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## 7th grade (Illustrative Mathematics)

### 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 $-7(-3)$ .

Before we do, make a prediction.

**What do you predict will be the value of**$-7(-3)$ ?

*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.

## 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.

## 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 $-2(-5)$ 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 $-5$ .

## 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.**

## Want to join the conversation?

- I am nearly a grown man struggling with 7th grade math(5 votes)
- Hey, at least ur giving it a try :)

Some end up actively ignoring math and never improving on or using it, even when it's quite beneficial like Algebra.

Happy learning.(2 votes)

- Find the answer for -3(-5)(3 votes)
- Yeah, these writing things he does instead of the videos just confuse me more and more every time I read them. Ugh.(4 votes)
- Normally, whenever two negative numbers decide to meet, the two negatives (- times -) decide to mash together to make a (+)

One negative stays same while the other turns side ways .(2 votes) - when you multiply two negative #sthe product will come out as positive. For example, -2(-45) will be 90(2 votes)
- -2(-45) means -(-45 + -45) = -(-90).

Let's think about it using debt. Debt would be 'negative money', so -3 is $3 in debt. If we had negative debt, that would be 'negative negative money' as if you're taking away your debt, so you'd have**more money**. So, -(-90) is like taking $90 away from your debt, so you'd have $90 more!

This is one reason why it equals 90.

Hope this helped.(1 vote)

- why am i bambozzed(2 votes)
- Yes I indeed have a question. I get confused on diving the pairs and finding the answer. I need a breakdown on how to do it. Please and thankyou.(1 vote)
- why is it so easy(0 votes)
- Equal groups of objects, the explain is not easy to understand. I seem to be missing something before taking away 2 group of -5.(0 votes)
- First question!(0 votes)
- hi hi hi hi hi hi hi hi hi(0 votes)