Multiplying and dividing integers

Here you will learn strategies on how to multiply and divide integers, including using visual models as well as using the number line.

Students will first learn about integers in 6th grade math as part of their work with the number system and expand that knowledge to operations with integers in the 7th grade.

What are multiplying and dividing integers?

Multiplying and dividing integers is when you multiply or divide two or more integers together to give a product or quotient that can be either positive or negative.

You can multiply and divide integers using visual models or a rule.

Do you notice a pattern or rule?

Rule for multiplying integers:

• If the integers have the same sign, the product will be positive.
• If the integers have different signs, the product will be negative.

Do you notice a pattern or rule?

Rule for dividing integers:

• If the integers have the same sign, the quotient will be positive.
• If the integers have different signs, the quotient will be negative.

What are multiplying and dividing integers?

Common Core State Standards

How does this apply to 6th grade math and 7th grade math?

• Grade 6: Number System (6.NS.C.6)
  Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

• Grade 7: Number System (7.NS.A.2)
  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

How to multiply and divide integers?

In order to add and subtract integers using counters:

1. If the integers have the same sign, the product or quotient is positive.  If not, go to step 2.
2. If the integers have different signs, the product or quotient is negative.
3. Find the product or quotient.

Multiplying and dividing integers examples

Example 1: multiplying integers with the same sign

Multiply: (-4) \times(-12)= \, ?

1. If the integers have the same sign, the product or quotient is positive. If they don’t go to step 2. 

-4 and -12 have the same sign so the product is positive.

2If the integers have different signs, the product or quotient is negative.

Integers have the same sign.

3Find the product or quotient.

(-4) \times(-12)=48

Example 2: multiplying integers with different signs

Multiply: (-13) \times 8= \, ?

Example 3: dividing integers with the same sign

Divide: \cfrac{(-18)}{(-3)}= \, ?

Example 4: dividing integers with different signs

Divide: -120 \div 3= \, ?

Example 5: multiplying integers word problem

From sea level, a submarine descends 25 \, ft. per minute (-25 \, ft.).

After 6 minutes, the submarine’s distance can be modeled by (-25) \times 6 = \, d , where d is the submarine in relation to sea level.

How far below sea level is the submarine?

Example 6: dividing integer word problems

On a certain winter day, the temperature changed at a rate of -4 degrees Fahrenheit per hour.

After a specific amount of time, the change in temperature was -36 degrees Fahrenheit, which is modeled by (-36) \div (-4) = \, h, where h represents the amount of hours.

How long did it take for the change in temperature to be -36 degrees Fahrenheit?

Teaching tips for multiplying and dividing integers

• Multiplying and dividing integers are foundational skills for Algebra 1. Using manipulatives helps students formulate conceptual understanding.

• Have students identify the patterns with multiplying and dividing integers so that they can figure out the rules on their own.

• Although practice integer worksheets have their place, have students practice problems through digital games or scavenger hunts around the room to make it engaging.

• Reinforce essential vocabulary such as dividend, divisor, quotient, factors,  and product.

Easy mistakes to make

• Mixing up the rules for multiplication and division
  For example, when multiplying integers with the same sign, you get a negative product, and when dividing integers with the same sign, you get a negative quotient.

• Mixing up the rules of multiplication and division with addition and subtraction
  For example, applying the rule of addition of integers to (15) \div(-3)=5 where the absolute value of 15 is greater than the absolute value of 3 so the answer must be positive.

Related multiplication and division lessons

This multiplying and dividing integers topic guide is part of our series on multiplication and division. You may find it helpful to start with the main multiplication and division topic guide for a summary of what to expect or use the step-by-step guides below for further detail on individual topics. Other topic guides in this series include:

• Multiplication and division
• Multiplying multi-digit numbers
• Understanding multiplication
• Multiplication and division within 100
• Long division
• Multiplying and dividing rational numbers
• Understanding division
• Dividing multi-digit numbers
• Multiplicative comparison

The next lessons are

• Types of numbers
• Rounding numbers
• Factors and multiples

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