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:

Sherwood out.

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.”

What I love about this was the links the students were making.  Equivalent fractions, calculations with decimals, terminating and recurring decimals.

What happened next was the best bit, mathematically.  One student asked me, “Miss, it’s not completely related to this, but how does 1.5/3 fit in, because that’s halfway between 1/3 and 2/3, isn’t it?”  I opened it up to the class, someone pointed out that 1.5/3 is equivalent to 3/6, which is 1/2.  “But that’s 0.5, Miss.  Why is it not recurring?”  I suggested that since the midpoint of 1/3 and 2/3 was 1.5/3, we should try finding the midpoint of the decimal equivalents (more connections – we learnt how to find midpoints when we were doing our units on the four operations).

Off they went: (0.333…. + 0.666…)/2 = 0.999…/2, and the grins appeared. “It only works if 0.999… is equal to 1.”  “Woah!”  “No way!”

It made my day.  We’re building up their knowledge carefully and incrementally, and they’re enjoying it so much.  One girl asked if she could write all her integers as recurring decimals instead now, which led to another interesting discussion, about why some representations of numbers are more useful than others.  Later on in the lesson, when we were going through the answers to some questions they were working on, one of the questions was 2/5 – 30% and the boy I picked on to tell me his answer grinned cheekily and said, “nine point nine recurring per cent”.

Year 7 out.

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