##### No 3 in a series of posts based on Thinking Mathematically (1985) by Mason, Burton, Stacey

This is a classic task for working systematically:

It was once claimed that there are 204 squares on an ordinary chessboard.
Can you justify this claim?

I like this way of stating the problem, rather than just “how many squares on a chessboard?” because it gets straight to the nub of the issue – we are looking at different sized squares, some of which overlap.

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
- Adding 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?

Here is a mixture of arithmetic questions to ponder with solutions (or here as a pdf in case the Equation editor in Word messes things up). They are all non-calc although that’s not immediately obvious when you look at them. I used this today with second set Year 9 and they seemed to appreciate that they achieved more than they originally thought they would when I put them all up at once.

Here is an example:

## Ideas for better maths teaching