When we teach that the multiplication of two negatives always produces a positive, we need to prove to pupils that this is so always and we don’t really want them taking our word for it without a good reason to back up…

# KS3

## Factorisation and expansion [2]

In the previous blog we considered how the teaching of expansion and factorisation might be better taught simultaneously. Looking at whole numbers only. In this blog we will continue by looking at different specific linear algebraic expressions. The first expression…

## What comes first expansion or factorisation?

My answer to the title question is expansion followed instantly by factorisation in order to make the links immediately between the two. As with all mathematics let us start with a simple case of expansion over brackets. 2(3+4) This example gives…

## Teaching, questioning but not telling. Angle at circumference of a semi-circle.

I wrote an earlier blog on this subject but thought it might be useful to break it down into a series of teacher to student questions and instructions to get a more active learning experience in the classroom. This can…

## Multiplying two improper fractions.

In this the third blog in the series of multiplying fractions we shall look at what is happening when two improper fractions are being multiplied. What we should not do at this stage is assume that what has been true…

## Commutativity and fraction problems

In this short blog we will look at how, by using the commutative property of multiplication it is possible to solve complex looking fraction multiplication problems mentally. Previously we looked at concrete approaches to conceptualising the process of multiplication of…

## Fraction multiplication – Making sense of multiplying improper fractions.

In the last blog we looked at why multiplication of proper fractions can be fluently completed by multiplying numerators then doing the same with denominators. At this stage learners will have begun to understand this procedure and the truth of it…

## Fraction multiplication – a concrete approach

Why is the following statement true? For those of us in the know we might be tempted to say on the left hand side if we multiply numerators by each other and do likewise with denominators we will…

## The angle inscribed in a semi-circle is a right angle.

Take any semi-circle, join two straight lines from each end of the diameter at an apex on the circumference and this angle will always be a right angle. This is one of the more accessible of the circle theorems to…

## Reasoning behind the shortcuts 1: Angles in any triangle sum to 180˚

Angles in any triangle sum to 180˚ I remember being given this piece of information at school and I have always remembered it. I was never told why it was true for all triangles however, I was simply required to…