Sarah, Benjie, and Sipple are biking up to the ski basin. They want to carry as little stuff as they can on their bikes because the weight makes it harder to bike up the mountain. They are trying to decide if they need to bring jackets, which each weigh a few pounds. It is \begin{align*}70^\circ\end{align*} in town. They decide they don't need jackets unless it drops below 65°. In town, it is about 7,000 ft. It is 12,000 ft at the ski basin. They know the temperature drops about 2° for every thousand feet they climb. Will they need their jackets?

In this concept, you will learn how to multiply integers.

### Multiplying Integers

An **integer** is the set of whole numbers and their opposites.

A **product** is the answer of a multiplication problem. Multiplication is repeated addition. When a value is being multiplied the multiplication symbol is a shorthand for this repeated addition.

**Commutative** means a group of quantities connected by a mathematical symbol that gives the same result whatever the order of the quantities involved. Multiplication is commutative.

Here is an example of a multiplication problem.

3 x 4 = ____

First, write out the multiplication as four 3s added together.

3 + 3 + 3 + 3 = 12

Next, because multiplication is commutative, write out the multiplication as three 4s added together.

4 + 4 + 4 = 12

Then, verify that the two answers are the same.

Here is another example.

6 x (-5) = ____

First, write out the multiplication as six -5s added together.

-5 + -5 + -5 + -5 + -5 + -5 = -30

The answer is -30.

If you know the multiplication table, you can simplify the process without converting to addition every time. Then, you just need to worry about the signs. Here are the sign rules for multiplication:

Positive \begin{align*}\times\end{align*} positive = positive

Negative \begin{align*}\times\end{align*} positive = negative(-->remember, because multiplication is commutative, this is the same as Positive X Negative)

Negative \begin{align*}\times\end{align*} negative = positive

Another way to think of it is: if there are an odd number of negatives in a multiplication problem, the answer is negative. If there are an even number, the answer is positive.

Here is another example.

(-5)(-2) = ____

First, multiply the integers ignoring their signs.

5 x 2 = 10

Next, use the sign rules for multiplication to determine the sign of the answer.

In this case, there was a negative times a negative. That equals a positive.

The answer is 10.

### Examples

#### Example 1

Earlier, you were given a problem about Sarah, Sipple, and Benjie.

Their ride starts in town at 7,000 ft, where it is 70°, and ends at 12,000 ft. The temperature drops 2° for every thousand feet they climb. They don't need their jackets unless it drops below 65°. Do they need to bring them?

First, calculate how many feet they climb in total.

12,000 - 7,000 = 5,000 ft. That is 5 thousand ft., or 5 x 1,000.

Next, they know that the temperature drops 2° for every thousand feet. So they can write a multiplication for the final temperature. Note that a "drop" in temperature is a decrease, or a negative.

5 x (-2) =

Then, they multiply the integers.

5 x 2 = 10

Then, they check the sign. A positive times a negative is a negative.

The temperature up at the ski basin is -10° from the temperature in town. That means, it is 10° colder.

Finally, they take this temperature from the initial temperature.

70° - 10° = 60°

It will be \begin{align*}60^\circ\end{align*} up at the ski basin.

They decided they need to bring their jackets if it is under 65°. Since 60° is colder than 65°, they must bring their jackets.

#### Example 2

Find the product.

(9)(-6)(5) = _____

First, multiply the first two integers ignoring their sign (because multiplication is commutative you can do the multiplications in any order).

9 x 6 = 54

Next, use the rule of signs to determine the sign.

In this case, it was a positive times a negative. That equals a negative.

So, the first product is -54.

Then, multiply that answer by the final integer, ignoring the signs.

54 x 5 = 270

Then, use the rule of signs to figure out the final sign.

This was a positive times a negative, too. So, the final answer is negative.

The answer is -270.

#### Example 3

Find the product.

-9(-8) = ____

First, multiply the integers together ignoring their sign.

9 x 8 = 72

Next, check the number of negative signs.

In this problem, there are two, which is even. When there are an even number of negative signs, the overall answer is positive.

The answer is +72.

#### Example 4

Find the product.

(4)(-12) = ____

First, multiply the integers together ignoring their sign.

4 x 12 = 48

Next, check the number of negative signs.

In this problem, there is one, which is odd. When there is an odd number of negative signs, the overall answer is negative.

The answer is -48.

#### Example 5

Find the product.

(5)(13) = ____

First, multiply the integers together ignoring their sign.

5 x 13 = 65

Next, check the number of negative signs.

In this problem, there are no negative signs, which means the answer is automatically positive.

The answer is +65.

### Review

Multiply each pair of integers to find a product.

- (-7)(-8)
- -3(4)
- 5(8)
- (-3)(-9)
- 6(12)
- -9(-9)
- 8(-4)
- -7(-2)
- -7(-3)
- 15(-2)
- -15(2)
- -2(-15)
- 12(-5)
- (-11)(-7)
- (-4)(-5)
- (-8)(-11)
- (2)(-3)
- -5(7)
- -13(-2)
- 14(2)

Evaluate each numerical expression.

- (-9)(2)(-1)
- (-3)(2)(-4)
- (-5)(9)(-1)
- (8)(-9)(-2)
- (2)(-3)(-5)

### Review (Answers)

To see the Review answers, open this PDF file and look for section 11.12.