Mathematics • Year 7 • Unit 1 • Lesson 2
Understanding Integers
Build fluency with the number line: plot positive and negative integers, find opposites, work out absolute values, and compare two integers by asking "which one is further to the right?".
1. I do — fully worked example
Read every line. Each step has a short reason on the right so you can see why, not just what.
Problem. Arrange these integers from smallest to largest: −4, 2, −7, 0, 3, −1.
Step 1 — Sketch a number line.
⟵ −8 · −7 · −6 · −5 · −4 · −3 · −2 · −1 · 0 · 1 · 2 · 3 · 4 ⟶
Reason: a quick visual makes ordering negatives much easier.
Step 2 — Plot each integer on the line.
Mark dots above: −7, −4, −1, 0, 2, 3.
Reason: each dot has only one home on the number line — no two integers share a position.
Step 3 — Read off the dots from left (smallest) to right (largest).
−7, −4, −1, 0, 2, 3
Reason: the rule of the number line is "right is larger". So the left-most dot is smallest.
Step 4 — Sanity check: −7 is the smallest because it is furthest from zero on the negative side.
Reason: for negatives, "further left = more negative = smaller". −7 < −4 < −1.
Answer: −7, −4, −1, 0, 2, 3.
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. Arrange these integers from smallest to largest, then state the opposite of −6: 5, −2, −6, 0, 1.
Step 1 — Sketch a number line covering at least −7 to +6.
Step 2 — Plot each integer:
Dots above: ____, ____, ____, ____, ____.
Step 3 — Read left to right:
____ , ____ , ____ , ____ , ____
Step 4 — The opposite of −6 is _________ (same distance from zero, other side).
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation, the middle two are standard, and the last two are extension.
Foundation — single step
3.1 Place > or < between: −3 ____ 2. 1 mark
3.2 Place > or < between: −8 ____ −3. 1 mark
3.3 What is the opposite of (a) 7 (b) −12? 1 mark
3.4 Find: (a) |−9| (b) |+4| (c) |0|. 1 mark
Standard — combine two ideas
3.5 Arrange from smallest to largest: −5, 3, 0, −10, 1, −2. 2 marks
3.6 Starting at 4 on a number line, what integer do you land on if you move 7 units to the left? 2 marks
Extension — push your thinking
3.7 Find all the integers n for which |n| = 5. How many are there, and what are they? 2 marks
3.8 List all integers that are greater than −4 AND less than 3. How many are there? 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (5, −2, −6, 0, 1)
Step 3 — order from smallest: −6, −2, 0, 1, 5.
Step 4 — the opposite of −6 is +6 (same distance from zero, on the positive side).
3.1 — Compare −3 and 2
−3 < 2. Any negative is smaller than any positive.
3.2 — Compare −8 and −3
On the number line, −3 is to the right of −8, so −8 < −3. (Common slip: thinking 8 > 3 means −8 > −3. For negatives, the opposite is true.)
3.3 — Opposites
(a) Opposite of 7 is −7.
(b) Opposite of −12 is +12.
3.4 — Absolute values
(a) |−9| = 9 (b) |+4| = 4 (c) |0| = 0. Absolute value = distance from zero, always positive (or zero).
3.5 — Order from smallest to largest
−10, −5, −2, 0, 1, 3. The negatives line up by "further left is smaller", so −10 is smallest.
3.6 — Starting at 4, move 7 units left
Each unit left decreases the value by 1. Start: 4. Move 7 left: 4 − 7 = −3. Quick check: from −3 to 4 you pass through −2, −1, 0, 1, 2, 3, 4 — that's 7 steps. ✓
3.7 — Integers with |n| = 5
|n| = 5 means "n is 5 units from zero". Two integers fit: n = 5 and n = −5. So there are 2 such integers.
3.8 — Integers between −4 and 3 (strictly)
Greater than −4 means we start at −3 (not −4). Less than 3 means we end at 2 (not 3). The integers are −3, −2, −1, 0, 1, 2. That's 6 integers.