More fun with counting and geometry

We finished section 3.3 of Art of Problem Solving’s Introduction to Counting and Probability today. One of the problems in the section asked about a few relatively basic integer sums. It made for a fun little morning project.

The first sum I asked the boys to look at was 1 + 2 + 3 + \ldots + n. They’d seen this sum already in the examples from the book, but I wanted to hear them talk through it on their own:

The next sum was the first n positive even integers 2 + 4 + 6 + \ldots + 2n. The tiny bit of extra difficulty for kids here is keeping track of what n is. Sorry about the garbage truck stopping by in the middle of this video . . . .

The next sum was the first n positive odd integers. There are a couple of ways to approach this sum, and the answer is a bit of a surprise:

Since we just found an amazingly simple formula for the sum off the first positive n odd integers, we brought out the snap cubes to see if we could find the connection to geometry:

So, a fun little project. A few basic sums that we encounter when learning to count give us some nice practice with both arithmetic and basic algebra, and we also get a surprising and pretty cool connection to geometry. Nice morning!

Approaching an elementary counting problem several different ways

Had sort of a busy day today with many hours driving. When we finally got home we planned on working through a few new counting problems, but everyone – me included – was pretty tired. Instead I had the boys work through a single problem from the new chapter in our Introduction to Counting and Probability book :

How many games are played in a round robin tournament with 8 different teams?

What caught my attention was that each kid’s approach was pretty different. We went through a few different approaches to the problem (with 5 people) for our project today:

First up, my younger son’s solution – he essentially lists out all of the games and is careful to avoid overlaps:

My older son’s approach was more geometric:

Next I showed them an approach where you do not avoid over counting:

Finally – to end with a bit of fun – is there a geometric way to understand this over counting idea:

So a fun and quick little project. One of the surprising things that comes up when you are learning to count is that all of these connections between geometry and numbers seem to come up all the time!

Exploring angles in platonic solids

The boys wanted to do a project with the Zometool set this morning, so I had them flip through Zome Geometry to find a project. They found a neat one about measuring dihedral angles in the platonic solids.

This project is a really great example of how useful the Zometool set is for exploring 3D geometry. Without the Zometool set this would be a pretty challenging project for kids. With it – we can build the shapes from scratch, measure the angles with a protractor, and – as an extra bonus – have every thing cleaned up and put away in an hour!

We started by building the shapes off camera and then having the kids talk about the project and talk about the shapes. Their initial thoughts about the various angles in the shapes hint at some of the ideas that they will learn about in this project:


First up, we explore the dihedral angle in a tetrahedron. It takes a while for the boys to figure out how to measure this angle (and even what angle it is). Once they see what the angle is, figuring out how to use the protractor to measure that angle takes a little bit of time. Luckily, though, once we solve this measuring problem for the tetrahedron, we can use the same idea to measure the dihedral angles in the remaining shapes.


Next up is measuring the dihedral angle in a cube. After the challenge of finding the correct angle in the tetrahedron, it is probably a lucky break that the next shape we looked at was fairly easy.


Next up was the octahedron. While holding the shape in his hands, my older son thinks that there is more than one angle we need to measure. He wondered about these angles at the beginning of the project, too. Luckily the Zometool set helps him see that, in fact, all of the dihedral angles are the same. Yet another educational win for the Zometool set!


The dodecahedron is a little larger than the other shapes we’ve looked at. I try to help out a little bit at the beginning, but wisely the boys ignore my help – always the right decision on any engineering problem!

For this shape we spent a little bit of time comparing the angles in the pentagons with the dihedral angles, too.


It was probably lucky that the icosahedron was the last shape we looked at. In retrospect I should have had the boys make this shape with side lengths equal to two long blue struts. The problem with the size of our shape was that the protractor was too big to fit inside the triangles to get a good measure of the angle. It took a little bit of playing around, but they were able to see that the angle was about 140 degrees.

We wrapped up this video with a quick review of what they learned during the project.


So a fun little geometry project. I’m really happy that we can use the Zometool set to introduce some basic ideas from 3D geometry to the kids. Hopefully seeing ideas like these one early on will take some of the mystery out of learning more 3D geometry when they encounter it later in school.

