Now to deduct the area of the small triangle base 1, which is half an isosceles triangle of base 2 and therefore has area half of 4/81 of the whole ie 2/81.

The difference is 23/81.

Of course this depends on understanding dimension – but that is a useful thing to know.

]]>O-level and CSEs.

Clearly the students need to be consulted and they need to leave such discussions knowing the risks and match these against their future hopes and their current mathematical prowess. If students know their options together with your opinion then they can take this information back to their parent(s)/guardian(s) and discuss with them what they think. For you to be the sole bearer of such decision-making is a situation you ought not to be placed in. ]]>

I could kind of imagine doing at least a count-the-dots sort of proof, like by the difference between

* * *

* * *

* * *

and

* * * * *

* * * * *

* * * * *

* * * * *

* * * * *

being (from rows) twice the smaller odd number, plus (from columns) twice the smaller odd number, plus the four dots in the square at the end. So that’s something that’s four more than a whole multiple of eight, plus four more. So, that’s a multiple of eight plus eight.

]]>3x + y = 17

3x + 2y = 22

and then ask what y is. Then that leads onto discussions about how 5 is the difference between the equations and that leads onto a discussion about subtraction. Then I would bring in addition later in the same way that you had thought to bring in subtraction by presenting a different problem.

That was the way I taught it to my 4/5 borderline class last year but then when we came back to SEs in revision and exam practice one or two of them had started using the multiplying by a negative method from their own initiative and then when they were helping their classmates solve problems THEY showed them that method and it spread through the class, becoming more popular (but they were multiplying by a negative so that they could create a subtraction, presumably because that is what I’d shown them first so it is what the default is in their minds, whereas for your pupils the default is presumably addition).

So interesting! Thanks for sharing 🙂 ]]>