Multiplying and Dividing Integers

Quick Review: What Are Integers? 🔢

Integers are the whole numbers and their opposites. There are no fractions or decimals.

$\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$

• Positive integers sit to the right of zero: $1, 2, 3, 4, \ldots$
• Negative integers sit to the left of zero: $-1, -2, -3, -4, \ldots$
• Zero is special — it is neither positive nor negative.

In Grade 6 you learned to add and subtract integers. Now you will master multiplying and dividing them. The good news? The number part of the answer is found exactly the way you already know. The only new skill is figuring out the sign of the answer.

The Rules for Multiplying Integers ✖️

To multiply integers, do two jobs:

1. Multiply the numbers as if both were positive.
2. Decide the sign using the rules below.

Rule 1 — Positive × Positive = Positive

This is normal multiplication: $5 \times 3 = 15$.

Rule 2 — Negative × Negative = Positive

Two negatives make a positive! $(-5) \times (-3) = 15$. A negative "reverses" a number, and reversing twice brings you right back to positive.

Rule 3 — Positive × Negative = Negative

A positive and a negative give a negative product: $5 \times (-3) = -15$.

Rule 4 — Negative × Positive = Negative

Order does not matter (that's the commutative property), so this is just like Rule 3: $(-3) \times 5 = -15$.

⭐ The Shortcut: Same vs. Different Signs

You do not need to memorize four separate rules. There is one simple idea:

So just ask yourself: "Are the signs the same or different?"

• Same signs → Positive ✅
• Different signs → Negative ❌

That's it! Whether you have $(+)\times(+)$ or $(-)\times(-)$, matching signs always give a positive answer. A mismatch always gives a negative answer.

Quick Concept Check ✅

Let's make sure the big idea is clear before we practice.

Multiplying Step by Step ✖️

Let's work two examples carefully so you see the pattern.

Example A: $(-7) \times 6$

• Step 1 — Multiply the numbers: $7 \times 6 = 42$.
• Step 2 — Choose the sign: one number is negative and one is positive, so the signs are different. Different signs → negative.
• Answer: $(-7) \times 6 = -42$

Example B: $(-9) \times (-4)$

• Step 1 — Multiply the numbers: $9 \times 4 = 36$.
• Step 2 — Choose the sign: both numbers are negative, so the signs are the same. Same signs → positive.
• Answer: $(-9) \times (-4) = 36$

Multiplying More Than Two Integers 🔁

When a problem has three or more factors, you don't need to track signs one pair at a time. Just count the negative signs:

• Even number of negatives → Positive result
• Odd number of negatives → Negative result

Example C: $(-2) \times (-3) \times (-4)$

• Count the negatives: there are 3 of them — that's odd → the answer is negative.
• Multiply the numbers: $2 \times 3 \times 4 = 24$.
• Answer: $-24$

Example D: $(-2) \times (-3) \times 4$

• Count the negatives: there are 2 — that's even → the answer is positive.
• Multiply the numbers: $2 \times 3 \times 4 = 24$.
• Answer: $+24$

Your Turn: Multiply These ✍️

First find the number part, then decide the sign. Type each answer as a signed number, like -42 or 36.

1. $(-7) \times 6 = \;?$
2. $(-9) \times (-4) = \;?$
3. $(-2) \times (-3) \times (-4) = \;?$

Dividing Integers ➗

Here is the best news of the whole lesson: division uses the exact same sign rules as multiplication!

• Same signs → Positive quotient
• Different signs → Negative quotient

The steps are the same as before: divide the numbers, then attach the sign. Now try the questions below.

Multiple Choice 🎯

Choose the Correct Sign 🔽

For each problem, the number part is already correct — you just pick the right sign of the answer.

Integers in the Real World 🌍

Negative numbers show up everywhere — temperatures dropping, money owed, and submarines diving. Multiplying and dividing helps us scale these changes up or down.

Temperature drop: A cold front makes the temperature fall $3$ degrees each hour. We write a drop as $-3$. After $4$ hours, the change is: $4 \times (-3) = -12 \text{ degrees}$

Sharing a debt: Four friends share a debt of $20 equally. A debt is negative, so the total is $-20$. Each person's share is: $(-20) \div 4 = -5 \Rightarrow \text{each owes } \$5$

Diving deeper: A submarine descends $25$ meters every minute. Going down is $-25$. After $3$ minutes its depth change is $3 \times (-25) = -75$ meters.

Watch the words: "drop," "owe," "below," and "descend" all point to negative numbers.

Word Problem Practice ✍️

Write each answer as a signed integer, like -12 or 6.

1. The temperature drops $3$ degrees each hour for $4$ hours. What is the total change? ( $4 \times (-3)$ )
2. Five friends equally share a debt of $30. What is each person's share as an integer? ( $(-30) \div 5$ )
3. A diver goes down $6$ meters each minute for $7$ minutes. What is the total change in depth? ( $7 \times (-6)$ )

Application Check 🎯

Putting It All Together 🧠

You now have one set of rules that works for both multiplication and division. Here is the full summary:

Remember these two key ideas:

• Same signs → Positive. Different signs → Negative.
• For three or more factors, count the negatives: even → positive, odd → negative.

Ready for the final challenge? It mixes everything together!

🏆 Mixed Challenge

Use the sign rules carefully — these mix multiplication and division.