If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: 7th grade (Illustrative Mathematics)>Unit 5

Lesson 9: Lesson 9: Multiplying rational numbers

# Multiplying two negative numbers

If 3(-8) can be 3 equal groups of -8, what does -3(-8) mean? What does it mean to multiply any two negative numbers? Let's use the distributive property and other properties of multiplication to find out.
When we multiply a positive number times a negative number, the product is the opposite of the product of the absolute values of the numbers. This means the result is always negative.
But what about when we multiply a negative number times a negative number? Let’s explore this idea using three different methods, starting with the distributive property.

## Multiplication with the distributive property: negative times negative

The distributive property works the same with negative numbers as with positive numbers and $0$. Let's use it to see what happens when we multiply two negative numbers, starting with the example $-7\left(-3\right)$.
Before we do, make a prediction.
What do you predict will be the value of $-7\left(-3\right)$?
This is an ungraded prediction, because we learn more when we make a guess before we get feedback.
Now let's use the zero-product property and the distributive property to reason about the product.
Fill each blank with a number to keep both sides of the equation equivalent.
$-7\left($
$\right)$
$=$$0$
$-7\left(-3+$
$\right)$
$=$$0$
$-7\left(-3\right)+\left(-7\right)\left(3\right)$$=$$0$
$-7\left(-3\right)+\left(-21\right)$$=$$0$
$+\left(-21\right)$
$=$$0$

## Multiplication by a negative as repeated subtraction from $0$‍

### Number lines

As a general trend, the symbol "$-$" changes the direction we move on a number line, whether we interpret it as a negative sign or a subtraction symbol.
Match the number lines to the expressions they represent.
Duplicate graphs will match to either equivalent expression.

Evaluate each expression.
$2\left(4\right)$$=$
$-2\left(4\right)$$=$
$-2\left(-4\right)$$=$

## Equal groups of objects

We represent multiplying by a positive number by adding equal groups of objects. We represent multiplying by a negative number by subtracting equal groups of objects.
So $-2\left(-5\right)$ is the value we have left after we take away $2$ groups of $-5$ objects. But how do we subtract groups of objects when we don't have any?
We can start with zero-pairs. The following diagram represents $0$ because there are $10$ positive integer chips and $10$ negative integer chips.
Now we can take away $2$ groups of $-5$.
Evaluate. $-2\left(-5\right)=$

## Conclusion

Now that we have explored multiplying a negative number times a negative number using three different methods, what conclusions can we draw?
Describe a general pattern for when we multiply two negative numbers.