Apply Properties of Operations to Expressions

Lesson 1 of 4: Expressions and Equations

In this lesson:

• Combine like terms with rational coefficients
• Expand using the distributive property
• Factor linear expressions and verify

Five Learning Objectives for This Lesson

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

1. Identify like terms with rational coefficients
2. Add and subtract linear expressions
3. Apply the distributive property to expand expressions
4. Factor a linear expression by extracting a common factor
5. Simplify multi-step expressions by expanding, then combining

What Do You Already Know?

You know how to combine like terms with whole numbers:

Today's question: What about rational coefficients?

Think about it — what operation do we apply to the coefficients?

Like Terms Share the Same Variable Part

• Like terms: same variable raised to the same power
• The coefficient is the numerical factor in front of the variable
• Only like terms can combine — variable parts must match exactly

Examples: and are like terms ✓ — but and are not ✗

Adding Rational Coefficients Step by Step

Coefficients are and — add them using a common denominator:

Five Combining Examples: Increasing Complexity

Sorting Variable and Constant Terms First

For :

Sort first — terms in different columns can never combine.

Quick Check: Which Terms Combine?

In , identify which terms can combine.

Answer: · stands alone ·

Result:

Try sorting before reading the answer above.

Subtracting Expressions: The Critical Step

Subtracting an expression means every term changes sign:

Strategy: Rewrite as adding the opposite

The minus sign flipped both signs inside the parentheses.

Worked Example: Subtracting Rational Coefficient Expressions

Step 1 — Flip signs:

Step 2 — Combine x terms:

Your Turn: Subtract These Expressions

Fill in the signs, then combine:

The minus sign flipped stays and becomes .

Error Analysis: Sign Distribution Mistake Explained

A student got

Error: Minus sign only reached , not

→ Correct:

Combining Like Terms: Three Key Rules

✓ Like terms: same variable, same exponent

✓ Combine by adding or subtracting coefficients only

✓ Subtracting an expression: every term inside changes sign

— variable and constant cannot combine

Distributive Property with Any Rational Factor

• — fraction factor
• — negative fraction
• — decimal factor

Arrow Diagram: Distribute to Every Term

• The factor outside connects to every term inside
• Count the terms — that many products to compute
• Three terms inside → three products

Worked Example: Positive Fraction Factor

Distribute — divide each term by 3:

Multiplying by is the same as dividing by 3.

Worked Example: Negative Fraction Factor

Distribute — apply the negative to each product:

Sign rule: negative × positive = negative

Worked Example: Subtracting a Parenthetical Expression

Distribute to both terms:

This is why the minus sign reaches every term inside.

Quick Check: Spot the Error

A student expanded and got:

Is this correct? If not, what's the error?

Think before advancing.

Quick Check: Expansion Error Revealed

Error: The student distributed to the first term but not the second.

Correct answer:

Worked Example: Multi-Step — Distribute Then Combine

Step 1 — Distribute each factor:

Step 2 — Combine like terms:

Your Turn: Distribute and Combine

Distribute:

The terms cancel — result is just the constant .

Try it before reading the answer above.

Practice: Expand Using the Distributive Property

Try each problem before looking across.

Factoring Reverses the Expanding Process

Expanding and factoring are inverse operations:

• Expand:
• Factor:

Three-step process:

1. Find the GCF of all terms
2. Factor it out: write GCF × (remaining terms)
3. Verify by expanding — mandatory

Finding the GCF of Fraction Coefficients

When coefficients are fractions, use:

Example: Find the GCF of and

Verify: ✓ and ✓

Factor Expand: Same Expression, Two Forms

These are the same expression — neither form is "more correct." Context determines which is more useful.

Worked Examples: Factoring (Part 1)

Worked Examples: Factoring (Part 2)

Quick Check: Which Factoring Is Correct?

Factor . Expand each to verify.

Your Turn: Factor and Verify

Factor .

1. Find the GCF: GCF of and
2. Write the factored form
3. Verify by expanding

Work through all three steps, then advance.

Your Turn: Factor and Verify Solution

GCF: numerators GCF(3,1) = 1; denominators LCM(8,4) = 8 → GCF =

Verify: ✓

Verify by Expanding: Why It Matters

The verification step catches sign errors with negative GCFs.

Factor :

Attempt: → Verify: ✗

Correct: → Verify: ✓

Five seconds of checking catches the most common error.

Practice: Factor and Verify Each Expression

Try each before looking at the GCF and factored form.

Lesson Summary: Key Rules and Warnings

✓ Combine like terms: add coefficients, variable stays

✓ Subtract expressions: minus sign reaches every term

✓ Distribute to every term — count before computing

— don't combine unlike terms

Always verify factoring by expanding

Coming Up Next: Equivalent Expression Forms

Next lesson: 7.EE.A.2 — Rewriting Expressions in Equivalent Forms

You'll apply today's skills to understand why:

Same expression — different information revealed by each form.