked at the incredible property that you can make any number in the interval [0,2] by adding two numbers in the Cantor set. This ideas here were a little harder for my younger son to understand than I was expecting, so I ended up breaking the discussion into two parts.

I think the ideas here are fun for kids to think through – how do I pick a number from one set and a second number (possibly from a different set) to add up to a specific number.

Here’s part 1:

and part 2:

Finally, we took a peek at the result from the paper -> how does multiplication work? This was also a fun discussion. The ideas necessary to see why you can find three numbers from the Cantor set that multiply to any number in [0,1] are obviously way out of reach for kids. However, seeing why the multiplication problem is difficult is within reach.

It is always a real treat to find math that is interesting to mathematicians to share with kids. I think talking through some of the ideas related to this new result about the Cantor set makes for an amazing math project for kids!
 

Sharing Kendra Lockman’s Desmos activity with my son

I saw this tweet from Kendra Lockman yesterday:

It looked like a fun activity to try, so we spent 20 minutes this morning going through it. It was nice to year what my son son thinking about fractions throughout the activity. The first 4 videos below show his work and the last is some quick thoughts from him on the morning:

Part 1:

Part 2:

Part 3:

Part 4:

Here’s his summary of the activity:

Exploring a fun number fact I heard on Wrong but Useful

I was listening to the latest episode of Wrong but Useful today:

The Wrong but Useful podcast on Itunes

During the podcast the following “fun fact” came up -> \ln(2)^5 \approx 0.16.

I thought exploring this fact would be a fun activity for the boys and spent the next 30 min daydreaming about how to turn it into a short project. I also wanted the project to be pretty light since today was the first day of school for them. Eventually I decided to explore various expressions of the form \ln(M)^N via continued fractions and see what popped up.

We started by looking at the approximation given in the podcast. During the course of the discussion we got to talk about the relationship between fractions and decimals:

Now we looked at some powers of \ln(3) until the phone rang. We found a neat relationship with the 5th power. This relationship was also mentioned in the podcast.

While I was on the phone I asked the boys to explore a little bit. Here’s what they showed me when I got back.

Oh, wait – EEEk – I just noticed writing this up that we counted back incorrectly in this video. Whoops! Here’s the number we thought we were exploring -> \ln(12)^{15} is very nearly equal to 850,454 + 19,118 / 28207.   The next approximation that is better is 850,454 + 33,481,089 / 49,398,529.

You can see in the pic below that the 19,118/28,207 is accurate to 12 decimal places!

Sorry for this mixup.

Continued Fraction

Next they showed me one more good approximations that they found -> \ln(8)^{18} is nearly an integer. After that I tried to show them one I found but we ran into a small technical problem, so no need to watch the rest of the video after we finish with \ln(8)^{18}.

Finally, I got the technical glitch fixed and showed them that \ln(11)^2 is approximately 5 3/4. The next better approximation is 5 + 1,907 / 2,543

So, a fun little number fact to study. Sorry for the bits of the project that went wrong, but hope the idea is still useful!

What a kid learning math can look like – fraction confusion

My younger son is currently in the review section on percents in his algebra book. Last night he chose a fairly standard problem on percents for our movie. The arithmetic with fractions tripped him up a little, though. The first video shows his struggle:

After we finished the problem I decided to propose an alternate solution just to get him a second round of fraction practice. His work here was really good, but the fraction arithmetic at the end still also gave him a tiny bit of trouble:

You never know what’s going to give a kid difficulty and it is interesting to watch them try to work through these unexpected struggles.

James Tanton’s candy dividing exercise

Yesterday we watched the “tie folding” part of James Tanton’s latest video:

The video led to a great project with the boys last night:

James Tanton’s tie folding problem

The boys knew from the video that the method could also be applied to sharing candy. Since we didn’t watch that part of the video I was wondering if the boys could figure out the connection on their own. Here’s the start:

Next we tried an example to see what would happen if our initial guess was a big over estimate of 1/3 of the Skittles:

Since we were struggling with our second time through the procedure in the last video, I thought it would be fun to try to be more precise in how we split the piles. That extra precision did lead to slightly better results.

So, a really nice math activity. It was really fun to see the procedure work when we couldn’t be totally sure we were actually dividing the piles in half. Such a great project for kids.

James Tanton’s tie folding problem

Saw a great new video from James Tanton today about folding a tie. The kids had spent yesterday hiking in New Hampshire and were a little tired, but Tanton’s project made for a perfect little afternoon project.

I’ll present the videos in the order that we did them, so Tanton’s video is the third one below. Showing his video later in the project will also give you a chance to think through the problem without spoilers.

