# 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.

# 2nd day of using Matt Parker’s Menger Sponge video

Disclaimer right at the beginning of this post – we made a mistake in the last video and I didn’t notice it until I was writing up our conversation. I decided to write up the project anyway because mistakes are a part of learning and talking about math.

So . . . yesterday we did a project inspired by Matt Parker’s Menger Sponge video. Here’s the project:

Using Matt Parker’s Menger Sponge video to talk about fractions with kids

and here’s Parker’s video:

Yesterday we studied the 2d version – the Sierpinski carpet – and today I wanted to talk about the 3d shape. Since the 2d shape was fairly complicated, I thought it would be worthwhile to review yesterday’s confirmation before diving into the new 3D shape:

After that review, we talked about the 3d version. For the first part of the 3d discussion, we talked about removing the middle “thirds” of each part of a cube. How many cubes do we remove? How big are they?

The 3d geometry ideas here are great for kids to talk through. The conversation about the shape is also a nice fraction conversation for kids.

The next part is where we started to make a mistake – I’ll get to that in a second.

Here we looked at the next step in the modified Menger Sponge process. We began to discuss what happens when you chop up our remaining cubes into 5 pieces (on each side) rather than 3 pieces. The kids were able to see this step fairly easily on the square, but it was a little more confusing on the cube. Luckily we had one of the Menger sponges to aid in the visualization.

The mistake we made was in counting the cubes that get removed when we chop the sides into 5 pieces. We forgot we were chopping each side into 5 pieces and counted as if we were chopping each side into three pieces.

When you chop the sides into thirds and “punch out” each middle third, you remove 3 small cubes with each “punch” and have to put back 2 because you punched out the middle cube 3 times.

When you chop the sides into fifths, though, you punch out 5 cubes with each punch, so you really are punching out 5 cubes with each punch and having to put back 2 at the end. So, rather than taking away 7 cubes, you take way 13. In general, chopping the sides into (2n + 1) pieces and removing the middles results in taking way 3*(2n+1) – 2 cubes. Unluckily we always said that we took away 7 😦

For the last part we extended our ideas to higher levels of the modified Menger sponge. Our mistake of taking away 7 cubes every time came along for the ride . . . We’ll have to correct this mistake in a new project:

So, a neat project even with the mistake. I really like using the geometric ideas from Parker’s video to talk a little geometry and talk a little fractions with kids.

# Using Matt Parker’s Menger Sponge video to talk fractions with kids

Saw Matt Parker’s latest video via an Evelyn Lamb tweet yesterday:

Here’s the video:

Watching the video last night I thought that there’d be a lots of different ways to use this one with kids. I chose to use it as a way to talk about fractions and scaling. Kids, I think, will be surprised by the result involving $\latex \pi$ but diving into that part is probably too much for kids. They can appreciate the result, though, and the discussion of fractions and scaling leads right up to the Wallis formula.

I started by asking the kids what they thought about Parker’s video. We talked about their thoughts as well as a few other fractal shapes that the knew:

Next we talked about Sierpinski’s carpet and walked through the calculations for the area. The boys saw the area essentially as a subtraction problem, so I spent a lot of time trying to help them understand how to see it as a multiplication problem:

I think that the boys still didn’t really see the area change from one step to the next as a product, so we spent a little more time talking about that idea in the modified Sierpinski’s carpet that Parker talks about in the video. The way the area changes from step to step is a great fraction problem for kids.

I’m sorry that we got a little bogged down in the calculations here, but I really wanted to be sure that the kids saw the relationship between the subtraction and multiplication approaches to calculating the area.

Finally, we looked at the complete calculation for the area of this modified Sierpinski carpet. The boys noticed a pattern in the products, which was cool, and we were able to transform our product into the famous Wallis product.

So, a fun project for kids. Parker’s video is great and serves as a great motivation for diving into the calculations. The calculations themselves are a great exercise in fractions for kids.

