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Mathematics › ⚡ Grade 6 › Expressions & Variables: The Machine That Waits for a Number
⚡ Grade 6 · Lesson 8 of 14

Expressions & Variables: The Machine That Waits for a Number

A variable is a blank box — plug in a number and the expression does the rest.

Grade 6Middle School
Expressions & Variables: The Machine That Waits for a Number — illustration
💡
The big idea: An algebraic expression is a recipe: it tells you exactly what operations to perform on an unknown input. The variable (x, n, or any letter) is just a placeholder — a blank slot waiting to be filled. Evaluating the expression means substituting the number and following the order of operations. Writing an expression means translating a word-problem rule into that compact recipe.
🎯 By the end, you'll be able to
  • Evaluate algebraic expressions by substituting given values for variables
  • Write algebraic expressions from word descriptions
  • Identify terms, coefficients, and constants in an expression
  • Simplify expressions by combining like terms
📎 You should already know
  • Order of operations (PEMDAS/BODMAS)
  • Multiplication of whole numbers

Terms, coefficients, and constants

In the expression 3x + 5:

  • 3x is a term — it involves the variable x. The number 3 in front is the coefficient (how many x's).
  • 5 is a constant term — it has no variable, its value never changes.
  • The two terms are separated by +.

Larger example: 4x² − 7x + 2. Three terms: coefficient 4 on x², coefficient −7 on x, constant 2.

🎮 Expression Builder & Evaluator LIVE
Choose one of the preset expressions, set the input value x using the slider, and watch the machine show the step-by-step calculation from INPUT through each operation to OUTPUT. Change expressions to compare behaviours.
🔑 Evaluating: substitute then follow PEMDAS

Evaluate 2x² − 3x + 1 at x = 4:

  1. Replace x with 4: 2(4)² − 3(4) + 1
  2. Exponents first: 2(16) − 3(4) + 1
  3. Multiply: 32 − 12 + 1
  4. Left to right: 20 + 1 = 21
⚠️ Like terms — only same variable AND same power

You can combine like terms: 3x + 5x = 8x (same variable x, same power 1).

You CANNOT combine: 3x + 5x² (different powers — x vs x²), or 3x + 5 (variable term + constant).

Think of it like units: apples + apples = apples, but apples + oranges stay separate.

✨ Writing expressions from words
  • '5 more than n' → n + 5
  • '3 times a number, decreased by 7' → 3n − 7
  • 'twice the sum of x and 4' → 2(x + 4) (the sum is grouped in parentheses)
  • 'a number divided by 6, plus 1' → n/6 + 1

Key: multiplication and 'of' usually signal ×; 'more than / increased by' signal +; 'less than / decreased by' signal − (watch the order: '5 less than n' = n − 5, not 5 − n).

📝 Worked example: Evaluate 4x − 3 at x = 5 and at x = −2.
  1. x = 5: 4(5) − 3 = 20 − 3 = 17.
  2. x = −2: 4(−2) − 3 = −8 − 3 = −11.
✓ At x = 5: <strong>17</strong>. At x = −2: <strong>−11</strong>.
📝 Worked example: Simplify: 5x + 3 − 2x + 7.
  1. Group like terms: (5x − 2x) + (3 + 7).
  2. Combine: 3x + 10.
✓ <strong>3x + 10</strong>.

Check your understanding

1. Evaluate 3n + 4 when n = 6.
3(6) + 4 = 18 + 4 = 22.
2. Which expression means 'twice the sum of a number and 3'?
'Twice the SUM of a number and 3' — the sum n + 3 is formed first (parentheses), then doubled: 2(n + 3).
3. Simplify: 7x − 2 + 3x + 8.
(7x + 3x) + (−2 + 8) = 10x + 6.
4. In the expression 6y² − 4y + 9, what is the coefficient of y?
The term with y (not y²) is −4y. Its coefficient is −4.
5. Evaluate x² + 2x − 1 at x = 3.
3² + 2(3) − 1 = 9 + 6 − 1 = 14.
✅ Key takeaways
  • A variable is a placeholder for a number; an expression is a recipe of operations applied to it.
  • To evaluate, substitute the value and follow PEMDAS (Parentheses, Exponents, ×/÷, +/−).
  • Terms with the same variable and same power are 'like terms' — only those can be combined.
  • Coefficients multiply the variable; constants are standalone numbers with no variable.
  • When writing expressions from words, 'less than' reverses order (5 less than n = n − 5).
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