Algebra — Class 6 Mathematics
'Algebra is arithmetic with a secret — the secret of the unknown, which we call x.'
1. Introduction
So far, all our mathematics has been with known numbers — 5 apples, 12 metres, Rs. 45. But what if we don't know a number? What if we say 'there are some mangoes in a basket'? In algebra, we use a variable (like x, y, n) to represent the unknown.
Why Algebra Matters
- Solves problems where information is missing
- Foundation for ALL higher mathematics
- Used in science, economics, computer programming
- AP context: calculating crop yield with unknown area, budgeting with unknown costs
Arithmetic vs Algebra
| Arithmetic | Algebra |
|---|---|
| 5 + 3 = ? | 5 + x = 12 |
| 12 ÷ 4 = 3 | n ÷ 4 = 3 |
| Only known numbers | Uses variables for unknown |
2. Variables
A variable is a letter or symbol that can take different values.
- Common variables: x, y, z, a, b, c, n, p, q
- A variable is NOT fixed — it can stand for any number
- Example: 'Let n be the number of students' — n could be 30, 35, 42, etc.
Where Variables Come From
- Unknown quantity: 'I bought some apples' → a apples
- Pattern: 2, 4, 6, 8, ... → 2n
- General rule: 'Perimeter of a square = 4s' — here s is a variable
Variable vs Constant
| Variable | Constant |
|---|---|
| Can change value | Fixed value |
| Example: n, x | Example: 5, 10, π |
| Represented by letters | Represented by numbers |
'Think of a variable as an empty box. You can put different numbers into it.'
3. Using Variables in Patterns
Pattern 1: Matchstick Pattern
□ □□ □□□ □□□□
1 2 3 4
Number of matchsticks needed for n squares: Rule = 3n + 1 (each square uses 3 new matchsticks + 1 for first)
- n = 1: 3(1) + 1 = 4 matches
- n = 2: 3(2) + 1 = 7 matches
- n = 5: 3(5) + 1 = 16 matches
Pattern 2: Number Patterns
| Pattern | Rule | Next Term |
|---|---|---|
| 3, 6, 9, 12, ... | 3n | 15 |
| 5, 10, 15, 20, ... | 5n | 25 |
| 1, 4, 9, 16, ... | n² | 25 |
| 2, 4, 6, 8, ... | 2n | 10 |
Pattern 3: Triangle Pattern
△ △△ △△△
Matchsticks for n triangles = 2n + 1
| n (triangles) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Matchsticks | 3 | 5 | 7 | 9 | 11 |
4. Expressions with Variables
An expression is a combination of variables and numbers using operations.
| Expression | Meaning | Value when x=5 |
|---|---|---|
| x + 3 | Add 3 to x | 8 |
| 2x | 2 times x | 10 |
| x/2 | x divided by 2 | 2.5 |
| x − 4 | Subtract 4 from x | 1 |
| 3x + 2 | 3 times x, then add 2 | 17 |
Forming Expressions from Statements
| Statement | Expression |
|---|---|
| Add 7 to x | x + 7 |
| Subtract 5 from y | y − 5 |
| 3 times p | 3p |
| One-fifth of m | m/5 |
| 2 more than 3 times z | 3z + 2 |
| 4 less than twice n | 2n − 4 |
Evaluating Expressions
Substitute the variable with a number, then calculate.
Example: Evaluate 3x + 5 when x = 7 3(7) + 5 = 21 + 5 = 26
Example: Evaluate 2y − 3 when y = 4 2(4) − 3 = 8 − 3 = 5
Example: Evaluate 4a + 2b when a = 3, b = 5 4(3) + 2(5) = 12 + 10 = 22
5. Using Expressions Practically
Perimeter Formulas
- Square: P = 4s (s = side)
- Rectangle: P = 2(l + b)
- Equilateral triangle: P = 3s
Area Formulas
- Square: A = s²
- Rectangle: A = l × b
Cost Calculations
Total cost = Number of items × Price per item + Fixed cost
Example: A plumber charges Rs. 200 as visit fee and Rs. 150 per hour. Total cost = 200 + 150h (where h = hours worked)
- h = 2: 200 + 150(2) = Rs. 500
- h = 5: 200 + 150(5) = Rs. 950
Age Problems
'Ravi is 4 years older than his sister.' Let sister's age = n Ravi's age = n + 4
- Sister is 8 → Ravi is 12
- Ravi is 14 → sister is 10
6. Equation — What Is It?
An equation is a mathematical statement with an equals sign (=) showing that two expressions are equal.
