Counting divisors and and some sneaky arithmetic practice

My younger son chose a neat problem to discuss from Art of Problem Solving’s Introduction to Number Theory book today. It is problem 3 from section 5.2 in the book.

I like the problem for two reasons. First, this entire book gives young kids a great chance to review arithmetic while learning some ideas in math that aren’t typically part of an elementary school curriculum. Second, many of those ideas are really cool!

Here’s the problem plus his arithmetic-based solution:

It turns out that this problem can be solved without listing all of the divisors, which is sort of amazing. The next section in the book goes through the details of how to do that, but I thought it would be fun to talk through the idea using this problem. It is a fun way to show off the power of math – it is really surprising (especially to a young kid!) that you can solve the problem without actually listing out all of the divisiors!

A geometric look at x^3 – y^3

Yesterday my older son started the section about factoring sums and differences of cubes in Art of Problem Solving’s Introduction to Algebra book. After he did some problems we talked about a geometric way to understand how to factor $x^3 - 1$.

In the evening we looked at $x^3 - y^3$:

It is neat to see these algebraic identities appear geometrically. The next thing we are going to look at is a geometric way to understand how $x^3 + 1$ factors.

Differences of squares and cubes

My older son is working his way through Art of Problem Solving’s Algebra book and has come to a section about factoring sums and differences of cubes.  This topic is new to him and I thought we’d work through a few introductory examples with numbers before diving in today.

There are a couple of surprises, I think.  First, although $x^2 - y^2$ is easy to factor, $x^2 + y^2$ is not.  Second, when you move up to cubes it turns out that $x^3 - y^3$ is reasonably easy to factor after you play around for a bit, and $x^3 + y^3$ is, too.

I wanted to show him a bit about what was going on before he dove in this morning.

Here are our short talks:

(1) We started by talking about $x^2 - 1$

(2) From there we moved on to $x^2 + 1$ and found a lot of primes

(3) Next up was $x^3 - 1$ which we were able to factor with a little work.

(4) Finally, we looked at $x^3 + 1$ which actually did factor in a very similar way to $x^3 - 1$!

So, hopefully a useful introduction. I’d like to do a few more projects over the course of the week to help give some different perspectives on factoring differences of squares and cubes equations.

Revisiting Equilibrio

Over the summer we did a project on a game called Equilibrio by Fox Mind Games:

A review of Equilibrio by Fox Mind Games

I originally learned about this game from this Tracy Johnston Zager tweet:

Since doing that original project we’ve lent the game to several family’s. It just returned to us yesterday so I decided to have the boys take a fresh look at the game this morning for today’s math project. The boys have a few different things going on today, though, so it made sense to have them work one at a time.

My younger son went first – here’s his thoughts on the game and the first puzzle he was planning to build:

Here are his thoughts after finishing that puzzle.

He wanted to build one more shape and picked a pretty challenging one:

Next up was my older son. Here are his thoughts about the game:

and here’s what he had to say after completing of the first shape:

Finally, he also chose a more challenging shape. The funny thing is that almost all of the shapes look pretty much impossible on paper, but the kids figure them out eventually (though we’ve not completed the book).

I really like this game and think it is an absolutely wonderful activity for kids. There are lots of great lessons about problem solving and perserverance the game. Can’t recommend Equilibrio enough.

Using AoPS’s “Intro. to Number Theory” to build number sense

I love the books from Art of Problem Solving.  Not just because of the subjects that the books cover, but also because the books give me a great chance to secretly review other subjects.

The Introduction to Number Theory book is amazing all by itself.  What makes me love it, though, is the opportunity it gives my younger son to build number sense.

Here’s a pretty typical example from this morning.  The problem (from the review section in Chapter 4) asks you to list the common divisors of 84 and 132.   I assume that the point of the problem is for the student to find the greatest common divisor, but instead my son lists all of the divisors and gets some great arithmetic practice.

Here’s the first part of the problem:

and here’s the 2nd half.

It is fun to watch him build up his number sense. There will be plenty of time to talk about greatest common divisor later!

The birthday problem with our 120-sided dice

At the end of yesterday’s project we talked about how many rolls of our 120-sided dice  it should take for us to see a duplicate number:

120-Sided Dice

Today we dove into that question a littler more deeply. Only a little bit more, though, as this was just a quick 10 minute project before the kids headed off to school.

We began by reviewing the question and then spending a bit of time talking about how you would even approach a question like this one. Eventually we landed on the complimentary counting approach:

Next we moved to Wolfram Alpha to look at the expected results with different numbers of rolls (and sorry if the screen grab for this video looks terrible, not sure what happened with the upload, but the video itself is fine):

I’m happy to say that my younger son didn’t quite believe the numbers and insisted on rolling the dice a few times after we finished up just to see what was going on.

