Patterns on the Coordinate Plane | Grade 5 Math

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What Are Coordinate Patterns?

A coordinate pattern is a set of ordered pairs (x, y) that follow a specific rule. When you plot these points on a coordinate plane, they often form recognizable shapes like lines, curves, or geometric figures.

Patterns help us predict where the next point will be and understand relationships between numbers!

Generating Ordered Pairs from Rules

You can create ordered pairs by applying a rule to a starting value. The most common patterns use a rule like: y = x + something or y = x × something

Example 1: y = x + 2

x	Rule: x + 2	y	Point (x, y)
0	0 + 2	2	(0, 2)
1	1 + 2	3	(1, 3)
2	2 + 2	4	(2, 4)
3	3 + 2	5	(3, 5)
4	4 + 2	6	(4, 6)

Example 2: y = 2x (y is double x)

x	Rule: 2 × x	y	Point (x, y)
0	2 × 0	0	(0, 0)
1	2 × 1	2	(1, 2)
2	2 × 2	4	(2, 4)
3	2 × 3	6	(3, 6)
4	2 × 4	8	(4, 8)

Pattern Tip: When you add a number (like y = x + 2), the line shifts up. When you multiply (like y = 2x), the line gets steeper!

Types of Patterns

📈 Linear Patterns

Points form a straight line. The y-value increases by the same amount each time x increases by 1.

Example: y = x + 1

📊 Skip Counting Patterns

Y-values skip by the same number. Common with multiplication rules like y = 3x.

Example: y = 2x (0, 2, 4, 6...)

🔷 Shape Patterns

Points form shapes like squares, triangles, or rectangles when connected.

Example: (1,1), (4,1), (4,4), (1,4) forms a square

Steps to Find a Pattern

1

Look at the X Values

Check how the x-coordinates change. Do they go up by 1 each time? By 2?

2

Look at the Y Values

Check how the y-coordinates change. What's the difference between each y-value?

3

Find the Relationship

Ask: How does y relate to x? Is it x + something? x times something?

4

Test Your Rule

Apply your rule to all the x-values. If it gives you the correct y-values, you found it!

Examples

Example: Finding the Rule

Given these points: (1, 4), (2, 5), (3, 6), (4, 7) — what's the rule?

Step 1: X goes up by 1 each time: 1, 2, 3, 4

Step 2: Y also goes up by 1 each time: 4, 5, 6, 7

Step 3: Compare x and y: When x = 1, y = 4. That's 1 + 3 = 4.

Step 4: Test: 2 + 3 = 5 ✓, 3 + 3 = 6 ✓, 4 + 3 = 7 ✓

Rule: y = x + 3

Example: Predicting the Next Point

A pattern follows the rule y = 2x + 1. If x = 5, what is y?

Apply the rule: y = 2(5) + 1 = 10 + 1 = 11

The next point is (5, 11)

Example: Creating Points for a Rectangle

Create a rectangle with corners at:

Bottom-left: (2, 1)
Bottom-right: (6, 1)
Top-right: (6, 4)
Top-left: (2, 4)

These 4 points form a rectangle that is 4 units wide and 3 units tall!

Interactive Pattern Builder

Generate and Plot Patterns

Choose a rule and see the pattern appear on the coordinate plane!

Rule:

Start x:

Points:

Generated Points

x	y	Point

Practice Problems

Problem 1

For the rule y = x + 4, what is the y-value when x = 3?

Problem 2

The points (0, 0), (1, 3), (2, 6), (3, 9) follow which rule?

Problem 3

If a pattern starts at (1, 5) and the rule is y = x + 4, what is the next point when x = 2?

Problem 4

For the rule y = 2x + 1, which point does NOT belong in the pattern?

Problem 5

Points (1, 2), (2, 4), (3, 6), (4, 8) form what kind of pattern?

Problem 6

To create a square with side length 3, starting at (1, 1), where would the top-right corner be?

Check Your Understanding

Question 1

What does it mean when we say points follow a "rule"?

Question 2

In the rule y = 2x, the number 2 tells us:

Question 3

If you want to predict the 10th point in a pattern with rule y = x + 5 starting from x = 0, what would the point be?

Summary

📝

Rules Create Patterns

Rules like y = x + 2 or y = 3x help generate ordered pairs.

📈

Linear Patterns

When plotted, these patterns form straight lines.

🔍

Finding Rules

Look at how x and y relate to discover the pattern rule.

🎯

Predicting Points

Once you know the rule, you can find any point in the pattern.

Remember: Patterns help us see relationships between numbers and make predictions. The coordinate plane is a powerful tool for visualizing these relationships!

Next Steps

Practice creating your own patterns with different rules
Try graphing patterns on paper to strengthen your skills
Explore how patterns connect to real-world situations (distance, time, cost)
Move on to the Data & Patterns unit to learn more about analyzing data