REVIEW LESSON: Integer Multiplication and Division

WATCH: Multiplying & Dividing Integers Video

Math Essentials: Multiplying & Dividing Integers Review from Teacher 2 Open High School on Vimeo.

Previous Material: Multiplying Integers

The material in each of the following subsections is the same material you were given in Quarter 2 on Multiplying Integers Integers. Please feel free to review them as needed.

WATCH: Modeling Multiplying Integers

READ: Multiplying Integers with Same and Different Signs

Multiplying Integers with Same and Different Signs

Now that you have learned how to add and subtract integers, it is time to tackle multiplying them. Remember that an integer is the set of whole numbers and their opposites. There are a few vocabulary words that help us when multiplying. The first is a factor. The numbers that are multiplied are called factors. The second is product. We multiply two or more factors to get a product.

Where do these rules come from?

Below are some multiplication facts for 5. Notice that the products show a pattern. Suppose you did not know the product of . How could you use the pattern shown below to determine that product?

Notice that each product shown is 5 less than the previous product. So, you can subtract 5 from the previous product, 5, to find the missing product. Since , the product of must be 0.

There are patterns that we can see when we multiply integers. Analyzing patterns like these can help us multiply positive and negative integers.

How do analyze these patterns?

First, we can notice that the pattern for the multiplication facts of 5 continues beyond zero. Up until this time in math, we have only looked at the positive products. But now, we know about the negative numbers, so we can look at continuing the facts. Let's see how patterns can help us find the products of integers.

Example

Use a pattern to find the missing products below.

You already know the pattern for this sequence of products. You can subtract 5 from the previous product to find the next product. Remember, subtracting 5 is the same thing as adding its opposite -5. Try adding -5 to the previous products to find the next products.

To find the product of , add

and , so subtract the lesser absolute value from the greater absolute value :

.

The integer with the greater absolute value is -5, so give the answer a negative sign.

, so

To find the product of , add -5 to the previous product, which is also -5.

In other words, add:

Both integers have the same sign, so add their absolute values.

, so add.

.

Give that answer a negative sign.

, so

Now we have our completed multiplication facts

Each product is 5 less than the previous product.

You may also notice that depending on what you are multiplying the sign changes.

Here are a few rules that we can conclude from the pattern.

• When 5, a positive integer, is multiplied by a positive integer, the product is positive.
• When 5, a positive integer, is multiplied by zero, the product is zero.
• When 5, a positive integer, is multiplied by a negative integer, the product is negative.

You may think that these rules apply only to the five table but they actually apply to ANY table when you are multiplying positive and negative integers.

Let’s look at another example that can help us finish figuring out the rules for multiplying integers.

Example

Use a pattern to find the missing products below.

To find the product of , add: . So, .

To find the product of , add: . So, .

This shows the completed multiplication facts.

Here are some other rules that you may notice.

• When -3, a negative integer, is multiplied by a positive integer, the product is negative.
• When -3, a negative integer, is multiplied by zero, the product is zero.
• When -3, a negative integer, is multiplied by a negative integer, the product is positive.

We can conclude the following rules.

EXAMPLE 1

EXAMPLE 2

EXAMPLE 3

READ: Multiplying Integers

Multiply Integers

Now that you understand the rules, we can work on applying them when actually multiplying integers.

Refer back to the rules as you work, but the key to becoming GREAT at multiplying integers is to commit these rules to memory!!

Example 1

Here we have negative four times a negative three. First, we multiply the terms, remember that a set of parentheses next to another set means multiplication.

Next, we figure out the sign.

A negative times a negative is a positive.

Our answer is 12.

Example 2

Here we have a negative five times a positive eight. Remember that a dot can also mean multiplication.

Next, we figure out the sign.

A negative times a positive is a negative.

Our answer is -40.

What about multiplying more than one term? We can do this easily. The key is to work from left to right and remember that the sign of each product can change with each factor.

Example

Here we have three negative terms being multiplied. First, let’s multiply the first two terms to get a product.

Now we multiply that product times the factor negative two.

Our answer is -48.

EXAMPLE 4

Check Yourself Quiz: Multiplying Integers

Previous Material: Dividing Integers

The material in each of the following subsections is the same material you were given in Quarter 2 on Dividing Integers. Please feel free to review them as needed.

WATCH: Dividing Integers

READ: Patterns of Quotients of Integers

Quotients of Integers with Same and Different Signs

Another important step in learning how to compute with integers is learning how to divide them. You can look for patterns in a sequence of quotients just as you looked for patterns in a sequence of products in an earlier lesson. These patterns will help you to understand the rules for dividing integers.

Where do they rules come from? Let’s look at some integer patterns with division. We are looking at quotients. A quotient is the answer in a division problem.

Example

Use a pattern to find the missing quotients below.

Look for a pattern among the quotients. Remember that a pattern has a rule that makes it repeat in a certain way. You will see that you can subtract 1 from the previous quotient to find the next quotient. Remember, subtracting 1 is the same thing as adding its opposite, -1. Try adding -1 to the previous quotients to find the next quotients.

To find the quotient of , add

and , so subtract the lesser absolute value from the greater absolute value.

The integer with the greater absolute value is -1, so give the answer a negative sign.

, so

To find the quotient of , add

Both integers have the same sign, so add their absolute values.

, so add

Give that answer a negative sign.

, so .

This shows the completed division facts.

Each quotient is still 1 less than the previous quotient.

What conclusions can we draw from this pattern?

You may notice the following.

• When a positive integer is divided by a positive integer, 2, the quotient is positive.
• When zero is divided by a positive integer, 2, the quotient is zero.
• When a negative integer is divided by a positive integer, 2, the quotient is negative.

These are the beginnings of our rules for dividing integers. Let’s look at another pattern to complete these rules.

Example

Look at the number facts below. Analyze the pattern of quotients shown.

What do you notice about these facts?

You may notice the following rules.

Based on the patterns, here are the rules for dividing integers.

READ: Dividing Integers

Divide Integers

Now we can use these rules to divide integers. Just like with the rules for multiplying, becoming great at dividing integers will require that you memorize these rules.

Next, let’s apply these rules to dividing integers.

Example 1

Find the quotient

To find this quotient, we need to divide two negative integers.

Divide the integers without paying attention to their signs. The quotient will be positive.

The quotient is 11.

Find the quotient .

To find this quotient, we need to divide two integers with different signs.

Divide the integers without paying attention to their signs. Give the quotient a negative sign.

, so .

The quotient is -4.

These problems used a division sign, but remember we can also show division using a fraction bar where the numerator is divided by the denominator.

Example 1

Example 2

Example 3

Check Yourself Quiz: Dividing Integers