How to Explain Equations to Sixth Graders

Solving equations is where algebra gets exciting—students become detectives uncovering the mystery value of x! This guide helps you teach equation-solving so students understand the reasoning, not just the steps.

Why Equations Matter for Sixth Graders

Equations are powerful tools for:

• Solving real problems: "I have $23 after spending $12. How much did I start with?"
• Reasoning backward: Finding unknown starting values
• Translating situations: Turning word problems into solvable math
• Building logical thinking: Following systematic procedures

Students who master equation-solving are prepared for:

• Multi-step equations in 7th grade
• Systems of equations in 8th grade
• All high school math courses

Key Concepts Broken Down Simply

What Is an Equation?

An equation is a mathematical statement that two expressions are equal.

Parts of an equation:

    2x + 5 = 13
    ↑       ↑
 expression expression
        ↑
   equals sign

The equation says: "2x + 5 has the same value as 13"

The Balance Concept

Think of an equation as a balanced scale:

         2x + 5           =           13

        ┌─────┐                   ┌─────┐
        │2x+5 │                   │ 13  │
     ═══╧═════╧═══════════════════╧═════╧═══
              \                   /
               \                 /
                \               /
                 \             /
                  \           /
                   \         /
                    ─────────
                        △

Whatever you do to ONE side, you MUST do to the OTHER side!

Inverse Operations

Inverse operations undo each other:

┌────────────────────┬────────────────────┐
│    OPERATION       │    INVERSE         │
├────────────────────┼────────────────────┤
│   Addition (+)     │   Subtraction (-)  │
│   Subtraction (-)  │   Addition (+)     │
│   Multiplication(×)│   Division (÷)     │
│   Division (÷)     │   Multiplication(×)│
└────────────────────┴────────────────────┘

Examples:
  +5 and -5 are inverses (they undo each other)
  ×3 and ÷3 are inverses

Solving One-Step Equations

Addition equation: x + 7 = 12

Goal: Get x alone (isolate the variable)

    x + 7 = 12

Step: Undo "+7" by subtracting 7 from BOTH sides

    x + 7 - 7 = 12 - 7
    x + 0 = 5
    x = 5

Check: 5 + 7 = 12 ✓

Subtraction equation: x - 4 = 9

    x - 4 = 9

Step: Undo "-4" by adding 4 to BOTH sides

    x - 4 + 4 = 9 + 4
    x = 13

Check: 13 - 4 = 9 ✓

Multiplication equation: 3x = 18

    3x = 18

Step: Undo "×3" by dividing BOTH sides by 3

    3x ÷ 3 = 18 ÷ 3
    x = 6

Check: 3(6) = 18 ✓

Division equation: x/5 = 4

    x/5 = 4

Step: Undo "÷5" by multiplying BOTH sides by 5

    x/5 × 5 = 4 × 5
    x = 20

Check: 20/5 = 4 ✓

Solving Two-Step Equations

Two-step equations require undoing two operations in reverse order.

Solve: 2x + 5 = 13

Think: What happened to x?
  1. x was multiplied by 2
  2. Then 5 was added

Undo in REVERSE order:
  1. First, undo the +5 (subtract 5)
  2. Then, undo the ×2 (divide by 2)

Step 1: Subtract 5 from both sides
    2x + 5 - 5 = 13 - 5
    2x = 8

Step 2: Divide both sides by 2
    2x ÷ 2 = 8 ÷ 2
    x = 4

Check: 2(4) + 5 = 8 + 5 = 13 ✓

Another example: (x - 3)/4 = 2

Think: What happened to x?
  1. 3 was subtracted from x
  2. Then the result was divided by 4

Undo in REVERSE order:
  1. First, undo the ÷4 (multiply by 4)
  2. Then, undo the -3 (add 3)

Step 1: Multiply both sides by 4
    (x - 3)/4 × 4 = 2 × 4
    x - 3 = 8

Step 2: Add 3 to both sides
    x - 3 + 3 = 8 + 3
    x = 11

Check: (11 - 3)/4 = 8/4 = 2 ✓

Order of Operations in Reverse

When solving, undo operations in reverse order:

Building an expression:    Undoing it:
Start with x               End with x
↓ multiply by 2           ↑ divide by 2
2x                         4
↓ add 5                   ↑ subtract 5
2x + 5                     9
↓ equals                  ↑ given
9                          2x + 5 = 9

Visual Examples and Diagrams

Balance Scale Model

Solve: x + 3 = 7

       ┌─────┐     ┌───┐
       │  x  │     │ 7 │
       │     │     │   │
       │ +3  │     │   │
    ═══╧═════╧═════╧═══╧═══
              \_____/
                 △      Balanced!

