Graphing and Distance in All Four Quadrants

In this lesson:

• Plot and read points in all four quadrants
• Find distances using coordinates and absolute value
• Solve real-world problems with the coordinate plane

What You Will Learn Today

By the end of this lesson, you will be able to:

1. Graph points with integer and rational coordinates in all four quadrants, and read coordinates of plotted points
2. Use coordinates and absolute value to find the distance between two points that share the same x- or y-coordinate
3. Solve real-world problems by graphing points and finding distances in the coordinate plane

Review: Signs, Quadrants, and Absolute Value

From 6.NS.C.6 and 6.NS.C.7 you know:

• The plane has four quadrants — sign patterns: (+,+), (−,+), (−,−), (+,−)
• Absolute value measures distance from 0

Today we apply both to plot, read, and measure.

The Coordinate Plane: All Four Quadrants

The sign of each coordinate tells direction; the magnitude tells how far.

Plotting Points in All Four Quadrants

How to plot any point :

• Start at the origin
• Move units right (positive) or left (negative)
• Move units up (positive) or down (negative)

→ 3 left, 4 up · → 2 right, 5 down

Read the Coordinates of Each Point

Your turn: Identify each point's coordinates.

• Point A: ___ , ___
• Point B: ___ , ___
• Point C: ___ , ___
• Point D: ___ , ___

Horizontal position first (x), then vertical (y)

Which Quadrant Contains the Point?

Which quadrant is the point in?

Think before the next slide:

• What does a negative tell you about direction?
• What does a negative tell you about direction?
• Which quadrant has both coordinates negative?

Finding Distance with a Shared Coordinate

Same -coordinate → vertical line → subtract -values

Same -coordinate → horizontal line → subtract -values

Example: Vertical Distance Crossing Zero

and share .

Count: to = 2; to = 5 → 7 total

Both orders work:

Example: Horizontal Pair with Shared Y

Points and share → horizontal line → subtract -values:

Count check: 4 units left of 0, 2 units right → total 6 ✓

Subtract the -values (they differ), not the -values (they match).

Example: Both Points in Negative Region

Points and share . Find the distance.

Same procedure — absolute value handles it automatically:

No special case needed — the formula works the same in every quadrant.

Your Turn: Find the Distance

Find the distance from to .

These points share a -coordinate — which coordinates do you subtract?

Pause and try before the next slide.

Answer: Distance from to

Check the other order too:

Key reminder: Absolute value ensures distance is always positive, regardless of order.

From Practice to Real-World Problems

So far: We can plot points and find distances between pairs that share a coordinate.

Next: Apply those tools to solve real-world problems — maps, missing vertices, and total distances.

The key habit: Ask "what does this coordinate mean?" and "what does this distance represent?"

City Grid: Map Problem Setup

A city grid is modeled on a coordinate plane:

• City Hall at
• Library at
• Park at

City Grid: City Hall to Library

City Hall → Library

Same -coordinate () → vertical distance, use -values:

The library is 7 blocks south of City Hall.

City Grid: Library to Park and Total Walk

Library → Park

Same -coordinate () → horizontal distance, use -values:

Total walk: City Hall → Library → Park = units

Finding the Missing Rectangle Vertex

Three vertices: , ,

• and share → top edge is horizontal
• and share → right edge is vertical
• Fourth vertex:

Calculate the Rectangle Perimeter Yourself

Rectangle vertices: , , ,

Find each length, then add for perimeter.

Rectangle Perimeter Answers Step by Step

Perimeter = units

What to Remember from This Lesson

• Plot : origin → horizontal (sign = direction) → vertical
• Same → vertical line →
• Same → horizontal line →

Watch out:

• Always use absolute value — distance is never negative
• Subtract the coordinates that differ, not the ones that match
• The axis is just a line — crossing it doesn't change the count

What's Next: Polygons in the Coordinate Plane

Coming up in 6.G.A.3:

• Draw polygons in the coordinate plane given coordinates for vertices
• Use coordinates to find perimeter and area of figures
• Connect coordinate geometry to real-world design problems

The distance skills from today are your foundation for all of that.