Rational Numbers on Lines and Planes

Understanding 6.NS.C.6

In this lesson:

• Locate any rational number on a number line
• Understand opposites and why
• Navigate all four quadrants of the coordinate plane

Learning Objectives for This Lesson

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

1. Locate and label any rational number on a number line
2. Recognize opposites are equidistant from 0; explain
3. Identify quadrant from signs; relate sign-difference pairs to reflections
4. Plot rational numbers on number lines and in the coordinate plane

Prior Knowledge: Activating What You Know

You've worked with number lines before:

• Integers:
• Positive fractions located by subdividing unit lengths
• Positive/negative as opposite directions (6.NS.C.5)

Today's question: Where do negative fractions and decimals live?

Sign and Magnitude Locate Every Rational Number

The number line extends infinitely in both directions.

• Every rational number sits at exactly one point
• The sign tells you which side of 0
• The magnitude tells you how far from 0

Same process as positive fractions: subdivide, count in the correct direction.

Rational Numbers Placed on the Number Line

and : same distance from 0, opposite sides.

Steps for Placing a Negative Fraction

Placing on the number line:

Step 1: Denominator 3 → subdivide each unit into 3 parts

Step 2: Negative sign → count to the left of 0

Step 3: Count 5 thirds left →

Between −1 and −2, closer to −2.

Opposites Are Reflections Across Zero

The opposite of a number is — same distance from 0, other side.

• Opposite of →
• Opposite of →
• Opposite of →

On the number line: opposites are reflections across 0.

Opposites on the Number Line

and : equal distance from 0, opposite directions.

Two Flips Return to the Original Number

What is ?

• The sign means "take the opposite"
• = opposite of 3 → flip once → land on
• = opposite of → flip again → land on

Two applications of "opposite" return the original.

Zero Is Its Own Unique Opposite

What is the opposite of 0?

• Zero sits at the reference point — flipping across itself leads nowhere
• Zero is its own opposite — unique among all numbers

Ask: Is there any number other than 0 whose opposite is itself?

Check-In: Place Two Rational Numbers

Place the following on a number line from −4 to 4:

Which integers does each lie between? What denominator tells you the subdivision?

Number Line Placement: Answers Revealed

• Subdivide into fourths; count 7 fourths left of 0
• Between −1 and −2, three-quarters past −1

• Halfway between −2 and −3 (count 2.5 units left of 0)

Extending the Number Line into a Plane

The number line is 1-dimensional — one axis.

The coordinate plane adds a perpendicular second axis:

• Horizontal: x-axis (extends left and right)
• Vertical: y-axis (extends up and down)

Every location needs two coordinates: .

The Full Coordinate Plane with Four Quadrants

Four quadrants numbered counterclockwise from Quadrant I (upper right).

Coordinate Signs Determine the Quadrant

Axis points (coordinate = 0) are not in any quadrant.

Quadrant Sign Chart: Visual Quick Reference

Read each coordinate separately: x tells left/right, y tells up/down.

Check-In: Identify the Quadrant from Signs

Without plotting, identify the quadrant for each point:

Use the sign pattern — not a graph.

Negating a Coordinate Reflects Across an Axis

When ordered pairs differ only in sign, the points are reflections:

• Negate → across the y-axis
• Negate → across the x-axis
• Negate both → across both axes (180° rotation)

Memory hook: x-axis reflection changes y. y-axis reflection changes x.

Reflection Family of : A Rectangle

, , , — all equidistant from each axis.

Guided Practice: Find Reflections of

Given the point in Quadrant II:

1. Reflect across the y-axis →
2. Reflect across the x-axis →
3. Reflect across both axes →

Identify the quadrant of each result.

Answers: Reflections of

1. Reflect across y-axis: → Quadrant I
2. Reflect across x-axis: → Quadrant III
3. Reflect across both axes: → Quadrant IV

Pattern: Negating one coordinate → adjacent quadrant; negating both → opposite quadrant.

Plotting Rational Coordinates in the Plane

Plot and label each point:

• → right , down
• → left , up
• → left , down

Use the same subdivision process as the number line — now in two directions.

Check-In: Reflections of a Given Point

For the point :

1. Which quadrant is in?
2. Reflection across the x-axis?
3. Reflection across the y-axis?
4. Reflection across both axes?

Answers: Reflections of

→ Quadrant II (, )

Summary: Key Takeaways from This Lesson

✓ Every rational number is a unique point — sign gives direction, magnitude gives distance

✓ Opposites are equidistant from 0 on opposite sides; ;

✓ Four quadrants determined by signs: I , II , III , IV

Watch out: means "opposite of " — if , then

Watch out: Across the x-axis changes y; across the y-axis changes x

Watch out: One negative coordinate alone does not mean Quadrant III

Preview: Ordering and Absolute Value Next

Next lesson: 6.NS.C.7 — Ordering and Absolute Value

• Compare and order rational numbers on the number line
• Absolute value as distance from 0
• Apply absolute value to real-world contexts

The number line you built today is the foundation for measuring distances.