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.