Lesson 4 ~30 min Unit 1 · Number +85 XP

Multiplying and Dividing Integers

Sign rules made simple. Plus times plus = plus. Minus times minus = plus. Everything else = minus.

Today's hook: If you lose $5 every day for 4 days, you're $20 behind. That's −5 × 4 = −20. But what if you stop losing $5 every day for 4 days? That's (−5) × (−4) = +20. Why does negative × negative = positive?

0/5QUESTS

Printable Worksheets

Print or save as PDF — or build a custom worksheet from any module's questions.

Build Apply Master Build custom

Think First

warm-up

Before you read on — quickly: What is (−3) × 4? What about (−3) × (−4)? Why does the sign change? Try to explain the pattern.

Record your answer in your workbook.

The Big Idea

+5 XP

Multiplication is repeated addition. Division is its inverse. The sign rules are simple: same signs → positive, different signs → negative. This applies to both multiplication and division.

Think of 3 × (−4) as (−4) + (−4) + (−4) = −12. Three groups of −4. For (−3) × (−4), use the pattern: 3 × (−4) = −12, 2 × (−4) = −8, 1 × (−4) = −4, 0 × (−4) = 0, (−1) × (−4) = 4. So (−3) × (−4) = 12.

Same signs = + Different signs = −

Count the negatives

Even number of negatives = positive answer. Odd = negative.

Odd number of negatives

1 negative or 3 negatives = negative answer.

Brackets around negatives

Always use brackets: (−3) × 4, not −3 × 4.

What You'll Master

objectives

Know

Sign rules for multiplying and dividing integers
That division sign rules match multiplication
Key vocabulary: product, quotient, fact family

Understand

Why negative × negative = positive (pattern argument)
That division is the inverse of multiplication
How multiple operations follow BODMAS

Can Do

Multiply and divide any integers confidently
Evaluate expressions with multiple operations
Check answers using inverse operations

Words You Need

vocabulary

MultiplyCombine by repeated addition. 3 × 4 means 4 + 4 + 4.

DivideSplit into equal groups. The inverse of multiplication.

ProductThe result of multiplying two numbers.

QuotientThe result of dividing one number by another.

Sign ruleSame signs → positive. Different signs → negative.

Fact familyA set of related multiplication and division facts. If 3 × 4 = 12, then 12 ÷ 3 = 4.

Spot the Trap

heads-up

Wrong: "Minus times minus = minus." No! Two negatives multiply to give a POSITIVE. (−3) × (−4) = 12, not −12.

Right: Memorise: (+)(+)=(+), (−)(−)=(+), (+)(−)=(−), (−)(+)=(−). Same signs = positive. Different = negative.

Wrong: Forgetting brackets: −3 × 4 is ambiguous. Is it (−3) × 4 or −(3 × 4)?

Right: Always use brackets around negative numbers in expressions: (−3) × 4 = −12.

Multiplying Integers

+5 XP

Multiplication is repeated addition. The sign rule tells us the sign of the answer. Multiply the absolute values, then apply the sign.

3 × (−4) = (−4) + (−4) + (−4) = −12. (−3) × (−4): use the pattern. Start with 3 × (−4) = −12. Each time the first factor decreases by 1, the product increases by 4. So 2 × (−4) = −8, 1 × (−4) = −4, 0 × (−4) = 0, (−1) × (−4) = 4, (−2) × (−4) = 8, (−3) × (−4) = 12.

|a| × |b|, then apply sign rule

Count negative signs

Even number of negatives = positive answer.

Multiply absolute values

Ignore signs first, multiply, then apply the sign rule.

Brackets prevent confusion

Always write (−3) × 4, never −3 × 4.

Dividing Integers

+5 XP

Division is the inverse of multiplication. The same sign rules apply. If (−3) × 4 = −12, then (−12) ÷ (−3) = 4 and (−12) ÷ 4 = −3.

Division sign rules are identical to multiplication. (−a) ÷ (−b) = a ÷ b (same signs = positive). (−a) ÷ b = −(a ÷ b) (different signs = negative). Always check your answer by multiplying back.

(−a) ÷ (−b) = a ÷ b (−a) ÷ b = −(a ÷ b)

Same rules as ×

Division sign rules are identical to multiplication.

Check by multiplying

(−12) ÷ 4 = −3? Check: 4 × (−3) = −12. Yes!

Never divide by zero

5 ÷ 0 is undefined. Always check the divisor.

Multiple Operations

+5 XP

When multiplying and dividing multiple integers, work left to right. Count the total number of negative signs to determine the final sign.

For [24 ÷ (−3)] × (−2) + (−4): Step 1: Brackets first: 24 ÷ (−3) = −8. Step 2: (−8) × (−2) = 16 (two negatives = positive). Step 3: 16 + (−4) = 12. Answer: 12.

Brackets → ×÷ → +− (left to right)

Brackets first

Always evaluate inside brackets before anything else.

× and ÷ equal priority

Work left to right for multiplication and division.

One step per line

Write each step clearly. Never skip steps.

Watch Me Solve It · 3 examples

Watch Me Solve It · Multiply integers

+15 XP per step

PROBLEM

Calculate (−6) × 7.

1
Check the signs
−6 and +7 → different signs
Different signs = negative answer.
2
Multiply the absolute values
6 × 7 = 42
3
Apply the sign
Different signs → negative: −42

Answer−42

Watch Me Solve It · Negative times negative

+15 XP per step

PROBLEM

Calculate (−8) × (−5).

