Confusing Things For Students 1

Sometimes a student's understanding is blocked by something that people who have made it past a certain level of maths don't even notice. In this series of posts I will highlight some of these blocks.

The first one has to do with implicit signs. Consider the following expression:

$$2(x+1)$$

There is an implicit multiplication in this expression, which can be made explicit by writing

$$2\times(x+1)$$

So far so good, for anyone who has made it past basic algebra. But now consider the fraction:

$$2\frac{1}{2}$$

Once again there is an implicit sign but this time it is different. The fraction can of course be broken up as

$$2+ \frac{1}{2}$$

So this can be extremely confusing for the beginning student of maths, who does not appreciate that implicit signs can be context dependent. Either mistakes can be made thinking that

$$2(x+1) = 2+x+1 = x+3$$

if the student is more used to seeing the implicit sign in fractions, or they can be mistaken in thinking things like

$$2\frac{1}{2} = 2\times\frac{1}{2} = 1$$

if the student has been doing a lot of beginning algebra and is rusty on their fractions.