# Do you understand reciprocals?

These questions test a lot of things so use them carefully.

Firstly there is what do we understand about reciprocals, namely:

These are tricky concepts to grasp. This is the order in which I teach them, but I don’t think the “flow” through these 4 concepts is particularly obvious and students need to be carefully led with lots of examples using Mini Whiteboards.

Other understanding required includes:

• Finding equivalent fractions
• Finding common denominators and using them to find the right equivalent fractions
• Converting between Mixed Number and Top-Heavy fractions
• Negative fractions

And the last question took me a good 3-4 minutes to convince myself I had the right answer.  Which always makes me stop to think – am I being fair to my students here?

# Generalising with Areas of Circles

So, at #MathsMeetGlyn last weekend Don Steward briefly put this picture up on the screen but then decided to talk about something else as time was pressing.

That’s a bit like a “Wet Paint” sign.  You have to touch it just to see if it really is wet…

So I had a little play with these. I couldn’t find anything about it on his blog but of course there is lots of other lovely stuff there.

A few different questions you could pose here:

• What is the ratio of red area to blue area?
• What is the red area as a fraction of the blue area?
• What is the red area as a fraction of the whole circle?
• What is the blue area as a fraction of the whole circle?

And then of course, can we generalise?  What about when there are n small circles.  What happens as n gets bigger? Why? Can we find a general expression for the area?

Have a go.  It actually turns out to be quite simple but depending on which question you start with, you can get into a lot of practice with ratio and dividing fractions.