# Multiplying integers using models

## Multiplying Integers

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#### Objectives

1. Understand how to use models to multiply integers
2. Evaluate integer multiplication problems using mental math
3. Develop methods for using integer tiles
4. Apply rules to multiplying two negative numbers

#### Warm up

• 1. Evaluate:
• a. - 15 + - 8
• b. 9 - 10
• c. 7 - (-2)
• d. 6 + (-4)
• 2. Writing: How is multiplication and addition related?
• 3. Writing: What is the commutative property?

#### Multiplying integers

Now that you recalled the relationship between multiplication and addition we are ready to use our integer tiles to multiply integers.

Evaluate: -3 x 4

If I wanted to change this into an addition problem this would mean -3 added together four times or visually:

So we have 12 red tiles or -12. That’s it!

So -3 x 4 = -12

Evaluate: 5 x -2

We can use the commutative property to rearrange this expression so that we can add 5 groups of -2. Our new expression looks like : -2 x 5 Using our tiles...

Therefore our final answer is -10 So, 5 x -2 = -10

RULE: When multiplying integers, a positive times a negative will always be a negative. Then just multiply the two numbers to get a final answer.

Unfortunately, tiles can not be used when we multiply a negative times a negative.

However, we do have a rule. When multiplying a negative times a negative the result is always a positive number.

However, there is a reason, as to why a negative times a negative is a positive.

Explanation: Each number has an "additive inverse" associated to it (a sort of "opposite" number), which when added to the original number gives zero. This is in fact the reason why the negative numbers were introduced: so that each positive number would have an additive inverse.

For example, the inverse of 3 is -3, and the inverse of -3 is 3.

Note that when you take the inverse of an inverse you get the same number back again: "-(-3)" means "the inverse of -3", which is 3 (because 3 is the number which, when added to -3, gives zero). To put it another way, if you change sign twice, you get back to the original sign.

Now, any time you change the sign of one of the factors in a product, you change the sign of the product:

(-something) × (something else) is the inverse of (something) × (something else), because when you add them (and use the fact that multiplication needs to distribute over addition), you get zero. For example, (-3) x (-4) is the inverse of (-3) x (4), because when you add them and use the distributive law, you get (-3) x (-4) + (3) x (-4) = (-3 + 3) x (-4) = 0 x (-4)=0 So (-3) x (-4) is the inverse of (3) x (-4), which is itself (by similar reasoning) the inverse of (3) x (4). Therefore, (-3) x (-4) is the inverse of the inverse of 12; in other words, the inverse of -12; in other words, 12. The fact that the product of two negatives is a positive is therefore related to the fact that the inverse of the inverse of a positive number is that positive number back again.

For example:

Evaluate: -5 x -2= 10

Evaluate: -4 x -8= 32

Evaluate: -9 x -6= 45

#### Assessment

• Visit this website for more practice
• Use mental math to perform the following operations
1. -9 x 2
2. -3 x -5
3. 6 x -8
4. 10 x 11