Lesson 3 ~30 min Unit 1 · Number +85 XP

Adding and Subtracting Integers

Master the rules for integer addition and subtraction. Same signs add, different signs subtract.

Today's hook: You have $50 but owe your friend $30. Your net worth is +$20. Now you borrow another $40. Can you figure out your new balance without counting on your fingers?

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Think First

warm-up

Before you read on — quickly: What is 5 + (−3)? What about (−4) + (−2)? Try to picture each on a number line.

Record your answer in your workbook.

The Big Idea

+5 XP

Adding integers means combining movements on the number line. Positive = move right. Negative = move left. Subtracting is the opposite of adding — or equivalently, "adding the opposite."

Think of addition as jumping on the number line. 5 + (−3) means start at 5, jump 3 units left (because of the minus), landing on 2. (−4) + (−2) means start at −4, jump 2 more left, landing on −6. Same signs = add the magnitudes. Different signs = subtract the magnitudes.

$a + (-b) = a - b$ (adding a negative = subtracting)

Positive = right

Adding a positive: jump right on the number line.

Negative = left

Adding a negative: jump left. Same as subtracting.

Start position matters

Always begin at the first number, then jump from there.

What You'll Master

objectives

Know

The sign rules for adding integers
That subtracting = adding the opposite
Key phrases: "add", "subtract", "sum", "difference"

Understand

Why adding a negative is the same as subtracting
Why subtracting a negative makes the answer larger
How the number line models addition and subtraction

Can Do

Add any two integers using the number line or sign rules
Subtract integers by "adding the opposite"
Solve combined addition and subtraction problems

Words You Need

vocabulary

AddCombine two numbers. On the number line: start at first, jump by second.

SubtractFind the difference. Same as adding the opposite: $a - b = a + (-b)$.

SumThe result of adding two numbers. The sum of 3 and −5 is −2.

DifferenceThe result of subtracting. The difference between 4 and 7 is −3.

OppositeThe number with the reverse sign. The opposite of −5 is +5.

Sign ruleSame signs: add the numbers, keep the sign. Different signs: subtract, keep the larger sign.

Spot the Trap

heads-up

Wrong: "−5 − 3 = −2" (subtracted the wrong way). You're subtracting 3 from −5, which makes it more negative.

Right: −5 − 3 = −5 + (−3) = −8. Subtracting 3 = adding −3. Both negative = add magnitudes, keep the negative sign.

Wrong: "7 − (−2) = 5" (subtracted instead of added). Subtracting a negative is adding!

Right: 7 − (−2) = 7 + 2 = 9. Two minuses make a plus. The answer gets larger, not smaller.

Adding Integers

+5 XP

There are two cases for adding integers: same signs and different signs. The sign rule tells you which operation to use on the magnitudes.

Same signs: add the magnitudes, keep the sign. (−4) + (−3) = −7. 5 + 3 = 8. Different signs: subtract the smaller magnitude from the larger, keep the sign of the larger. 5 + (−3) = 2 (positive wins). (−5) + 3 = −2 (negative wins).

Same: add, keep sign. Different: subtract, keep larger sign.

Same sign = add

(−6) + (−2) = −8. Both negative, so add and keep minus.

Different sign = subtract

6 + (−2) = 4. Subtract 2 from 6, keep the + (larger mag).

Memorise the rhyme

"Same sign add and keep, different sign subtract. Keep the sign of the bigger one, and you'll get it exact!"

What to write in your book

Same signs → add the magnitudes, keep the sign.
Different signs → subtract the magnitudes, keep the sign of the larger.
Example: $(-6) + (-2) = -8$, but $6 + (-2) = 4$.

Quick check — which one is right?

Subtracting Integers

+5 XP

Every subtraction can be rewritten as adding the opposite. This is the key trick: $a - b = a + (-b)$. Once rewritten, use the addition sign rules.

