Lesson 5 ~30 min Unit 2 · Linear Relationships +85 XP

Gradients (Slope) of a Line

Discover how gradient measures both the steepness and direction of a line. Master positive, negative, zero and undefined gradients.

Today's hook: Which line is steeper — one that goes up 4 for every 2 across, or one that goes up 3 for every 6 across? The answer tells you everything about gradient.

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Think First

warm-up

Which line is steeper? Line A goes up 4 units while going across 2 units. Line B goes up 3 units while going across 6 units.

Your challenge: Which line looks steeper? What does "steepness" mean in maths?

Jot your prediction in your workbook before reading on.

Reveal answer

Line A is steeper! It rises 2 units for every 1 unit across (gradient = 2), while Line B rises only 1 unit for every 2 units across (gradient = 0.5). Steepness is the ratio of rise to run — this is the gradient.

What is Gradient?

+5 XP

The gradient of a line describes how steep it is and which direction it slants. We calculate it by comparing the vertical change (rise) to the horizontal change (run) between any two points on the line.

The gradient formula is always rise over run. The rise is how far up (positive) or down (negative) the line goes. The run is always measured left to right — it is always positive.

$$\text{Gradient} = \frac{\text{rise}}{\text{run}}$$

Always left to right

Read the line from left to right. Run is always positive.

Up = positive rise

Rise is positive going up, negative going down.

Also written as

Gradient = change in $y$ ÷ change in $x$

Positive Gradient

+5 XP

A line has a positive gradient when it goes uphill from left to right. As $x$-values increase, $y$-values also increase.

Positive gradient — line goes uphill left to right

Watch Me Solve It · 3 examples

Watch Me Solve It · Positive gradient

+15 XP per step

PROBLEM

Find the gradient of the line passing through $(2, 1)$ and $(6, 4)$.

1

Find the rise

Rise $= 4 - 1 = 3$

Moving from $(2,1)$ to $(6,4)$: the $y$-value goes from 1 to 4. Positive, so the line goes up.
2

Find the run

Run $= 6 - 2 = 4$

The $x$-value goes from 2 to 6. Always measured left to right, so always positive.
3

Calculate gradient

Gradient $= \dfrac{3}{4} = 0.75$

Positive gradient confirms the line goes uphill from left to right.

AnswerGradient $= \dfrac{3}{4} = 0.75$ (positive — line goes uphill)

Negative Gradient

+5 XP

A negative gradient goes downhill from left to right. As $x$ increases, $y$ decreases. The rise is negative because the line falls.

Negative gradient — line goes downhill left to right

Memory trick

Trace with your finger left to right. Down = negative. Up = positive.

Watch Me Solve It · Negative gradient

+15 XP per step

PROBLEM

Find the gradient through $(1, 5)$ and $(4, 2)$.

1

Find the rise

Rise $= 2 - 5 = -3$

Negative rise: the line goes down from left to right.
2

Find the run

Run $= 4 - 1 = 3$

Always positive — measured left to right.
3

Calculate gradient

Gradient $= \dfrac{-3}{3} = -1$

For every 1 unit across, the line falls 1 unit.

AnswerGradient $= -1$ (negative — line goes downhill)

Zero and Undefined Gradient

+5 XP

Zero Gradient

A horizontal line has gradient zero. Rise = 0, so $\dfrac{0}{\text{run}} = 0$. All horizontal lines have equation $y = c$.

Undefined Gradient

A vertical line has an undefined gradient. Run = 0, so $\dfrac{\text{rise}}{0}$ is undefined (division by zero). All vertical lines have equation $x = c$.

Wrong: Thinking zero gradient and undefined gradient are the same thing.

Right: Zero = flat horizontal line. Undefined = straight up vertical line. They are opposites!

Comparing Steepness

+5 XP

When comparing steepness, we look at the absolute value of the gradient. The sign only tells direction — ignore it for steepness comparisons.

Line A has gradient 2. Line B has gradient −3. Which is steeper? Take absolute values: $|2| = 2$ and $|-3| = 3$. Since $3 > 2$, Line B is steeper, even though it goes downhill.

