Letter to a student – Fibonacci numbers and Lucas Numbers

Last spring I received a letter from a student who wanted to know more about me, and more, especially, about Fibonacci numbers. I wrote him back, and shared a bit more about Fibonacci numbers and their twin sibling, the Lucas numbers. Fibonacci numbers get a lot more attention, but, like real and imaginary numbers, there are many things about them that remain invisible until you put both together.

I’m including the letter below, and I hope you enjoy reading it, or sharing it with students in your life who might be interested in the Fibonacci numbers. Eddie Woo recently included some beautiful Fibonacci-related images in his TEDx Talk. These images, the “magical” connections are the bait we mathematician-educators use to draw people toward our topic. (Mystics do this too.) But you really see the magic and the meaning when you dig into them and do the math.

 

Here’s the letter.

_______________________________________________

Dear M-,

Thanks for writing. I’m happy to hear about your passion for Fibonacci numbers! They deserve your excitement.

I grew up in Olympia, Washington, and have loved math for as long as I can remember. I’m a big fan of graph theory and combinatorics, which is the mathematics of how things connect, and counting. I’m quite familiar with Fibonacci numbers, though, and I’ve explored and even taught about them.

Here’s something really important about the Fibonacci numbers: they’re only half of the picture. There’s another sequence called the Lucas numbers that go like this:

1, 3, 4, 7, 11, 18, 29, …

(see what’s happening there?)

The Lucas numbers are like the twin sibling of the Fibonacci numbers. They really belong together, and you can see things when you put them together that are invisible with just one.

For example, have you ever added every other Fibonacci number together?

The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, …
The sum of every other Fibonacci number is: 1 + 2 = 3, 1+ 3 = 4, 2 + 5 = 7, 3 + 8 = 11.
It’s the Lucas sequence! It was in there all along.

What if we add every other Lucas number together?
1 + 4 = 5, 3 + 7 = 10, 4 + 11 = 15, 7 + 18 = 25, 11 + 29 = 40, …

Notice anything about the sequence 5, 10, 15, 25, 40, … ?

For starters, all those numbers are multiples of 5. If we divide them by 5, we would get… (drumroll) the Fibonacci sequence again.

Once you have these two sequences, you can play away in almost any way you want to, and there’s some connection to find. Take the sums of every third, fourth, or fifth Fibonacci number or Lucas number. Or try the differences! Or square the numbers and add them together. Or cube them and take the difference. You won’t always be able to find a connection, but you’ll be surprised how often it works out.

Here’s one more to notice, just so you see how weird this all is. Write both sequences, one above the other, and then take the product of each number on top by each number on the bottom.

1,  1,  2,  3,  5,  8,  13
1,  3,  4,  7,  11,  18,  29
1,  3,  8,  21,  55,  144,  377

Notice anything about that bottom row? It’s every other Fibonacci number.

This is all fun, but I’m not explaining why this happens. I don’t have a book to provide that does, but if you feel like you can hack it, there’s a great video/essay series from James Tanton that you can get at. He talks fast, and some of the math is a little higher level, but it’ll give you a lot of cool ideas and connections to explore.

As for me, I’m working on all kinds of puzzles and games, most of which are about math but not about the Fibonacci numbers. There are a lot more neat connections to discover, though, and if you’re willing to put in the time to follow your passion, and don’t mind being stuck for a while when the going gets tough, I think you’ll be happy with what you get out of the exploration. Some of the connections related to Fibonacci numbers seem almost like magic. For example, if you want to know, say, the 50th Fibonacci number, just take the golden ration (approximate 1.618), raise it to the 50th power, divide by the square root of 5 (approximately 2.236), and round off your answer to the nearest whole number. It seems like magic, but it all makes sense once you develop the right tools, and put the time in. Once you do, you get to be the magician.

All the Best,
Dan

Can’t let it go

I had a lovely experience last spring after a workshop with elementary teachers in a nearby school district. I thought I’d share.

It began with saying yes to a teacher’s idea, and is such a perfect example of how saying yes can plant a seed that grows into a problem people can’t let go.

Early in the workshop I shared some unit chats, which remain my favorite opener. (You can see my writeup on unit chats here, along with images). One of the unit chats I shared was from Lee Dawson; it’s an arrangement of 21st Century Pattern Blocks (from Christopher Danielson) that gives a lot of things to see and count. In unit chats, students say how many they see, which requires that they identify a unit as well. How many triangles/rhombuses/holes/blocks/etc. are there? Depending on your choice of unit, there are lots of choices of what to count.

One teacher, perhaps jokingly, said he saw “1 square” in the picture. The whole thing is a square!

Or is it?

 

It’s easy to wave away these kinds of comments. After all, I wanted people to warm up by counting something that was tricky to count. If there’s just 1 square, there’s not much counting to do.

But we should beware saying no and shutting down the contribution. Students often test teachers to see how they will or won’t accept ideas. There’s an enormous strength in taking even silly contributions seriously, because it sets the tone for what you expect, how you intend to react, and how students are expected to react. So I said: “One square… very interesting! Except now I’m wondering… is that really a square? How do you know?”

