Patterns & Graphing
How can patterns help us predict what comes next?
📋 Standards & Objectives
- Identify and extend number patterns
- Find rules for input/output tables
- Describe relationships between two patterns
- Graph ordered pairs on a coordinate grid
🚀 Patterns Are Everywhere!
You save $5 every week. After 1 week: $5. After 2 weeks: $10. After 3 weeks: $15. Can you predict how much you'll have after 10 weeks?
A player scores 3 points per game. After 1 game: 3 total. After 2 games: 6 total. What's the pattern? How many after 8 games?
A tree grows 2 feet taller each year. It starts at 6 feet. After 1 year: 8 feet. After 2 years: 10 feet. What comes next?
📖 Key Vocabulary
A set of numbers that follows a rule. Example: 2, 4, 6, 8 follows the rule "add 2."
The operation that tells you how to get from one number to the next. Rules can be addition, subtraction, multiplication, or division.
A table where one number goes IN, a rule is applied, and a new number comes OUT. The rule connects every input to its output.
Two numbers written as (x, y) that show a point's location on a grid. The first number is across, the second is up.
✨ Number Patterns
A pattern is a sequence of numbers that follows a rule. To find the pattern, look at what happens from one number to the next.
Example: What's the pattern?
Each number is 3 more than the one before. The next number is 18!
📌 Finding the Pattern Rule
Follow these steps to find any pattern's rule:
Look at Pairs
Compare each number to the one right after it. What operation connects them?
Find the Operation
Is it +, −, ×, or ÷? Find the number that connects each pair.
Verify
Check your rule with every pair in the sequence. Does it work for ALL of them?
State the Rule
Write it as a clear statement: "Add 5" or "Multiply by 3"
📌 Writing the Rule
We can write pattern rules as math expressions using variables:
Addition Pattern
Input: 1, 2, 3, 4 → Output: 6, 7, 8, 9
y = x + 5
"Add 5 to each input"
Multiplication Pattern
Input: 1, 2, 3, 4 → Output: 3, 6, 9, 12
y = 3x
"Multiply each input by 3"
Subtraction Pattern
Input: 10, 20, 30 → Output: 4, 14, 24
y = x − 6
"Subtract 6 from each input"
Division Pattern
Input: 10, 20, 30 → Output: 5, 10, 15
y = x ÷ 2
"Divide each input by 2"
Remember: x is the input (what goes in) and y is the output (what comes out)!
👁️ Problem 1
👁️ Problem 2
✍️ Problem 3
✍️ Problem 4
✏️ Problem 5
✏️ Problem 6
✨ Input/Output Tables
An input/output table organizes numbers using a rule. A number goes IN, the rule is applied, and a new number comes OUT.
Example: Rule is "Add 3"
| Input (x) | Output (y) |
|---|---|
| 1 | 4 |
| 2 | 5 |
| 3 | 6 |
| 4 | 7 |
| 5 | 8 |
Every output is 3 more than the input. The rule connects every pair!
📌 Finding Table Rules
Follow these steps to find the rule from a table:
Compare Each Pair
Look at each input and its output. What operation turns the input into the output?
Find the Operation
Is the output bigger (add or multiply)? Or smaller (subtract or divide)? By how much?
Write the Rule
Express as an equation: y = x + 5 or y = 3x
Verify ALL Rows
Test your rule with every row. If it works for all of them, you've got it!
👁️ Problem 7
| Input (x) | Output (y) |
|---|---|
| 2 | 10 |
| 5 | 13 |
| 7 | 15 |
| 10 | 18 |
| 12 | 20 |
👁️ Problem 8
| Input (x) | Output (y) |
|---|---|
| 1 | 2 |
| 3 | 6 |
| 5 | 10 |
| 7 | 14 |
| 9 | 18 |
✍️ Problem 9
| Input (x) | Output (y) |
|---|---|
| 3 | 5 |
| 6 | 8 |
| 10 | 12 |
| 15 | 17 |
| 20 | 22 |
✍️ Problem 10
| Input (x) | Output (y) |
|---|---|
| 1 | 5 |
| 2 | 10 |
| 3 | 15 |
| 4 | 20 |
| 6 | 30 |
✏️ Problem 11
| Input (x) | Output (y) |
|---|---|
| 4 | 10 |
| 7 | 13 |
| 11 | 17 |
| 15 | 21 |
| 20 | 26 |
✏️ Problem 12
| Input (x) | Output (y) |
|---|---|
| 2 | 8 |
| 3 | 12 |
| 5 | 20 |
| 8 | 32 |
| 10 | 40 |
✨ Two Patterns, One Relationship
We can create two patterns using different rules and then compare how their terms relate to each other.
Example:
Pattern A: Start at 0, add 1
Pattern B: Start at 0, add 3
Compare corresponding terms: 0→0, 1→3, 2→6, 3→9, 4→12. Each B term is 3 times its A term! The relationship is B = 3 × A.
📌 From Table to Graph
Once we have two patterns, we can form ordered pairs and graph them!
Generate Both Patterns
Use each rule to generate at least 5 terms for both patterns.
Form Ordered Pairs
Match corresponding terms: (A₁, B₁), (A₂, B₂), etc. Pattern A is x, Pattern B is y.
Plot on the Grid
Start at the origin (0,0). Go right for x, then up for y. Place a point.
Describe the Relationship
What do you notice about the points? Do they form a line? What's the relationship?
📊 Parts of a Coordinate Grid
The horizontal line going left to right. We move across first.
The vertical line going up and down. We move up second.
The point (0, 0) where the x-axis and y-axis cross. Every point starts from here!
Two numbers: x = across, y = up. Example: (3, 5) means go right 3, up 5.
👁️ Problem 13
Generate 5 terms for each. What's the relationship?
👁️ Problem 14
✍️ Problem 15
Generate 5 terms, form ordered pairs, and graph.
✍️ Problem 16
Generate 5 terms, form ordered pairs, and graph.
✏️ Problem 17
Generate 5 terms, find the relationship, and graph.
✏️ Problem 18
| x | y |
|---|---|
| 1 | 4 |
| 2 | 5 |
| 3 | 6 |
| 5 | 8 |
| 7 | 10 |
🏆 Challenge Problem
Pattern A: 0, 4, 8, 12, 16 | Pattern B: 0, 8, 16, 24, 32
Ordered pairs: (0,0), (4,8), (8,16), (12,24), (16,32)
Relationship: B = 2 × A — B is always double A!
10th term: A₁₀ = 0 + (9 × 4) = 36 | B₁₀ = 0 + (9 × 8) = 72
Check: 72 = 2 × 36 ✓ — The relationship still holds!
💬 Turn & Talk
"How can you tell the relationship between two patterns just by looking at the rules?"
Think About:
If both patterns add the same number, what happens? If one adds more, what happens?
Use an Example:
Compare "add 2" and "add 6." What would the relationship between those patterns be?
📝 Key Takeaways
🎫 Exit Ticket
Answer these 3 questions on your exit ticket slip.
Find the rule: 8, 16, 24, 32, 40, ___. What is the rule? What comes next?
Complete the table: Input 3→9, 5→15, 7→21, 10→?. What is the rule?
Using the rule y = x + 5, what ordered pair do you get when x = 4? Plot it: go right ___, up ___.
Great Work! 🎉
You can now find patterns, use tables, and graph ordered pairs!