Mathematics Standard • Year 11 • Module 1 • Lesson 10

Gradient as Rate of Change

Build fluency calculating gradient from two points using m = (y₂ − y₁) / (x₂ − x₁), with correct units and sign.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 Write the gradient formula using output and input changes.

m = ____________ ÷ ____________

Q1.2 What does each sign of gradient mean? Match each to "increasing", "decreasing" or "constant".

positive = ____________ negative = ____________ zero = ____________

Q1.3 A car covers 120 km in 2 hours. State the rate (with units). m = ____________ km/h

Stuck? Revisit lesson § Gradient Is a Rate of Change — m = output change / input change.

2. Worked example — savings rate

Follow each line of working. The units come from the context: dollars per week.

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

Step 1 — Identify the two ordered pairs.

(x₁, y₁) = (0, 120) (x₂, y₂) = (6, 210)

Step 2 — Output change (rise).

y₂ − y₁ = 210 − 120 = 90

Reason: how much the balance grew.

Step 3 — Input change (run).

x₂ − x₁ = 6 − 0 = 6

Reason: how many weeks passed.

Step 4 — Divide and add units.

m = 90 / 6 = 15

Reason: output units (dollars) over input units (weeks) gives dollars per week.

Conclusion. The balance increases by $15 per week.

3. Faded example — fill in the missing steps

A car has travelled 40 km after 0.5 h and 160 km after 2 h. Find the average speed. 4 marks

Step 1 — Pairs:

(x₁, y₁) = (____, ____) (x₂, y₂) = (____, ____)

Step 2 — Output change (rise):

y₂ − y₁ = ____ − ____ = ____________ km

Step 3 — Input change (run):

x₂ − x₁ = ____ − ____ = ____________ h

Step 4 — Divide and add units:

m = ____ / ____ = ____________ km/h

Conclusion sentence. The average speed is ____________ km/h.

Stuck? Revisit lesson § Worked Example 2 — Calculate speed from two points.

4. Graduated practice — gradient from two points

Show working below each part. Always include units in the final gradient.

Foundation — single-step calculations (4 questions)

Q	Problem	Answer (with units)
4.1 1	(0, 0) and (2, 80). Find m. Output is distance (km), input is time (h).
4.2 1	(0, 100) and (5, 150). Find m. Output is litres, input is minutes.
4.3 1	(0, 30) and (4, 70). Find m. Output is $, input is weeks.
4.4 1	(0, 50) and (3, 50). Find m. State what this means in plain English.

Standard — typical HSC difficulty (6 questions)

Show output change, input change, then m with units. Interpret in a sentence where asked.

4.5 A tank fills from 20 L to 95 L in 5 minutes. Find the gradient. 2 marks

4.6 A trip records (0.5 h, 30 km) and (2.5 h, 150 km). Find the speed in km/h. 2 marks

4.7 A bank balance changes from $500 to $380 over 4 weeks. Find m and interpret the sign. 2 marks

4.8 Temperature is 22 °C at 8 am and 22 °C at 11 am. Find m and explain what it tells you. 2 marks

4.9 Volume of water in a tank: 40 L at minute 0 and 0 L at minute 8. Find m and interpret. 2 marks

4.10 A car has travelled 24 km at 0.25 h and 144 km at 1.75 h. Find the average speed. 2 marks

Extension — common error and a missing-point trick (2 questions)

4.11 A student calculates speed as 0.5 h / 30 km = 0.0167 (and writes "h/km"). State what mistake was made and write the correct calculation with units. 3 marks

4.12 A savings table is linear with m = $25 per week. The balance at week 0 is $80. Use the gradient to fill in the balances at weeks 1, 2, 3, 4 without rewriting the formula each time. 3 marks

Stuck on 4.11? Revisit lesson § Use Output Change Over Input Change — output (km) on top, input (h) on bottom, not the other way around.

5. Self-check the easy 3

Tick the first three once you've checked your method works.

For 4.1 I wrote m = (80 − 0) / (2 − 0) = 40 km/h.

For 4.4 I got m = 0, meaning the balance stays constant.

For 4.7 my m is negative because the balance decreased.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Gradient formula

m = output change ÷ input change (or rise / run, or (y₂ − y₁) / (x₂ − x₁)).

Q1.2 — Sign meanings

positive = increasing, negative = decreasing, zero = constant.

Q1.3 — Speed

m = 120 / 2 = 60 km/h.

Q3 — Faded speed example

(x₁, y₁) = (0.5, 40); (x₂, y₂) = (2, 160).
Rise = 160 − 40 = 120 km.
Run = 2 − 0.5 = 1.5 h.
m = 120 / 1.5 = 80 km/h.
Conclusion: the average speed is 80 km/h.

Q4.1 — (0, 0) and (2, 80)

m = (80 − 0) / (2 − 0) = 80 / 2 = 40 km/h.

Q4.2 — (0, 100) and (5, 150)

m = (150 − 100) / (5 − 0) = 50 / 5 = 10 L/min.

Q4.3 — (0, 30) and (4, 70)

m = (70 − 30) / (4 − 0) = 40 / 4 = $10 per week.

Q4.4 — (0, 50) and (3, 50)

m = (50 − 50) / (3 − 0) = 0 / 3 = 0. The output stays the same — no change over time.

Q4.5 — Tank 20 → 95 L in 5 min

m = (95 − 20) / 5 = 75 / 5 = 15 L/min.

Q4.6 — (0.5, 30) and (2.5, 150)

m = (150 − 30) / (2.5 − 0.5) = 120 / 2 = 60 km/h.

Q4.7 — Balance 500 → 380 over 4 weeks

m = (380 − 500) / 4 = −120 / 4 = −$30/week. Negative sign means the balance is decreasing by $30 each week.

Q4.8 — Constant temperature

m = (22 − 22) / 3 = 0 °C/h. The temperature didn't change — it stayed at 22 °C for the 3-hour period.

Q4.9 — Tank emptying

m = (0 − 40) / 8 = −40 / 8 = −5 L/min. Tank loses 5 L every minute (draining).

Q4.10 — (0.25, 24) and (1.75, 144)

m = (144 − 24) / (1.75 − 0.25) = 120 / 1.5 = 80 km/h.

Q4.11 — Common-error fix

The student wrote input ÷ output (h ÷ km), giving 0.5 / 30 = 0.0167. Gradient must be output ÷ input. Correct: m = 30 / 0.5 = 60 km/h.

Q4.12 — Use the gradient to fill the table

Each week adds $25 (because m = $25/week). Starting at $80: week 1 = $105, week 2 = $130, week 3 = $155, week 4 = $180.