$a - b = a + (-b)$. Subtracting a positive = adding a negative (answer gets smaller). $a - (-b) = a + b$. Subtracting a negative = adding a positive (answer gets bigger!). Two minuses always make a plus.

$a - b = a + (-b)$ (subtract = add opposite)

Rewrite first

Always change subtraction to "+ (opposite)" before calculating.

Two minuses = plus

− (−) always becomes +. This is the most important rule.

Check with number line

If unsure, draw it. 3 − (−5) = 8 means start at 3, jump 5 right.

What to write in your book

Every subtraction can be rewritten as adding the opposite: $a - b = a + (-b)$.
Two minuses next to each other always make a plus.
Once rewritten, use the addition sign rules from Card 5.

True or false?

$5 - (-3) = 5 + 3 = 8$. Two minuses next to each other become a plus.

Combined Operations

+5 XP

When you have multiple operations, work left to right, rewriting each subtraction as you go. Brackets first (if any), then left to right.

For −8 + 12 − (−5) − 7: Step 1: Rewrite subtractions. −8 + 12 + 5 − 7. Step 2: Work left to right. (−8 + 12) = 4. (4 + 5) = 9. (9 − 7) = 2. Answer: 2.

Work left to right, rewrite − as +(−)

Left to right

Don't jump around. Work systematically from left to right.

Rewrite first

Change all subtractions to additions before calculating.

One step at a time

Write each step on a new line. Never skip steps.

Watch Me Solve It · 3 examples

Watch Me Solve It · Adding negatives

+15 XP per step

PROBLEM

Calculate −7 + (−4) using the number line.

1
Identify the operation
Both numbers are negative (same sign)
Same sign = add the magnitudes, keep the negative sign.
2
Add the magnitudes
7 + 4 = 11
3
Apply the sign
Both negative, so answer is negative: −11
On the number line: start at −7, jump 4 left, land on −11.

Answer−11

Watch Me Solve It · Subtracting a negative

+15 XP per step

PROBLEM

Calculate 5 − (−3).

1
Rewrite the subtraction
5 − (−3) = 5 + (+3)
Subtracting a negative = adding the opposite. Two minuses make a plus.
2
Add
5 + 3 = 8
3
Verify on number line
Start at 5, jump 3 right (because −(−3) = +3), land on 8.
The answer is larger than 5 because we subtracted a negative!

Answer8

Watch Me Solve It · Combined operations

+15 XP per step

PROBLEM

Evaluate −8 + 12 − (−5) − 7.

1
Rewrite all subtractions
−8 + 12 − (−5) − 7 = −8 + 12 + 5 + (−7)
−(−5) = +5. −7 = +(−7).
2
Work left to right
−8 + 12 = 4
4 + 5 = 9
9 + (−7) = 2
3
Alternative: group positives and negatives
Positives: 12 + 5 = 17. Negatives: −8 + (−7) = −15. 17 + (−15) = 2.
Both methods give the same answer — use whichever you prefer!

Answer2

Common Pitfalls

heads-up

Forgetting that subtracting a negative = adding

7 − (−3) is NOT 4. Two minuses make a plus: 7 − (−3) = 7 + 3 = 10. This is the most common error in integer subtraction.

Fix: Whenever you see − (−), immediately rewrite as +. Circle it in your working.

Mixing up sign rules

(−6) + 4 = −2 (different signs: subtract 4 from 6, keep the −). But students sometimes write −10 (adding instead of subtracting) or +2 (wrong sign).

Fix: Ask "which magnitude is larger?" The answer keeps that sign. 6 > 4, so keep −.

Ignoring brackets

−3 − (−2) without brackets looks like −3 − −2, which is confusing. Always use brackets around negatives: (−3) − (−2).

Fix: Write brackets around every negative number. It prevents sign confusion.