Introduction to geometric series

Earlier in the week we looked at a problem that involved the series 1 + 2 + 2^2 + 2^3 + \ldots + 2^n. Each kid had a different and interesting approach to summing that series. For today’s project I wanted to review each of their approaches and then explore a few other simple geometric series to see if we could use the same ideas to add them up, too.

We started with a review of my older son’s approach to the original series. His approach was essentially mathematical induction, which was pretty cool since we’ve never talked about mathematical induction. Oddly, though, telling him that his approach had a name seems to have confused him a little bit, and it takes a little while for him to remember what he did. That little bit of confusion made me happy that we decided to revisit this topic today:


Next I had my younger son explain his approach to summing the series. His approach came from noticing a connection to binary numbers when he first saw the series. That connection to binary is a really clever way to think about this sum:


With that introduction and review of their prior ideas, we decided to see if we could apply those ideas to a new series: 1 + 3 + 3^2 + \ldots + 3^n. My older son tries his approach first – the connection isn’t obvious, but then we compare a few of the sums to the powers of 3 and make some progress.


Now my younger son takes a shot a the new series with his “trinary” ideas.


Finally, we wrap up the project by looking at the series 1 + 9 + 9^2 + \ldots + 9^n At this point they’ve seen each of the two prior sums in two different ways and they are able to see how the prior ideas apply to the new series. They even speculate about what a general formula would be!


So, a fun little exploration. I’m happy that the boys had a chance to review their prior solutions, especially since some of the ideas weren’t quite in the front of their mind anymore. Hopefully that review was helpful even if it wasn’t intended to be the main point of the project. By the end of the discussion today they seemed much more comfortable applying their ideas to a few new (though obviously similar) series. It is fun to show them how two ideas that seem pretty different help you work through these problems.

Quick question about kids and numbers

This morning in our little summer tour through Art of Problem Solving’s Introduction to Counting and Probability we came across the question:

How many different ways are there to arrange the letters of the word MISSISSIPPI?

The question has a couple of potential ways to leave the answer. You could write:

(1) \frac{11!}{4! 4! 2!}, or

(2) 34,650

for example. However, the boys told me that the answer was:

(3) 11*10*9*7*5.

So, they were sort of in the middle of (1) and (2) – not wanting to leave the factorials, but also not wanting to do the multiplication. Ha!

I actually wanted them to work out the multiplication simply to continue to build up their numbers number sense. At the same time, though, answer (1) above is really want you want to build up their counting sense (for lack of a better term).

So I was just curious what people thought was the best way to express the answer for this question? Also, I suppose, if the answer to my first question would vary based on the age of the students working on the problem?


Why I love working with my kids

Saw this problem yesterday via a Fawn Nguyen tweet:

Thought it would be a fun problem to talk through with the boys, but then my older son had to run an errand with wife so it ended up just being my younger son.

His approach just blew me away. I’m so glad to have the opportunity to have conversations like this with my kids:


[update from the day after originally publishing follows]

This morning I went through the question with my older son. He struggled with the problem much more than my younger son did. One of the contributing factors to his struggle is that he chose to keep track of the number of leaves on the ground rather than the number that fell each minute.

It was interesting to watch him work through the difficulties. He looked for lots of different patterns before finding the right one. I broke up his solution into 3 parts:




After he finished we had a neat discussion about how to add up the series:

1 + 2^1 + 2^2 + \ldots + 2^n

He had a neat approach, which was essentially mathematical induction. You know that 1 + 2 is 4 – 1. That helps you see that 1 + 2 + 4 is 8 – 1, and so on. That was a neat way to think about the sum and led to a fun (and basic) conversation about mathematical induction.

Thanks to the Math Forum for this fun problem.

Larry Guth’s “No Rectangles” problem

[note: sorry for the rushed write up – I wanted to get this written up before 9:00 this morning because of a work project that’s going to have me tied up for most of the rest of the day]

Yesterday I was lucky to attend a lecture that Larry Guth gave at MIT:

Larry Guth’s “No Rectangles” problem

It was sort of doubly lucky because I’d already set up a meeting in Cambridge and a little bit of juggling with some times allowed me to have the meeting and attend the lecture. The problem that Guth discussed has the wonderful property of being simultaneously accessible to kids and interesting to research mathematicians. His talk began with a discussion about a simple 3×3 grid and ended with a discussion of intersecting lines in the field Z mod p! Today I used some of the basic ideas in Guth’s talk as the basis for a fun little project with the boys.

