Patterns and Relationships

Welcome to the world of patterns! Learning to recognize, describe, and extend patterns is a key skill that helps you think like a mathematician and prepares you for algebra.

What Is a Pattern?

A pattern is a sequence that follows a rule. Patterns can be found in numbers, shapes, colors, or even sounds. When you can identify the rule, you can predict what comes next.

Types of Number Patterns

Repeating Patterns

A repeating pattern is a pattern that repeats the same sequence over and over.

Example: 2, 5, 8, 2, 5, 8, 2, 5, 8...

• The pattern 2, 5, 8 repeats
• The next numbers would be 2, 5, 8

Growing Patterns (Arithmetic Sequences)

A growing pattern increases or decreases by the same amount each time. The amount of change is called the "common difference."

Example: 3, 7, 11, 15, 19...

• Each number increases by 4
• The common difference is +4
• The next number would be 19 + 4 = 23

Example: 50, 45, 40, 35, 30...

• Each number decreases by 5
• The common difference is -5
• The next number would be 30 - 5 = 25

Multiplicative Patterns (Geometric Sequences)

A multiplicative pattern changes by multiplying or dividing by the same number each time.

Example: 2, 6, 18, 54...

• Each number is multiplied by 3
• The pattern rule is "multiply by 3"
• The next number would be 54 × 3 = 162

Example: 80, 40, 20, 10...

• Each number is divided by 2 (or multiplied by 1/2)
• The pattern rule is "divide by 2"
• The next number would be 10 ÷ 2 = 5

Finding the Pattern Rule

To find the pattern rule, ask yourself these questions:

Step 1: How are the numbers changing?

• Are they getting bigger or smaller?
• By how much are they changing?

Step 2: Is the change the same each time?

• If yes, it's likely an arithmetic pattern (adding or subtracting)
• If no, check if it's multiplicative (multiplying or dividing)

Step 3: Test your rule

• Apply your rule to each number
• If it works for all the numbers, you've found the pattern!

Example: Find the rule for 5, 9, 13, 17, 21...

• Each number increases by 4
• Rule: Add 4 to get the next term
• Check: 5 + 4 = 9, 9 + 4 = 13, 13 + 4 = 17 ✓

Pattern Rules with Variables

We can describe patterns using variables like n to represent the position in the pattern.

Example: Pattern is 4, 7, 10, 13, 16...

• Position 1: 4 = 3(1) + 1
• Position 2: 7 = 3(2) + 1
• Position 3: 10 = 3(3) + 1
• Position 4: 13 = 3(4) + 1
• Pattern rule: 3n + 1, where n is the position number

This means for any position n, the value equals 3 times that position plus 1.

To find the 10th term: 3(10) + 1 = 30 + 1 = 31

Input-Output Tables

Input-output tables (also called function tables) show relationships between two sets of numbers.

Example table showing the relationship "multiply by 4 and add 1":

Input (x) | Output (y) 1 | 5 2 | 9 3 | 13 4 | 17

The rule is: y = 4x + 1

• When x = 1, y = 4(1) + 1 = 5
• When x = 2, y = 4(2) + 1 = 9

Graphing Patterns

Patterns can be shown on a coordinate plane! Each pair of numbers from an input-output table becomes a point (x, y).

Example: For the rule y = 2x + 3

• When x = 0, y = 3 → Plot (0, 3)
• When x = 1, y = 5 → Plot (1, 5)
• When x = 2, y = 7 → Plot (2, 7)
• When x = 3, y = 9 → Plot (3, 9)

When you plot these points and connect them, they form a straight line! This is called a linear relationship.

Real-World Patterns

Patterns appear everywhere in real life:

Money:

• Saving $5 per week: 5, 10, 15, 20, 25...
• Rule: 5n (where n is the number of weeks)

Age:

• You are 3 years older than your sister
• If she's 7, you're 10; if she's 8, you're 11
• Rule: Your age = Sister's age + 3

Distance:

• A car travels 60 miles per hour
• After 1 hour: 60 miles, after 2 hours: 120 miles, after 3 hours: 180 miles
• Rule: Distance = 60 × hours

Geometry:

• Perimeter of squares with different side lengths
• Side 1: Perimeter 4; Side 2: Perimeter 8; Side 3: Perimeter 12
• Rule: Perimeter = 4 × side length

Extending Patterns

Once you know the rule, you can extend the pattern forward or backward:

Example: Pattern is 15, 12, 9, 6...

• Rule: Subtract 3
• Extend forward: 6, 3, 0, -3, -6
• Extend backward: 18, 21, 24

Common Mistakes to Avoid

1. Looking at only two terms: Check the pattern with at least three numbers to be sure of the rule
2. Assuming all patterns add or subtract: Some patterns multiply or divide
3. Forgetting negative numbers: Patterns can include negatives and zero
4. Not testing the rule: Always check that your rule works for ALL numbers in the pattern
5. Mixing up input and output: In tables, make sure you know which is x and which is y

Pattern Recognition Tips

• Write the differences: Write the difference between consecutive numbers above them
• Look for multiplication: If differences aren't the same, try division or multiplication
• Use a table: Organize the information in a table to see relationships
• Check for special patterns: Square numbers (1, 4, 9, 16...), powers of 2 (2, 4, 8, 16...)
• Think about position: Sometimes the rule relates to the position number

Advanced Patterns

Square Numbers: 1, 4, 9, 16, 25...

• These are 1², 2², 3², 4², 5²
• Rule: n²

Triangle Numbers: 1, 3, 6, 10, 15...

• These represent dots in triangular arrangements
• Each term adds the next counting number

Fibonacci Pattern: 1, 1, 2, 3, 5, 8, 13...

• Each number is the sum of the two previous numbers
• Found in nature (sunflower seeds, pinecones, shells)

Practice Strategy

To master patterns and relationships:

• Practice identifying patterns in everyday situations
• Create your own patterns and challenge friends to find the rule
• Use input-output tables to organize pattern information
• Graph patterns on coordinate planes
• Work backward from a rule to create the pattern
• Practice both extending patterns and finding missing terms

Understanding patterns is the foundation for algebra! When you recognize relationships between numbers, you're thinking algebraically and developing problem-solving skills that will help you throughout mathematics.