Fun with Fibonacci

This is an old one but fun, and a good way to use algebra to show why a trick works.  It’s a similar to showing how Magic Squares work.  It’s not a formal proof as such, but I think it’s a good way to introduce the topic.

Once students have grasped the basic concept of a Fibonacci Series (something which, in my experience they often see at Primary School even if they can’t remember what it is called), then you are ready to start the trick.

Fibonacci series don’t have to start with a 1 and a 1 as in the diagram above.  You start by asking students which two numbers they want to start with.

Then they get ready to be wowed with your powers of mental arithmetic. Tell them that you will be able to add up the first 10 digits of this sequence in your head faster than they can on calculators. Get one student up to the board to write down the numbers one by one.  TOP TIP here: make sure you have the numbers 1-10 written in a vertical column and that the chosen student writes down each term in the sequence against the numbers.  You should end up with something like this on your board:

As soon as term 7 goes up on the board, you start calculating.  You should be able to find the sum of the first 10 terms before they even get to term 10 and this is why:

I quite like doing the calculation on a miniwhiteboard, then writing the answer face down on a students’ desk and then walking to the other side of the room.  Once they have finally totalled the column of numbers on their calculator, you ask a student to have a look under the whiteboard.

And like all good magicians, you DO then go on to reveal the secrets of your trick!

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1001 is a lovely number!

I like to use this as a sort of a crescendo to teaching prime factor decomposition which is itself a very satisfying experience.

Although it sometimes feels a bit procedural it’s a nice way of:

1. Practicing times tables
2. Getting to know your primes
3. Appreciating the commutativity of multiplication.

Anyway, here’s the trick (everyone needs a calculator in front of them)

1. Chose any three digit number. Write it down somewhere.
2. Type your number into your calculator and divide by 7.
3. Hands up if you got an integer answer. Opportunity here for a nice discussion that we might expect 1 in 7 hands to be up at this point.
4. Press clear and divide the same number by 11. Repeat again with 13. Right, now they’ve appreciated that not many numbers are divisible exactly by 7, 11 and 13.  Time to blow their minds…
5. This time type your 3 digit number into your calculator twice so you have a 6-digit number. e.g.
   
6. Divide by 7. Hands up if you have a whole number. Wow, everyone. Now don’t press clear, but divide by 11. And then 13. Wow. Gets you exactly back to your original 3-digit number.

How much you chose to explain this will depend on the ability of the class, but the points are:

1. Whatever 3-digit number you chose, the 6-digit number is 1001 times the 3-digit.
2. 1001=7x11x13. Weird but true. And this is why it works.

If your students seem to like it, I always ask them to try it out on their family when they get home. I love the idea that I just might have created a discussion about maths around the dinner table – you never know…