I really loved talking through Guth’s problem with them. As often happens in our projects, we ended up going in a slightly different direction than I’d anticipated – but the discussion we had as a result was also much better than I’d anticipated!

We started with a quick introduction to Guth’s “no rectangles” problem and the boys tried to work through the 3×3 case:


The first part of the project ended with the boys finding a way to place 6 snap cubes on a 3×3 square without forming a rectangle. Now we try to see if 6 is the maximum number or is there a way to place 7?

They approached this question in a way that surprised me a little. The first thing they wanted to do was count the total number of rectangles in the grid (which has one subtle point). This approach turned out to be a pretty interesting way to look at the original problem. It is also a pretty interesting counting problem for kids all by itself:


So, having counted 9 rectangles in the 3×3 grid, the boys then tried to see if they could eliminate all 9 of these rectangles by taking away just two of the snap cubes. The conversation here was super fun – lots of great opportunities for the kids to talk about patterns and explore different ideas. For example, my younger son noticed that you are forced to take away one of the cubes on a corner of the 3×3 grid.

In the 2nd half of this video I show them a different way to see that you can’t have 7 cubes on the grid without forming a rectangle. This approach – which is the way Guth explained it in his talk – involves the pigeonhole principle.


For the next part of the project I thought it would be fun for the kids to try to figure out the total number of different ways to place snap cubes onto an NxN grid. Thinking through this problem helps you understand that trying to work through the “no rectangles” problem by brute force with a computer is going to take a long, long time.

Just like the rectangle counting problem the boys came up with in the earlier part of the project, this problem is a nice counting problem for kids all by itself.


Now that we’ve found that the total number of ways to put the snap cubes onto an NxN square is 2^{n^2}, we try to understand the size of some of these large powers of 2. We start by finding an approximation for 2^{25} – the number of ways to put snap cubes on a 5×5 grid – and then try to see if a computer could handle a number this large.

What about a 10×10 grid, though? This problem ends up bringing to the surface a little problem understanding the exponential notation, but that’s one of the great things about this project – you have a nice opportunity to talk about / review exponents while talking about a really interesting problem:


For the last part of the project we try to understand how long 10^{21} seconds is. It is actually so long that there’s probably no way for kids to have an intuitive understand of that many seconds. However, we can try to convert it to other units of time, and years seemed like a pretty natural choice. We found that if you can check 1 billion squares per second, searching through all of the possible ways to put snap cubes on a 10×10 grid would take about 300 trillion years!!

Since that’s more time than we’d probably want to spend on that problem, you need to look for a different approach. One really cool thing about Guth’s problem is that there’s a surprising (to me!) connection to geometry and number theory hiding beneath the surface. Guth ended his talk by explaining how to solve the problem using those ideas. That approach, by the way, is truly marvelous but there wasn’t enough room in the margin of my white board to explain it . . . . 🙂


I love being able to share projects like this one with the boys. There aren’t too many problems that are both interesting to research mathematicians and accessible to kids. The kids were able to understand the problem right way and had lots of interesting ideas about the 3×3 case. Exploring the 4×4 case would probably be lots of fun for kids, too. It always amazes me how much fun you can have and how far math conversations with kids can go when the problems keep them engaged. It really was a great bit of luck to have had the opportunity to attend Guth’s lecture yesterday.

A pretty neat counting problem from Mathcounts

The boys came back from a 2 day camping trip today. I had some afternoon meetings, but luckily they got back early enough for us to do a little counting project.

Without anything in particular planned, I just picked the first problem in the challenge problem section at the end of chapter 2 in our Introduction to Counting and Probability book. The problem is a pretty neat case by case counting problem (even though it looks fairly dull at first glance):

How many positive integers between 24 and 125 have a digit sum that is a multiple of 7?

The boys did a nice job of breaking the problem up into cases right away. In the first video we look at the cases where we have a 2 digit integer. I really liked hearing my younger son talk about the two digit numbers whose digit sum is 14 – it is fun to hear a young kid bring together the ideas you need to work through this problem. One of the nice surprises about this problem is that there’s some interesting number sense ideas hiding inside of it!


