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Course: 7th grade > Unit 5

Lesson 1: Multiply with negatives
Math
7th grade
Negative numbers: multiplication and division
Multiply with negatives
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Multiplying two negative numbers

Google Classroom
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.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
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(
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
)		=0
−7(−3+
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
)		=0
−7(−3)+(−7)(3)=0
−7(−3)+(−21)=0
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
+(−21)		=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.
1

Evaluate each expression.
2(4)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
−2(4)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
−2(−4)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

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.
4 rows of integer chips. The first row has 5 positive chips. The second row has 5 negative chips. The third row has 5 positive chips. The fourth row has 5 negative chips. Each positive integer chip is grouped with a negative integer chip. There are 10 groups and no chips left over.
Now we can take away 2 groups of −5.
4 rows of integer chips. The first row has 5 positive chips. The second row has 5 negative chips. The third row has 5 positive chips. The fourth row has 5 negative chips. All 10 negative chips are crossed out.
Evaluate. −2(−5)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

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.

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