These family math project are just for fun. These projects tend to cover either fun math we find around the house – see the paper folding example from all the way back in Family Math 1:

or, if not stuff from around the house, they are intended to be a fun overview of some advanced math. The overview of fractions and decimals in binary was supposed to be in the second category, but it led to a really great surprise this morning.

Today with my younger son we moved on to a new chapter in our book – repeating decimals. A few days ago we had started off talking about decimals and fractions by reviewing why .9999…. = 1, so I was hoping to play off of that to show why 1/3 = 0.33333…. However, when I sat down and asked my son what he thought the decimal expansion for 1/3 would be I got a little surprise:

“I don’t know, but I know what it is in binary.”

So fun that he remembered this talk from the Family Math project from a few weeks ago:

With that, we started down a totally new path – how does knowing what 1/3 is in binary help you understand the decimal expansion?

Since my kids love playing with these cubes, I thought there must be a great Family Math project hiding in here somewhere. After thinking about what might be a fun project for kids, I decided that counting the number of different arrangements of a 2x2x2 cube might work well.

We really haven’t done much counting / combinatorics yet, so it seemed like the best place to start was with a much more straightforward problem – counting the number of arrangements of blocks in a line. That meant instead of Rubik’s cubes, the math starts with counting arrangements of snap cubes. I think kids will enjoy trying to figure out the two problems we pose in this video – it was definitely fun for me to hear my son’s reasoning. Also, sorry for the glare in the first two videos, I didn’t notice it until I was turning off the camera after the 2nd video.

The next step before we get to the cube is counting the arrangements of snap cubes in a square rather than in a line. This problem allows us to talk a little about symmetry. The problem here is still pretty easy to understand, but understanding the symmetries makes it a little bit harder than counting arrangements in a line. My son struggled a little bit here, but hopefully those struggles are actually helpful in understanding why this problem is a little harder:

With that background we moved on to counting the possible arrangements in a 2x2x2 cube. We used the snap cube counting as the a starting point, and were also lucky enough to have a broken cube handy to help us see how to build one from the pieces. Counting the symmetries here is a little bit more difficult here, and made even more complicated by the fact that some of the arrangements of a 2x2x2 cube cannot be solved:

The very last step is figuring out which arrangements can be solved and which can’t. A rigorous solution to this problem is a little bit outside of what I think my kids can understand, but from playing around with these cubes they to have some idea about the answer. If you build up a 2x2x2 cube from scratch, you’ll either be able to solve it, or the best you’ll be able to do is nearly solve it with just one piece being rotated. Since there are three possible rotations for each piece, 1 out of 3 arrangements is solvable, so for our final answer for the number of arrangements, we have to divide by 3.

This was a really fun project. I’m a little sorry that I had to squeeze it in quickly since I’ve leaving for a short trip in 10 minutes, but we still had fun. Lots of neat math hiding in these cubes!

I’m not even sure the title is asking the right question, but hopefully it is close enough to the right question!

Work has been really busy for me lately and both of the kids have also been a bit sick. Because of these two little problems I’ve been doing a lot of review work with the boys rather than trying to cover too much new material. We are still plodding along, though, and I started Chapter 20 in Art of Problem Solving’s Algebra book with my older son earlier this week. The first section was a review of radicals.

Maybe not the most interesting topic in general, but there were a few neat sample problems. For instance, we had a really nice discussion about the problem: “Explain why when ” Watching my son struggle with how to explain this fact was fascinating.

We’ve spent the last couple of days on the problems at the end of this section, and with only one problem left to do today, I’d actually planned to jump to the next section on absolute value. For no particular reason, though, I changed my mind and got a great surprise.

Here’s the problem:

20.1.6. Solve the equation

This turned out to be one of the most interesting 45 minute discussions that we’ve had on a math problem in a long time. I’m so happy that I didn’t skip over this one.

As I’ve mentioned several times already on the blog I’ve never taught any elementary math before and I’m just learning as I go. Sometimes I feel as though I’m covering the material better with my younger son for the laughable reason that I’ve already been through the topics once, but with my older son teaching just about everything is new to me. That was certainly the case here.

