Thursday, 13 February 2020
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 teacher asked me about it recently and we talked about a nice connection to Valentines Day.

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

Why not use the scale of emotion shown on the number line to help us describe calculating with integers.

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

### Addition and subtraction with negative numbers

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

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

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

Try this with the others:

- 8 - - 5 = ?

You are -8 on the HATE 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 -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.

### Teaching tip

When calculating with negative numbers, 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.

### Multiplying with negative numbers

The context works for the saying:

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

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.

LOVE x LOVE = LOVE            positive x positive = positive

HATE x LOVE = HATE           negative x positive = negative

HATE x HATE = LOVE           negative x negative = positive