How to Explain Expressions to Seventh Graders

"Let x equal the number of apples..."

When students first encounter variables, many wonder: "Why are we putting letters in math?" The answer is simple but powerful: variables let us describe patterns, relationships, and unknowns that numbers alone cannot express.

Seventh grade is when algebraic expressions become central to mathematics. Let's explore how to build understanding from the ground up.

Why Expressions Matter in Seventh Grade

Expressions are the language of algebra:

• Generalizing patterns: Instead of "3+3+3+3," write "4 × 3" or "4n"
• Representing unknowns: "Some number plus 5" becomes "n + 5"
• Modeling situations: "Cost = $5 per ticket times number of tickets" → C = 5t
• Simplifying calculations: Complex problems become manageable

Mastering expressions prepares students for:

• Solving equations
• Working with functions
• Graphing linear relationships
• All higher mathematics

Parts of an Expression

Vocabulary Essentials

Expression: 3x² + 5x - 7

Terms:     3x²    5x    -7
           ↓      ↓      ↓
         term   term   term
         (connected by + or -)

Coefficient: The number multiplied by a variable
           3x² → coefficient is 3
           5x  → coefficient is 5
           x   → coefficient is 1 (invisible)

Constant: A term without a variable
          -7 is a constant

Variable: A letter representing an unknown value
          x is the variable

Identifying Terms

Terms are separated by addition or subtraction signs:

4a + 3b - 2c + 7

Terms: 4a, 3b, -2c, 7
(Note: -2c, not 2c—the sign stays with the term)

Coefficients vs. Exponents

Don't confuse them!

5x³

5 = coefficient (multiplied)
3 = exponent (power)

"5 times x cubed"

Writing Expressions from Words

Key Translation Words

Addition: sum, plus, more than, increased by, total
"5 more than x" → x + 5

Subtraction: difference, minus, less than, decreased by
"7 less than y" → y - 7  (careful: y - 7, not 7 - y!)

Multiplication: product, times, of, twice, double
"twice a number" → 2n
"product of 3 and x" → 3x

Division: quotient, divided by, per, ratio
"n divided by 4" → n/4 or n ÷ 4

Watch for "Less Than" Trap

"5 less than x" is NOT "5 - x"

"5 less than x" → x - 5

Think: Start with x, then take away 5.

"x less than 5" → 5 - x

Think: Start with 5, then take away x.

Complex Expressions

"Three times the sum of x and 4"
= 3(x + 4)
(NOT 3x + 4—the sum must be found first)

"The product of 5 and x, decreased by 2"
= 5x - 2

"Half of y plus 3"
= y/2 + 3  or  (1/2)y + 3

Like Terms

What Makes Terms "Like"?

Like terms have the SAME variable(s) raised to the SAME power(s).

Like terms:           Unlike terms:
5x and 3x            5x and 5y (different variables)
4x² and -2x²         4x² and 4x (different exponents)
7 and -3             3xy and 3x (different variable combinations)
2xy and -5xy

Why Like Terms Matter

Only like terms can be combined:

• You can add 5 apples + 3 apples = 8 apples
• You can NOT add 5 apples + 3 oranges = 8 ??? (they're different things!)

Same with variables:

• 5x + 3x = 8x ✓
• 5x + 3y = 5x + 3y (cannot simplify) ✗

Combining Like Terms

The Process

Add or subtract the coefficients; keep the variable part the same.

7x + 4x = (7 + 4)x = 11x

9y - 3y = (9 - 3)y = 6y

-2a + 8a = (-2 + 8)a = 6a

With Multiple Terms

Simplify: 5x + 3y + 2x - y

Step 1: Identify like terms
        5x and 2x are like terms
        3y and -y are like terms

Step 2: Combine each group
        5x + 2x = 7x
        3y - y = 2y

Step 3: Write simplified expression
        7x + 2y

With Exponents

Simplify: 3x² + 5x - 2x² + 4x

Group like terms:
x² terms: 3x² - 2x² = 1x² = x²
x terms: 5x + 4x = 9x

Answer: x² + 9x

The Distributive Property

Expanding Expressions

a(b + c) = ab + ac

Multiply the outside term by EACH term inside the parentheses.

