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Module 1 · L10 of 13 ~45 min ⚡ +90 XP available

Gradient as Rate of Change

Calculate gradient from two points and interpret it as a practical rate such as dollars per week, kilometres per hour or litres per minute. Gradient is not just a graph slope, it is a real-world measurement with units.

Today's hook, A savings balance increases from $120 to $210 over 6 weeks. How much is the balance increasing per week? Can you find it without a graph?

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The key idea: gradient as a rate

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Worksheets

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Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

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Think First, your gut answer first

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A savings balance increases from $120 to $210 over 6 weeks. How much is the balance increasing per week?

Without a formula write the rate and explain how you found it. Make a prediction before the lesson walks through the steps.

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The key idea: gradient as a rate

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Gradient measures how much the output changes for each 1-unit change in input. It is always a rate, and that rate must be expressed with context units.

Gradient = rise ÷ run on a graph. In a practical context it becomes dollars per week, kilometres per hour, or litres per minute. The sign matters: positive means increasing, negative means decreasing, zero means constant.

A slope triangle on the line shows gradient = rise ÷ run.

$$m = \frac{\text{change in output}}{\text{change in input}} = \frac{y_2 - y_1}{x_2 - x_1}$$

Always include units

Gradient without units is incomplete. $m = 15$ alone is meaningless; $m = 15$ dollars per week tells the full story.

Sign tells the story

Positive gradient: output increases. Negative gradient: output decreases. Zero gradient: output stays constant.

Output change over input change

The numerator is always the output change, denominator is input change. Reversing gives a different (wrong) rate.

What you'll master

Know

Key facts

Gradient measures change in output divided by change in input.
Gradient has units from the context.
Positive, negative and zero gradients describe different trends.

Understand

Concepts

Gradient is a practical rate of change, not just a graph calculation.
The sign of the gradient tells whether the output increases, decreases or stays constant.
Units make the rate meaningful.

Can do

Skills

Calculate gradient from two points.
Interpret gradient in context.
Identify positive, negative and zero gradients.

Key terms

Gradient ($m$)The rate of change of the output relative to the input: $m = \frac{\Delta y}{\Delta x}$.

RiseThe vertical change (change in output) between two points on a graph.

RunThe horizontal change (change in input) between two points on a graph.

Rate of changeHow quickly one quantity changes relative to another, expressed with context units.

Positive gradientOutput increases as input increases, an upward-sloping line.

Negative gradientOutput decreases as input increases, a downward-sloping line.

Gradient is a rate of change

Gradient tells how much the output changes for each 1-unit change in the input.

Gradient is a rate of change

core concept

Gradient tells how much the output changes for each 1-unit change in the input. It is not just a slope on a graph, it is a meaningful quantity tied to the real-world context.

If savings increase by $\$90$ over 6 weeks, the rate is $\frac{90}{6} = 15$. The gradient is $\$15$ per week.

$$m = \frac{\text{change in output}}{\text{change in input}} = \frac{y_2 - y_1}{x_2 - x_1}$$

Key idea: Always describe gradient with context units, such as dollars per week or kilometres per hour.

Gradient = change in output ÷ change in input (m = Δy/Δx). Positive gradient: line slopes up left to right. Negative gradient: slopes down. Gradient has units (e.g. $\$$/km) and a real-world meaning, always state what it represents.

Pause, copy the gradient formula m = Δy/Δx = (y₂ − y₁)/(x₂ − x₁), the real-world meaning (gradient = rate of change in the y-unit per unit of x), and the sign interpretation (positive: increasing; negative: decreasing) into your book.

Quick check: A savings balance changes from $\$300$ at week 0 to $\$450$ at week 5. What is the gradient with correct units?

Use output change over input change

We just saw that gradient m = Δy/Δx is a rate of change with a real-world meaning, e.g.

Use output change over input change

core concept

We just saw that gradient m = Δy/Δx is a rate of change with a real-world meaning, e.g. dollars per kilometre, metres per second. That raises a question: gradient is a fraction, and fractions can be written two ways, does it matter which quantity goes on top? This card answers it → yes, order always matters: gradient = rise (vertical change) ÷ run (horizontal change); writing run ÷ rise gives a different and meaningless quantity that costs marks.

Gradient is change in output divided by change in input. Reversing this gives a different quantity and usually wrong units.

Situation	Gradient	Meaning
Water drains from a tank	−4 L/min	Volume decreases by 4 litres each minute
Temperature stays constant	0 °C/h	Temperature is not changing
Savings grow	+$25/week	Savings increase by $25 each week

Interpretation habit: A negative gradient is not automatically wrong. It means the output is decreasing as the input increases.

Common error: Do not calculate input change over output change. For speed, use kilometres divided by hours, not hours divided by kilometres.

Gradient = (output change) ÷ (input change), order matters. Reversing (input ÷ output) gives a different and meaningless quantity. From a table: pick any two rows, calculate Δy/Δx. From a graph: rise (vertical) over run (horizontal).