Remove 3 from BOTH sides:

       ┌─────┐     ┌───┐
       │  x  │     │ 4 │
    ═══╧═════╧═════╧═══╧═══
              \_____/
                 △      Still balanced!

x = 4

Tape Diagram Model

Solve: 3x = 12

3x means "three x's"

┌─────┬─────┬─────┐
│  x  │  x  │  x  │  = 12
└─────┴─────┴─────┘

If all three parts equal 12...
Each part equals 12 ÷ 3 = 4

┌─────┬─────┬─────┐
│  4  │  4  │  4  │  = 12
└─────┴─────┴─────┘

x = 4

Bar Model for Two-Step Equations

Solve: 2x + 3 = 11

┌─────────────────────┬─────┐
│         2x          │  3  │ = 11
└─────────────────────┴─────┘

Remove the 3: 2x = 11 - 3 = 8

┌─────────────────────┐
│         2x          │ = 8
└─────────────────────┘

Split 2x into 2 equal parts:

┌──────────┬──────────┐
│    x     │    x     │ = 8
└──────────┴──────────┘

Each x = 8 ÷ 2 = 4

x = 4

Cover-Up Method

Solve: 5x - 3 = 17

Cover the "5x" with your hand:

[COVERED] - 3 = 17

What minus 3 equals 17?
Answer: 20

So 5x = 20

Now cover the "5":

[COVERED] × x = 20

What times x equals 20?
5 × 4 = 20

x = 4

Hands-On Activities

Activity 1: Balance Scale Practice

Materials: Pan balance (or ruler balanced on pencil), small objects of equal weight

Setup: Use coins or cubes where each represents "1"

Task: Model equations like x + 3 = 7

• Put 3 objects on one side with a mystery box (x)
• Put 7 objects on the other side
• Remove objects equally from both sides to find x

Activity 2: Equation Card Match

Materials: Cards with equations and cards with their solutions

Examples:

• x + 5 = 12 ↔ x = 7
• 3x = 15 ↔ x = 5
• x/4 = 3 ↔ x = 12

Students match equations to solutions.

Activity 3: Working Backward

Game: "I'm thinking of a number..."

"I'm thinking of a number.
 I multiplied it by 4 and got 20.
 What's my number?"

Equation: 4x = 20
Solution: x = 5

Have students create their own puzzles for partners.

Activity 4: Equation Building

Materials: Expression cards, equals sign cards, number cards

Task: Build valid equations and solve them

Cards: [3x] [+] [2] [=] [14]

Arrange: 3x + 2 = 14
Solve: x = 4

Activity 5: Real-World Equation Hunt

Find situations that create equations:

1. "Movie tickets cost $9 each. Total cost was $36. How many tickets?"

   • Equation: 9t = 36

2. "After spending $15, you have $23 left. How much did you start with?"

   • Equation: m - 15 = 23

Common Mistakes and How to Fix Them

Mistake 1: Operating on Only One Side

Wrong:

x + 5 = 12
x = 12 - 5    (only subtracted from right side)
x = 7         (correct by accident, but wrong process!)

Better process:

x + 5 = 12
x + 5 - 5 = 12 - 5  (subtract from BOTH sides)
x = 7

Fix: Always show the operation on both sides until it becomes automatic.

Mistake 2: Using Wrong Inverse Operation

Wrong:

3x = 12
3x - 3 = 12 - 3  (should divide, not subtract!)

Correct:

3x = 12
3x ÷ 3 = 12 ÷ 3
x = 4

Fix: Identify what operation is applied to x. "3x" means x is MULTIPLIED by 3, so DIVIDE to undo.

