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Numbers not adding up? Be thankful

To err is divine as mistakes offer a golden opportunity to deepen understanding

To err is divine as mistakes offer a golden opportunity to deepen understanding

Should I confess to this? I love it when my students make mistakes. If I can explain the problem well, it will hopefully mean one less error when they come to the exam. What's more, a mistake can be hugely revealing.

I'm not suggesting that anyone should deliberately make an error for my benefit, but the challenge of unravelling snagged thought processes trumps that of ticking perfect answers every time. A classroom where mistakes are always accepted and never ignored is a classroom where students are less afraid.

Decimal difficulties

Take the following lesson for the GCSE non-calculator paper. Vicky tackles the question: "Write 116 as a decimal."

She writes "0.16", a mistake (the answer is 0.0625), but there are good things about her answer. It's better than 1.6, I would say, although 0.016 would have been closer to the truth.

It's time to direct a plenary to the whole class: is Vicky's rule for converting fractions into decimals always, sometimes or never true? We experiment a little.

"Hey, 110 is 0.10, so the rule does work," says Lawrence.

"What about 12?" I ask.

"A half is 0.5, so it doesn't work there," comes the reply.

"Does it work for 15?"

"15 = 0.2, so no."

On the board, I write:

"Might there be somewhere in the middle, between 5 and 2, where Vicky's rule works?" I ask.

Experimentation begins again.

"13 goes to 0.33333, so the number we want must be between 3 and 3.33333," Rosemary concludes.

Collective chipping away

The problem is now turning into an illuminating exercise in trial and improvement.

"13.2 = 0.3125," offers Nicky - we've now abandoned the "non-calculator" aspect of the lesson. "And 13.125 = 0.32."

"Can anyone give me a number between 3.2 and 3.125? Jessica?" I ask. It's an effort for her, but she succeeds.

"3.175? And 13.175 = 0.31496."

And so on until the class is open to trying a different tack.

Pushing the principle

"OK, so how about some algebra on this?" I say, to an inevitable chorus of groans, but we stick with it.

"We need 1x = 0.x = x10," I say, becoming a little freer with my notation.

"So, 1x = x10. Multiply both sides by 10x and we get x2 = 10," says Damien.

"How do we undo the square?" I ask.

"Square root: that gives x = 10," offers Lisa, "which is 3.16227."

Everyone tries 13.16227 and finds it equal to 0.316227.

Simple stuff, but this final calculation takes on the appearance of a magic trick.

"So was the algebra useful or not?" I ask, trying not to sound too smug. I am met with grudging but conciliatory mumbles.

Jonny Griffiths is a secondary maths teacher

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