Tag: Cut The Knot

A challenging but worthwhile probability problem for kids

Alexander Bogomolny shared a great problem from the 1982 AHSME yesterday:

I remember this problem from way back when I was studying for the AHSME back in the mid 1980s. I thought it would be fun to talk through this problem with my older son – it has some great lessons. One lesson in particular is that there is a difference between counting paths and calculating probabilities. It was most likely this problem that taught me that lesson 30+ years ago!

So, here’s my son’s initial reaction to the problem:

Next we talked through how to calculate the probabilities. This calculation gave him more trouble than I was expecting. He really was searching for a rule for the probabilities that would work in all situations – but the situations are different depending on where you are in the grid!

Despite the difficulty, I’m glad we talked through the problem.

(also, sorry about the phone ringing in the middle of the video!)

So, definitely a challenging problem, but also a good one to help kids begin to understand some ideas about probability.

Playing with geometric transformations

This week I bought a book by Greg Frederickson on the recommendation of Alexander Bogomolny:

Though I’ve hardly even scratched the surface of the book, just flipping through it showed me dozens of ideas to share with the boys.

Last night I had them skim through the book to find one idea each that they thought would be interesting to study.

My older son picked a project about hexagrams that we were able to study with our Zometool set (sorry that I forgot to adjust the focus on the camera . . . ):

The second project was one my younger son selected, but luckily he forgot how the dissection worked after picking it out last night. You can see from this part of the project that reconnecting the pieces into smaller shapes is a challenging project for kids even when they know it can be done.

Thanks to Alexander Bogomolny for making me aware of this book. I think we’ll have lots of fun playing around with the ideas we find!

A terrific probability problem for kids shared by Alexander Bogomolny

Saw this tweet from Alexander Bogomolny yesterday and knew immediately what today’s project was going to be 🙂

The problem is, I think, accessible to kids without much need for additional explanation, so I just dove right in this morning to see how things would go.

My first question to them was to come up with a few thoughts about the problem and some possible strategies that you might need to solve it. They had some good intuition:

Next we attempted to use some of the ideas from the last video to begin to study the problem. Pretty quickly they saw that the initial strategy they chose got complicated, and a more direct approach wasn’t actually all that complicated:

I intended to have them solve the 4x4x4 problem with one of our Rubik’s cubes as a prop, but we could only find our 5x5x5 cube. So, we skipped the 4x4x4 case, solved the 5x5x5 case and then jumped to the NxNxN case:

Finally, I wanted the boys to see the “slick” solution to this problem – which is really cool. You’ll hear my younger son say “that’s neat” if you listen carefully 🙂

Definitely a fun problem – would be really neat to share this one with a room foll of kids to see all of the different strategies they might try.

Comparing a tetrahedron and a pyramid with theory and experiment

We’ve done a few projects on pyramids and tetrahedrons recently thanks to ideas from Alexander Bogomolny and Patrick Honner. Those projects are collected here:

Studying Tetrahedrons and Pyrmaids

One bit that remained open from the prior projects was sort of a visual curiosity. When you hold the zome Tetrahedron and zome Pyramid in your hand, it doesn’t look at all like the pyramid has twice the volume. Today’s project was an attempt to dive in a bit more into this puzzle.

We started by reviewing the ideas that Alexander Bogomolny and Patrick Honner shared:

Next we reviewed the geometric ideas that lead you to the fact that the volume of the square pyramid is double the volume of the tetrahedron.

Now we moved to the experiment phase – we put packing tape around the tetrahedron and the pyramid and filled them with water (as best we could). We then dumped that water into a bowl and used a scale to measure the amount of water. Our initial experiment led us to conclude that there was roughly 1.8 times as much water in the pyramid.

After that we repeated each of the measurements to get a total of 5 measurements of the volume of water in each of the shapes. Here are the results:

Definitely a fun project. I wish that we’d have gotten measurements that were closer to the correct volume relationship, but it is always nice to see that experiments don’t always match the theory!

Sharing a great Alexander Bogomolny probability problem with kids

[note: I’m trying up this post at my son’s karate class. It is loud and unfortunately I forgot my headphones. I’m left having to describe the videos without being able to listen to them . . . . ]

I saw a really great problem today from Alexander Bogomolny:

By coincidence I heard the recent Ben Ben Blue podcast yesterday which had a brief mention / lament that it was hard to share mistakes in videos.

This problem is probably a good challenge problem for my older son and definitely above the level of my younger son. But listening to both of them try to work through the problem was really interesting.