In the next video we look at the three digit numbers from 100 to 125 whose digit sum is a multiple of 7. There’s a little surprise in this part, too, since there are no numbers in the range with a digit sum is 14. It takes a moment for the boys to realize that the largest digit sum isn’t necessarily the largest number, so that’s a another neat math idea hiding in this problem:


So a nice problem with a couple of neat ideas beyond the case by case counting. Fun little project to do in the lucky extra 15 minutes I had with the kids today.

10 number 10’s

[not a math post – this one is about ultimate]

Yesterday I had a 6 hour drive and for part of it was sort of day dreaming about how great it would be if women’s ultimate could get as much great publicity as the US women’s soccer team did. After seeing post after post about Carli Lloyd I thought about all of the great #10’s in women’s ultimate. Saw lots of them play this past weekend. In no particular order:

(1) Fury’s Gen Laroche

7 club championships + 3 world championships. Oh, and did you know she’s been on Sportcenter’s top 10?


(2) Riot’s Shira Stern:

She’ll be representing the US on the U23 team in London next week, and this is her 2nd time on the U23 team! You can follow the U23 teams at the World Championships here:

Skyd Magazine’s U23 coverage

And then 30 minutes after publishing this post, Shira’s grab in the US Open finals gets a Sportscenter top 10 nomination:


(3) Brute Squad’s Amber Sinicrope

So happy to see her returning to Brute this season. Even with the year off, I think she’s the active “most seasons with Brute” player. I have her Amherst high school jersey hanging on the wall of my study – from her pre- #10 days 🙂


(4) Scandal’s Jenny Fey

Guess who led the US Open in combined goals (13) and assists (23). Come on, take a wild guess . . . .

US Open stats on the right hand side

(5) Phoenix’s Michelle Ng

Before Qxhna’s All Star Ultimate Tour the biggest fundraiser I can remember in ultimate was this:


I saw Liên Hoffmann wearing her Without Limits jersey at the Harvard stadium on Monday! There are not enough ways to thank Michelle for all she’s done for ultimate.

(6) Speaking of Qxhna – she wore #10 in college, for the Beach World’s team, and on Brute Squad last year:

Please support her All Star Tour if you haven’t yet:

The All Star Ultimate Tour

and watch her incredible Callahan video, too:


(7) Another #10 super star from college who is crushing it in club right now is Ozone’s Sophie Darch. Here’s her awesome Callahan video from last year:


(8) One #10 we’ve not been able to see yet this year is former Molly Brown captain Lindsey Cross.

She had a nice interview a few years back on Ultimate Interviews:

Lauren Boyle and Lindsey Cross Interview

and also kicked off Molly Brown’s awesome twitter roster announcement this year:

(9) One other #10 (who may not be wearing #10 this year, I’m not sure) that we’ve not seen yet is former #10 for the Capitals and current Traffic player Kathryn Pohran. I’m very happy to see her back after an awful knee injury at ECC last year.


(10) Who’s the 10th #10? Who did I forget on my drive last night? You tell me!

A challenging case by case counting problem

We were finishing up Chapter 2 of Art of Problem Solving’s Introduction to Counting and Probability this morning and ran across a pretty challenging counting problem. It made for an excellent little project with the boys since there are multiple ways to approach the problem.

Here’s a quick introduction to the problem:


First up was my younger son using the “b’s” as the cases. We’d already spent time going through this approach, but I wanted to make sure that the kids had understood the cases.


Next up was my older son counting using the “c’s” as cases. Both kids really struggled with this approach when we talked through the problem earlier in the morning, and my son struggled a little to get going here. Once he understood that the case we were looking at first was the case c = 2, though, he was able to work through the problem.


In the last video we didn’t quite get to the end after 5 minutes, so we broke the discussion into two pieces. Here we finished up the “c” case and found that our answer matches the answer we got when we looked at the cases involving the “b’s”. Adding up the various “c” cases was a little challenging for the boys the first time around, so that was another reason that I wanted to go slowly for this last part of the project.


So, although a fairly contrived counting problem, I like this project a lot. The kids got some nice practice working with both algebraic expressions and with numbers. They also had to understand a little bit about inequalities to work thorough the problem. Of course, from the point of view of learning about counting techniques, the nice thing about this problem is that there are multiple ways to break the problem into cases. Seeing that those to approaches lead to the same answer hopefully helps them gain a better understanding of case by case counting.