Mistake 3: Wrong Order for Two-Step Equations

Wrong:

2x + 5 = 13
Divide by 2 first:
x + 5 = 6.5 (WRONG!)

Correct:

2x + 5 = 13
Subtract 5 first:
2x = 8
Then divide by 2:
x = 4

Fix: Remember "PEMDAS in reverse"—undo addition/subtraction before multiplication/division.

Mistake 4: Sign Errors with Negatives

Wrong:

x - 7 = 3
x - 7 - 7 = 3 - 7  (should ADD, not subtract!)

Correct:

x - 7 = 3
x - 7 + 7 = 3 + 7
x = 10

Fix: "To undo subtraction, add" — even though it might feel counterintuitive.

Mistake 5: Forgetting to Check

Missing step: Always substitute back!

Solve: 2x + 3 = 11
Solution: x = 4

CHECK: 2(4) + 3 = 8 + 3 = 11 ✓

The check catches errors!

Practice Ideas for Home

One-Step Equation Practice

Addition/Subtraction:
  x + 8 = 15    →  x = 7
  n - 4 = 9     →  n = 13
  y + 12 = 20   →  y = 8
  m - 7 = 7     →  m = 14

Multiplication/Division:
  4x = 28       →  x = 7
  n/3 = 6       →  n = 18
  5y = 35       →  y = 7
  m/8 = 4       →  m = 32

Two-Step Equation Practice

Level 1:
  2x + 3 = 11   →  x = 4
  3n - 5 = 10   →  n = 5
  4y + 2 = 18   →  y = 4

Level 2:
  x/2 + 4 = 7   →  x = 6
  (n - 3)/5 = 2 →  n = 13
  5x - 8 = 22   →  x = 6

Word Problem Practice

Translate and solve:

1. "Five more than a number is 18. Find the number."

   • x + 5 = 18, x = 13

2. "A number divided by 4 equals 9. Find the number."

   • n/4 = 9, n = 36

3. "Twice a number plus 7 equals 19. Find the number."

   • 2x + 7 = 19, x = 6

Create Your Own

Have students write equations for:

• Their age-related problems
• Money scenarios
• Measurement situations

Connection to Future Math Concepts

7th Grade: Multi-Step Equations

3(x + 2) - 4 = 14

Same principles, more steps:
  3x + 6 - 4 = 14
  3x + 2 = 14
  3x = 12
  x = 4

8th Grade: Equations with Variables on Both Sides

5x + 3 = 2x + 12

Move variables to one side:
  3x + 3 = 12
  3x = 9
  x = 3

8th Grade: Systems of Equations

x + y = 10
2x - y = 5

Solve for multiple variables!

High School: Quadratic Equations

x² + 5x + 6 = 0

Different methods, same goal: find x

Quick Reference

┌────────────────────────────────────────────────────┐
│           EQUATIONS QUICK REFERENCE                │
├────────────────────────────────────────────────────┤
│ GOAL: Isolate the variable (get x alone)          │
│                                                    │
│ GOLDEN RULE: Whatever you do to one side,         │
│              do to the other side!                │
│                                                    │
│ INVERSE OPERATIONS:                               │
│   + ↔ -    (add undoes subtract)                 │
│   × ↔ ÷    (multiply undoes divide)              │
│                                                    │
│ TWO-STEP STRATEGY:                                │
│   1. Undo addition/subtraction FIRST              │
│   2. Undo multiplication/division SECOND          │
│                                                    │
│ ALWAYS CHECK: Substitute answer back in!          │
│   If both sides equal → correct!                  │
└────────────────────────────────────────────────────┘

Tips for Teaching Success

1. Start with the balance: The visual model builds lasting understanding
2. Emphasize inverse operations: Name them explicitly every time
3. Require checking: Make it part of every solution
4. Progress gradually: Master one-step before two-step
5. Connect to real life: "What's the missing value?" is everywhere

Equation-solving is a cornerstone of algebra. When students understand that solving means "undoing operations to find the unknown," they have a framework that extends through all of mathematics. With consistent practice using the balance model and inverse operations, your sixth grader will develop the problem-solving confidence they need.