So, a fun (and quick) project. One other fun thing personally with this particular project is that many years ago I had to study 10-sided dice really carefully. It is sort of a long story, but this video tells part of it 🙂

120-sided dice!!

Our d120s arrived!!

Here’s the unboxing:

We were pretty lucky to have gotten our order in before all of the publicity caused the number of orders to explode. Before the boys ran off to show their friends the new dice, we did a few projects. I asked the boys to think of a question that they thought would be interesting to study with the dice. They didn’t have a lot of time to think about it, but I just wanted their gut reactions anyway.

My older son thought it would be fun to see how far they rolled. We have several different types of dice around the house, plus a few 3d printed shapes, so we saw how far the different shapes rolled:

The “winner” wasn’t actually one of the d120s, but rather a pentagonal hexecontahedron that we’d printed from Laura Taalman’s blog:

Day 194 of Laura Taalman’s Makerhome blog – the Pentagonal Hexecontahedron

The Dice Lab actually makes a d60 in the shape of a deltoidal hexecontahedron, so – no surprise, really – they are way ahead of us!

My younger son wanted to use them to make binary codes. I didn’t quite understand what he meant, but we eventually decided that the odd numbers would represent a 1 and the even numbers would represent a 0. We rolled the dice to create some 5-digit binary numbers. Strangely, we rolled lots of even numbers:

Finally, we did a project that my wife suggested – how many rolls to you think it will take until we see a number that we’ve already rolled? Fun! We had a great time exploring this question.

So, some fun little projects with the dice. Now the boys are off showing their friends and using them for some advantage playing Magic: The Gathering 🙂

Oh, and just in case you’ve not seen the video about these new dice, here it is:

A fun coincidence with our “1/3 in binary” project

Saw this tweet from Matt Henderson (via a Steven Strogatz retweet) today:

It first reminded me of one of Patrick Honner’s blog posts from a few years ago:

Honner’s post plus a lucky coincidence with a Numberphile video inspired a fun project with the boys:

Numberphile’s Pebbling the Chessboard game and Mr. Honner’s square

It wasn’t until later in the day that a different thought about this graphic hit me – we did a project about this shape LAST WEEK!!

Revisiting 1/3 in Binary

Since I was looking for a quick little something to do with the boys tonight, I head each one of them take a look at the tweet and tell me what they thought.

My younger son went first – he had lots of neat thoughts about the shape and we eventually found our way to the connection between this shape and writing 1/3 in binary:

My older son went next – he didn’t see the connection right away, but we eventually got there, too (oh, and ugh, just listening to this video I realize that I misunderstood my son when he was talking about a geometric series – whoops, he did say the right definition).

Thank you internet – what a fun coincidence!

Revisiting 1/3 in binary

Last night we talked about writing $pi$ in base 3.   A long long long time ago we talked about writing 1/3 in binary.  Here are those two projects:

Pi in base 3

Writing 1/3 in Binary

I suspected that the boys wouldn’t remember the project about writing 1/3 in binary, so I thought it would make a good follow up to last night’s project.

I started by just posing the question and seeing where things went. They boys had lots of ideas and we eventually got most of the way there:

At the end of the last video they boys figured out that if our number was indeed 1/3, if we multiplied it by 3 we should get 1. That reminded them of the proof that 0.9999…. (repeating forever) = 1.

We reviewed that proof and applied it to the situation we had now.

Just one little problem . . . what if we apply the idea in this proof to a different series, say 1 + 2 + 4 + 8 + 16 + . . . . ?

We’ve looked at the idea in this video before:

Jordan Ellenberg’s “Algebraic Intimidation”

We felt pretty comfortable believing that 0.9999…. = 1 and that we’d found the correct series for 1/3 in binary, but do we believe the results when we apply the exact same ideas to a new series?

I love projects like this one 🙂

Pi in base 3

There’s a new – and amazing – video out about the seemingly crazy math fact that 1 + 2 + 3 + . . . . = -1/12:

At 16:53 in the video something amazing happens – a t-shirt with $\pi$ in base 3!! I ordered it immediately 🙂

I’m happy that I did because the boys were curious about how to do the calculation. Tonight we talked about it starting with a few easier examples first:

Next we moved to Wolfram Alpha to finish up the calculations:

It was nice that we were able to get 6 digits after the decimal point so easily. Fun little project coming from an awesome shirt 🙂