Subtracting Rational Numbers

Lesson 2 of 2: Subtraction and Properties

In this lesson:

• Rewrite any subtraction problem as addition:
• Find the distance between two rational numbers using
• Use properties strategically to simplify multi-term expressions

What You Will Learn Today

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

1. Rewrite subtraction as adding the inverse:
2. Determine the distance between two rationals as
3. Apply properties strategically to simplify multi-term expressions

Reconnecting: Five Minus Eight on Number Line

Start at , subtract — move 8 units left — land at .

A negative result from subtraction. Does subtracting always make things smaller?

Not always — it depends on the sign of what you subtract.

Subtraction = Addition of the Opposite

Side-by-side comparison on the number line:

Both produce identical movements — they land at the same point.

Therefore:

Subtracting a Negative Number Increases the Value

Rewrite:

Start at , move 4 units right — land at .

Result: — subtracting a negative increased the value.

Removing a Debt Makes You Richer

You have . A debt is canceled — you gain .

Removing a negative obligation is equivalent to gaining that amount. Subtracting negative = adding positive.

Second Example: Two Negatives in the Problem

Rewrite:

Number line: Start at , move right — land at .

Check-In: Rewriting Subtraction as Addition

Fill in the blank, then evaluate:

Fill in the blank first, then compute.

Distance Between Numbers: Absolute Value Formula

Distance between two numbers = absolute value of their difference:

Distance is always non-negative — it's a length.

Example: Distance between and :

Absolute Value Does Not Distribute: Common Error

Incorrect:

Absolute value does NOT distribute over subtraction.

Correct: Compute inside first, then take absolute value:

Real World: Death Valley to Mount Whitney

Death Valley: m. Mount Whitney: m.

Elevation difference:

Subtraction-as-addition makes the computation natural.

Quick Check: Finding the Temperature Range

Low: — High:

What is the temperature range?

Range is always positive — it's a distance.

Your Turn: Rewrite and Evaluate Each Problem

Rewrite each as addition, then evaluate:

4. Distance between and

Show the rewriting step. Try all four before advancing.

Practice Answers: Four Subtraction Problems

3. — subtracting increased the value
4. units — distance is always positive

Properties Hold for All Rational Numbers

These properties extend from positives to all rationals:

• Commutative:
• Associative:
• Inverse:

Strategy: Scan for inverse pairs before computing.

Strategy in Action: Finding the Inverse Pairs

Scan first: and are a pair; and are a pair.

Applying the Strategy to Fractions

Spot: and are additive inverses.

No common denominator needed — the pair cancels first.

Real World: Profits and Losses

A store's quarterly profit and loss figures:

Net result:

Quick Check: Use the Properties

Evaluate using properties — do not compute left to right:

Hint: scan for additive inverse pairs first.

What pairs do you see? What's left after they cancel?

Lesson 2 Summary: Key Ideas to Remember

• — subtraction is adding the opposite
• Distance: — always non-negative
• Subtracting a negative increases the value
• Scan for inverse pairs before computing

Watch out: — compute inside bars first

What's Coming Next in 7.NS.A.2

You've mastered 7.NS.A.1: Adding and subtracting rational numbers

Next standard: Multiplying and dividing rational numbers

• Why does ?
• Division as multiplying by the reciprocal
• Additive inverse → multiplicative inverse (reciprocal)