Adding Rational Numbers

Lesson 1 of 2: Additive Inverses and Number Line Addition

In this lesson:

• Explain what additive inverses are and why opposites sum to zero
• Represent addition of rational numbers on a number line

What You Will Learn Today

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

1. Explain additive inverses and show that
2. Represent on a number line: move in the direction of
3. Apply inverse properties to add rational numbers fluently

Opposites Cancel in the Real World

A bank account tells a clear story about opposites:

• Deposit $50: balance rises by $50
• Withdraw $50: balance falls by $50
• Net change: $0 — back where you started

Football: gain 8 yards, lose 8 yards → net gain: 0.

Defining the Additive Inverse Property

Every rational number has an additive inverse :

• Opposite of is because
• Opposite of is because
• Opposite of is — zero is its own inverse

Opposites Are Mirror Images of Zero

• is 7 units to the right of zero
• is 7 units to the left of zero
• Equal distance, opposite direction — sum:

Check: Find Each Number's Opposite

What is the additive inverse of each?

1. The opposite of is ___
2. The opposite of is ___
3. The opposite of is ___

Verify: Sum each pair — does it equal ?

Think through each one before the next slide.

Additive Inverses Extend to All Rationals

Additive inverses exist for every rational number:

• Integers:
• Decimals:
• Fractions:

means "opposite of " — if , then

Addition Is Movement on the Number Line

To compute : start at , then move units.

• → move right
• → move left

The sign gives direction. The absolute value gives distance.

Case 1: Positive + Positive

Start at , move 4 units right — land at

Case 2: Positive + Negative

Start at , move 5 units left — land at

Case 3: Negative + Positive

Start at , move 4 units right — land at

Case 4: Negative + Negative

Start at , move 3 units left — land at

Quick Check: Predict Before Computing

Use the number line model to predict the result:

• Where do you start?
• Which direction do you move?
• How far?
• Where do you land?

Reason through the movement before computing numerically.

Reading the Pattern Across All Four Cases

Same signs → that sign, larger absolute value:

• and

Different signs → sign of the larger :

• since
• since

Real World: Scuba Diver Ascends 20 Feet

A scuba diver at ft ascends ft.

Start at , move 20 units up on a vertical number line — land at .

She is 5 feet above sea level.

Real World: Temperature Drops at Midnight

Temperature at noon: . Drops another by midnight.

Start at , move 8 units left — land at .

Midnight temperature:

Connecting Addition Back to Additive Inverses

Verify using the number line model:

Start at , move 6 units right — land at exactly .

Additive inverses: equal distance, opposite direction → always land at .

Your Turn: Three Number Line Problems

Compute each sum using the number line model.

For each: identify start, direction, distance, and landing point.

Try all three before the next slide.

Practice Answers: Number Line Check

1. — move right
2. — move left, cross zero
3. — move more left

1, 2: different signs → larger wins. 3: same sign → more negative.

Lesson 1 Summary: Key Ideas to Remember

• Additive inverse: for all rational
• Addition = movement: positive → right; negative → left
• means "opposite" — if , then
• Two negatives added: , more negative

Preview of Lesson 2: Subtraction and Properties

In Lesson 2, we extend the number line model to subtraction:

• Subtraction as addition:
• Subtracting a negative increases the value — here's why
• Distance formula: between any two rationals
• Properties: spot inverse pairs to simplify efficiently