What is the sign of the product of two positive integers?
If you are looking at the product 3 \cdot \left(-4\right), how many tiles are in 3 groups of -4 tiles?
What is the sign of the product of one positive and one negative integer?
Is the product of 3 \cdot \left(-4\right) the same as -4 \cdot 3? Check this for other products.
What is the sign of the product of two negative integers?

Unlike adding and subtracting integers, where we can use the number line or counters, multiplication comes down to looking at the sign of factors.

A positive times a positive equals a positive.

We have seen that the product of two positive integers is a positive integer.

A positive times a negative equals a negative. A negative times a positive equals a negative.

The product of a positive integer and a negative integer is a negative integer.

A negative times a negative equals a positive.

The product of two negative integers is a positive integer.

Examples

Example 1

Find the value of: -4 \cdot 5

Worked Solution

Create a strategy

Determine the sign of the product, then multiply 4 and 5 and apply the sign.

Apply the idea

The sign of the product is negative because we are multiplying a negative and a positive integer.

\displaystyle 4 \cdot 5

\displaystyle =

\displaystyle 20

Evaluate

We determined that the sign of the product should be negative, so the answer is -20.

Reflect and check

Let's visualize the multiplication of a negative and a positive integer using an array. In this case, we are multiplying -4 and 5.

In the array, we have 4 rows and 5 columns, which represent the multiplication of -4 and 5. Each row has 5 orange tiles, representing negative integers. When we count the total number of orange tiles, we have 20 orange tiles, which represent the product of -4 and 5. Since orange tiles represent negative integers, our product is -20 which matches our previous answer.

Example 2

Find the value of: -7 \cdot \left(-5\right)

Worked Solution

Create a strategy

We have the product of two negative integers, so the product will be positive.

Apply the idea

\displaystyle -7 \cdot \left(-5\right)

\displaystyle =

\displaystyle 35

Evaluate

Reflect and check

Let's visualize the multiplication of the two negative integers by using an array representation with blue tiles for positive integers and orange tiles for negative integers.

In the array, there are 35 blue tiles, which represent positive integers. This shows that when we multiply two negative integers, like -7 and -5, the product is a positive integer, in this case, 35.

Example 3

A submarine dives 22 \text{ m} each minute for 16 minutes. What integer represents the total depth of the dive after 16 minutes?

Worked Solution

Create a strategy

Diving 22 meters is represented by the integer -22. We can think of this as repeated addition because an extra 22 \text{ m} is added for each minute the diver is under. That means we can also think of this as multiplication.

Apply the idea

\displaystyle \text{Depth of the dive}	\displaystyle =	\displaystyle -22 \cdot 16	Set up the equation
	\displaystyle =	\displaystyle -352 \text{ m}	Evaluate

Reflect and check

We can check the reasonableness of our answer by thinking about the signs. We knew diving 22 \text{ m} was a negative integer because the diver is going under water. And time was positive because time is always moving forward.

Multiplying a positive and negative integer results in a negative integer, which is what we got.

This makes sense because the diver ends up deeper underwater.

Idea summary

We have seen that the product of two positive integers is a positive integer.

The product of a positive integer and a negative integer is a negative integer.

The product of two negative integers is a positive integer.

Divide integers

The same principles that help us to multiply integers also apply to divide.

Exploration

The applet allows you to select two integers. The horizontal slider selects the divisor (number to divide by), the vertical slider selects the quotient (result of the division), and the checkboxes change the sign of the integers.

Blue tiles represent positive integers.
Orange tiles represent negative integers.
The dividend (number being divided) is shown as an array.
The divisor is on the top edge.
The quotient is on the left edge.

Loading interactive...

What is the sign of the quotient of two positive integers?
What is the sign of the quotient of two negative integers?

A positive divided by a positive equals a positive.

We have seen that the quotient of two positive integers is a positive number.

A positive divided by a negative equals a negative. A negative divided by a positive equals a negative.

The quotient of a positive integer and a negative integer is a negative number.

A negative divided by a negative equals a positive.

The quotient of two negative integers is a positive number.

The same sign properties apply to both multipication and division. However, unlike multiplication, division of two integers does not always result in another integer.

Examples

Example 4

Find the value of: 48 \div \left(-6\right)

Worked Solution

Create a strategy

We have the quotient of one positive and one negative number, so the quotient will be negative.

Apply the idea

\displaystyle 48 \div \left(-6\right)

\displaystyle =

\displaystyle -8

Evaluate

Reflect and check

We can check if we have determined the correct sign, by using multiplication. Would -8 times -6 equal 48?

Example 5

Evaluate: \dfrac{-60}{-10}

Worked Solution

Create a strategy

We have the quotient of two negative integers, so the quotient will be positive.

Apply the idea

\displaystyle \dfrac{-60}{-10}

\displaystyle =

\displaystyle 6

Evaluate

Reflect and check

We can check if we have determined the correct sign, by using multiplication. Would 6 times -10 equal -60?

Idea summary

We have seen that the quotient of two positive integers is a positive number.

The quotient of a positive integer and a negative integer is a negative number.

The quotient of two negative integers is a positive number.

Outcomes

6.CE.2

The student will estimate, demonstrate, solve, and justify solutions to problems using operations with integers, including those in context.

6.CE.2a

Demonstrate/model addition, subtraction, multiplication, and division of integers using pictorial representations or concrete manipulatives.*

6.CE.2b

Add, subtract, multiply, and divide two integers.*

6.CE.2d

Estimate, determine, and justify the solution to one and two-step contextual problems, involving addition, subtraction, multiplication, and division with integers.

4.02 Multiply and divide integers

Ideas

Multiply integers

Exploration

Examples

Example 1

Example 2

Example 3

Divide integers

Exploration

Examples

Example 4

Example 5

Outcomes

6.CE.2

6.CE.2a

6.CE.2b

6.CE.2d

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