Tag counting

Sharing John Cook’s Fibonacci / Prime post with kids

Saw a neat post from John Cook about using a fun fact about the Fibonacci numbers to prove there are an infinite number of primes:

Infinite Primes via Fibonacci numbers by John Cook

Funny enough, we’ve played with the Fibonacci idea before thanks to Dave Radcliffe:

Dave Radcliffe’s Amazing Fibonacci GCD post

That project was way too long ago for the kids to remember, so today we started by just trying to understand what the Fibonacci identity means via a few examples:

Next we looked at the idea from Cook’s post that we need to understand to use the Fibonacci identity to prove that there are an infinite number of primes. The ideas are a little subtle, but I think the are accessible to kids with some short explanation:

We got hung up on one of the subtle points in the proof (that is pointed out in the first comment on Cook’s post). The idea is that we need to find a few extra prime numbers from the Fibonacci sequence since the 2nd Fibonacci number is 1. Again, this is a fairly subtle point, but I thought it was worth trying to work through it so that the boys understood the point.

Finally, we went upstairs to the computer to explore some of the results a bit more using Mathematica. Luckily Mathematica has both a Fibonacci[] function and a Prime[] function, so the computer exploration was fairly easy.

One thing that was nice here was that my older son was pretty focused on the idea that we might see different prime numbers in the Fibonacci list than we saw in the list of the first n primes. We saw quickly that his idea was, indeed, correct.

This project made me really happy 🙂 If you are willing to take the Fibonacci GCD property for granted, Cook’s blog post is a great way to introduce kids to some of the basic ideas you need in mathematical proofs.

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 🙂

Sharing a “visual pattern” triangular number identity with kids

Saw a fun tweet last night from Matt Enlow:

Here’s the underlying tweet since it doesn’t show up in wordpress:

Shortly after seeing the tweet my younger son and I were playing Othello. The combination gave me the idea for today’s project.

We started by talking about the triangular numbers and why consecutive triangular numbers might sum up to be a perfect square. My older son’s idea of how to think about triangular numbers was computational rather than geometric.

Now we moved to the Othello board and looked at the geometry. My younger son found two different geometric ideas which was fun.

Finally, I gave the kids a challenge to try to find another geometric version of the identity. This question was a bit more challenging that I intended it to be, but we eventually got there and even saw how our new picture related to the sum formula that my older son used in the first video:

Sharing Kelsey Houston-Edwards’s binary video with kids

Kelsey Houston-Edwards released a new math video last week:

So far we’ve been able to use all of her videos for great weekend projects. This video had a fun little surprise because we’d seen the problem she talks about in a (seemingly) totally different context – an old magic set! Once we dug out that out magic set from under my younger son’s bed we started the project 🙂

Here’s what the boys had to say about the video:

Before jumping to the challenge problems, we looked at the old magic trick:

Finally, we tried to answer the two challenge problems from Houston-Edwards’s video. I’m sorry this got a little rushed at the end – I’d not noticed that we were out of batteries! We finished with about 10 seconds to spare!

The two challenges are great problems for kids to think through – the boys found a few interesting patterns even though the relationship with powers of 2 was a little hard for them to see.

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

A nice Simpson’s Paradox example for kids

Last night I was looking for a project for today and grabbed this book off the shelf:

In the middle of the book I found a really nice Simpson’s Paradox example and tried it out today with the kids. For more on Simpson’s Paradox, the Wikipedia page is actually great:

Simpson’s Paradox on Wikipedia

So, here’s the first part of today’s project – we have 4 boxes that have fixed amounts of red and blue cubes inside of them. First we divide them into two groups of 2 and ask which one in each group gives you a better chance of selecting a red block. It turns out that this is also a good introductory fraction exercise for kids, too!

Next we see the “paradox.” We combine the two winning boxes into one box and combine the two losing boxes into one box. Now which of the two remaining boxes gives you a better chance of selecting a red block?

So a fun and strange example for kids to see. Again, the Wikipedia page linked above gives a few more fun (and famous) examples. Really happy to have found this example in Moscovich’s book last night!

Sharing Kelsey Houston-Edwards’s Pigeonhole Principle video with kids

The 3rd video in Kelsey Houston-Edwards’s amazing new series was published last week. I’ve already used the first two videos for projects with the boys – I love this series so much!

