Patterns and Expressions
Grade 5 · Math · Study Guide
Discover number patterns and learn to write and evaluate simple algebraic expressions.
A numerical pattern is a sequence of numbers that follows a rule. Finding the rule lets you predict any term in the pattern. Patterns can involve adding, subtracting, multiplying, or dividing by the same number each time.
An algebraic expression uses numbers, operation symbols, and variables (letters that stand for unknown numbers) to describe a math relationship. For example, 'n + 3' means 'a number plus 3.'
To evaluate an expression, substitute a specific value for the variable and then calculate. For example, if n = 4, then n + 3 = 4 + 3 = 7.
| Sequence | Rule | Next Two Terms |
|---|---|---|
| 2, 4, 6, 8, ... | Add 2 | 10, 12 |
| 3, 9, 27, 81, ... | Multiply by 3 | 243, 729 |
| 100, 90, 80, 70, ... | Subtract 10 | 60, 50 |
| Expression | Value of n | Evaluated Result |
|---|---|---|
| n + 5 | n = 3 | 3 + 5 = 8 |
| 4 × n | n = 7 | 4 × 7 = 28 |
| n - 6 | n = 10 | 10 - 6 = 4 |
Key terms
- Pattern
- A sequence of numbers or shapes that follows a rule.
- Variable
- A letter (like n or x) used to represent an unknown or changing number.
- Expression
- A math phrase using numbers, variables, and operations — no equals sign.
- Evaluate
- To find the value of an expression by substituting a number for the variable.
Practice questions
What is the next number in the pattern: 5, 10, 20, 40, ___?
- 60
- 80
- 75
- 50
Show answer
Answer: 80
The rule is multiply by 2. 40 × 2 = 80.
If n = 6, what is the value of 3 × n + 2?
- 11
- 20
- 18
- 26
Show answer
Answer: 20
3 × 6 = 18, then 18 + 2 = 20.
Write an expression for 'a number n decreased by 7': ___
Show answer
Answer: n - 7
'Decreased by' means subtraction. The expression is n - 7.
Evaluate 5n - 3 when n = 4: ___
Show answer
Answer: 17
5 × 4 = 20, then 20 - 3 = 17.
What is the rule for the pattern: 1, 3, 9, 27, 81?
- Add 2
- Add 8
- Multiply by 3
- Multiply by 2
Show answer
Answer: Multiply by 3
Each term is multiplied by 3: 1×3=3, 3×3=9, 9×3=27, 27×3=81.
Fun facts
- The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13...) appears in sunflower seeds, pinecones, and even your own hand!
- Variables were first used by the mathematician Francois Viete in the 1590s to represent unknown quantities.
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