203r. Recognise Linear Change

Learning Intentions

• Identify situations involving constant rates of change.
• Represent linear change Use tables, rules and graphs.
• Interpret gradient and intercept in context.

Pre-requisite Summary

• Know that a rate of change describes how one quantity changes compared with another quantity.
• Know that a constant rate of change means the same amount is added or subtracted each time.
• Know that a table can show paired values, such as time and distance.
• Know that a linear rule can be written in the form , where is the gradient and is the vertical intercept.
• Know that the gradient describes the rate of change.
• Know that the vertical intercept describes the starting value when .

Worked Examples

Worked Example 1

Decide whether each situation involves a constant rate of change.

a) A taxi fare starts at $ and increases by $ for every kilometre travelled.

b) A plant grows cm on Monday, cm on Tuesday and cm on Wednesday.

c) A water tank loses litres every minute.

Worked Example 2

For each table, decide whether the relationship shows a constant rate of change.

a)

b)

Worked Example 3

Complete the table for the rule .

Worked Example 4

Write a linear rule for the table.

Worked Example 5

Use the rule to create a table and describe how the graph would look.

Use .

Worked Example 6

A gym membership costs $ to join and $ per week.

a) Write a rule for the total cost after weeks.

b) State the gradient.

c) State the vertical intercept.

Worked Example 7

A candle is initially cm tall and burns down by cm each hour.

a) Write a rule for the height after hours.

b) Interpret the gradient in context.

c) Interpret the vertical intercept in context.

Worked Example 8

A linear graph has equation .

a) State the gradient.

b) State the vertical intercept.

c) Describe what the gradient and intercept could mean in a real-world context.

Problems

Problem 1

Decide whether each situation involves a constant rate of change.

a) A car rental costs $ plus $ for every hour.

b) A student saves $ in week , $ in week and $ in week .

c) A swimming pool fills by litres every minute.

Problem 2

For each table, decide whether the relationship shows a constant rate of change.

a)

b)

Problem 3

Complete the table for the rule .

Problem 4

Write a linear rule for the table.

Problem 5

Use the rule to create a table and describe how the graph would look.

Use .

Problem 6

A streaming service costs $ to join and $ per month.

a) Write a rule for the total cost after months.

b) State the gradient.

c) State the vertical intercept.

Problem 7

A tank initially contains litres of water and drains by litres each minute.

a) Write a rule for the amount of water after minutes.

b) Interpret the gradient in context.

c) Interpret the vertical intercept in context.

Problem 8

A linear graph has equation .

a) State the gradient.

b) State the vertical intercept.

c) Describe what the gradient and intercept could mean in a real-world context.

Exercises

Understanding and Fluency

Exercise 1

Decide whether each situation involves a constant rate of change.

a) A phone plan costs $ plus $ for each gigabyte of data.

b) A runner travels km in the first hour, km in the second hour and km in the third hour.

c) A tree grows cm, then cm, then cm over three months.

d) A bank account earns different amounts of interest each month.

Exercise 2

For each table, decide whether the relationship shows a constant rate of change.

a)

b)

c)

Exercise 3

Complete each table.

a)

b)

Exercise 4

Write a linear rule for each table.

a)

b)

Exercise 5

For each rule, state the gradient and vertical intercept.

a)

b)

c)

d)

Exercise 6

For each rule, describe whether the graph rises or falls from left to right.

a)

b)

c)

d)

Exercise 7

Use each rule to create a table for .

a)

b)

c)

Exercise 8

Match each context to a rule.

a) A taxi costs $ plus $ per kilometre.

b) A tank starts with litres and loses litres per minute.

c) A worker earns $ per hour with no starting fee.

Rules:

Exercise 9

For each context, state the gradient and interpret it.

a) A car travels km each hour.

b) A candle loses cm of height each hour.

c) A savings account increases by $ each week.

Exercise 10

For each context, state the vertical intercept and interpret it.

a) A taxi fare is , where is kilometres travelled.

b) A tank volume is , where is time in minutes.

c) A plant height is , where is weeks.

Reasoning

Exercise 11

Explain why a situation with a constant rate of change can be represented by a straight-line graph.

Exercise 12

A student says the table below is linear because all the -values increase.

Explain the mistake.

Exercise 13

A student writes the rule for a bike hire that costs $ to start and $ per hour.

Explain the mistake and write the correct rule.

Exercise 14

Explain why the gradient in is negative and what this means for the graph.

Exercise 15

A table starts at when and increases by each time increases by .

Explain why the rule is .

Exercise 16

Decide whether each statement is true or false. Justify your answer.

a) In , the gradient is .

b) In , the graph rises from left to right.

c) In , the vertical intercept is .

Problem-solving

Exercise 17

A cinema charges $ for a membership card and $ per movie ticket.

a) Write a rule for the total cost after buying tickets.

b) State and interpret the gradient.

c) State and interpret the vertical intercept.

Exercise 18

A tank contains litres of water and drains at a constant rate of litres per minute.

a) Write a rule for the amount of water after minutes.

b) Create a table for .

c) Describe the graph.

Exercise 19

A student is comparing two phone plans.

Plan A:

Plan B:

a) Interpret the gradient and intercept for Plan A.

b) Interpret the gradient and intercept for Plan B.

c) Which plan is cheaper when ?

Exercise 20

Create your own real-world situation involving linear change.

Your response must include:

• a context with a constant rate of change
• a table of at least four values
• a linear rule
• a sentence interpreting the gradient
• a sentence interpreting the vertical intercept

Potential Misunderstandings

• Students may think any increasing pattern is linear, even when the increase is not constant.
• Students may identify the largest number in a table as the rate of change instead of comparing equal changes in and .
• Students may ignore decreasing patterns and assume linear change must always increase.
• Students may write the rule with the gradient and intercept reversed.
• Students may forget that the vertical intercept is the value of when .
• Students may create a graph from a table but plot the - and -values in the wrong order.
• Students may interpret the gradient as a starting value rather than a rate.
• Students may interpret the intercept as a rate rather than an initial amount.
• Students may not connect the sign of the gradient with whether the graph rises or falls from left to right.

Next: 204. Modelling Simple Financial Growth