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7th grade (Illustrative Mathematics)

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

Lesson 9: Lesson 9: Multiplying rational numbers
Math
7th grade (Illustrative Mathematics)
Rational number arithmetic
Lesson 9: Multiplying rational numbers
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Multiplying a positive and a negative number

Google Classroom
Discover what it means to multiply by negative numbers by using repeated addition and using the distributive property.

Extending multiplication to more numbers

Whether we use whole numbers, fractions, or integers, multiplication still has the same meanings.
Let's explore how different properties of multiplication help us make sense of multiplication with negative numbers.

Multiplication as repeated addition

Multiplication can be a shortcut for adding equal groups of the same size. For example, we can show 5 groups of 7 these ways:
  • 7+7+7+7+7
  • 5×7
We can make equal groups with negative values, too.
Rewrite the expression 3(−4) as repeated addition.
3(−4)=

Evaluate.
3(−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

This makes sense on a number line, too.
Number line from negative 12 to 2. There are 3 arrows, each labeled negative 4. The first arrow starts at 0 and ends at negative 4. The second arrow starts at negative 4 and ends at negative 8. The third arrow starts at negative 8 and ends at negative 12.
Starting from 0, taking 3 steps of 4 units each in the negative direction leads us to −12.

Multiplication with the distributive property: positive times negative

The properties of multiplication work the same with negatives as with positive numbers and 0. Let's use the distributive property to convince ourselves of the value of 6(−10).
Fill each blank with a number to keep both sides of the equation equivalent.
6(
  • 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
6(
  • 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
+(−10))		=0
60+6(−10)=0
60+
  • 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

Evaluate.
6(−10)=
  • 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

Describe a general pattern for when we multiply a positive number times a negative number.

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