$|-3| = 3 > |2| = 2$ ⇒ Line B is steeper

Watch Me Solve It · Comparing steepness

+15 XP per step

PROBLEM

Which line is steeper: gradient $-4$ or gradient $3$?

1

Sign = direction only

Use absolute values for steepness

The sign tells us uphill or downhill — not how steep.
2

Find absolute values

$|-4| = 4$ and $|3| = 3$
3

Compare

$4 > 3$, so gradient $-4$ is steeper

It falls 4 units per 1 unit across; gradient 3 only rises 3 units per 1 unit across.

AnswerGradient $-4$ is steeper because $|-4| = 4 > |3| = 3$

Four Gradient Types

summary

PositiveGoes up left-to-right. Rise > 0. e.g. gradient = 2

NegativeGoes down left-to-right. Rise < 0. e.g. gradient = −1

ZeroPerfectly flat horizontal line. Rise = 0. $y = c$

UndefinedStraight vertical line. Run = 0. $x = c$

Real-World Gradients

+5 XP

Gradients appear everywhere. Road signs express gradients as percentages — the rise per 100 units of run.

Roads & ramps

Highways: below 6%. Wheelchair ramps: ~7%. Steep ski runs: 40%+.

Did you know?

Baldwin Street, NZ: world's steepest residential street at ~35% (gradient ≈ 0.35).

Common Pitfalls

heads-up

Confusing negative gradient direction

Thinking negative gradient means the line goes down right-to-left.

Fix: Always read left to right. Negative = goes DOWN left-to-right. Trace it with your finger!

Thinking zero gradient = "no line"

Gradient = 0 means no vertical change, but the line still exists.

Fix: Zero gradient = horizontal line. Equation: $y = c$. The line is flat, not absent.

Confusing undefined with zero

Students sometimes say vertical lines have gradient 0 or "infinity".

Fix: Vertical = undefined (run = 0, division by zero impossible). Zero = flat horizontal. They are opposites!

Comparing steepness without absolute values

Saying gradient $-5$ is "less steep" than gradient 3 because $-5 < 3$.

Fix: Sign = direction only. Compare steepness using absolute values: $|-5| = 5 > |3| = 3$, so $-5$ is steeper.

Copy Into Your Books

Gradient Formula

Gradient $= \dfrac{\text{rise}}{\text{run}}$
Always read line left to right
Rise = vertical change (up = +, down = −)
Run = horizontal change (always +)

Four Types

Positive — goes UP left-to-right
Negative — goes DOWN left-to-right
Zero — flat horizontal ($y = c$)
Undefined — vertical ($x = c$, run = 0)

Comparing Steepness

Use absolute values: ignore the sign
Larger $|\text{gradient}|$ = steeper line
Example: $|-3| = 3 > |2| = 2$ ⇒ $-3$ is steeper

Real-World

Road gradients expressed as % (rise per 100m run)
5% = 0.05; 10% = 0.1; 20% = 0.2
Australian highways: below 6% gradient

Brain Trainer · 10 problems

Brain Trainer · Gradient Drills

10 problems

Speed drill — answer each question, then reveal the answer.

1 A line goes up 5 units and across 5 units. What is its gradient?

Gradient = $\frac{5}{5} = 1$1
2 What type of gradient does a line have if it goes DOWN from left to right?

Negative gradient
3 Which is steeper: gradient $4$ or gradient $-5$?

Gradient $-5$: $|-5| = 5 > |4| = 4$Gradient −5 is steeper
4 What is the gradient of a horizontal line through $(2, 7)$ and $(8, 7)$?

Rise = 0, so gradient = $\frac{0}{\text{run}} = 0$0
5 What is the gradient of a vertical line? Explain why.

Undefined — run = 0 and division by zero has no value.Undefined
6 A line has gradient $\frac{2}{3}$. What does this mean?

Goes up 2 units for every 3 units across. Positive, shallow gradient.Up 2 for every 3 across
7 Order by steepness (least to steepest): gradient $-1$, $3$, $0$, $-2$.