Pause, copy the rise-over-run method (from a graph: count vertical rise and horizontal run between two clear grid points; gradient = rise ÷ run) and the reversed-fraction error warning (run ÷ rise is wrong and gives a different answer) into your book.

True or false: A gradient of −6 L/min for a water tank means the tank is losing 6 litres every minute.

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · DOLLARS PER WEEK

A savings balance is $\$120$ at week 0 and $\$210$ at week 6. Find the gradient.

Change in savings: $210 - 120 = 90$ dollars.

Identify the change in output (savings, $y$). Use the later value minus the earlier value.

PROBLEM 2 · SPEED FROM TWO POINTS

A car has travelled 40 km after 0.5 h and 160 km after 2 h. Find the average rate of change.

Change in distance: $160 - 40 = 120$ km.

Output change: use the two distance values. The two points are $(0.5, 40)$ and $(2, 160)$.

PROBLEM 3 · NEGATIVE AND ZERO GRADIENTS

Interpret each gradient in context.

Situation	Gradient	Meaning
Water drains from a tank	−4 L/min	Volume decreases by 4 litres each minute
Temperature stays constant	0 degrees per hour	Temperature is not changing
Savings grow	+$25/week	Savings increase by $25 each week

Negative gradient (−4 L/min): output is decreasing.

A negative sign means the quantity is going down as time increases. This is not an error, it correctly models the draining tank.

Fill the gap: A tank volume increases from 15 L to 75 L over 4 minutes. The change in output is L and the change in input is min, giving a gradient of L/min.

Common errors · the 3 traps that cost marks

Trap 01

Dividing input change by output change

Gradient is always $\Delta y \div \Delta x$ (output over input). For speed, use kilometres divided by hours. Writing $\Delta x \div \Delta y$ gives hours per kilometre, a different (and usually wrong) quantity.

Trap 02

Forgetting units in the answer

Writing "$m = 15$" without units is an incomplete answer. The question asks you to interpret gradient, so always write "$m = 15$ dollars per week" or "$m = 80$ km/h".

Trap 03

Treating negative gradient as an error

A negative gradient is not wrong, it means the output is decreasing. A draining tank, a falling balance, or a cooling temperature all have negative gradients. State this clearly in your interpretation.

Revisit your thinking

Quick-fire practice · 4 questions

A tank fills from 20 L to 95 L in 5 minutes. Find the gradient and interpret it.

A distance changes from 30 km at 0.5 h to 150 km at 2.5 h. Find the rate in km/h.

A bank balance changes from $500 to $380 over 4 weeks. Find and interpret the gradient.

Explain what a zero gradient would mean for a temperature graph.

Odd one out: Three of these are correct interpretations of gradient. Which one is wrong?

Revisit your thinking

Earlier you estimated the savings rate for a balance that went from $120 to $210 over 6 weeks. Let's confirm:

Change in savings: $210 - 120 = 90$ dollars. Change in time: $6 - 0 = 6$ weeks.

$$m = \frac{90}{6} = \$15 \text{ per week}$$

The savings balance increased by $15 per week. This is the gradient, and it describes the rate of change in context.

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Final check: True or false: to find gradient you always divide the output change by the input change, and the result carries context units.

Show what you have learned

Multiple choice, then short answer under exam conditions.

Multiple choice

+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence. That tells the system what to drill next. Each retry pulls a fresh mix from the bank.

Short answer

ApplyBand 33 marks

Q1. A tank volume increases from 15 L to 75 L over 4 minutes. Find the gradient and interpret it. (3 marks)

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ApplyBand 43 marks

Q2. A car travels from 20 km at 0.25 h to 140 km at 1.75 h. Find the average speed. (3 marks)

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AnalyseBand 42 marks

Q3. Explain what a gradient of −6 L/min means for a water tank. (2 marks)

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📖 Answers (click to reveal)

Q1 (3 marks): $\Delta y = 75 - 15 = 60$ L [1]. $\Delta x = 4$ min [1]. $m = 60 \div 4 = 15$ L/min. The tank fills at 15 litres per minute [1].

Q2 (3 marks): $\Delta y = 140 - 20 = 120$ km [1]. $\Delta x = 1.75 - 0.25 = 1.5$ h [1]. $m = 120 \div 1.5 = 80$ km/h [1].

Q3 (2 marks): The volume of water in the tank is decreasing [1] at a rate of 6 litres per minute [1].

Drill 1: $\Delta y = 75$, $\Delta x = 5$, $m = 15$ L/min (tank fills at 15 L/min) · Drill 2: $\Delta y = 120$, $\Delta x = 2$, $m = 60$ km/h · Drill 3: $\Delta y = -120$, $\Delta x = 4$, $m = -30$ $\$$/week (balance falling $\$30$/week) · Drill 4: Zero gradient means temperature is constant, not changing over time.

Consolidate and move on

Sit the module quiz, then close the lesson off.

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For each situation, identify the output change, input change and units before calculating the gradient. Beat the boss to bank a tier, gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.

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Module overview · Year 11 Maths Standard