With that background, I often struggle trying to understand how my son sees a problem that requires essentially no thought from me to solve. This one was a real eye opener. Among the interesting points of discussion:

(1) How do you deal with all of these square roots?
(2) Once you figure out that you should square them, now what – there are still square roots?

(3) Ah ha – rearrange and square again to get a quadratic equation.

(4) Wait – how come one of the solutions of that quadratic equation doesn’t solve the original problem?

(5) Really, seriously, wait, how could that be, didn’t we always do the same thing to both sides of the equation?

Anyway, though it was hardly perfect from start to finish, the discussion we had was super fun. I’m going to take another full morning to talk through this problem one more time tomorrow because I think there are so many lessons hiding in it. But, if you’ve read this far, what I’m really interested to hear is what I wrote in the title – how would you talk about this problem?

** update on 4/24/2014 **

Here’s how we talked it through this morning:

** update 2 on 4/25/2014

Talking through the algebraic solution part 1:

Talking about why only one of the two “solutions” we found in part 1 actually solve the original equation

I have my kids write short reports every day on chapters they select in Cifford Pickover’s amazing “Math Book.” (sorry I don’t know Latex well enough to format the title correctly).

These reports give them a chance to see fun math outside of the standard stuff covered in their school books. Last week my younger son stumbled across the section on the Banach-Tarski theorem and it really intrigued him. I finally got around to talking a little more about that theorem with the boys today, though it obviously isn’t the easiest subject to cover with younger kids!

The first thing we talked through was the two different statements of the theorem. A short, and excellent as usual, summary of the two theorems can be found on the Cut the Knot website:

I covered the the two different statements of the theorem and moved on to a much easier to understand example of infinite sets – why there are the “same number” of positive integers and positive even integers.

With the example with integers and even integers showing us how to compare infinite sets, I moved on to showing them that a line segment of length 1 has the “same number” of points as a line segment of length 2. The ideas in this proof at least let you see how one object could somehow be the “same size” as something that seems to be twice as large.

The next thing we talked about was how we could see that a line segment of finite length could have the “same number” of points as an infinitely long line. We approach this idea using stereographic projection:

Next we moved on to 3D and I showed them that the sphere has the same number of points as the plane. The idea here was also to look at stereographic projection, though luckily for this example we have a special prop designed by Henry Segerman that we found on Laura Taalman’s 3D printing blog:

Goes without saying that holding the model in your hand is quite an improvement over a sketch on the board!

So, by this time we’d seen that a line segment has the “same number” of points as the whole line, and a sphere has the “same number” of points as the plane. Now we show something really amazing – a line can fill up a square, and hence the plane. That means that a line segment has the “same number” of points as the whole plane. Wow.

So, the 5 minutes on camera was actually preceded by a couple of hours of printing and drawing Hilbert curves on our own. It made for a really fun morning:

Lots of people to thank for this one – Clifford Pickover, Kerry Mitchell, Alexander Bogomolny, Laura Taalman, and Henry Segerman. So glad to have resources like theirs online to help kids learn about this kind of fun math.

** Addendum **
After finishing up this post we were playing around with our 3D printed Hilbert curve and took it off the base. After we did that, we found that we could stretch it out into almost a line. Cool!! I think it helps kids get a better feel for the fact that it is all one long line segment twisted up into a curve:

In my jet-lagged state I came up with what seemed like a great response, but not too surprisingly, it didn’t seem so great once I wasn’t so jet-lagged! However, Pershan’s post stayed with me for a couple of weeks. I’m not sure why – there wasn’t anything specific that bothered me, or at least nothing that I was able to articulate well. I still couldn’t shake it though, and because I’ve spent the week with my older son talking about logs and imaginary numbers I really wanted to talk to the boys about i.

By lucky coincidence just this morning Steven Strogatz linked to another blog post on twitter that helped me collect my thoughts on the subject of i (even though the blog post has nothing to do with complex numbers):

The part of this post that got my attention was this line: “Well, I’ve finally got my answer, and it only takes eleven words: Math is big ideas, approached from as many angles as possible.”

