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Lesson 2 ~25 min Unit 1 · Number +85 XP

Understanding Integers

Positive and negative numbers, the number line, and real-world contexts where integers are everywhere.

Today's hook: The temperature drops to −5 degrees. Your bank account goes to −$20. A lift goes to basement level −2. Integers are just whole numbers that can be negative — and they rule our world.
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Think First
warm-up

Before you read on — quickly: What numbers are less than zero? Where do you see negative numbers in real life? List as many examples as you can.

Record your answer in your workbook.
1
The Big Idea
+5 XP

Integers are whole numbers that can be positive, negative, or zero. They include ..., −3, −2, −1, 0, 1, 2, 3, ... The set of integers is written as $\mathbb{Z}$. Every integer has an opposite (the same distance from zero but on the other side).

Think of integers as positions on a number line. Zero is the centre. Positive integers go to the right (1, 2, 3...). Negative integers go to the left (−1, −2, −3...). The further from zero, the larger the absolute value — but negatives get smaller as you move left.

+ − 0 −5 −3 +2 +4 origin left < 0 < right
$\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$
Zero is an integer
It's neither positive nor negative. It's the origin.
Opposite numbers
3 and −3 are opposites. Same distance from zero, opposite sides.
No fractions or decimals
Integers are whole numbers only. 0.5 and ½ are NOT integers.
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What You'll Master
objectives

Know

  • What integers are and how they're written
  • The symbols for positive and negative
  • Key vocabulary: opposite, absolute value, origin

Understand

  • Why numbers to the left of zero are negative
  • Why −5 is less than −2 (counter-intuitive!)
  • How the number line shows order and magnitude

Can Do

  • Plot integers on a number line
  • Compare integers using < and >
  • Find the opposite and absolute value of any integer
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Words You Need
vocabulary
IntegerA whole number that can be positive, negative, or zero. No fractions or decimals.
PositiveGreater than zero. Written with or without a + sign. Found to the right of zero.
NegativeLess than zero. Always written with a − sign. Found to the left of zero.
Number lineA line showing numbers in order. Zero is the centre. Left is smaller, right is larger.
OppositeTwo numbers the same distance from zero but on opposite sides. 4 and −4 are opposites.
Absolute valueThe distance from zero. Always positive. Written as |−5| = 5.
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Spot the Trap
heads-up

Wrong: "−5 is bigger than −2 because 5 is bigger than 2." On the number line, −5 is further left, so it's smaller.

Right: −2 > −5. The number closer to zero (on the right) is always larger for negatives.

Wrong: "Zero is not an integer." Zero IS an integer. It's the only integer that's neither positive nor negative.

Right: $\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$. Zero is the origin and belongs to the set of integers.

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What Are Integers?
+5 XP

Integers extend the counting numbers in both directions. Counting numbers (1, 2, 3...) go right. Their opposites (−1, −2, −3...) go left. Zero sits in the middle as the dividing line.

Natural numbers = $\{1, 2, 3, \ldots\}$ (counting numbers). Whole numbers = $\{0, 1, 2, 3, \ldots\}$ (add zero). Integers = $\{\ldots, -2, -1, 0, 1, 2, \ldots\}$ (add negatives). Each set is a subset of the next.

INTEGERS ..., −2, −1, 0, 1, 2, ... WHOLE 0, 1, 2, 3, ... NATURAL 1, 2, 3... −3 −1 +4
Natural $\subset$ Whole $\subset$ Integer
Natural = counting
1, 2, 3... The numbers you first learned as a child.
Whole = natural + 0
Add zero to the counting numbers.
Integer = whole + negatives
The complete set. Both directions from zero.
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The Number Line
+5 XP

The number line is the most powerful visual tool for integers. Right is larger, left is smaller. Zero is the origin. Every integer has a unique position.