I’d like to try to work through the 3d version, too, so maybe we’ll do that tomorrow. My back of the envelope calculation tells me that the 4d version doesn’t have the same property of having the “volume” of the 4-d sphere (which is $\pi^2 / 2$, though I may not have done the calculations correctly the first time around and will revisit them later this weekend, too.

# Ben Orlin’s Fraction Problem

Earlier in the week I saw this tweet from Ben Orlin:

I looked like a pretty interesting problem and wanted to try it out with the boys. Today I finally got around to it. I won’t provide much of any commentary, but just leave the videos below as examples of kids working through this problem.

First up was my older son (6th grade). The problem was – as Ben predicted – difficult for him. After lots of thinking and a couple of false starts he did find a good approach to the problem:

After we finished up we checked the answer on Wolfram Alpha:

Next up was my younger son (4th grade). The problem was very difficult for him, but by the end of this video he’s found an approach he likes:

The process of finding a common denominator was challenging for him. I think this video is a good example of what a kid learning fractions can look like:

Finally, my younger son almost landed on a “common numerator” approach in the last video. That’s a nice approach to the problem, too, so I wanted to show both boys what that approach looked like:

So, a really productive morning talking about fractions. Thanks to Ben Orlin for posting this problem.

# The UK Intermediate Mathematics Challenge part 2

Yesterday I saw this tweet about the UK Intermediate Mathematics Challenge:

and had my kids work through one problem each:

https://mikesmathpage.wordpress.com/2016/02/15/a-couple-of-problems-from-the-uk-intermediate-mathematical-challenge/

Today we took a look at the last problem. The kids are in 4th and 6th grades so they are at pretty different levels in mathematical development. I decided to have them work individually on the problem so that each kid could use his own ideas to solve it.

Here’s the problem (from the link above):

First up was my younger son. It was a struggle for him to see the pattern that was repeating. The first part of the project was primarily him working to see if the pattern that he thought repeated actually did repeat.

When the first pattern didn’t quite work, we tried again – this time he found a pattern that, indeed, did repeat:

At the end of the last part we needed to figure out how many of our equilateral triangles were part of the repeating shape. His approach was to tile the hexagon with equilateral triangles – a different approach than my older son used later:

Next up (about an hour later, so better lighting!) was my older son. He also struggled to find the repeating pattern. He even wanted to go to the white board to try to work there, but I wanted him to see the pattern in the Zometool set. Eventually he was able to reason that there must be two triangles and one hexagon in the repeating block. It was interesting to me that he came to that conclusion via calculation rather than seeing the pattern.

Finally, having seen the pattern we still needed to calculate the ratio of area of the triangles to the area of the repeating pattern. Again he approached this problem via calculation rather than through geometric ideas:

So, an interesting problem. The calculation itself isn’t that interesting, but finding the repeating pattern in the tiling is an interesting exercise for kids – as is finding the number of triangles in the repeating pattern itself.

# Dividing -24 into -2 groups

My younger son had a little trouble with this problem today:

This is problem 8 from the 1991 AJHME, which you can find here:

The 1991 AJHME on Art of Problem Solving’s website

When we sat down to talk through the problem I asked him how you could make a fraction large? His answer was that you wanted to divide by 1, and he thought that 8 divided by 1 would lead to the largest value here.

However, he knew that 8 was not the correct answer and he then wondered what would happen if you divided a negative number by a negative number – specifically -24 divided by -2.

I asked him what division meant in this case and he said “you’d have to divide -24 into -2 groups.”

Interesting.

I thought it would be fun to show him an old video from 2013 – in this video he and I were talking about division and specifically the definition of division:

Of course, we talked about division in many different ways over the course of that year:

and looking at some of these old videos helped me get a better understanding of today’s problem.

Thinking about the importance of understanding the definition of division reminded me a little of the first couple of minutes of Jacob Lurie’s Breakthrough prize talk (specifically from about 1:00 to 3:00):

As with Lurie’s point about an imaginary number of apples, dividing -24 into -2 groups is hard to understand. Though the definition of division is a little abstract, I think it is useful for kids to see and will help them down the road as they try to understand more difficult problems.