Absolute values: 0, 1, 2, 3 → order: $0$, then $-1$, then $-2$, then $3$0, −1, −2, 3
8 A line falls 8 units while running 4 units across. What is its gradient?

Gradient = $\frac{-8}{4} = -2$−2
9 True or False: A line with gradient $-0.5$ goes uphill from left to right.

False — negative gradient means downhill left-to-right.False
10 A road has a 10% gradient. If you travel 200 m horizontally, how far do you rise?

$10\% \times 200 = 0.10 \times 200 = 20$ metres20 m

Quick Check · 5 questions

What type of gradient does this line have?

+10 XP

Which line has a negative gradient?

+10 XP

What is the gradient of a horizontal line?

+10 XP

Which line is steeper: gradient $2$ or gradient $-3$?

+10 XP

A line goes up 3 units for every 2 units across. What is its gradient?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

SAQ 1. Describe the gradient of each line (positive, negative, zero, or undefined) and identify the steepest.

Answer in your workbook.

Understand Easy 3 MARKS

SAQ 2. Explain why a vertical line has an undefined gradient. (Hint: think about run)

Answer in your workbook.

Apply Medium 3 MARKS

SAQ 3. Line A: gradient 2, Line B: gradient −1, Line C: gradient 0. Sketch and label each on the same axes.

Sketch in your workbook.

Comprehensive Answers

Quick Check

1. B — Positive gradient. The line goes uphill from left to right.

2. B — A line that goes downhill from left to right has negative gradient.

3. C — Gradient = 0. Horizontal line: rise = 0, so 0/run = 0.

4. B — Gradient $-3$ is steeper: $|-3| = 3 > |2| = 2$.

5. A — Gradient = rise/run = 3/2 = 1.5.

Model Answers

SAQ 1: Line P (green): positive. Line Q (red): negative. Line R (gold): zero. Line S (purple): undefined. Lines P and Q appear equally steep but in opposite directions.

SAQ 2: A vertical line has run = 0 (no horizontal change). The gradient formula gives rise/0, and division by zero is undefined. "Undefined" means no numerical value exists — it is not correct to say "infinite".

SAQ 3: Line A (gradient 2): steep uphill, rises 2 per 1 across. Line B (gradient −1): moderate downhill, falls 1 per 1 across. Line C (gradient 0): flat horizontal. Line A ($|2|=2$) is steeper than Line B ($|-1|=1$). Line C is flattest.

Stretch Challenge · +25 XP, +10 coins

Gradient Mastery Challenges

Challenge 1: Find the gradient for each description:

(a) Rises 6 units, runs 2 units. (b) Falls 4 units, runs 8 units. (c) Rises 5 units, runs 5 units.

Challenge 2: Order from least to steepest: gradients $0.5$, $-2$, $1$, $-4$, $0$.

Challenge 3: A road sign shows 12% gradient. (a) Write as a fraction. (b) How far do you rise travelling 500 m horizontally? (c) Is this gentle, moderate, or steep?

Reveal solutions

Challenge 1: (a) $\frac{6}{2} = 3$ (b) $\frac{-4}{8} = -0.5$ (c) $\frac{5}{5} = 1$

Challenge 2: By absolute value: $0 < 0.5 < 1 < 2 < 4$. Order: $0$, $0.5$, $1$, $-2$, $-4$.

Challenge 3: (a) $\frac{12}{100} = \frac{3}{25}$ (b) $0.12 \times 500 = 60$ metres (c) Steep (most roads stay below 6%)

Quick Review

Gradient formula

$\text{Gradient} = \dfrac{\text{rise}}{\text{run}}$

Positive gradient

Goes uphill left-to-right. Rise is positive.

Negative gradient

Goes downhill left-to-right. Rise is negative.

Zero gradient

Flat horizontal line. Rise = 0. Equation $y = c$.

Undefined gradient

Vertical line. Run = 0. Division by zero.

Comparing steepness

Use absolute value. Larger $|m|$ = steeper line.

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