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Teacher Guide: Multiplying and Dividing Integers

Learn the rules for multiplying and dividing positive and negative numbers.

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Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Negative Numbers. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Apply sign rules to multiply integers correctly
  • Apply sign rules to divide integers correctly
  • Determine the sign of a product or quotient before calculating
  • Solve real-world problems involving integer multiplication and division
Prerequisites
  • • Understanding of negative numbers and the number line
  • • Addition and subtraction of integers
  • • Basic multiplication and division facts
Discussion Starters
  • 1. Why do you think mathematicians decided that negative times negative should be positive?
  • 2. Can you think of a real-life situation where you multiply a negative by a negative?
  • 3. What patterns do you notice when multiplying three or more negative numbers?
  • 4. How can you quickly tell if a product of many integers will be positive or negative?
Common Misconceptions

Two negatives always make a positive

Remediation: Clarify this only applies to multiplication and division, NOT addition. (−3)+(−4)=−7(-3) + (-4) = -7 is still negative!

Division sign rules are different from multiplication

Remediation: Emphasize that the same sign rules apply to both operations. Create a unified sign rule chart.

Differentiation Ideas

For Struggling Students:

  • • Use color coding: green for positive results, red for negative
  • • Start with single-digit integers only
  • • Provide a sign rule reference card for every problem

For On-Level Students:

  • • Include mixed multiplication and division problems
  • • Add word problems with temperature and money contexts
  • • Practice with two and three integer products

For Advanced Students:

  • • Explore patterns with longer chains of multiplication
  • • Introduce division with remainders
  • • Challenge with algebraic expressions using variables
Standards Alignment
  • 7.NS.A.2 (CCSS.MATH.CONTENT.7.NS.A.2)

    Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers

  • 7.NS.A.2a (CCSS.MATH.CONTENT.7.NS.A.2a)

    Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations

Lesson Resources
  • visualSign Rule Chart

    Color-coded reference showing all sign combinations

  • activityInteger Product Game

    Match expressions to their correct products

  • worksheetTemperature Word Problems

    Apply integer operations to real scenarios

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

When multiplying or dividing integers, the sign of the answer depends on the signs of the numbers involved.
The Sign Rules:
OperationResult
(+)×(+)(+) \times (+)Positive
(−)×(−)(-) \times (-)Positive
(+)×(−)(+) \times (-)Negative
(−)×(+)(-) \times (+)Negative
The same rules apply to division!
Simple Rule:
  • Same signs → Positive result
  • Different signs → Negative result
(−3)×(−4)=12(same signs = positive)(-3) \times (-4) = 12 \quad \text{(same signs = positive)}
(−3)×4=−12(different signs = negative)(-3) \times 4 = -12 \quad \text{(different signs = negative)}

Worked Examples

Calculate: (−6)×(−4)(-6) \times (-4)
1

Identify the signs

Both numbers are negative: (−6)(-6) and (−4)(-4) → Same signs
2

Apply the sign rule

Same signs → positive result → Answer will be positive
3

Multiply the absolute values

6×4=246 \times 4 = 24
4

Apply the sign

Positive result → +24+24

Common Mistakes

Thinking (−3)×(−4)=−12(-3) \times (-4) = -12

Why it's wrong: Students forget that multiplying two negatives gives a positive result.

Correct: Same signs → positive: (−3)×(−4)=+12(-3) \times (-4) = +12

Confusing (−5)×3(-5) \times 3 with 5×(−3)5 \times (-3)

Why it's wrong: Both give the same answer! Different signs → negative in both cases.

Correct: (−5)×3=−15(-5) \times 3 = -15 and 5×(−3)=−155 \times (-3) = -15. Order does not matter.

Forgetting the sign when the answer is positive

Why it's wrong: When two negatives multiply, students may write −12-12 out of habit.

Correct: Always check: same signs = positive, different signs = negative.

Why It Matters

Multiplying and dividing integers appears everywhere:
  • Temperature changes: If temperature drops 3 degrees per hour for 5 hours, the total change is (−3)×5=−15(-3) \times 5 = -15 degrees
  • Finance: A monthly debt payment of 200 euros over 6 months means (−200)×6=−1200(-200) \times 6 = -1200 euros total
  • Gaming: Losing 5 points per wrong answer over 4 questions gives (−5)×4=−20(-5) \times 4 = -20 points
  • Elevation: Descending 10 meters per minute for 8 minutes means (−10)×8=−80(-10) \times 8 = -80 meters
These operations help us calculate repeated gains or losses accurately!

Real World Applications

Temperature Changes Over Time

Meteorologists calculate temperature changes using integer multiplication.

Example:

If the temperature drops 2 degrees every hour, after 6 hours the change is (−2)×6=−12(-2) \times 6 = -12 degrees.

1Try It Yourself

A freezer cools food by dropping 5 degrees every 10 minutes. How much does the temperature change in 40 minutes?

What is the total temperature change?

Step 1: Write the mathematical expression

Calculate: (−5)×(-5) \times number of 10-minute periods

Stock Market Changes

Investors track daily gains and losses over multiple days.

Example:

A stock loses 3 euros per day for 5 days: (−3)×5=−15(-3) \times 5 = -15 euros total loss.

2Try It Yourself

An investor reverses a bad trade. They undo a loss of 8 euros that happened 3 times. What is the net effect?

What is the total change when reversing these losses?

Step 1: Write the mathematical expression

Undoing a loss is like: (−1)×(−8)×3(-1) \times (-8) \times 3

Key Takeaways

  • 1Same signs (both positive or both negative) → positive result
  • 2Different signs (one positive, one negative) → negative result
  • 3These rules work for both multiplication AND division
  • 4When multiplying many integers: odd number of negatives → negative, even → positive
  • 5Always calculate with absolute values first, then apply the sign rule

Frequently Asked Questions

Why does a negative times a negative equal a positive?

Think of it as reversing a reversal. If going backward (−-) is negative, then going backward while facing backward (−×−- \times -) means you actually go forward (positive)!

Does the order matter when multiplying integers?

No! Multiplication is commutative. (−3)×5=5×(−3)=−15(-3) \times 5 = 5 \times (-3) = -15. The sign rules give the same answer either way.

What about zero?

Zero times any integer is zero: 0×(−5)=00 \times (-5) = 0. Zero has no sign, so the result is simply 00.

Glossary

Integer
A whole number that can be positive, negative, or zero
Absolute value
The distance from zero, always positive. ∣−7∣=7|-7| = 7
Product
The result of multiplication
Quotient
The result of division
Sign rule
The rule that determines if the result is positive or negative based on the signs of the operands

Formula Card

Positive × Positive

(+)×(+)=+(+) \times (+) = +

Positive times positive equals positive

Negative × Negative

(−)×(−)=+(-) \times (-) = +

Negative times negative equals positive

Positive × Negative

(+)×(−)=−(+) \times (-) = -

Positive times negative equals negative

Negative × Positive

(−)×(+)=−(-) \times (+) = -

Negative times positive equals negative

More in This Topic

Introduction to Negative Numbers

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Adding and Subtracting Integers

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Comparing Integers

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Ordering Integers

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