# Integers

## Multiply and Divide Integers


### Multiply Integers

Since multiplication is mathematical shorthand for repeated addition, our counter 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.
We remember that  means add  times. Here, we are using the model shown in  just to help us discover the pattern.
Now consider what it means to multiply  by  It means subtract  times. Looking at subtraction as taking away, it means to take away  times. But there is nothing to take away, so we start by adding neutral pairs as shown in .
In both cases, we started with  neutral pairs. In the case on the left, we took away  times and the result was  To multiply  we took away  times and the result was  So we found that
Notice that for multiplication of two signed numbers, when the signs are the same, the product is positive, and when the signs are different, the product is negative.
When we multiply a number by  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

Division is the inverse operation of multiplication. So,  because  In words, this expression says that  can be divided into  groups of  each because adding five three times gives  If we look at some examples of multiplying integers, we might figure out the rules for dividing integers.
Division of signed numbers follows the same rules as multiplication. When the signs are the same, the quotient is positive, and when the signs are different, the quotient is negative.
Remember, you can always check the answer to a division problem by multiplying.
Just as we saw with multiplication, when we divide a number by  the result is the same number. What happens when we divide a number by  Let’s divide a positive number and then a negative number by  to see what we get.
When we divide a number by,  we get its opposite.

### Simplify Expressions with Integers

Now we’ll simplify expressions that use all four operations–addition, subtraction, multiplication, and division–with integers. Remember to follow the order of operations.

### Evaluate Variable Expressions with Integers

Now we can evaluate expressions that include multiplication and division with integers. Remember that to evaluate an expression, substitute the numbers in place of the variables, and then simplify.

### Translate Word Phrases to Algebraic Expressions

Once again, all our prior work translating words 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.

### Key Concepts

1. Multiplication of Signed Numbers


2. Division of Signed Numbers


3. Multiplication by 


4. Division by 




### Practice Makes Perfect

Multiply Integers 
In the following exercises, multiply each pair of integers.
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 Word Phrases to Algebraic Expressions 
In the following exercises, translate to an algebraic expression and simplify if possible.

### 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?
