Wednesday, 16 April 2014

Adding and subtracting negative numbers is easy enough to put into a context, but it can be tricky to find a good way of representing the multiplication of negative numbers so that it makes sense. A trainee teacher asked me about it  last week and it made me think afresh about teaching this.

I remember seeing this quote from some time ago, so cannot recall where from (although it reads as rather Shakespearean!):

“When you love love there is love, if you hate love there is hate, but if you hate hate there is love.”

Why not use this scale of emotion to help us describe calculating with integers.

8 + - 5 = ?                        - 8 - - 5 = ?                        8 - - 5 = ?

- 8 + 5 = ?                        - 8 + - 5 = ?                        - 8 – 5 = ?

#### Teaching tip

With these, it is a good idea to call the integers negative and positive numbers, so 8 + - 5 is said as ‘positive 8 add negative 5’. Giving one more example, 8 – 5 would be positive 8 subtract positive 5. This helps distinguish the operation from the integer.

In this context, hating more is a negative move to the left so you become more negative. Conversely, loving more is positive and you increase your love (lovely…).

You can follow the first example using the number line.

8 + - 5 = ?     You are 8 on the LOVE scale, then you add 5 negative HATE moves to the left, which leaves you with 3 on the scale (still showing some love!)

Try this with others:

- 8 - - 5 = ?  You are -8 on the HATE scale, you take away some of your HATE which is the same as adding some LOVE.  Add 5 to -8 and you are on -3 (not quite so hateful…).

8 - - 5 = ?    You are 8 on the LOVE scale and you take away some of your HATE which is the same as adding some LOVE.  Add 5 to 8 and you are on 13 (a whole lotta love…).

You get the idea – try it with other examples.

### So what about multiplying negative numbers?

The context works from the saying:

“When you love love there is love,
if you hate love there is hate,
but if you hate hate there is love.”

LOVE x LOVE = LOVE         ⇒   positive x positive = positive

HATE x LOVE = HATE         ⇒   negative x positive = negative

HATE x HATE = LOVE        ⇒    negative x negative = positive

Unfortunately, it isn’t an easy one to model in this context using, for example, repeated addition or arrays.

Looking at patterns on a multiplication grid is a better model, starting with the well-known positive numbers and then extending each range to include negative numbers and continuing the pattern.

If you can think of a way of linking this to love and hate, then do let me know.