I started with my older son – he initially approached the problem by comparing the individual probabilities:

After his initial work, I talked with him about comparing the probabilities of the complete events described in the problem. Initially there was a little confusion on his part, but eventually he understood the idea:

Next up was my younger son – not surprisingly, he had a hard time getting started with the problem. His initial approach was similar to what my older son had done – he looked at the one head and two heads events separately to see which one was more likely for each coin:

As I did with my older son, I asked him to look at the two events as a single event and see which one was more likely when each coin went first:

So, a nice project and an opportunity to see a few mistakes and as well as how kids approach a challenging probability problem.

Studying Tetrahedrons and Pyrmaids

[had to write this in more of a hurry than usual as 30 min of my morning was spent fishing for a dropped retainer that fell through a gap in our bathroom floor . . . . so sorry for the quite write up, but this project is a really fun way to get to hear a younger kid think about 3d geometry]

There were two really nice math ideas shard on twitter this week and I had no idea that they were related.

The first was a famous problem shared by Alexander Bogomolny:

I did a fun Zometool project with my younger son using the problem:

Zome

That project is here:

Alexander Bogomolny shared one of my all time favorite problems this morning

Then came Patrick Honner’s appearance on the My Favorite Theorem podcast:

I shared some of the ideas from the podcast and subsequent twitter follow up with my older son:

Tet

Sharing Patrick Honner’s My Favorite Theorem appearance with kids

Today – with just my younger son – I looked at a surprising connection between these two projects. We started by reviewing the Pyramid / Tetrahedron problem and then trying to guess the relationship between the volume of the two shapes.

Sorry that the lighting is so awful in these videos – unfortunately I only noticed after we were done.

Next I showed him the larger Tetrahedron with the inscribed octahedron. Although the main point of today’s project wasn’t Varignon’s theorem, I explained the theorem and asked my son to find some of the inscribed squares.

This connection was pointed out by Graeme McRae in this tweet:

At the end of the last video my son was starting to think about how volume scales. Since that’s an important point for this project I wanted to have all of those thoughts in one video.

It is interesting to hear how he tries to reconcile his mathematical thoughts about the volume of the two shapes with what he sees right in front of him.

Finally, we wrapped up by trying to find the relationship between the volume of the small tetrahedron and the volume of the pyramid.

I’m happy that my son is not convinced that the mathematical scaling arguments are correct. I can also say that holding these two objects in your hand it really does not look like the pyramid has twice the volume. Can’t wait to follow up on this.

Alexander Bogomolny shared one of my all time favorite problems this morning

This tweet brought a big smile to my face this morning:

This is an absolute treasure of a 3d geometry problem, so if you’ve not seen it before definitely take some time to ponder it.

I asked my younger son to play around with the problem using our Zometool set. Here’s what he found:

I love that the Zometool set helps make this problem accessible to kids.

An instructive counting / probability question from Alexander Bogomolny

I saw this question from Alexander Bogomolny today through a tweet from Nassim Taleb:

It reminded me a bit of a fun question I saw from Christopher Long a few years ago:

A great introductory probability and stats problem I saw from Christopher Long

The first solution that came to mind for the dice question involved generating functions. Here’s the code I wrote in Mathematica:

Mathematica Code

The general idea was to look at powers of the polynomial x + x^2 + x^3 + x^4 + x^5 + x^6 and keep track of the coefficients for powers greater than x^{12}. The one tiny bit of difficulty is that you also have to strip off the powers of x greater than x^{12} after each stage (since you only want to count the first roll giving you a sum greater than 12). Here’s that polynomial cubed, for example:

Coefficient Example

The coefficients for x^{13} through x^{18} tell us the number of different ways to get a 13 through an 18 with three rolls.

The results of counting the polynomial coefficients are given below (columns give the number of ways to roll 13 through 18, and the rows are the dice rolls 3 through 13):

Counts at each stage

These counts don’t have the right weights, though, since 21 ways of getting a 13 on the 3rd roll have a much higher chance of happening than the 1 way of getting an 18 on the 13th roll. In fact each row has a weight 6x greater than the row immediately below it. Weighting the rows properly we get the following counts:

Total Counts.jpg

Now we just have to add up the columns and divide by 6^{13} to get the probabilities of ending on 13 through 18 (and do a weighted sum to get the expected final number) -> the probability of ending on a 13 is about 27.9% and the expected value of the final number is roughly 14.69:

Probabilities

This is a nice problem to show how generating functions can help you find exact answers to problems that seem to require simulations. It was fun to think through this one.

