Multiplying Integers

3 x 7 rows of soldiers
Image Source:

Can you work out how many soldiers are in the photo ?

We can see they are standing “three deep” : there is a soldier in the front, one in the middle, and one at the back.

There are three rows of soldiers.

We can also see that there are seven soldiers when we look across the front row of the photo.

So in the soldiers photo we have the following situation:

Diagram of 3 lots of 7 by addition
Image Source:

We can add up the seven lots of three and obtain an answer of 21.

Or we can multiply and say that 3 x 7 = 21 or 7 x 3 = 21.

Multiplication involves adding up how many lots of something we have.

For example 2 x 3 means we have “two lots of 3″, or 3 + 3, which equals 6.

4 x 3 means we have “four lots of 3″, or 3 + 3 + 3 + 3 which is 12.

When we have big numbers to multiply like 12 x 3, then we have to add up the three twelve times like this:

3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3

to obtain an answer of 36.

In primary school we learn our multiplying tables and know that 12 x 3 = 36, 7 x 3 = 21, 2 x 3 = 6, and so on.

It is much quicker and easier to know the multiplication tables than to write out long addition sums and add them up.

Multiplying Using Negative Numbers

But what about negative numbers ?

If we have 2 x -3

then we have two lots of -3

which is -3 + -3

which equals -6.

4 x -3 = ?

is four lots of negative 3

= -3 + -3 + -3 + -3

= -12

But what about something like:

-2 x 3 = ?

We can think of the negative sign at the start of the sum as meaning “the opposite of”.

So for -2 x 3 we need to find the opposite of 2 x 3.

Opposite of 2 x 3

= opposite of 6

= -6

So -2 x 3 = -6

Now here is the hardest one to get our head around:

-2 x -3

= opposite of 2 x -3

= opposite of 2 lots of -3

= opposite of -3 + -3

= opposite of -6

= +6 but we usually leave the positive sign off normal numbers

= 6

That was a very long lot of working out!

Integer Multiplication Rules

All the thinking and working out of how many lots of, and opposites of, and opposites of opposites, can get quite challenging.

But thankfully, Mathematicians noticed a pattern that Integer multiplications always follow, which goes like this:

Longhand Multiplying Rules

These Integer Multiplication Rules can be summarised as follows.

Mult Rules summary

In the summary diagram above:

+ means a positive number and does not mean addition.

– means a negative number and does not mean subtraction.

the “x” means multiply, and does not stand for the algebra variable “x”.

Integer Multiplying Rules Song

Here is a great short soulful song all about Integer Multiplying Rules.

Love Hate Rules for Multiplying Integers

These rules came from the webpage:

and they are a helpful way of remembering the multiplying rules.

It is good (+) to love (+), and it is bad (-) to hate (-)!

If you love to love, that is good. (positive x positive = positive)

If you love to hate, that is bad. (positive x negative = negative)
If you hate to love, that is bad. (negative x positive = negative)
If you hate to hate, that is good. (negative x negative = positive)

Same Sign Positive “SSP” Rule

The previous Integer Rules diagram we had was as follows:

Int Mult Rules Two

This diagram can be summarised even further:

Notice that the following always happens in the above diagram when multiplying integers:

Positive x Positive = Positive Answer

Negative x Negative = Positive Answer

If we multiply two items that have the same sign, we always get a positive answer.

If the items are not the same sign we get a negative answer when multiplying.

This is summarised into the sentence:

When MULTIPLYING : Same Sign Positive, Different Signs Negative.

Some people like to remember this rule in terms of relationships:

Two people the same will get along okay, and have a Positive relationship.

Two people that are completely different will probably not get along and have a negative relationship.

Multiplying Integers Video

The following video shows some examples that use the “same Sign Positive” rule.

Multiplying Integers Using the Triangle Rule

Some people like to summarise the integer multiplying rules into a Triangle, and draw this triangle next to their working out for each multiplying question that they do.

The following video shows how to do this.

Note that Americans use a dot instead of a multiplying “X” sign.

Also note that the same rules for multiplying also work for dividing, which we will be covering in a later lesson.

Summary of Multiplying Rules

We hope we have not confused everyone by supplying so many versions of the Integer Multiplying Rules.
Here at Passy World we like to use “SSP” : “Same Sign Positive, otherwise Negative”, because we find it quick and easy.

However, we recommend that you pick the Rule method which works best for you.

The “+ – – Triangle”, or the “Love Hate” rules work just as well as “SSP”.

Example 1 : 4 x -3

The question contains different signed numbers multiplied together,
so using the “SSP” rule the answer needs to be Negative.

Mult Q1

Example 2 : -2 x -5

Here we have the same signs, and so using “SSP” rule, the answer will be Positive.

Mult Q2

Example 3 : -7 x 3

Here we have different signs, and so using “SSP” rule, the answer will be Negative.

Mult Q3

Example 4 : 2 x 9

Here we have the same signs, (both numbers are positive), and so using “SSP” rule the answer will be Positive.

Mult Q4

Integer Warp Multiplying Integers Game

To play this game We need to know our multiplying rules which are:

- x – = a Positive answer (Same Sign = Positive answer)


+ x – = a Negative answer (Different signs = Negative answer)

Click on the image below, or the text link which follows, to play this game.

Integers Warp Multiply Game

Related Items

Introduction to Integers
Arranging Integers in Order
Adding Integers Using Number Lines
Adding Integers Using Zero Pairs
Subtracting Integers
Dividing Integers
Integers Order of Operations
Directed Number Integers Games
Integers in the Real World
Integers in Drag Racing

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