💼 Hours Worked and Wages
Hours 4↑Wages $12
4 hours → $12
Learning Objectives
You should be able to:
- explain variation in plain English;
- identify two quantities that are connected;
- distinguish a relationship from a coincidence;
- recognise direct and inverse behaviour informally;
- describe how one quantity changes when another changes.
Introduction
This lesson develops Introduction to Variation from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
Visual Exploration
Notice→Connect→Explain
Use the diagrams, tables or examples in this lesson to identify what changes, what stays fixed and which relationship connects the values.
🤔 Think About This...
Suppose one exercise book costs US$2. One book costs US$2, two books cost US$4 and five books cost US$10.
What happens to the total cost when the number of books doubles?
The cost doubles too. The two quantities are not changing randomly: they are connected by a clear rule.
What Is Variation?
Variation describes how one quantity changes when another quantity changes. The two quantities may increase together, decrease together, or move in opposite directions.
Key idea: Variation is about a relationship between
changing quantities.
Number of books changes
→
Total cost changes
Everyday Relationships
Shopping
Buying more identical items usually increases the total cost.
Travel
At the same speed, travelling for longer increases the distance.
Work
More workers may reduce the time needed to finish a job.
Recipes
Making twice as much food requires twice as much of each ingredient.
Two Important Patterns
Moving together
As one quantity increases, the other also increases in a predictable ratio. This leads to direct variation.
Moving oppositely
As one quantity increases, the other decreases in a predictable way. This leads to inverse variation.
At this stage, focus on the behaviour of the quantities before thinking
about formulas.
Worked Example 1: Recognising a Connection
A taxi charges US$1 per kilometre.
1
The changing quantities are distance and cost.
2
Each extra kilometre adds US$1.
3
Travelling twice as far costs twice as much.
4
The quantities move together in a direct pattern.
Worked Example 2: Moving in Opposite Directions
Six workers complete a task in 8 hours. More workers are added, while everyone works at the same rate.
1
The quantities are number of workers and time taken.
2
More workers should complete the same job in less time.
3
One quantity increases while the other decreases.
4
This is the behaviour associated with inverse variation.
🌟 Did You Know?
Scientists use variation to describe how measurable quantities are
connected. For example, the distance travelled at a constant speed
varies with time.
🔍 How Do I Recognise Variation?
Look for two changing quantities.
- What is changing?
- Does changing one quantity affect the other?
- Do they move together or in opposite directions?
- Is the relationship predictable?
⚠️ Common Mistakes
Mistake: Assuming that any two increasing quantities
are directly proportional.
Two quantities may both increase without increasing in the same ratio. Variation requires a consistent mathematical relationship.
Two quantities may both increase without increasing in the same ratio. Variation requires a consistent mathematical relationship.
✅ Quick Check
A recipe is doubled. What should happen to every ingredient?
More taps fill the same tank. What should happen to the time?
Name the two changing quantities when buying identical pens.
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Practice Questions
Foundation- Explain variation using your own words.
- Name the two changing quantities when a shopper buys several identical loaves of bread.
-
State whether the following quantities generally move together or
in opposite directions:
- litres of fuel purchased and total cost;
- number of workers and time to complete the same task;
- time travelled at constant speed and distance covered.
- Explain why “age and height” are not necessarily directly proportional, even though children generally grow taller as they age.
- Give one original example of quantities that move together and one example of quantities that move in opposite directions.
- A company’s sales and advertising spending both increased during one month. Is that enough information to prove variation? Explain.
✅ Check Your Work
Attempt every question before opening the answers. Learning happens when you think first.
Show practice answers
Foundation
- Variation describes how one changing quantity is connected to another changing quantity.
- Number of loaves bought and total cost.
-
- They move together.
- They move in opposite directions.
- They move together.
Developing
- Children do not grow by the same fixed ratio whenever their age changes. For example, doubling a child’s age does not generally double the child’s height.
- Answers vary. Example: quantity of rice and cost move together; number of identical machines and completion time for one fixed job may move oppositely.
Challenge
- No. The two values increasing together once does not prove a consistent mathematical relationship. More data would be needed to test whether one quantity changes predictably with the other.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Variation describes a relationship between changing quantities.
- Some quantities move together.
- Other quantities move in opposite directions.
- A true variation relationship must be consistent and predictable.