Coordinate Plane - Fifth Grade
Complete Notes & Formulas
1. Describe the Coordinate Plane
What is a Coordinate Plane?
A coordinate plane (also called Cartesian plane) is a two-dimensional surface formed by two perpendicular number lines that intersect at zero.
Parts of the Coordinate Plane
| Part | Description |
|---|---|
| X-axis | The horizontal number line (left to right) |
| Y-axis | The vertical number line (up and down) |
| Origin | The point where x-axis and y-axis meet (0, 0) |
| Quadrants | Four regions created by the axes (I, II, III, IV) |
| Ordered Pair | Location of a point written as (x, y) |
The Four Quadrants
| Quadrant | Location | Signs (x, y) | Example |
|---|---|---|---|
| I | Upper right | (+, +) | (3, 5) |
| II | Upper left | (−, +) | (−3, 5) |
| III | Lower left | (−, −) | (−3, −5) |
| IV | Lower right | (+, −) | (3, −5) |
Understanding Ordered Pairs
Ordered Pair Format: (x, y)
x = horizontal distance from origin
y = vertical distance from origin
Remember: The x-coordinate always comes first in an ordered pair!
2. Objects on a Coordinate Plane
Locating Points
Every point on the coordinate plane has a unique location described by an ordered pair (x, y).
How to Find the Coordinates of a Point
Step 1: Start at the point
Step 2: Look down (or up) to find the x-coordinate on the x-axis
Step 3: Look left (or right) to find the y-coordinate on the y-axis
Step 4: Write the ordered pair as (x, y)
Special Points
Points on the X-axis: Have y-coordinate of 0
Example: (5, 0), (−3, 0), (7, 0)
Points on the Y-axis: Have x-coordinate of 0
Example: (0, 4), (0, −2), (0, 9)
The Origin: Has both coordinates equal to 0
Origin = (0, 0)
Example
Find the coordinates of Point A:
If Point A is 4 units to the right and 6 units up from the origin:
x-coordinate: 4 (right = positive)
y-coordinate: 6 (up = positive)
Answer: Point A is at (4, 6)
3. Graph Points on a Coordinate Plane
Steps to Plot a Point
Step 1: Start at the origin (0, 0)
Step 2: Move along the x-axis (horizontal)
• Positive x → Move RIGHT
• Negative x → Move LEFT
Step 3: From that position, move along the y-axis (vertical)
• Positive y → Move UP
• Negative y → Move DOWN
Step 4: Mark the point with a dot
Step 5: Label the point with its letter name
Direction Memory Aid
Think: "X before Y" (Run before you jump!)
Move horizontally (x) first, then vertically (y)
Examples
Example 1: Plot point A (5, 3)
Step 1: Start at origin (0, 0)
Step 2: Move 5 units to the RIGHT
Step 3: Move 3 units UP
Step 4: Mark and label Point A
Point A is in Quadrant I
Example 2: Plot point B (−4, 2)
Step 1: Start at origin (0, 0)
Step 2: Move 4 units to the LEFT (negative x)
Step 3: Move 2 units UP (positive y)
Step 4: Mark and label Point B
Point B is in Quadrant II
Example 3: Plot point C (3, −5)
• Start at origin
• Move 3 units RIGHT
• Move 5 units DOWN (negative y)
Point C is in Quadrant IV
4. Graph Triangles and Quadrilaterals
Creating Shapes on a Coordinate Plane
Triangles and quadrilaterals can be drawn on a coordinate plane by plotting vertices (corner points) and connecting them with line segments.
Steps to Graph a Shape
Step 1: Plot all the vertices (points)
Step 2: Label each vertex with its letter name
Step 3: Connect the points with straight lines in order
Step 4: Close the shape by connecting the last point to the first
Important Shapes
Triangle: 3 vertices, 3 sides
Example: Points A, B, C
Quadrilateral: 4 vertices, 4 sides
Types: Rectangle, Square, Parallelogram, Trapezoid
Example: Points A, B, C, D
Examples
Example 1: Plot a triangle with vertices A (2, 5), B (6, 5), C (4, 2)
• Plot Point A at (2, 5)
• Plot Point B at (6, 5)
• Plot Point C at (4, 2)
• Connect A to B, B to C, C back to A
Triangle ABC is formed!
Example 2: Plot a rectangle with vertices P (1, 1), Q (5, 1), R (5, 4), S (1, 4)
• Plot all four points
• Connect P to Q (bottom side)
• Connect Q to R (right side)
• Connect R to S (top side)
• Connect S back to P (left side)
Rectangle PQRS is formed!
Finding Length
Horizontal Distance = |x₂ − x₁|
Vertical Distance = |y₂ − y₁|
5. Graph Points from a Table
Understanding Tables
A table shows the relationship between x and y values. Each row represents one ordered pair.
Steps to Graph from a Table
Step 1: Read each row as an ordered pair (x, y)
Step 2: Plot each point on the coordinate plane
Step 3: Label each point if requested
Step 4: Look for patterns (optional: connect points)
Example
Graph the points from this table:
| x | y |
|---|---|
| 1 | 2 |
| 2 | 4 |
| 3 | 6 |
| 4 | 8 |
Ordered Pairs:
(1, 2), (2, 4), (3, 6), (4, 8)
Pattern:
y = 2x (y is always double x)
When graphed, these points form a straight line!
6. Use a Rule to Complete a Table and Graph
What is a Rule?
A rule is an equation that describes the relationship between x and y values.