3(x + 4) = 3·x + 3·4 = 3x + 12

5(2y - 3) = 5·2y - 5·3 = 10y - 15

-2(a + 6) = -2·a + (-2)·6 = -2a - 12

Visual Model

3(x + 4) means "three groups of (x + 4)"

[x + 4] + [x + 4] + [x + 4]

= x + x + x + 4 + 4 + 4

= 3x + 12

With Negative Signs

Be careful with negatives!

-4(x - 5) = -4·x - (-4)·5
          = -4x - (-20)
          = -4x + 20

Remember: negative × negative = positive

Distributive Property with Variables

x(x + 3) = x·x + x·3 = x² + 3x

2y(y - 4) = 2y·y - 2y·4 = 2y² - 8y

Factoring Expressions

What Is Factoring?

Factoring is the reverse of distributing—finding what was multiplied.

Distributing:  3(x + 4) = 3x + 12

Factoring:     3x + 12 = 3(x + 4)

Finding the Greatest Common Factor (GCF)

Factor: 6x + 18

Step 1: Find GCF of coefficients
        GCF of 6 and 18 is 6

Step 2: Check for common variables
        First term has x, second doesn't
        No common variable

Step 3: Factor out the GCF
        6x + 18 = 6(x + 3)

Check: 6(x + 3) = 6x + 18 ✓

With Variables

Factor: 8x² + 12x

Step 1: GCF of 8 and 12 is 4
Step 2: Both terms have x (common variable)
        8x² has x², 12x has x
        Common variable factor: x

Step 3: Factor out 4x
        8x² + 12x = 4x(2x + 3)

Check: 4x(2x + 3) = 8x² + 12x ✓

Factoring Checklist

1. Find the GCF of ALL coefficients
2. Find the LOWEST power of any common variables
3. Divide each term by the GCF
4. Write as: GCF(remaining expression)
5. CHECK by distributing

Adding and Subtracting Expressions

Adding Expressions

(3x + 5) + (2x - 3)

Remove parentheses (no sign change):
= 3x + 5 + 2x - 3

Combine like terms:
= 5x + 2

Subtracting Expressions

Distribute the negative sign!

(3x + 5) - (2x - 3)

Distribute the negative:
= 3x + 5 - 2x + 3
       ↑      ↑
    changed signs!

Combine like terms:
= x + 8

Common Error

WRONG: (3x + 5) - (2x - 3) = 3x + 5 - 2x - 3 = x + 2

RIGHT: (3x + 5) - (2x - 3) = 3x + 5 - 2x + 3 = x + 8

The -3 becomes +3 because you're subtracting the whole expression.

Evaluating Expressions

Substitution

Replace the variable with the given value, then calculate.

Evaluate 3x + 7 when x = 4

Step 1: Substitute
        3(4) + 7

Step 2: Calculate
        12 + 7 = 19

With Multiple Variables

Evaluate 2a - 3b when a = 5 and b = -2

Step 1: Substitute
        2(5) - 3(-2)

Step 2: Calculate
        10 - (-6)
        10 + 6 = 16

With Exponents

Evaluate x² - 4x + 3 when x = -2

Step 1: Substitute (use parentheses!)
        (-2)² - 4(-2) + 3

Step 2: Calculate
        4 - (-8) + 3
        4 + 8 + 3 = 15

Hands-On Activities

Expression Sort

Write terms on cards. Students sort into groups of like terms:

Cards: 5x, 3y, -2x, 4, 7y, -1, x², 2x², -3

Groups:
x terms: 5x, -2x
y terms: 3y, 7y
x² terms: x², 2x²
constants: 4, -1, -3

Algebra Tiles

Use physical or virtual algebra tiles:

• Large square = x²
• Rectangle = x
• Small square = 1

Build expressions and combine like terms visually.

