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.

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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.