Copy Into Your Books

Adding Integers

Same sign: add magnitudes, keep sign
Different signs: subtract magnitudes, keep larger sign
Use number line to verify

Subtracting Integers

$a - b = a + (-b)$
Subtracting negative = adding positive
Two minuses always make a plus

Sign Rules

(+) + (+) = (+)
(−) + (−) = (−)
(+) + (−) = subtract, keep larger
$a - (-b) = a + b$

Combined Operations

Rewrite all subtractions first
Work left to right
Or group all positives and negatives

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Integer Operations

4 problems

Four drill problems to master integer addition and subtraction. Work each, then reveal the answer.

1 −15 + 8

Different signs: subtract 8 from 15 = 7. The larger magnitude is 15 (negative), so keep the minus.−7
2 6 − (−9)

Subtracting a negative = adding the opposite. 6 − (−9) = 6 + 9.15
3 −4 − 7 + (−3) − (−10)

Rewrite: −4 + (−7) + (−3) + 10 = −14 + 10 = −4.−4
4 Temperature is −6°C. It drops 8°C, then rises 5°C. What's the final temperature?

−6 − 8 + 5 = −6 + (−8) + 5 = −14 + 5 = −9.−9°C

Complete in your workbook.

Quick Check · 5 questions

(−9) + 6 = ?

+10 XP

8 − (−5) = ?

+10 XP

(−10) + (−6) = ?

+10 XP

−3 − (−8) + 2 = ?

+10 XP

A diver is at −20 metres. She rises 15 m, then descends 8 m. Where is she now?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. Evaluate (−5) + 8 + (−12) − (−3). Show all steps.

Answer in your workbook.

UnderstandMedium3 MARKS

Q7. Explain why 10 − (−4) gives a larger answer than 10 − 4. Use the number line or the "add the opposite" rule in your explanation.

Answer in your workbook.

AnalyseHard3 MARKS

Q8. Find the value of $a$ if $a + 7 = −2$. Then find $b$ if $−5 − b = 3$. Explain your method for each.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — −9 + 6 = −3 (different signs, subtract, keep larger sign).

2. C — 8 − (−5) = 8 + 5 = 13 (two minuses = plus).

3. A — −10 + (−6) = −16 (same sign, add, keep minus).

4. D — −3 − (−8) + 2 = −3 + 8 + 2 = 7.

5. B — −20 + 15 − 8 = −5 − 8 = −13 metres.

Show Your Working Model Answers

Q6 (3 marks): Rewrite: −5 + 8 + (−12) + 3 [1]. −5 + 8 = 3 [0.5]. 3 + (−12) = −9 [0.5]. −9 + 3 = −6 [1].

Q7 (3 marks): 10 − (−4) = 10 + 4 = 14 [1]. 10 − 4 = 6 [1]. 14 is larger because subtracting a negative means adding, which increases the value [1].

Q8 (3 marks): $a + 7 = −2$, so $a = −2 − 7 = −9$ [1.5]. $−5 − b = 3$, so $−b = 3 + 5 = 8$, thus $b = −8$ [1.5].

Stretch Challenge · +25 XP, +10 coins

The Magic Sum

Using each of the integers −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5 exactly once, fill in the boxes to make the equation true: $\square + \square + \square + \square + \square = 0$. How many different solutions can you find?

Reveal solution

One simple solution: (−5) + (−4) + 0 + 4 + 5 = 0. The key insight: choose pairs of opposites that cancel, plus zero. There are many solutions — try (−3) + (−2) + 0 + 2 + 3 = 0, or (−5) + (−3) + 1 + 2 + 5 = 0. Any combination where the positives and negatives cancel out works.

Quick Review

Adding

Same sign: add, keep sign

Adding

Different: subtract, keep larger

Subtracting

$a - b = a + (-b)$

Key rule

−(−) = + (two minuses)

Number line

Right = +, left = −

Combined

Rewrite, then left to right

Interactive: Integer Number Line Jumper

Set your starting number and jump amount, then watch the jumper land. Green = positive, red = negative.

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