And of course, there followed a spirited debate, since determining whether this shape is actually a square is beyond the purview of elementary mathematics, meaning people had hunches but didn’t have immediate access to the kinds of tools that would convincingly answer the question. After a few minutes I cut off the conversation to focus us in another direction, but I left the question open. (See: Don’t be the answer key.) And because it was unresolved, there were a few people who couldn’t let it go.

The fundamental issue is whether the height of three rhombuses (each made from two equilateral triangles) is the same as the width of five rhombuses. A table of teachers recreated a version of the image from pattern blocks later on. 

 

And just to be clear, we’re done with the session. All the other teachers have gone home. But there’s a group that can’t let it go. They’re pulling out mathematical tools that they haven’t used in a long time, like algebra and the Pythagorean theorem, and they’re attacking this problem with everything they’ve got. They want to know if this thing is a square or not. And they’re not leaving until they figure it out.

With their permission, I taped them as they worked through the problem. Here they are translating the problem into algebraic equations.

And here they are deciding that, as long as they’re choosing a number to set the scale in their drawings, 2 might be a better choice than 10.

I’m happy to say that they solved the problem to their satisfaction in the end. I’ll leave it open in case you’d like to try too.

The deeper takeaway for me here is that I like it when people decide that some problem is theirs. I like when it becomes personal. And look at how focused they are, and how they’re valuing and testing each others’ ideas as they try to work it out. Look how they’re having fun even as they’re struggling!

Part of doing this is to realize that this level of “stay-in-from-recess”/”don’t go home at the end of the sessions”-type focus isn’t something you can guarantee. You have to find the opportunities when they arise, and nurture those seeds as they get planted. I’ve realized that I’m always trying to notice those seeds when they get scattered in my classes, and trying to help them find soil to take root. Because when you can’t let it go, that means it belongs to you.

12

Math Game Short List 2018

In 2011 I posted a “short list” of favorite math resources for parents of young children. Since then, a lot of fantastic new materials have emerged! I thought the list was due for an update.

There are so many resources around that it’s easy to make the list too long. I’m breaking it up into separate lists—games, books, websites, etc.—and I’ll keep each one relatively short, and include only materials I think are absolutely top notch. I’ll be leaving a lot off, but if you feel I’ve missed something vital, please write a comment.

I’m also leaving off a lot of classics: Chess, Go, Backgammon, Checkers, Chinese Checkers, Yahtzee, Pente, Connect 4, Sorry, Othello, Mastermind, and so on. These games are all fantastic for encouraging math thinking. They’re also quite well known.

With that said, I give you the updated…

Short List of Math Games

Pencil and paper games

Image from wikipedia

  • Nim — I spent many car trips playing nim with my little brother. Later, I read about “nimbers” and other contributions to game theory and set theory based on the game.
    Hex     — Phenomenal game, which I’ve used in math classes and with students for years. There are some beautiful geometric and topological questions that arise from the game, but the strategy alone is enough to make it worth playing. You’ll need to print up a board to play.
  • Dots and Boxes — Kids can play this game endlessly. Some beautiful strategy, and also good exposure to arrays and multiplication.

Card games

  • Cribbage — I played cribbage incessantly when I was a kid. I credit it with my skill in adding and finding combinations.
  • Casino — Another classic from my youth. Really fun, and also great practice for adding and subtracting, and subtler strategies. I recommend Royal Casino, with Sweeps.

Games involving special decks

  • Tiny Polka Dot – we invented this card deck to make the mathematics more explicit and inviting. Perfect for building meaningful numeracy with kids age 3 – 8. (buy on Amazon)
  • Set – a classic family game. Takes a little to develop the instinct for finding matches, but super fun once you get the hang of it. (buy on Amazon)
  • 24 –  Another classic. If the speed aspect doesn’t work for your kids, it’s pretty easy to remove. (buy on Amazon)
  • Rat-a-tat Cat – Okay, technically this is just a dressed up version of the game of Golf. Still, kids like the cards, and it’s a great game. (buy on Amazon)

Board Games

  • Prime Climb – this one is ours too, and we’re really proud of it. The board is color-coded to give a completely different insight into how multiplication and division work, and it supports a much deeper understanding and flexibility of arithmetic. (buy on Amazon)
  • Gobblet Gobblers – I wish I thought of this game first. Such a simple, clever strategy game. Like Tic-Tac-Toe, but way, way better. For the harder version, check out Gobblet. (buy on Amazon)

Puzzle Games

ThinkFun stands out as the top puzzle game publisher. Their entire portfolio is excellent, but I especially recommend Rush Hour, Chocolate Fix, and Laser Maze.

Other Resources

Any I’m missing? Of course! But other folks folks have great lists too. Some good ones to check out:

Let me know what else I should know about! And remember…

What books are to reading, play is to mathematics.