Unit 9 — Patterns, Variables, and Introduction to Algebra

Welcome back!

Mathematics is more than calculating numbers.

It is also about discovering patterns, describing relationships, and solving unknown quantities.

These ideas form the foundation of algebra, one of the most important branches of mathematics.

Today you will begin thinking like an algebraist.

Finding Patterns

Patterns appear throughout mathematics and nature.

Examples include:

• 2, 4, 6, 8, 10 ...
• 5, 10, 15, 20, 25 ...
• 1, 4, 9, 16, 25 ...

Ask yourself:

• How does the pattern change?
• What comes next?
• Can I describe the rule?

Recognizing patterns helps us predict future values.

Variables

A variable is a symbol that represents an unknown or changing value.

The most common variable is:

x

Example:

x + 7 = 15

The variable stands for the missing number.

Variables allow mathematicians to write general rules that apply to many situations.

Writing Expressions

An expression uses numbers, variables, and operations.

Examples:

• x + 5
• 3n
• y − 8
• 4(a + 2)

Expressions do not contain an equals sign.

They describe mathematical relationships.

Equations

An equation states that two expressions are equal.

Examples:

x + 4 = 12

3y = 18

An equation has one or more unknown values that can be solved.

Solving Simple Equations

Solve by finding the value of the variable.

Example:

x + 9 = 17

Ask:

"What number plus 9 equals 17?"

Answer:

x = 8

Always check your solution by substituting it back into the equation.

Using Variables in Real Life

Variables help describe many everyday situations.

Examples:

Distance traveled:

d = rt

where:

• d = distance
• r = speed
• t = time

Perimeter of a square:

P = 4s

where:

• P = perimeter
• s = side length

Variables make formulas easier to understand and use.

Graphing Simple Relationships

Tables help organize information.

Example:

| x | y |

|---|---|

| 1 | 3 |

| 2 | 6 |

| 3 | 9 |

| 4 | 12 |

Look for the relationship between x and y.

Then predict additional values.

Recognizing relationships prepares students for graphing in later grades.

Solving Word Problems

Example:

A theater charges $8 per ticket.

Write an expression for the total cost of x tickets.

Answer:

8x

If a family buys 5 tickets:

8 × 5 = $40

Variables allow us to describe many similar situations using one expression.

Activity

Complete these tasks.

• Extend five number patterns.
• Write five algebraic expressions.
• Solve ten simple equations.
• Create one real-world problem using a variable.
• Explain your reasoning for each solution.

Practice

Answer these questions.

What is a variable?

How is an expression different from an equation?

Why should you check your solution?

How do patterns help mathematicians?

Give one example of a real-world formula.

Why are variables useful?

Review

Today you learned:

• mathematics contains many patterns
• variables represent unknown values
• expressions describe mathematical relationships
• equations can be solved
• formulas use variables to represent general rules
• algebra helps solve many real-world problems

Algebra allows us to solve problems that would be difficult using numbers alone.

Looking Ahead

Next we will begin Grade 5 Science by exploring the scientific method, experimental design, observation, measurement, and the importance of evidence in scientific discovery.