Coordinate Plane - Sixth Grade
Complete Notes & Formulas
1. Parts of the Coordinate Plane
Definition
A coordinate plane is a 2D surface
formed by two perpendicular number lines
Key Parts
| Part | Description | Symbol/Location |
|---|---|---|
| X-axis | Horizontal number line | Runs left-right |
| Y-axis | Vertical number line | Runs up-down |
| Origin | Where axes intersect | (0, 0) |
| Quadrants | Four regions divided by axes | I, II, III, IV |
Visual Diagram
2. Ordered Pairs (Coordinates)
What is an Ordered Pair?
(x, y)
x = x-coordinate (horizontal position)
y = y-coordinate (vertical position)
Important Rules
• Order matters! (3, 5) ≠ (5, 3)
• First number (x) → move left or right
• Second number (y) → move up or down
• Always start from the origin (0, 0)
Example: (4, 3)
Step 1: Start at origin (0, 0)
Step 2: Move 4 units RIGHT (x = 4, positive)
Step 3: Move 3 units UP (y = 3, positive)
Step 4: Plot the point
Point (4, 3) is in Quadrant I
3. Graphing Points on the Coordinate Plane
Steps to Graph
Step 1: Start at the origin (0, 0)
Step 2: Move horizontally based on x-coordinate
• If x is positive → move RIGHT
• If x is negative → move LEFT
Step 3: Move vertically based on y-coordinate
• If y is positive → move UP
• If y is negative → move DOWN
Step 4: Place a dot and label the point
Example: Graph (−3, 2)
Start at (0, 0)
x = −3 (negative) → Move 3 units LEFT
y = 2 (positive) → Move 2 units UP
Point (−3, 2) is in Quadrant II
4. The Four Quadrants
Quadrant Signs
| Quadrant | X-coordinate | Y-coordinate | Example |
|---|---|---|---|
| I (upper right) | Positive (+) | Positive (+) | (3, 4) |
| II (upper left) | Negative (−) | Positive (+) | (−2, 5) |
| III (lower left) | Negative (−) | Negative (−) | (−4, −3) |
| IV (lower right) | Positive (+) | Negative (−) | (5, −2) |
Memory Trick: Quadrants are numbered counter-clockwise starting from upper right!
I (top-right) → II (top-left) → III (bottom-left) → IV (bottom-right)
5. Reflecting Points Over Axes
Reflection Rules
Reflection over X-axis:
(x, y) → (x, −y)
Change the sign of y-coordinate
Reflection over Y-axis:
(x, y) → (−x, y)
Change the sign of x-coordinate
Examples
Example 1: Reflect (3, 4) over x-axis
Original: (3, 4)
Keep x, change y: (3, −4)
Answer: (3, −4)
Example 2: Reflect (−2, 5) over y-axis
Original: (−2, 5)
Change x, keep y: (2, 5)
Answer: (2, 5)
6. Distance Between Two Points
Distance Formula
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Where (x₁, y₁) and (x₂, y₂) are two points
Special Cases (Easier!)
Horizontal line (same y-coordinates):
d = |x₂ − x₁|
Vertical line (same x-coordinates):
d = |y₂ − y₁|
Example: Distance between (1, 2) and (4, 6)
Step 1: Identify coordinates
(x₁, y₁) = (1, 2) and (x₂, y₂) = (4, 6)
Step 2: Find differences
x₂ − x₁ = 4 − 1 = 3
y₂ − y₁ = 6 − 2 = 4
Step 3: Apply formula
d = √[(3)² + (4)²]
d = √[9 + 16]
d = √25 = 5
Answer: 5 units
7. Area & Perimeter on Coordinate Plane
Formulas for Rectangles
Length (L) = |x₂ − x₁|
Width (W) = |y₂ − y₁|
Area = L × W
Perimeter = 2L + 2W
Steps to Find Area & Perimeter
Step 1: Identify the corners (vertices)
Step 2: Find length (horizontal distance)
Step 3: Find width (vertical distance)
Step 4: Use formulas
Example: Rectangle with corners (1, 2), (1, 5), (4, 2), (4, 5)
Length: |4 − 1| = 3 units
Width: |5 − 2| = 3 units
Area: 3 × 3 = 9 square units
Perimeter: 2(3) + 2(3) = 12 units
Area: 9 sq units, Perimeter: 12 units
8. Coordinate Planes as Maps
Using Coordinates for Locations
• Each location is represented by an ordered pair
• Objects can be placed at specific coordinates
• Distance between locations can be calculated
• Useful for real-world navigation
Example: City Map
Problem: On a city map, the library is at (2, 3) and the park is at (5, 7). How far apart are they?
d = √[(5−2)² + (7−3)²]
d = √[3² + 4²]
d = √[9 + 16] = √25 = 5
Answer: 5 blocks apart
9. Following Directions on Coordinate Plane
Direction Keywords
Right: Add to x-coordinate
Left: Subtract from x-coordinate
Up: Add to y-coordinate
Down: Subtract from y-coordinate
Example: Following Directions
Problem: Start at (2, 3). Move 4 units right and 2 units down. Where are you?
Start: (2, 3)
Right 4: x becomes 2 + 4 = 6
Down 2: y becomes 3 − 2 = 1
Answer: (6, 1)
Quick Reference: Coordinate Plane
| Concept | Formula/Rule |
|---|---|
| Ordered Pair | (x, y) |
| Origin | (0, 0) |
| Reflect over X-axis | (x, y) → (x, −y) |
| Reflect over Y-axis | (x, y) → (−x, y) |
| Distance Formula | d = √[(x₂−x₁)² + (y₂−y₁)²] |
| Rectangle Area | A = Length × Width |
💡 Important Tips to Remember
✓ Order matters! (x, y) ≠ (y, x)
✓ X first, Y second - always!
✓ Start at origin (0, 0) when plotting
✓ Quadrant signs: I (+,+), II (−,+), III (−,−), IV (+,−)
✓ Reflection over x-axis: Change y sign
✓ Reflection over y-axis: Change x sign
✓ Distance formula comes from Pythagorean theorem
✓ Horizontal/vertical lines make distance easier!
✓ Right/Up = Add, Left/Down = Subtract
✓ Label your points clearly on graphs
🧠 Memory Tricks & Strategies
Ordered Pairs:
"X comes before Y in the alphabet, just like in (x, y)!"
Axes:
"X-axis crosses (X marks the spot!), Y-axis climbs high!"
Quadrants:
"I starts top-right, goes counter-clockwise!"
Reflections:
"X-axis flip? Change the Y! Y-axis flip? Change the X!"
Distance:
"Subtract, square, add, then square root - distance formula route!"
Graphing:
"Run before you jump! (X horizontal, then Y vertical)"
Master the Coordinate Plane! 📍 📊 🎯
Remember: (x, y) - X first, Y second!