To plot an integer: draw a horizontal line, mark zero in the centre, mark equal spaces to the right (positive) and left (negative). −4 is 4 units left of zero. +3 is 3 units right. The distance from zero is called the absolute value: |−4| = 4.

larger → ← smaller 0 −5 −4 −1 +1 +3 +5 −4 +3 |−4| = 4, |+3| = 3 Both 4 units from zero
Right → larger    Left → smaller
Zero is the origin
It's the reference point. Everything is measured from zero.
Symmetry
For every positive, there's a matching negative. The line is a mirror.
Equal spacing
Each tick mark is the same distance apart. The scale never changes.
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Comparing Integers
+5 XP

On the number line, any number to the right is greater than any number to its left. This simple rule handles all comparisons — but it's especially tricky with negatives.

To compare −7 and −3: on the number line, −3 is to the right of −7. So −3 > −7. The absolute value of −7 is larger (7 > 3), but −7 itself is smaller. Think temperature: −7°C is colder than −3°C.

0 −8 −6 −4 +2 +4 −7 −3 −3 is RIGHT of −7 −3 > −7
−3 > −7   (counter-intuitive!)
Bigger absolute value ≠ bigger number
For negatives, a larger absolute value means a SMALLER number.
Temperature trick
−10°C is colder than −5°C. So −10 < −5.
Draw the number line
If unsure, sketch it. Right is always larger.

Watch Me Solve It · 3 examples

Watch Me Solve It · Arrange in order
+15 XP per step
Q1
PROBLEM
Arrange from smallest to largest: −4, 2, −7, 0, 3, −1.
  1. 1
    Plot on a number line
    Draw a line: −7, −4, −1, 0, 2, 3
    Visualising makes ordering trivial.
  2. 2
    Read left to right
    −7 is furthest left (smallest), 3 is furthest right (largest)
  3. 3
    Write the order
    −7, −4, −1, 0, 2, 3
Answer−7, −4, −1, 0, 2, 3
Watch Me Solve It · Number line distance
+15 XP per step
Q2
PROBLEM
What integer is 5 units to the left of 2 on the number line?
  1. 1
    Start at 2
    Position: 2 on the number line
  2. 2
    Move 5 units left
    2 − 5 = −3
    Left means subtract. Each unit left decreases by 1.
  3. 3
    Check
    From −3 to 2: −3 → −2 → −1 → 0 → 1 → 2. That's 5 units. ✓
Answer−3
Watch Me Solve It · Opposite and absolute value
+15 XP per step
Q3
PROBLEM
What is the opposite of −8? What is |−5|?
  1. 1
    Opposite of −8
    The opposite of −8 is the number the same distance from zero on the other side: +8
  2. 2
    Absolute value of −5
    |−5| = 5
    Absolute value = distance from zero. Distance is always positive.
  3. 3
    Verify
    Opposite of −8 = 8, and |−5| = 5
AnswerOpposite of −8 is 8; |−5| = 5
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Common Pitfalls
heads-up
Thinking −10 is greater than −5
This is the #1 integer error. Students see 10 > 5 and assume −10 > −5. But on the number line, −10 is further left. −10 < −5.
Fix: Think temperature. −10°C is colder than −5°C. Colder = smaller number.
Forgetting zero is an integer
Zero is neither positive nor negative, but it IS an integer. It's the origin on the number line.
Fix: $\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$. Zero is right there in the middle.
Confusing absolute value with the number itself
|−7| = 7 (the distance), but −7 itself is still a negative number less than zero. Absolute value strips the sign; the original number keeps it.
Fix: |−7| = 7 (distance) but −7 < 0 (position). Two different concepts.
Copy Into Your Books

Integers

  • $\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$
  • Whole numbers + their negatives
  • No fractions or decimals

The Number Line

  • Zero is the origin (centre)
  • Right = positive = larger
  • Left = negative = smaller
  • Equal spacing throughout

Comparing

  • Right is always greater than left
  • −3 > −7 (counter-intuitive!)
  • Use temperature or money to check

Absolute Value

  • |−5| = 5 (distance from zero)
  • Always positive or zero
  • Opposite of $a$ is −$a$

How are you completing this lesson?

Brain Trainer · 4 problems

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Brain Trainer · Integer Skills
4 problems

Four drill problems to build your integer fluency. Work each, then reveal the answer.

