Here’s something for those encountering integers.

In Mathematics you can’t get far without negatives. We all know that if you’re multiplying and you want the end result to be negative you are going to have to use a negative and a positive.
[Tip: Encourage students to regard x + 2x – 5x as x positive 2x negative 5x.]
So we teach:

+ x + = +
– x – = +          2 of the same will give a positive

+ x – = –
– x + = –          2 different will give a negative

Students generally catch on to multiplying easily but with dividing there is often an uncertainty about whether the outcome will be a negative or a positive.

It’s easy – if you show them one thing. Look above at where any negatives are used. What do you notice?

Wherever any negatives were used there was always one positive out of the 3 signs involved.
When we divide and negatives are present there will have to be a positive used also.
Consider the following:

+6 ÷ +2 = +3
-6 ÷ -2 = +3

+6 ÷ -2 = -3
-6 ÷ +2 = -3

So when a student has a negative divided by a positive and wonders what the outcome will be, they can confidently expect a second negative. In the same way, if they are dividing a negative by a negative they can be assured they should get a positive answer.

The application of this reaches beyond the integers work often seen in the middle years at school. An example of the above being helpful is when students are doing such things as solving.

In 3x = -12 you need to divide the -12 by a [positive] 3. Result: x = -4

If the equation was -3x = -12 you’d be dividing the -12 by a -3 [negative]. Result: x = 4 [positive]

You can be positive in the way you use negatives; and to good effect. Try it.