Patterns, Variables, and Algebraic Thinking
2. Variables and Unknowns
Learning outcomes
- I can explain what a variable represents in mathematics.
- I can use letters to represent unknown quantities.
- I can distinguish between known and unknown values.
- I can interpret variables in mathematical expressions and equations.
- I can use variables to model simple situations.
Introduction
Mathematics often involves quantities whose values are not yet known or may change. Instead of writing a different calculation for every possible value, mathematicians use variables. A variable is a letter or symbol that represents a number. Variables allow us to describe patterns, write formulas, solve problems, and model real-world situations.
Variables are one of the foundations of algebra. They help us move from working with specific numbers to thinking about general mathematical relationships. Understanding variables is an important step toward solving equations and using mathematics to describe the world around us.
What Is a Variable?
A variable is a letter or symbol that represents a number.
The value of a variable may:
- Be unknown.
- Change.
- Represent any number.
Common variables include:
- x
- y
- n
- a
- b
For example:
x = 5
Here, the variable x represents the number 5.
Figure 1. Variables use letters to represent numbers in mathematical expressions and equations.
Why Do We Use Variables?
Variables make mathematics simpler and more flexible.
They allow us to:
- Represent unknown numbers.
- Describe patterns.
- Write formulas.
- Solve problems.
- Model real-life situations.
Instead of writing many similar calculations, one expression with a variable can represent them all.
For example:
The perimeter of a square can be written as:
4 × s
where s is the length of one side.
Unknown Quantities
An unknown is a value that has not yet been determined.
Example:
x + 7 = 12
The value of x is unknown.
By solving the equation:
x = 5
The unknown value has now been found.
Variables often begin as unknowns but become known once they are solved.
Known and Unknown Values
A mathematical statement may contain both known and unknown values.
Example:
3 + x = 10
Known values:
- 3
- 10
Unknown value:
- x
Recognising which quantities are known and which are unknown helps us solve problems correctly.
Figure 2. Equations often contain both known values and unknown variables.
Variables in Mathematical Expressions
An expression is a mathematical phrase containing numbers, variables, and operations.
Examples:
- x + 4
- 5y
- 2a − 3
- 8 ÷ n
Expressions do not contain an equals sign.
Variables allow one expression to represent many different values.
Example:
If:
x = 6
Then:
x + 4 = 10
If:
x = 12
Then:
x + 4 = 16
The same expression produces different answers depending on the value of the variable.
Variables in Equations
An equation states that two expressions are equal.
Equations always contain an equals sign (=).
Examples:
- x + 3 = 8
- 2y = 14
- a − 5 = 9
The goal is to find the value of the variable that makes the equation true.
Example:
x + 3 = 8
Subtract 3 from both sides.
x = 5
Figure 3. Expressions contain variables and operations, while equations also include an equals sign.
Using Variables to Model Real Situations
Variables can represent quantities in everyday situations.
Example 1: Age
If Mia is x years old:
Next year she will be:
x + 1
Example 2: Cost
If one notebook costs n dollars:
Three notebooks cost:
3n
Example 3: Distance
If a cyclist travels d kilometres:
After another 5 kilometres:
d + 5
Variables make it easy to describe many possible situations with one mathematical rule.
Choosing Variable Names
Mathematicians often choose letters that remind them of the quantity being represented.
Examples:
| Variable | Represents |
|---|---|
| t | Time |
| d | Distance |
| m | Mass |
| l | Length |
| p | Price |
| n | Number |
However, almost any letter can be used as a variable.
Figure 4. Variables can represent quantities such as time, distance, age, or cost in real-world situations.
Variables Help Describe Patterns
Variables allow us to describe mathematical patterns.
Example:
2, 4, 6, 8, 10, ...
The rule is:
Value = 2 × Term Number
Using a variable:
Value = 2n
where:
n represents the term number.
Variables help transform patterns into mathematical rules.
Why Variables Are Important
Variables are used throughout mathematics and science.
They help us:
- Solve equations.
- Write formulas.
- Describe relationships.
- Make predictions.
- Build mathematical models.
Nearly every topic in algebra depends on understanding variables.
Figure 5. Variables are used throughout mathematics to describe patterns, formulas, and relationships.
Worked Example
Question
A movie ticket costs $12.
Let n represent the number of tickets purchased.
Write an expression for the total cost.
Solution
Each ticket costs $12.
For n tickets:
Total cost = 12n
If:
n = 4
Then:
12 × 4 = $48
Real-World Connection
Variables are used in almost every profession that involves mathematics. Scientists use variables to represent quantities such as temperature, mass, and time during experiments. Engineers use variables when designing bridges and machines, economists use them to model prices and profits, and computer programmers use variables to store and manipulate information in software applications.
Did You Know?
The use of letters to represent unknown numbers became common in the 1600s through the work of the French mathematician René Descartes. He popularised using letters near the end of the alphabet, such as x, y, and z, for unknown quantities—a convention that is still widely used today.
Key Terms
Equation – A mathematical statement showing that two expressions are equal.
Expression – A mathematical phrase made from numbers, variables, and operations but without an equals sign.
Known value – A quantity whose value is given or already determined.
Model – A mathematical representation of a real-world situation.
Unknown – A value that has not yet been found.
Variable – A letter or symbol that represents a number that may be unknown or may change.
Key Takeaways
- A variable is a letter or symbol used to represent a number.
- Variables can represent unknown values or quantities that change.
- Expressions contain variables and operations, while equations also contain an equals sign.
- Variables help describe patterns, write formulas, and solve mathematical problems.
- Using variables allows mathematicians to model real-world situations efficiently.
- Understanding variables is one of the most important foundations of algebra.