Expression Auction

Students bid on expressions. After purchasing, they simplify. Whoever has the greatest simplified value wins!

Expression 1: 3(x + 2) when x = 5
Expression 2: 4x - 7 when x = 5
Expression 3: x² - 10 when x = 5

Simplify and evaluate to find the winner!

Create Your Own Word Problems

Give an expression; students write a real-world situation:

Expression: 2.50x + 15

Student story: "A gym membership costs $15 per month plus
$2.50 per fitness class. The total cost for x classes is 2.50x + 15."

Expression Telephone

Like the telephone game, but with algebra:

1. First student writes an expression
2. Second student simplifies it
3. Third student writes an equivalent expression
4. Continue—does it come back the same?

Common Mistakes and How to Fix Them

Mistake 1: Combining Unlike Terms

Error: 3x + 5 = 8x

Fix: Check if terms are "like." 3x has a variable; 5 is a constant. They're different "types" and cannot be combined. Answer: 3x + 5 (already simplified).

Mistake 2: Forgetting to Distribute to All Terms

Error: 4(x + 3) = 4x + 3

Fix: The 4 must multiply EVERY term inside. 4(x + 3) = 4x + 12. Use arrows to show each multiplication.

Mistake 3: Sign Errors When Distributing Negatives

Error: -2(x - 5) = -2x - 10

Fix: -2 times -5 is +10, not -10. Write it out: -2(x - 5) = (-2)(x) + (-2)(-5) = -2x + 10.

Mistake 4: Wrong Order for "Less Than"

Error: "6 less than n" written as 6 - n

Fix: Read it backwards: "6 less than n" = "n minus 6" = n - 6. The number after "less than" comes first.

Mistake 5: Not Using Parentheses When Substituting Negatives

Error: Evaluating x² when x = -3 as -3² = -9

Fix: -3² means -(3²) = -9, but we want (-3)². Use parentheses: (-3)² = (-3)(-3) = 9.

Connecting to Other Concepts

Expressions to Equations

Add an equals sign to create an equation:

Expression: 3x + 5
Equation: 3x + 5 = 14

Simplifying expressions is essential for solving equations.

Expressions and Functions

Functions are expressions with specific input-output relationships:

f(x) = 2x + 3

This expression gives outputs for any x input.

Expressions and Area

Area formulas are expressions:

Rectangle: A = lw (expression with two variables)
Square: A = s² (expression with one variable)

Expressions and Proportional Relationships

Proportional relationships use expressions:

y = kx

"y equals k times x" — an expression showing the relationship

Practice Ideas for Home

Daily Expression Writing

Turn daily activities into expressions:

• "You read x pages per day. After 5 days, you've read ___"
• "Each apple costs $0.75. For a apples, you pay ___"

Simplification Races

Who can simplify fastest (correctly)?

5x + 3 - 2x + 7 - x = ?

Evaluate for Different Values

Give one expression, evaluate for multiple values:

Expression: 2x² - x + 1

When x = 0: ___
When x = 1: ___
When x = 2: ___
When x = -1: ___

Equivalent Expression Hunt

Find different expressions equal to 6x + 12:

6(x + 2) ✓
3(2x + 4) ✓
2(3x + 6) ✓

Real-World Expression Problems

• Calculate total cost with expression: items × price + tax
• Distance formula: rate × time
• Perimeter of rectangle: 2l + 2w or 2(l + w)

The Bottom Line

Expressions are the building blocks of algebra. They let us describe mathematical relationships in general terms, work with unknown quantities, and simplify complex problems.

Key takeaways:

1. Terms are separated by + or - signs
2. Like terms have the same variable(s) to the same power(s)
3. Distributive property: a(b + c) = ab + ac
4. Factoring reverses distribution
5. When subtracting expressions, distribute the negative sign

When seventh graders master expressions, they've opened the door to all of algebra. They can describe patterns, model real situations, and prepare for equations—the next step in their mathematical journey.