1
Check the signs
−8 and −5 → same signs (both negative)
Same signs = positive answer.
2
Multiply the absolute values
8 × 5 = 40
3
Apply the sign
Same signs → positive: +40

Answer40

Watch Me Solve It · Combined operations

+15 XP per step

PROBLEM

Evaluate [24 ÷ (−3)] × (−2) + (−4).

1
Brackets first
24 ÷ (−3) = −8
Different signs: 24 ÷ 3 = 8, keep minus.
2
Multiply
(−8) × (−2) = 16
Same signs (both negative) = positive. 8 × 2 = 16.
3
Add
16 + (−4) = 12
16 − 4 = 12.

Answer12

Common Pitfalls

heads-up

Sign errors

The #1 mistake: getting the sign wrong. Memorise: same signs → +, different signs → −. Write the sign rule at the top of your page until it sticks.

Fix: Before calculating, ask "are the signs the same or different?" Circle the signs first.

Forgetting brackets around negatives

−3 × 4 looks like you're subtracting 3 × 4. Use brackets: (−3) × 4.

Fix: Every negative number in an expression gets brackets. No exceptions.

Doing multiplication before division at same priority

12 ÷ 3 × 2: if you multiply first (3 × 2 = 6, then 12 ÷ 6 = 2), you get the wrong answer. Work left to right: 12 ÷ 3 = 4, then 4 × 2 = 8.

Fix: × and ÷ have equal priority. Always work left to right.

Copy Into Your Books

Sign Rules

(+) × (+) = (+)
(−) × (−) = (+)
(+) × (−) = (−)
(−) × (+) = (−)

Division

Same rules as multiplication
Check by multiplying back
Never divide by zero

Multiple Operations

Brackets first
× and ÷ left to right
Count negatives for sign

Brackets

Always use (−a) not −a
Prevents sign confusion
Makes working clearer

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Integer Multiplication and Division

4 problems

Four drill problems to sharpen your skills. Work each, then reveal the answer.

1 (−9) × 4

Different signs: 9 × 4 = 36, keep minus.−36
2 (−7) × (−6)

Same signs: 7 × 6 = 42, keep plus.42
3 (−48) ÷ 6

Different signs: 48 ÷ 6 = 8, keep minus.−8
4 100 ÷ [(−5) × (−4)]

Brackets first: (−5) × (−4) = 20. Then 100 ÷ 20 = 5.5

Complete in your workbook.

Quick Check · 5 questions

(−3) × (−5) = ?

+10 XP

24 ÷ (−6) = ?

+10 XP

(−2)³ = ?

+10 XP

Which gives a positive answer?

+10 XP

(−6)² ÷ (−3) = ?

+10 XP

Show Your Working · 3 questions

Show Your Working

10 marks total

ApplyMedium3 MARKS

Q6. Evaluate (−5) × 8 × (−2). Show each step and explain the sign at each stage.

Answer in your workbook.

ApplyMedium4 MARKS

Q7. The temperature drops 3°C each hour for 6 hours starting from 5°C. What is the final temperature? Show working with integers.

Answer in your workbook.

ApplyHard3 MARKS

Q8. If $a$ = −4 and $b$ = 6, find the value of $a$ × (−b) ÷ 2.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — (−3) × (−5) = 15 (same signs = positive).

2. B — 24 ÷ (−6) = −4 (different signs = negative).

3. A — (−2)³ = (−2) × (−2) × (−2) = −8.

4. B — (−3) × (−7) = 21 (two negatives = positive).

5. A — (−6)² = 36, then 36 ÷ (−3) = −12.

Show Your Working Model Answers

Q6 (3 marks): (−5) × 8 = −40 [1] (different signs). −40 × (−2) = 80 [1] (same signs = positive). Final answer: 80 [1].

Q7 (4 marks): Starting temp = 5°C [1]. Drop = 3 × 6 = 18°C (or −18 as integer) [1]. Final = 5 − 18 = 5 + (−18) = −13°C [2].

Q8 (3 marks): $a$ × (−b) = (−4) × (−6) = 24 [1]. Then 24 ÷ 2 = 12 [2].

Stretch Challenge · +25 XP, +10 coins

The Factor Pairs Puzzle

Find all integer pairs $(x, y)$ where $x \times y = −24$ and $x < y$. How many pairs are there? List them all with proper working.

Reveal solution

The factor pairs of 24 are: 1×24, 2×12, 3×8, 4×6. For each pair, one must be negative to get −24. With $x < y$: (−24,1), (−12,2), (−8,3), (−6,4), (−4,6), (−3,8), (−2,12), (−1,24). That's 8 pairs.

Quick Review

Same signs

(+)(+)=(+) (−)(−)=(+)

Different signs

(+)(−)=(−) (−)(+)=(−)

Multiply abs values

Then apply the sign rule

Division = inverse

Same sign rules as ×

Check by × back

Verify every division answer

Brackets

Always around negatives

Interactive: Integer Multiplication Explorer

Drag integers onto the grid to explore multiplication and division patterns visually.

Your Badges

0 of 6

First Steps

3-Day Streak

3 in a Row

Lesson Ace

Stretch Seeker

Daily Warrior

Mark lesson as complete

Tick when you've finished Learn, Practice and the Stretch. Earns +85 XP and +25 coins.

Unit overview · Maths subject page · Unit quiz