GM Lesson 002 Comparing Quantities Using Rates

Learning Intentions

• Identify rates in practical financial contexts.
• Calculate simple rates from given information.
• Interpret rates using appropriate units and context.

Prerequisites

Students should already be able to:

• Understand that a rate compares two quantities with different units.
• Recognise common financial quantities such as cost, income, time, distance, weight and quantity.
• Divide whole numbers and decimals using a calculator.
• Interpret units such as dollars, hours, kilograms, litres and kilometres.

Key Idea Summary

A rate compares two quantities with different units.

Common financial rates include:

• dollars per hour
• dollars per kilogram
• dollars per litre
• dollars per kilometre
• kilometres per litre
• items per dollar

Rates often use the word “per”, meaning “for each one”.

To calculate a simple rate:

The units are just as important as the number. For example, $18 per hour and $18 per kilogram have different meanings.

Direct Instruction and Worked Examples

Time Allocation

Time Allocation

Time Allocation

• Introduction, warmup and vocabulary: 5 minutes
• Direct instruction: 15 minutes
• Understanding checks: 5 minutes
• Exercises: 20 minutes
• Homework: 20 to 30 minutes outside the lesson it was taught in.

Link to original

Review and Introduction

Ask students to describe what each statement means:

• Petrol costs $2.10 per litre.
• A worker earns $24 per hour.
• Apples cost $5 per kilogram.
• A car travels km on L of fuel.

Emphasise that each statement compares two different units.

Direct Instruction

A rate tells us how much of one quantity corresponds to one unit of another quantity.

For example:

• $24 per hour means $24 for hour.
• $5 per kilogram means $5 for kilogram.
• km per litre means km for litre.

When calculating a rate, identify:

1. The total quantity.
2. The number of units.
3. The meaning of the final unit.

Worked Example 1: Hourly Rate

A student earns $96 for working hours at a weekend job.

Teacher prompts:

• What two quantities are being compared?
• Which quantity should be divided by which?
• What does “per hour” mean?
• What unit should the final rate have?

Set-up:

Students should interpret the answer as dollars per hour.

Worked Example 2: Unit Price

A kg bag of rice costs $9.90.

Teacher prompts:

• What is the total cost?
• How many kilograms are bought?
• What does “cost per kilogram” mean?
• Why is this useful when comparing products?

Set-up:

Students should interpret the answer as dollars per kilogram.

Worked Example 3: Fuel Rate

A car uses L of petrol to travel km.

Teacher prompts:

• What two quantities are being compared?
• What would kilometres per litre tell us?
• What would litres per km tell us?
• Why might different rates be useful in different situations?

Set-up for kilometres per litre:

Set-up for litres per km:

Students should interpret each rate in context.

Understanding Checks

Check 1

A bag of oranges costs $8 for kg.

What rate would help compare this bag with other bags of oranges?

Check 2

A person earns $150 for hours of work.

What are the two units being compared?

Check 3

A taxi fare is $36 for a km trip.

Write the calculation needed to find the cost per kilometre.

Check 4

A car travels km using L of fuel.

Write the calculation needed to find the number of kilometres travelled per litre.

Check 5

A rate is calculated as:

In a taxi context, explain why the answer must include units.

Exercises

Simple Familiar Exercises

Exercise 1

A worker earns $84 for hours of work.

Calculate the hourly rate.

Exercise 2

A kg bag of potatoes costs $6.40.

Calculate the cost per kilogram.

Exercise 3

A L bottle of cleaning liquid costs $17.50.

Calculate the cost per litre.

Exercise 4

A delivery driver is paid $120 for travelling km.

Calculate the payment per kilometre.

Exercise 5

A box of cans costs $18.

Calculate the cost per can.

Complex Familiar Exercises

Exercise 6

A shopper compares two bags of rice.

• Bag A: kg for $7.80
• Bag B: kg for $18.50

Calculate the cost per kilogram for each bag and identify the better value option.

Exercise 7

A casual worker is offered two shifts.

• Shift A: $108 for hours
• Shift B: $156 for hours

Calculate the hourly rate for each shift and compare the offers.

Exercise 8

A car travels km using L of petrol.

Calculate the fuel efficiency in kilometres per litre.

Exercise 9

A printer prints pages in minutes.

Calculate the printing rate in pages per minute.

Exercise 10

A family buys two different packs of bottled water.

• Pack A: bottles for $9.60
• Pack B: bottles for $21.00

Calculate the cost per bottle for each pack and decide which is better value.

Homework Problems

Problem 1

A worker earns $210 for hours of work.

Calculate the hourly rate.

Problem 2

A kg bag of flour costs $11.20.

Calculate the cost per kilogram.

Problem 3

A car travels km using L of petrol.

Calculate the fuel efficiency in kilometres per litre.

Problem 4

A gym offers two casual passes.

• Pass A: $18 for visit
• Pass B: $75 for visits

Calculate the cost per visit for each option and decide which is better value.

Problem 5

A family compares two internet plans.

• Plan A: $60 per month for GB
• Plan B: $80 per month for GB

Calculate the cost per GB for each plan and decide which has the better rate.

Problem 6

A courier earns $132 for delivering packages.

Calculate the earnings per package.

Problem 7

A supermarket sells two packets of cheese.

• Packet A: g for $8.50
• Packet B: g for $11.25

Calculate the cost per g for each packet and decide which is better value.

Problem 8

Write a short explanation of why rates are useful in consumer arithmetic.

Next: GM Lesson 003 Salary Pay Cycles