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

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

the latest video is about the Pigeonhole Principle and begins with the question – Do any two human beings have exactly the same number of body hairs:

Before diving into the video I asked the boys what they thought about the hair question – fortunately I got two different answers!

Next we watched Houston-Edwards’s new video:

Here’s how the boys reacted to the video:

(1) They were excited about the hair result and were also able to understand and explain it.

(2) They gave a nice summary of the Pigeonhole Principle.

(3) They really liked the example about 5 points on a sphere, so we took a really close look at that example. One of the tricky parts of that problem is understanding *why* you can draw an equator through any two points – both kids gave nice explanations of that idea.

Now I moved on to a couple of fun Pigeonhole Principle examples that weren’t covered in the video. I wanted to show the boys that the idea comes up in lots of different situations, including some that are not at all obvious Pigeonhole Principle situations!

The first example comes from my college combinatorics textbook – Applied Combinatorics with Problem Solving by Jackson and Thoro:

Screen Shot 2016-12-03 at 8.47.57 AM.png

Small twitter math world fun fact – the professor for this class (~25 years ago!) was Jim Propp!

Here’s the problem (which is example 5 on page 35 of the book):

Suppose that we are given a set X of 10 positive integers, now of which is greater than 100. Show that there are two disjoint nonempty subsets of this set whose elements have the same sum.

I had to do a little bit of work on the fly to translate the problem into something that the boys could understand (and also explain quickly why there are 1024 subsets), but it seemed like they enjoyed this example:

The last problem is one I remembered when reading through some of the other examples in Jackson and Thoro’s book and is one that I talked about with the boys last year:

A challenging arithmetic / number theory problem

Here’s the problem:

Show that every positive integer has a multiple whose base 10 representation consists of only 1’s and 0’s.

It certainly isn’t obvious at all at the start why this is a Pigeonhole Principle problem!

As I said at the beginning – I love this new series from Kelsey Houston-Edwards. I’m so happy to be able to use these videos to explore fun mathematical ideas with my kids!

Sharing Kelsey Houston-Edwards’s philosophy of math video with kids

Kelsey Houston-Edwards is making a series of math videos and the first two are outstanding. We looked at the first one last week:

Sharing Kelsey Houston-Edward’s video with kids

This week’s video is about philosophy and math. A deep subject, for sure, but one which the kids thought was interesting. Here’s the video (and the twitter link so you know when the new videos appear!):

Here’s my older son’s reaction and a few things he thought were interesting:

and here’s what caught my younger son’s eye:

It is so great to see someone doing such an incredible math outreach program. I’m so excited about this video series!

Revisiting our Zometool Snowman

When we first moved into our house we did a couple of fun and large Zometool projects because we didn’t have any furniture 🙂

This week I saw a fun tweet from Eli Lubroff that reminded me of one of those projects:

Here’s a part of that old project 🙂

Snowman

Today we revisited that old snowman and had the boys talk about each of the Archimedean solids in the shape. This is a fun project – not just because the shapes themselves are cool – but you get a nice opportunity to talk about counting and symmetry. You’ll see in the videos that my older son is a bit more comfortable with the idea, but my younger son seems to catch on by the 3rd video.

Here’s a link to all of the Archimedean solids on Wikipedia:

The Archimedean Solid page on Wikipedia

And here’s our project:

First the bottom of the snowman – the Truncated Icosidodecahedron

Next was the Rhombicosidodecahedron

Next was the Icosidodecahedron

Finally the Archimedean Solid Snowman 🙂 Two years later and he still fits!

Definitely one of my all time favorites and a really fun way to discuss counting and symmetry!

A challenging counting problem from the 2011 AMC 8

My older son has been preparing for the AMC8 and this problem from 2011 gave him a little trouble:

 

2011

We talked through it this morning and he was still a little confused about why his original answer isn’t correct. The error is pretty subtle – especially for a test that gives you roughly 2 min per problem:

So, we talked for a bit more and he was able to find some numbers that he counted that did not fit the requirements of the problem:

Finally, when he got home from school tonight we revisited the problem and counted the number of solutions directly:

I like this problem a lot – it is a great one for helping you learn how to count carefully!