# Foundations

## Multiply and Divide Integers


### Multiply Integers

Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.
We remember that  means add a, b times.
The next two examples are more interesting.
What does it mean to multiply 5 by  It means subtract 5, 3 times. Looking at subtraction as “taking away,” it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away 5 three times.
In summary:
Notice that for multiplication of two signed numbers, when the:
1. signs are the same, the product is positive.
2. signs are different, the product is negative.
We’ll put this all together in the chart below.
When we multiply a number by 1, the result is the same number. What happens when we multiply a number by  Let’s multiply a positive number and then a negative number by  to see what we get.
Each time we multiply a number by  we get its opposite!

### Divide Integers

What about division? Division is the inverse operation of multiplication. So,  because  In words, this expression says that 15 can be divided into three groups of five each because adding five three times gives 15. Look at some examples of multiplying integers, to figure out the rules for dividing integers.
Division follows the same rules as multiplication!
For division of two signed numbers, when the:
1. signs are the same, the quotient is positive.
2. signs are different, the quotient is negative.
And remember that we can always check the answer of a division problem by multiplying.

### Simplify Expressions with Integers

What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?
Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.
The next example reminds us to simplify inside parentheses first.

### Evaluate Variable Expressions with Integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

### Translate Phrases to Expressions with Integers

Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.
When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.
Be careful to get a and b in the right order!
Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is “product” and for division is “quotient.”

### Use Integers in Applications

We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

### Key Concepts

1. Multiplication and Division of Two Signed Numbers


2. Strategy for Applications



### Practice Makes Perfect

Multiply Integers
In the following exercises, multiply.
Divide Integers
In the following exercises, divide.
Simplify Expressions with Integers
In the following exercises, simplify each expression.
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression.
Translate English Phrases to Algebraic Expressions
In the following exercises, translate to an algebraic expression and simplify if possible.
Use Integers in Applications
In the following exercises, solve.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?