Extending our Alexander Bogomolny / Nassim Taleb project from 3 to 4 dimensions

Last week I saw really neat tweet from Alexander Bogomolny:

The discussion about that problem on Twitter led to a really fun project with the boys:

A project for kids inspired by Nassim Taleb and Alexander Bogomolny

That project reminded the boys about a project we did at the beginning of the summer that was inspired by this Kelsey Houston-Edwards video:

Here’s that project:

One more look at the Hypercube

For today’s project I wanted to have the boys focus on the approach that Nassim Taleb used to study the problem posed by Alexander Bogonolny. That approach was to chop the shape into slices to get some insight into the overall shape. Here’s Taleb’s tweet:

Taleb1

So, for today’s project we followed Taleb’s approach to study a 4d space similar to the space in the Bogomolny tweet above. The space is the region in 4d space bounded by:

|x| + |y| + |z| \leq 1,

|x| + |y| + |w| \leq 1,

|x| + |w| + |z| \leq 1, and

|w| + |y| + |z| \leq 1,

To start the project we reviewed the shapes from the project inspired by Kelsey Houston-Edwards’s hypercube video. After that we talked about the equations we’d looked at in the project inspired by Alexander Bogomolny’s tweet and the shape we encountered there:

Next we talked a bit about the equations that we’d be studying today and I asked the boys to take a guess at some of the shapes we’d be seeing. We also talked a little bit about absolute value which briefly caused a tiny bit of confusion.

The next part of the project used the computer. First we reviewed Nassim Taleb’s approach to studying the problem posed by Alexander Bogomolny. I think it is really useful for kids to see examples of how people use mathematical ideas to solve problems.

The 2d slicing was a fascinating way to approach the original 3d problem. We’ll use the same idea (though in 3d) to gain some insight on the 4d shape.

One fun thing about this part of the project is that we encountered a few shapes that we’ve never seen before!

Finally, I revealed 3d printed copies of the shapes for the boys to explore. They immediately noticed some similarities with the hypercube project. It was also really interesting to hear them talk about the differences.

At the end, the boys think that the 4d shape we encountered in this project will be the 4d version of the rhombic dodecahedron. We’ve studied that shape before in this project inspired by a Matt Parker video:

Using Matt Parker’s Platonic Solid video with kids

I don’t know if we are looking at a 4d rhombic dodecahedron or not, but I’m glad that the kids think we are 🙂

It amazes me how much much fun math is shared on line these days. I’m happy to have the opportunity to share all of these ideas with my kids!

A project for kids inspired by Nassim Taleb and Alexander Bogomolny

I woke up yesterday morning to see this problem posted on twitter by Alexander Bogomolny:

About a two months ago we did a fun project inspired by a different problem Bogomolny posted:

Working through an Alexander Bogomolny probability problem with kids

It seemed as though this one could be just as fun. I started by introducing the problem and then proposing that we explore a simplified (2d) version. I was excited to hear that the boys had some interesting ideas about the complicated problem:

Next we went down to the living room to explore the easier problem. The 2d version, |x| + |y| \leq 1, is an interesting way to talk about both absolute value and lines with kids:

Next we returned to the computer to view two of Nassim Taleb’s ideas about the problem. I don’t know why the tweets aren’t embedding properly, so here are the screen shots of the two tweets we looked at in this video. They can be accessed via Alexander Bogomolny’s tweet above (which is embedding just fine . . . .)

Taleb1

Taleb2

The first tweet reminded the boys of a different (and super fun) project about hypercubes inspired by a Kelsey Houston-Edwards video that we did over the summer:

One more look at the Hypercube

The connection between these two projects is actually pretty interesting and maybe worth an entire project all by itself.

Next we returned to the living room and made a rhombic dodecahedron out of our zometool set. Having the zometool version helped the boys see the square in the middle of the shape that they were having trouble seeing on the screen. Seeing that square still proved to be tough for my younger son, but he did eventually see it.

After we identified the middle square I had to boys show that there is also a cube hiding inside of the shape and that this cube allows you to see surprisingly easily how to calculate the volume of a rhombic dodecahedron:

Finally, we wrapped up by using some 3d printed rhombic dodecahedrons to show that they tile 3d Euclidean space (sorry that this video is out of focus):

Definitely a fun project. I love showing the boys fun connections between algebra and geometry. It is also always tremendously satisfying to find really difficult problems that can be made accessible to kids. Thanks to Alexander Bogomolny and Nassim Taleb for the inspiration for this project.