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.

45 min Algebra Linear relationships Lesson 10 of 13
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Printable worksheet

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Use the printable version for calculating gradients, attaching units and interpreting positive, negative and zero rates.

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Think First

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

Type the rate and explain how you found it.

Write the rate and explanation in your book.

Write your rate in your book
Saved

Know

  • 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

  • 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

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

$m = \frac{\text{change in output}}{\text{change in input}}$
Often called rise over run on a graph.
$m = \frac{y_2-y_1}{x_2-x_1}$
Use two ordered pairs $(x_1,y_1)$ and $(x_2,y_2)$.

Core Content

1. Gradient Is a Rate of Change

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

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

Key idea: Always describe gradient with context units, such as dollars per week or kilometres per hour.
Worked Example 1

Calculate 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.

Change in time: $6 - 0 = 6$ weeks.

$m = \frac{90}{6} = 15$.

Answer: The balance increases by $15 per week.

Worked Example 2

Calculate 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.

Change in time: $2 - 0.5 = 1.5$ h.

$m = \frac{120}{1.5} = 80$.

Answer: The average speed is 80 km/h.

Worked Example 3

Interpret negative and zero gradients

SituationGradientMeaning
Water drains from a tank-4 L/minVolume decreases by 4 litres each minute
Temperature stays constant0 degrees per hourTemperature is not changing
Savings grow+25 dollars/weekSavings increase by $25 each week
Interpretation habit: A negative gradient is not automatically wrong. It means the output is decreasing as the input increases.

2. Use Output Change Over Input Change

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

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

Gradient Practice

  1. A tank fills from 20 L to 95 L in 5 minutes. Find the gradient and interpret it.
  2. A distance changes from 30 km at 0.5 h to 150 km at 2.5 h. Find the rate in km/h.
  3. A bank balance changes from $500 to $380 over 4 weeks. Find and interpret the gradient.
  4. Explain what a zero gradient would mean for a temperature graph.
Complete the gradient practice in your book.

Revisit the Savings Rate

The savings balance increased by $90 over 6 weeks, so the gradient is $15 per week. This describes the rate of change.

Explain the units in your book.

Assessment

MC

Multiple Choice

Random questions from the lesson bank - feedback appears immediately.

SA

Short Answer

Calculate gradient and interpret it with units.

1. A tank volume increases from 15 L to 75 L over 4 minutes. Find the gradient and interpret it. 3 MARKS

Answer in your book.

2. A car travels from 20 km at 0.25 h to 140 km at 1.75 h. Find the average speed. 3 MARKS

Answer in your book.

3. Explain what a gradient of -6 L/min means for a water tank. 2 MARKS

Answer in your book.

Rate Reader

For each situation, identify the output change, input change and units before calculating the gradient.

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