Common Rules
| Rule | Meaning | Example |
|---|---|---|
| y = x + 3 | Add 3 to x | If x = 5, then y = 8 |
| y = 2x | Multiply x by 2 | If x = 5, then y = 10 |
| y = x − 4 | Subtract 4 from x | If x = 5, then y = 1 |
| y = x ÷ 2 | Divide x by 2 | If x = 10, then y = 5 |
Steps to Use a Rule
Step 1: Look at the given rule
Step 2: Substitute each x-value into the rule
Step 3: Calculate the y-value
Step 4: Write the ordered pair (x, y)
Step 5: Plot the points on a graph
Example
Rule: y = x + 2
Complete the table for x = 0, 1, 2, 3, 4
| x | Rule: y = x + 2 | y |
|---|---|---|
| 0 | 0 + 2 | 2 |
| 1 | 1 + 2 | 3 |
| 2 | 2 + 2 | 4 |
| 3 | 3 + 2 | 5 |
| 4 | 4 + 2 | 6 |
Plot points: (0, 2), (1, 3), (2, 4), (3, 5), (4, 6)
7. Analyze Graphed Relationships
What Does "Analyze" Mean?
To analyze a graph means to look at the pattern of points and understand the relationship between x and y values.
What to Look For
1. Pattern of Points
• Do they form a line?
• Do they curve?
2. Direction
• Does the line go up (increasing)?
• Does the line go down (decreasing)?
• Is it horizontal (constant)?
3. Rate of Change
• How much does y change when x increases by 1?
Types of Relationships
Linear (Straight Line): Points form a straight line
Example: y = 2x, y = x + 3
Constant Relationship: y stays the same
Example: y = 5 (horizontal line)
Proportional Relationship: Passes through origin (0, 0)
Example: y = 3x
Real-World Examples
Example 1: Distance and Time
If you walk at 3 miles per hour, the rule is: Distance = 3 × Time
This creates a straight line going up from left to right
Example 2: Cost of Items
If each apple costs $2, the rule is: Total Cost = 2 × Number of Apples
This also creates a straight line through the origin
8. Coordinate Planes as Maps
Real-World Application
Coordinate planes can be used as maps to locate buildings, streets, or objects in a city or neighborhood.
How Maps Use Coordinates
• Each location has a unique ordered pair
• Streets can run along grid lines
• Distance between places can be calculated
• Directions can be given using coordinates
Finding Distance on a Map
Horizontal Distance = |x₂ − x₁| units
Vertical Distance = |y₂ − y₁| units
Example
City Map:
• Library is at (3, 7)
• School is at (3, 2)
• Park is at (8, 2)
Question: How far is the Library from the School?
Both at x = 3 (same vertical line)
Distance = |7 − 2| = 5 units
Question: How far is the School from the Park?
Both at y = 2 (same horizontal line)
Distance = |8 − 3| = 5 units
9. Follow Directions on a Coordinate Plane
Types of Directions
| Direction | How to Move | Effect |
|---|---|---|
| Right | Add to x-coordinate | x increases |
| Left | Subtract from x-coordinate | x decreases |
| Up | Add to y-coordinate | y increases |
| Down | Subtract from y-coordinate | y decreases |
Steps to Follow Directions
Step 1: Start at the given point
Step 2: Follow each direction one at a time
Step 3: Adjust x or y coordinate based on direction
Step 4: Write the new ordered pair
Examples
Example 1: Start at (4, 5). Move 3 units right and 2 units up. Where are you now?
Starting point: (4, 5)
Move right 3: x = 4 + 3 = 7
Move up 2: y = 5 + 2 = 7
New position: (7, 7)
Example 2: Start at (6, 8). Move 2 units left and 5 units down. Where are you now?
Starting point: (6, 8)
Move left 2: x = 6 − 2 = 4
Move down 5: y = 8 − 5 = 3
New position: (4, 3)
Example 3: Path with multiple steps
Start: (2, 3) → Right 4 → Up 2 → Left 1 → Down 3
Start: (2, 3)
After right 4: (6, 3)
After up 2: (6, 5)
After left 1: (5, 5)
After down 3: (5, 2)
Final position: (5, 2)
Quick Reference Guide
Key Formulas
Ordered Pair: (x, y)
Origin: (0, 0)
Horizontal Distance: |x₂ − x₁|
Vertical Distance: |y₂ − y₁|
Movement Rules
Right: x + __
Left: x − __
Up: y + __
Down: y − __
Quadrant Signs
I: (+, +) | II: (−, +) | III: (−, −) | IV: (+, −)
💡 Important Tips to Remember
✓ Always write x first in ordered pairs: (x, y)
✓ The origin (0, 0) is where the axes meet
✓ X-axis is horizontal (like the horizon)
✓ Y-axis is vertical (goes up and down)
✓ Move horizontally first, then vertically when plotting
✓ Positive numbers go right and up
✓ Negative numbers go left and down
✓ Quadrants are numbered counterclockwise (I, II, III, IV)
✓ Points on axes are not in any quadrant
✓ Check your work by counting from the origin!
🧠 Memory Tricks
Remember "X before Y":
Think: "X comes before Y in the alphabet!"
Axes Direction:
X-axis is like a cross (horizontal)
Y-axis asks "Why?" - reaches up high!
Quadrant Signs:
Quadrant I: All Positive (both +)
Quadrant II: X is negative, Y is positive
Quadrant III: All Negative (both −)
Quadrant IV: X is positive, Y is negative
Movement Trick:
"Right and Up are positive (+)"
"Left and Down are negative (−)"
Master the Coordinate Plane! 📍🗺️
Coordinates help us locate anything - from buried treasure to our favorite places!