  1. 1 Is −3 an integer? Is 0.5? Is 0? Explain each answer.

    −3 IS an integer (it's a whole number, negative). 0.5 is NOT (it's a decimal). 0 IS an integer (it's the origin).−3: yes   0.5: no   0: yes

  2. 2 Insert < or >: −12 ___ −8

    On the number line, −12 is further left than −8. So −12 is smaller.−12 < −8

  3. 3 What is the distance between −4 and 6 on the number line?

    From −4 to 0 is 4 units. From 0 to 6 is 6 units. Total: 4 + 6 = 10 units.Distance = 10 units

  4. 4 List all integers $x$ where −3 < $x$ ≤ 4.

    −3 < $x$ means $x$ is greater than −3 (not including −3). $x$ ≤ 4 means up to and including 4.$x \in \{-2, -1, 0, 1, 2, 3, 4\}$
Complete in your workbook.

Quick Check · 5 questions

1
Which is NOT an integer?
+10 XP
2
Which is the largest?
+10 XP
3
What is |−9|?
+10 XP
4
What is the opposite of the opposite of −6?
+10 XP
5
Arrange from largest to smallest: −1, −7, −4
+10 XP

Show Your Working · 3 questions

Show Your Working
8 marks total
UnderstandEasy2 MARKS

Q6. Explain why −3 is an integer but ⅓ is not. Use the definition of integers in your answer.

Answer in your workbook.
ApplyMedium3 MARKS

Q7. The temperature was −5°C at 6am. By noon it had risen to 4°C. Draw a number line showing both temperatures and calculate how many degrees the temperature rose.

Answer in your workbook.
AnalyseHard3 MARKS

Q8. If |$x$| = 7, what could $x$ be? If |$x$| < 3 and $x$ is an integer, list all possible values of $x$. Explain your reasoning.

Answer in your workbook.
Comprehensive Answers

Quick Check

1. B — 0.5 is a decimal, not a whole number, so not an integer.

2. C — 3 is furthest right on the number line.

3. A — |−9| = 9. Absolute value = distance from zero.

4. B — Opposite of −6 is 6. Opposite of 6 is −6. You end up back where you started!

5. B — −1 > −4 > −7. Closer to zero = larger for negatives.

Show Your Working Model Answers

Q6 (2 marks): Integers are whole numbers (positive, negative, or zero) with no fractional or decimal part [1]. −3 is a whole number, so it is an integer. ⅓ is a fraction, not a whole number, so it is not an integer [1].

Q7 (3 marks): Number line drawn from at least −5 to +4 with both points marked [1]. From −5 to 0 is 5 degrees, and from 0 to 4 is 4 degrees [1]. Total rise = 5 + 4 = 9 degrees [1].

Q8 (3 marks): If |$x$| = 7, then $x$ = 7 or $x$ = −7 (both are 7 units from zero) [1]. If |$x$| < 3, then $x$ is less than 3 units from zero, so $x$ ∈ {−2, −1, 0, 1, 2} [2].

Stretch Challenge · +25 XP, +10 coins

The Integer Maze

Start at 0 on a number line. Move 4 steps right, then 7 steps left, then 5 steps right, then 3 steps left. Where do you finish? What is the total distance you travelled (counting every step)?

Reveal solution

Final position: 0 + 4 − 7 + 5 − 3 = −1. Total distance: 4 + 7 + 5 + 3 = 19 units. Note: final position and total distance are different!

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Quick Review

Integers

$\{\ldots, -2, -1, 0, 1, 2, \ldots\}$

Number Line

Zero at centre, + right, − left

Comparing

Right is always larger

Negatives

−3 > −7 (closer to zero)

Absolute Value

|−5| = 5 (distance from 0)

Opposite

Same distance, other side

Interactive: Power Tower Builder

Build a tower to discover what exponents really mean. Is 2³ the same as 2 × 3? Find out!

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List all integers between −10 and +10 that have an absolute value greater than 6, in under 60 seconds for double coins!
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