# Adventures in Mastery 5: Making Connections

I taught my year 7 class today and had the most wonderful time.  I really love my year 7s, they’re so enthusiastic.  So far this year we’ve done place value, rounding, four operations (with natural numbers and decimals), powers, roots and primes, negative numbers, order of operations, fractions (including four operations) and are early into our unit on percentages.  Today it was common FDP conversions (quarters, eighths, fifths, thirds, ninths, etc).

We looked at 1/3 and 2/3, which led us to the fact that 0.999999….. = 3/3, which is, of course, 1.  I love teaching this fact, I tell them I’m about to blow their minds, and when I show them the initial reaction is always something like, “but it can’t be 1, it’s less than 1″.  We talk about the idea of infinitely many 9s (“I could keep saying 9s till I die, then someone else could take over, then someone else, and when the whole world ends and the last person dies saying 9s, the 9s still carry on”).  They have so many questions and objections.

“But, can’t you just add 0.00000000(loads of 0s)00001 to make it up to 1?  Then it’s just a bit less.”  That one I like, I show them what we add to numbers to make 1.  To 0.9 we add 0.1, to 0.99 we add 0.01, to 0.999 we add 0.001, and so on, so to 0.9999999999… we add 0.0000000000…, and if the 9s never end then the 0s never end, so we are adding 0.

“But, it just can’t be, it’s always a bit less”.  To which I try and explain that the second you say it’s a bit less than 1, you have stopped the 9s and terminated the decimal, so you’re not talking about 0.999… any more.

At one point today I paused, they all went silent and were clearly just pondering the insanity of it all when one boy at the back of the room made me crease up.  He dropped his pen the way Obama dropped the mic:

Once they’ve made their objections, they start to think about what I’ve shown them, really think about it, and you can see them getting convinced.

“So, it works with 3/3, but it works if you do it with ninths too: 0.999… is 9/9 as well, and that’s equivalent to 1/1, which is just 1”.

“So that means that 1.999… is the same as 2.”