Multiplying and Dividing Signed Numbers

Lesson 1 of 2: Sign Rules and Applications

In this lesson:

• Prove why negative × negative = positive
• Apply sign rules to multiply and divide all rationals
• Interpret products in real-world contexts

What You Will Learn Today

By the end of this lesson, you should be able to:

1. Explain why the product of two negatives is positive
2. Apply sign rules: same signs → positive; different signs → negative
3. Interpret products of rational numbers in context
4. Divide rationals using the multiply-by-reciprocal relationship

Two Negatives — But Why?

You've probably heard: "two negatives make a positive."

• But is it just a rule someone invented?
• Or does it have to be true?

We can prove it — using a property you already know.

Using the Distributive Property as Proof

We start with something we know for certain:

Distribute the 3, then use :

So . Positive × negative = negative. ✓

Proving Negative × Negative = Positive

Same logic, now with a negative multiplier:

Distribute; since :

Therefore . Negative × negative = positive. ✓

Seeing the Pattern in a Table

The pattern in each row continues naturally — it cannot stop or reverse.

Sign Rules: The Complete Picture

Same signs → Positive product

• Positive × Positive = Positive
• Negative × Negative = Positive

Different signs → Negative product

• Positive × Negative = Negative
• Negative × Positive = Negative

Key shortcut: Count the negative factors.

• Even count → positive
• Odd count → negative

Check: Apply the Sign Rules

What is ?

Use the distributive property argument if you need to:

Think through the sign before advancing for the answer.

Answer:

• Sign: negative × negative = positive
• Absolute value: 4 × 6 = 24
• Result: +24

Verify:

So ✓

From Integers to All Rationals

Sign rules extend to all rational numbers: integers, fractions, and decimals.

Two-step process:

1. Determine the sign (same signs → +, different → −)
2. Compute the absolute value (ignore signs, just multiply)

Sign Rules Applied to Integer Operations

Multiplication (sign first, then magnitude):

• : different signs → −56
• : same signs → +36

Division (same sign rules apply):

• : same signs → +8
• : different signs → −4

Applying Sign Rules to Signed Fractions

Step 1: Determine the sign
Step 2: Multiply the absolute values

Division: Flip the Divisor, Not Both

Division = multiply by the reciprocal of the divisor.

Step 1: Change ÷ to ×; flip the divisor (the number after ÷)

Step 2: Sign (neg × neg = pos), then value

The Two-Step Process with Signed Decimals

Same two-step process applies:

: sign = pos; value = 0.4 × 2.5 = 1.0 → Result: 1.0

: sign = neg; value = 3.6 ÷ 0.9 = 4 → Result: −4

Quick check for decimal division: Does ? ✓

Check: Multiply Two Signed Fractions

Compute .

Show your work in two steps:

• Sign: _____ × _____ = _____
• Value: = _____

Write both steps before advancing.

Answer:

• Sign: negative × negative = positive
• Value:
• Result:

Cancel before multiplying: — the 2 and 4 share factor 2; the 3 and 9 share factor 3.

The Even/Odd Negative Factor Rule

Count the negatives — the sign alternates:

Example: : 3 negatives → odd → −24

Real World: Modeling Submarine Descent

A submarine descends at meter per second.

How far from its starting point after 8 seconds?

• Rate: m/s (negative = downward)
• Time: 8 s (positive)

Interpretation: The submarine is 6 meters below its starting point.

Real World: Going Backward in Time

The temperature was decreasing at 4°F per hour.

What was the temperature 3 hours AGO, relative to now?

• Rate: °F/hour (decreasing)
• Time: hours (3 hours in the past)

Interpretation: Three hours ago it was 12°F warmer than now.

Your Turn: Mixed Signed Number Practice

Solve each. Show sign and value separately.

3. A diver descends feet per minute. How far from the surface after 6 minutes?

Work through all three, then advance for answers.

Answers: Checking Your Signed Calculations

1. — same signs, 12 × 5 = 60

2. — neg ÷ pos = neg; cancel 5s to get

3. ft — 15 feet below the surface

Lesson 1 Key Ideas to Remember

• neg × neg = pos: distributive property forces it
• Sign rule: same signs → pos; different → neg
• Even/odd: even negatives → pos; odd → neg
• Two-step: sign first, then magnitude

Watch out: Two negatives → positive; flip only the divisor

Coming Up: Properties and Decimals

Lesson 2 covers:

• Three equivalent forms of a negative fraction:
  
• Using properties of operations to multiply efficiently
• Converting rational numbers to decimals using long division
• Terminating vs. repeating decimals and bar notation