Anyway, here’s how we started -> what do you have to do to fold a tie in half?

I was super happy with how the introductory problem went because at the end of the last video my older son said that he thought folding the tie into thirds would be hard. Well . . . that’s exactly what we are going to try to figure out!

Next we watched Tanton’s video. He talks about both folding ties and sharing candy, but for today at least we are just focused on the tie folding part:

Now we tried to replicate Tanton’s procedure. My 5th grader had a little bit harder of a time understanding the procedure than my 7th grader did, but they both eventually got it.

At the end we talked about why they thought the procedure worked.

So, a super fun project and a really easy one to implement, too. So many potential extensions, too – might be neat to see how kids approached folding into 5 parts after seeing Tanton’s video, for example.

Thanks for another great project, James!

Dave Radcliffe’s “unit fraction” tweet

Saw a neat tweet from Dave Radcliffe a few weeks ago:

I’d played around with it a bit on Mathematica and the code was still up on my computer screen when we were playing with base 3/2 yesterday, so the kids asked about it.

Radcliffe’s proof is a bit too difficult for kids, I think, but the general idea is still fun to explore. I stumbled through a few explanations throughout this project (forgetting to say the series should be finite, and saying “denominator” rather than “numerator” at one point), but hopefully the videos are still clear.

I started by explaining the problem and looking at a few simple examples:

Next we looked at how it could be possible for a finite sum of distinct numbers of the form 1 / (an integer) could add up to 100, or 1000, or some huge number:

Now that we understood a bit about the Harmonic series, we jumped to Mathematica. I sort of half explained / half skipped over the “greedy algorithm” procedure that Radcliffe uses in his paper. I thought seeing the results would explain the procedure a bit better.

We played around with adding up to 3 and then a couple of numbers that the boys picked.

After playing around with a sum adding up to 3, we tried 4 and the boys got a big surprise. We then tried 5 and couldn’t get to then end!

After we turned off the camera we played around with the sum going up to 5 a bit more sensibly and found that there are (from memory) 102 terms and “n” in the last 1/n term has 142,548 digits!

So, a little on the complicated side, but still a fun math fact (and computer project!) for kids to explore.

An introduction to the Mandelbrot set for kids

Last night I was writing about “beautiful math” for kids:

When I asked my younger son what he thought was the most beautiful math he’d seen, he replied “fractals” and specifically mentioned the Mandelbrot set.  We haven’t done a project about the Mandelbrot set, so it seemed like a good idea to talk about it today.

As I say in the introduction below – it isn’t just a pretty picture, there is some really cool math.

So, I started the project by talking (or probably more accurately stumbling) through an explanation of the map that defines the Mandelbrot set.  After that, we worked through a few examples:

 

Having looked at 0, -1, and 1, we now moved on to looking at some complex numbers. The next numbers we tried were i and -i. It turns out that both of these numbers are part of the Mandelbrot set, but the calculations are slightly (really, just slightly) more complicated.

At the middle of this video we produced a crude map of the Mandelbrot set with snap cubes, and then at the end we discussed a little bit about how a computer program to plot the Mandelbrot set would work.

 

Now we moved to the computer to study the Mandelbrot set in Mathematica. Luckily Mathematica has a function – MandelbrotSetPlot[] – that makes this part of the project pretty easy for us. In this part we talked a little bit about what happens when you vary the number of iterations, and also what happens when you zoom in.

Determining the coordinates for zooming in was also a nice little mathematical discussion with the boys.

 

The coordinates to zoom in on turned out to be a more interesting topic than I was expecting. With the camera off we had a long discussion about the coordinates of the location that they wanted to see more carefully. I’m sorry I turned the camera off, actually, but oh well.

I love the discussion and general thoughts from the kids about the shapes we were seeing here. My younger son is right – this really is beautiful math for kids to see!

 

“fence post” problems

Saw this conversation on Twitter last week:

Once of the most surprising lessons I learned teaching my kids math came from my older son struggling with this type of problem. He has struggled with them for *years* and seemingly no amount of discussion / practice / reading / and etc has made this problems easier.

I’m baffled, but one thing for sure is that I understand that this type of problem can be difficult for kids.

I decided to try out the problem with my kids this morning. My younger son went first and didn’t have too much trouble:

 

My older son, on the other hand, stumbled a little. In fact, you’ll see that his initial reaction is to label the dots with 1/7ths:

 

So, I assume that lots of kids will have little trouble with problems like this one, but some kids will struggle. For those kids, this type of problem is far more difficult than you can imagine.