Thursday, 19 May 2011

How do the Chinese do it? part 4

S,D,P,Q  7,7012

Try it - it's hard!  Students will rapidly write:
Sum 7019
Difference 7005, -7005   (I like to ask them to fully justify that -7005, remember that the trick:  it's the negative of the other difference, which they will have discovered by now, only gets them half a mark)

The multiplication will keep some busy a bit longer.  Many will naturally split the 7000 and the 12.  It's nice to get them to explain and notice what they are doing here too.

But the divisions?  Nasty!  The obvious division is hard enough but the reciprocal?  They will wrestle with it.

I often give full credit for a numerical answer which is very nearly right and has a correct method and a lot of effort behind it.  It helps to emphasise the aspects of their work which I value and to keep them engaged.
They will more deeply consider their thinking about reciprocals because it is challenged. 

The answers aren't the point yet. It's all about the method and their thinking.  If they get that right then the answers will come and the methods won't be forgotten so rapidly as they are when we teach them abstracted algorithms.  Because they have discovered their method for themselves they will have the confidence to believe they can do so again as and when they need to.

One nice aspect of this pair of numbers is that they encourage students to estimate.  One of the answers is going to be about 1000 isn't it.  Does that mean the other will be close to 1/1000?  Can we be more accurate than that?  When students get the hang of writing divisions as fractions (which is a really useful skill), change the rules - allow only mixed numbers or allow only decimals to at least 3dp.  It's your classroom - you can have different rules for different students at different times provided the underlying theme is fair - that all students are being stretched.

Choose another couple of examples like this yourself.  That takes us to about an hour of classroom time doesn't it?  What have the students learned in this time?  What have you the teacher discovered?

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