Opposites and Plotting on a Number Line

In this lesson:

• See the number line stretch into negatives
• Name opposite as same distance, other side
• Plot fractions and decimals on horizontal and vertical lines

What You Will Be Able to Do

By the end, you will:

1. Name the opposite of any rational number
2. Recognize on a number line
3. Recognize as its own opposite
4. Plot fractions and decimals — including negatives
5. Read horizontal and vertical number lines

What's the Same About 3 and –3?

Look at the number line. Plot and .

• Both points are units from
• One sits on the right, the other on the left

What word should we use for this relationship?

The Number Line Is Symmetric Around 0

Same distance from , on opposite sides.

Opposite: Same Distance, Other Side

The opposite of a number is the point the same distance from on the other side of the number line.

• Opposite of is
• Opposite of is

The leading minus sign reads as "opposite of."

Read Opposites Off the Line

Plot each pair on the line — opposites always sit symmetric across .

A Special Case: Is Its Own Opposite

The opposite of is .

• sits at the origin — not to the right, not to the left
• It is the only number with no other side to flip to

Every other number has a distinct opposite; does not.

The Opposite of an Opposite

Each leading minus is one trip across . Two trips return you home.

Two Readings of — Which Is Right?

Compute .

• A. "Make negative twice" →
• B. "Opposite of " →

Read it on the line: opposite of is .

Evaluate Two More: and

: opposite of is

: opposite of is

Two minus signs cancel — write the answer with no leading sign.

Check-In: Evaluate

Compute and explain what each minus sign does on the number line.

Pause and write your answer before advancing.

Now Plot Numbers Between the Ticks

So far we plotted whole numbers. What about or ?

• We need to subdivide the interval between integer ticks
• The negative side subdivides the same way as the positive side — in mirror image

Endpoints first, then the marks between them.

Negative Fourths Move Away From 0

is closest to . is closest to .

Plot Between and

sits halfway between and .

• Plot first; plot next
• Halfway between them — closer to the more negative number
• Result: is to the left of

Plot and

: between and , three-quarters of the way toward .

: between and , four-tenths of the way toward .

Same procedure on either side; "away from " tells you the direction.

Find the Error: Wrong Subdivision

A student labeled the fourths between and as:

Where did they go wrong? Re-label the marks correctly.

Vertical Number Lines: Same Math, Rotated

Negatives go down; positives go up. Used for temperature, elevation, floors.

Plot Three Vertical Readings

Your Turn: Plot and

On a clean horizontal line from to :

1. Plot
2. Plot
3. Which is more negative?

No pre-marked subdivisions — you place every mark.

Mixed Practice: Plot Three Values

Plot each on the indicated line:

1. on a horizontal line from to
2. on a horizontal line from to
3. on a vertical thermometer from to

Mark the integer endpoints first.

Key Takeaways for Opposites and Plotting

• Opposite = same distance from , other side
• : each minus is one flip across
• is its own opposite
• Subdivide away from :
• Vertical lines: down is negative, up is positive

Coming Up: Two Number Lines, One Plane

You can place every signed rational on a single line.

In Lesson 2, you will:

• Combine a horizontal and a vertical line into a coordinate plane
• Plot ordered pairs in all four quadrants
• See reflections across the axes as sign changes