Coordinate Plane
Complete Notes & Formulas for Grade 5
1. Describe the Coordinate Plane
What is a Coordinate Plane?
A coordinate plane is a two-dimensional surface formed by two perpendicular number lines that intersect at a point called the origin.
Key Parts of the Coordinate Plane
- X-axis: The horizontal number line
- Y-axis: The vertical number line
- Origin: The point where the axes meet, written as (0, 0)
- Ordered Pair: A pair of numbers (x, y) that describes a point's location
The Four Quadrants
The axes divide the plane into 4 regions called quadrants:
| Quadrant | Location | Signs |
|---|---|---|
| I | Upper Right | (+x, +y) |
| II | Upper Left | (-x, +y) |
| III | Lower Left | (-x, -y) |
| IV | Lower Right | (+x, -y) |
2. Objects on a Coordinate Plane
Finding Coordinates of Objects
To find the location of any object on a coordinate plane:
- Start at the origin (0, 0)
- Count how many units right or left (x-coordinate)
- Count how many units up or down (y-coordinate)
- Write the location as an ordered pair: (x, y)
📍 Important Rules
- X always comes first: (x, y) not (y, x)
- Right is positive, Left is negative for x-values
- Up is positive, Down is negative for y-values
- The origin is always (0, 0)
Example
If a point is 4 units to the right and 3 units up from the origin:
Coordinates = (4, 3)
3. Graph Points on a Coordinate Plane
Steps to Plot a Point (x, y)
- Step 1: Start at the origin (0, 0)
- Step 2: Move along the x-axis
- Move RIGHT if x is positive
- Move LEFT if x is negative
- Step 3: Move along the y-axis
- Move UP if y is positive
- Move DOWN if y is negative
- Step 4: Plot a dot and label the point
📝 Plotting Formula
Point (x, y) = (horizontal distance, vertical distance)
x = distance from origin along x-axis
y = distance from origin along y-axis
✏️ Practice Examples
| Point | Instructions |
|---|---|
| (3, 5) | Right 3, Up 5 |
| (6, 2) | Right 6, Up 2 |
| (0, 4) | Stay at origin for x, Up 4 |
| (5, 0) | Right 5, Stay at origin for y |
4. Graph Triangles and Quadrilaterals
Creating Shapes on the Coordinate Plane
To draw shapes, you need to:
- Plot each vertex (corner point) of the shape
- Connect the points in order with straight lines
- Label each vertex with its coordinates
🔺 Triangle
A triangle needs 3 vertices (3 ordered pairs)
Example: Triangle ABC
A = (1, 1), B = (4, 1), C = (2, 4)
Plot these 3 points and connect them to form a triangle.
◼️ Quadrilaterals
Quadrilaterals need 4 vertices (4 ordered pairs)
| Shape | Number of Sides | Special Properties |
|---|---|---|
| Rectangle | 4 | Opposite sides are equal and parallel |
| Square | 4 | All sides are equal |
| Trapezoid | 4 | One pair of parallel sides |
| Parallelogram | 4 | Two pairs of parallel sides |
Example: Rectangle ABCD
A = (2, 2), B = (6, 2), C = (6, 5), D = (2, 5)
Plot all 4 points and connect them in order: A→B→C→D→A
5. Graph Points from a Table
Using Tables to Plot Points
Tables organize x and y values to make plotting easier:
- Read the x-value from the first column
- Read the corresponding y-value from the second column
- Create an ordered pair (x, y)
- Plot each ordered pair on the coordinate plane
📊 Example Table
| x | y |
|---|---|
| 0 | 1 |
| 1 | 3 |
| 2 | 5 |
| 3 | 7 |
Points to Plot:
(0, 1), (1, 3), (2, 5), (3, 7)
🔍 Pattern Recognition
After plotting, connect the points to see if they form a pattern or line!
6. Use a Rule to Complete a Table and Graph
What is a Rule?
A rule is a mathematical pattern or formula that shows how x and y are related.
