Multiplying Two Negative Numbers

Calculate: $(-6) \times (-4)$

Identify the signs: Both numbers are negative: $(-6)$ and $(-4)$ = Same signs

Apply the sign rule: Same signs → positive result = Answer will be positive

Multiply the absolute values: $6 \times 4 = 24$ = $24$

Apply the sign: Positive result = $+24$

Answer: $(-6) \times (-4) = 24$

Multiplying Positive and Negative

Calculate: $7 \times (-5)$

Identify the signs: $7$ is positive, $(-5)$ is negative = Different signs

Apply the sign rule: Different signs → negative result = Answer will be negative

Multiply the absolute values: $7 \times 5 = 35$ = $35$

Apply the sign: Negative result = $-35$

Answer: $7 \times (-5) = -35$

Dividing Integers with Different Signs

Calculate: $(-36) \div 9$

Identify the signs: $(-36)$ is negative, $9$ is positive = Different signs

Apply the sign rule: Different signs → negative result = Answer will be negative

Divide the absolute values: $36 \div 9 = 4$ = $4$

Apply the sign: Negative result = $-4$

Answer: $(-36) \div 9 = -4$

Dividing Two Negative Numbers

Calculate: $(-48) \div (-8)$

Identify the signs: Both numbers are negative: $(-48)$ and $(-8)$ = Same signs

Apply the sign rule: Same signs → positive result = Answer will be positive

Divide the absolute values: $48 \div 8 = 6$ = $6$

Apply the sign: Positive result = $+6$

Answer: $(-48) \div (-8) = 6$

Multiple Operations

Calculate: $(-2) \times (-3) \times (-4)$

Multiply the first two numbers: $(-2) \times (-3)$: same signs → positive = $6$

Multiply by the third number: $6 \times (-4)$: different signs → negative = $-24$

Alternative: Count negatives: 3 negative signs (odd) → negative result = Confirms $-24$

Answer: $(-2) \times (-3) \times (-4) = -24$

Mistake: Thinking $(-3) \times (-4) = -12$

Why: Students forget that multiplying two negatives gives a positive result.

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

Mistake: Confusing $(-5) \times 3$ with $5 \times (-3)$

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

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

Mistake: Forgetting the sign when the answer is positive

Why: When two negatives multiply, students may write $-12$ out of habit.

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

Temperature Changes Over Time

Meteorologists calculate temperature changes using integer multiplication.

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

Stock Market Changes

Investors track daily gains and losses over multiple days.

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

Q: Why does a negative times a negative equal a positive?

A: 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)!

Q: Does the order matter when multiplying integers?

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

Q: What about zero?

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