So, a few weeks of thinking things over in my head combined with a little jolt from the Math With Bad Drawing’s post led to this morning’s Family Math conversation:

(1) The first thing I wanted to talk about was the hardest – structure. I dug up my old copies of Mike Artin’s and van der Waerden’s “Algebra” books and talked about rings. I don’t think I did a good job here, but all I wanted to do was point out some of the things that mathematicians think are important about the number system. A lot of this was sort of review – they know the words “associative” and “commutative” though they may not have ever seen all of this structure on the board all at the same time. They certainly had never heard the word “ring” in this context before:

(2) With the background structure out of the way, and in particular with the mention of 0 and 1, I wanted to start talking about some specific numbers. We started with an “easy” number – 0. What are the important properties of 0? Why doesn’t 0 seem strange to anyone any more?

(3) Next up was -1. Same sort of idea behind this talk as the talk about 0. -1 is a little hard to understand, but we seem to be pretty comfortable with it and I don’t think many people would think -1 is a number that mathematicians just made up:

(4) Next up was . I wanted to talk through some of the ideas behind irrational numbers and hint at some of the confusion that they had caused over the years. We talk about a simple right triangle and show how comes up pretty naturally. We also then walk through the proof of why it is irrational:

(5) Now for two famous irrational numbers and e. We just touch on a few simple properties of these two numbers and talk about they are in some sense even more strange than .

(6) Finally we get to i. Following Artin and van der Waerden, I introduce the complex numbers by looking at a specific quotient ring in a polynomial ring . . . . Ha!

Actually, we first quickly review a few of the interesting numbers that we’ve talked about already and then point out that we still do not know about any number that satisfies a pretty simple equation – . I call that solution “i” and then we talk about a few interesting properties ranging from the famous to Gauss’s Fundamental Theorem of Algebra. I also mention a few applications that come up in physics.

So, thanks to Michael Pershan and to Ben Orlin for both causing and helping me to think through talking about i. Definitely a fun morning.

By lucky coincidence I’ve just been talking about logs and exponentials with my older son. By unlucky coincidence he’s been sick for a few days, but he was feeling well enough this morning to talk a little. I thought it would be fun to try to answer the question on twitter with a few non-standard log examples for kids.

The goal was just to show a few examples that don’t appear to have anything whatsoever to do with logs. I’m not trying to go into any detail, just wanted to show some fun and unusual math results.

The first bit of the talk was to switch from log base 10 to log base e. The first result we talked about was that the area under the curve y = 1/x from 1 to x is equal to ln(x), followed by a couple of surprising results related to that:

The next part of the talk was about the distribution of prime numbers. There are so many amazing results here that it was hard to even understand how to limit the talk to 5 minutes. I ended up choosing three ideas – the approximate number of primes less than n, the approximate number of twin primes less than n, and the amazing Erdos-Kac theorem:

The last part of the talk was about the famous equation . We’ve spent a little time in the past talking about this equation, so it wasn’t the first time he’s seen it. The nugget I wanted to extract out of the equation this time around was that this equation allows us to talk about ln(-1). After the video I explained some of the complications that arise when you talk about logs of negative numbers, but those details weren’t important for this talk. For now, all I wanted to show is that ln(-1) might be equal to .

So, the nice little question on twitter made for a fun morning. Not sure this is exactly the answer that anyone was looking for, or if it is even an answer at all, but it was neat to have these conversations with my son this morning.

Yesterday we had quite an adventure with Graham’s number. I wanted to have one more talk about powers before the weekend was over, so the topic I chose for today was Fermat’s Little Theorem. I wasn’t planning on going into too much depth on the number theory side, but I did want to emphasize a little bit of logical thinking and just walk through a few basic computations with powers.

We started with a quick introduction to Fermat’s Little Theorem and then computed an easy example with the number 5. After that we talked a little bit about the logic – prime numbers satisfy Fermat’s Little Theorem, so if a number fails to satisfy Fermat’s Little Theorem, then it must not be prime:

Next we moved on to a couple of slightly more complicated examples. First we took a look at 21 and tried to find the remainder when is divided by 21. I told them that we only needed to worry about the remainders, which is fine for today. We’ll cover the details behind that idea sometime later on when we study number theory. It was fun, though, just to show that you could compute the remainder when is divided by 21 without even knowing what the exact value of is!