📐 Common Rules (Formulas)
| Rule | What it Means | Example |
|---|---|---|
| y = x + 2 | Add 2 to x to get y | If x = 3, then y = 5 |
| y = 2x | Multiply x by 2 to get y | If x = 4, then y = 8 |
| y = x - 1 | Subtract 1 from x to get y | If x = 5, then y = 4 |
| y = 3x + 1 | Multiply x by 3, then add 1 | If x = 2, then y = 7 |
Steps to Use a Rule
- Choose x-values (usually 0, 1, 2, 3, 4)
- Apply the rule to find each y-value
- Create ordered pairs (x, y)
- Fill in the table with your values
- Plot all points on the coordinate plane
- Connect the points to see the pattern
✏️ Complete Example
Rule: y = x + 3
| x | Rule (x + 3) | y | Ordered Pair |
|---|---|---|---|
| 0 | 0 + 3 | 3 | (0, 3) |
| 1 | 1 + 3 | 4 | (1, 4) |
| 2 | 2 + 3 | 5 | (2, 5) |
7. Analyze Graphed Relationships
Reading and Interpreting Graphs
When analyzing graphs, look for these patterns:
📈 Types of Relationships
| Relationship | What It Looks Like | Description |
|---|---|---|
| Increasing | Line goes up ↗ | As x increases, y increases |
| Decreasing | Line goes down ↘ | As x increases, y decreases |
| Constant | Horizontal line → | y stays the same as x changes |
| Linear | Straight line | Points form a straight pattern |
🔍 Questions to Ask
- Do the points form a straight line?
- Is the pattern increasing or decreasing?
- What is the relationship between x and y?
- Can you find a rule that describes the pattern?
- What happens when x = 0?
Real-World Applications
- Distance vs. Time: How far you travel over time
- Cost vs. Items: Total cost based on number of items
- Temperature vs. Time: How temperature changes throughout the day
- Money Saved vs. Weeks: Savings growth over time
8. Coordinate Planes as Maps
🗺️ Using Coordinates Like a Map
Coordinate planes work just like city maps! Each location has specific coordinates that help us find places.
How to Read a Coordinate Map
- Find the x-coordinate: This tells you how far left or right to go
- Find the y-coordinate: This tells you how far up or down to go
- Locate the intersection: Where these meet is your destination
- Label landmarks: Each place has its own coordinate address
📍 Map Example
Imagine a town where:
| Location | Coordinates |
|---|---|
| School | (3, 4) |
| Library | (5, 2) |
| Park | (1, 6) |
| Store | (7, 3) |
🧭 Finding Distance
To find the distance between two points horizontally or vertically:
Horizontal Distance = |x₂ - x₁|
Vertical Distance = |y₂ - y₁|
(The | | symbols mean "absolute value" - always take the positive distance)
9. Follow Directions on a Coordinate Plane
🧭 Movement on the Coordinate Plane
Starting from any point, you can move in four directions:
⬆️⬇️⬅️➡️ Direction Guide
| Direction | How It Changes | Example |
|---|---|---|
| Right → | Add to x-coordinate | (3, 2) + 2 right = (5, 2) |
| Left ← | Subtract from x-coordinate | (5, 3) - 2 left = (3, 3) |
| Up ↑ | Add to y-coordinate | (2, 1) + 3 up = (2, 4) |
| Down ↓ | Subtract from y-coordinate | (4, 6) - 2 down = (4, 4) |
📐 Movement Formulas
• Moving Right: (x, y) → (x + n, y)
• Moving Left: (x, y) → (x - n, y)
• Moving Up: (x, y) → (x, y + n)
• Moving Down: (x, y) → (x, y - n)
(where n = number of units moved)
🎯 Practice Problem
Start at point (2, 3)
Follow these directions:
- Move 3 units right
- Move 2 units up
- Move 1 unit left
Solution:
Start: (2, 3)
After right 3: (2+3, 3) = (5, 3)
After up 2: (5, 3+2) = (5, 5)
After left 1: (5-1, 5) = (4, 5)
Final Position: (4, 5)
📚 Quick Reference: Key Formulas
| Concept | Formula/Rule |
|---|---|
| Ordered Pair | (x, y) |
| Origin | (0, 0) |
| Horizontal Distance | |x₂ - x₁| |
| Vertical Distance | |y₂ - y₁| |
| Move Right | (x, y) → (x + n, y) |
| Move Left | (x, y) → (x - n, y) |
| Move Up | (x, y) → (x, y + n) |
| Move Down | (x, y) → (x, y - n) |
💡 Important Tips to Remember
✓ X comes first, Y comes second
Always write coordinates as (x, y)
✓ Start at the origin
Begin at (0, 0) when plotting points
✓ Move horizontally first
Find x-coordinate before y-coordinate
✓ Label your points
Mark each point with its letter and coordinates
🌟 Practice makes perfect! Keep plotting points and exploring the coordinate plane! 🌟