We also discussed a number that satisfies the conditions of Fermat’s Little Theorem that isn’t prime – 1,729. We didn’t go into the calculations, but did tell a fun and famous story about 1,729.

Finally, we couldn’t spend the morning talking about Fermat’s Little Theorem without mentioning Fermat’s Last Theorem. We walk through a few examples of triples that satisfy the Pythagorean Theorem and talk through what Fermat’s Last Theorem says. We conclude by mentioning a few famous problems in math that remain unsolved.

Between yesterday’s discussion of Graham’s number and this one on Fermat’s Little theorem, it was definitely a fun weekend of math.

I’d never head of Graham’s number before, but Lamb’s piece (and the Numberphile piece linked in the piece) really grabbed me. Here’s their video for completeness:

I spent the week thinking of how to talk about Graham’s number with the boys. It was a fun and challenging week, and one conclusion for sure is that Lamb’s title is indeed the correct summary – Graham’s number really is too big to even talk about!

After a couple of false starts earlier in the week I choose a pretty well-known problem involving a chess board as the starting point. The problem involves putting one penny on the first square, 2 pennies on the 2nd square, 4 pennies on the 3rd square, and proceeding all the way to the 64th square. How many pennies are on the 64th square? Seemed like a good starting point for kids because you got to talk about powers as well as some really large numbers:

The next step was trying to introduce the arrow notation used in describing Graham’s number. It took me a really long time to understand the notation, and those struggles made me want to cover the notation with the boys in only a superficial way. Lamb’s piece above links to the Wikipedia page on Graham’s number – that’s as good a place as any to start if you really want to dig into the notation. So, this part on Graham’s number is a little bit of notation and a little bit on powers of 3:

The second part of the discussion of Graham’s number is understanding how to compute the actual number. The up arrow notation makes things look easy, but the computations themselves aren’t so easy. In fact, almost immediately you encounter numbers that are virtually impossible to describe because they are so large. Off camera I explained that as you created towers of powers of 3, the number of digits in each successive number was roughly equal to half of the previous number. I’m not sure how much that helps, though, when you are talking about numbers that have 10 to the 3.6 trillion digits. Ha ha:

This is all pretty heavy stuff, so I wanted to come back a little bit closer to reality, so we wrapped up with a question that Steven Strogatz posted on twitter last year:

Consider the sequence a, a^a, a^(a^a), etc., where x[1]=a and x[n+1]=a^(x[n]), n>1. For which a>0 does this sequence converge?

This is a pretty neat question and since it involves an infinite tower of powers it is sort of connected to what we were talking about. Strogatz’s question essentially asks if there are numbers greater than 1 for which the infinite tower of powers does not go to infinity. (Also, I just started logs with my older son last week, so it was also a nice example with logs. )

Finally, since we weren’t going to solve the equation that came up in the last video on our own, we moved over to Mathematica and found the neat surprise that the answerr to Strogatz’s question involves the number e:

So, special thanks to Evelyn Lamb, Numberphile, and Steven Strogtaz for providing the inspiration for another fun Family Math day. Talking about Graham’s number was really fun.

As I was putting my younger son to bed last night he asked me about decimals in base 4. I don’t know why – our previous conversation had been about how to play the golf game on the Wii – but something about the weekend talk had obviously stuck with him.

This morning we had some fun with a specific number – 0.111111….. (repeating forever). What does this number equal in different bases? It was a fun and unusual way to review decimals:

the Vi Hart video we reference at the end is here:

Last night my younger son asked me about . I got so caught up in thinking about what to talk about that I actually forgot the context that gave rise to the question. Oh well.

It certainly is true that comes up less frequently than plain old , but there were still a few fun ideas.

First was the obvious one – squaring the circle:

Finally, a fun suggestion from twitter:

@mikeandallie Which is larger, the square root of pi or the pi-th root of two?