Equation vs Expression
| Expression | Equation |
|---|---|
| 3x + 5 | 3x + 5 = 20 |
| No equals sign | Has equals sign |
| Cannot be solved | Can be solved |
Parts of an Equation
3x + 5 = 20
│ │ │
│ │ └── Right-hand side (RHS)
│ └── equals
└── Left-hand side (LHS)
True or False Equations
- 3 + 5 = 8 → TRUE (both sides equal)
- 2 × 6 = 10 → FALSE (12 ≠ 10)
- x + 3 = 7 → depends on x (TRUE if x=4, FALSE otherwise)
7. Solution of an Equation
The solution of an equation is the value of the variable that makes the equation TRUE.
Trial and Error Method
Try different values until you find the one that makes LHS = RHS.
Example: Solve x + 5 = 12
| x | LHS: x+5 | RHS: 12 | LHS = RHS? |
|---|---|---|---|
| 5 | 10 | 12 | No |
| 6 | 11 | 12 | No |
| 7 | 12 | 12 | YES ✓ |
Solution: x = 7
Example: Solve 2x − 3 = 7
| x | LHS: 2x-3 | RHS: 7 | LHS = RHS? |
|---|---|---|---|
| 3 | 3 | 7 | No |
| 4 | 5 | 7 | No |
| 5 | 7 | 7 | YES ✓ |
Solution: x = 5
'Start with a reasonable guess. If the LHS is too small, try a larger number. Too big? Try a smaller one.'
8. Simple Word Problems
Problem 1 (Fruit)
A basket has mangoes. After adding 12 more, there are 28. How many mangoes were there initially?
Solution: Let initial mangoes = x x + 12 = 28 By trial: x = 16 (16 + 12 = 28 ✓) Answer: 16 mangoes
Problem 2 (Age)
Ravi's father is 45 years old. He is 3 times as old as Ravi. How old is Ravi?
Solution: Let Ravi's age = n 3n = 45 By trial: n = 15 (3×15 = 45 ✓) Answer: Ravi is 15 years old
Problem 3 (AP Context — Mangoes per Basket)
A farmer packs mangoes into baskets. Each basket holds 24 mangoes. She fills 5 baskets and has 12 mangoes left. How many mangoes did she have?
Solution: Let total mangoes = y y = 24 × 5 + 12 y = 120 + 12 = 132 Answer: 132 mangoes
Problem 4 (Money)
Neha and Priya together have Rs. 500. Neha has Rs. 150. How much does Priya have?
Solution: Let Priya's money = p 150 + p = 500 By trial: p = 350 (150 + 350 = 500 ✓) Answer: Rs. 350
9. Common Mistakes — Fix Them Now
| # | Mistake | Correction |
|---|---|---|
| 1 | Writing 2x as x + x (fine) but also as 2 + x | 2x means 2 × x, not 2 + x |
| 2 | Thinking x means only one value | x can be MANY values in expressions; only ONE in an equation |
| 3 | Writing n/2 as 2/n | n/2 = n ÷ 2, not 2 ÷ n |
| 4 | Forgetting to use trial systematically | Start with a sensible guess, adjust based on whether LHS too big/small |
| 5 | Writing 3z + 2 when statement is '2 more than z times 3' | 3z + 2 is correct for '2 more than 3 times z' |
10. Exam Focus
Marks Blueprint
| Question Type | Marks | Topic |
|---|---|---|
| MCQ | 1 | Variable identification |
| Short answer | 2 | Form expression from statement |
| Short answer | 2 | Evaluate expression |
| Word problem | 3 | Form and solve equation |
| Pattern | 2 | Matchstick pattern rule |
Quick Self-Test (5 Questions)
Q1: Write an expression: '7 less than twice a number n'.
<details><summary>Answer</summary>2n − 7</details>Q2: Evaluate 4a + 3 when a = 6.
<details><summary>Answer</summary>4(6)+3 = 24+3 = 27</details>Q3: Solve by trial: x − 8 = 15
<details><summary>Answer</summary>x = 23 (23−8 = 15 ✓)</details>Q4: A number multiplied by 6 gives 54. Find the number.
<details><summary>Answer</summary>6n = 54, n = 9</details>Q5: Write the rule for the pattern: 7, 14, 21, 28, ...
<details><summary>Answer</summary>7n (or 7 × n)</details>11. Chapter Summary
- Variable: a letter representing an unknown (can change value)
- Expression: combination of variables and numbers (no equals sign)
- Equation: expression = expression (has equals sign)
- Solution: value of variable that makes the equation true
- Trial and error: try values systematically until LHS = RHS
- Patterns: algebra helps describe patterns with rules
'Algebra is the language of patterns and the key to solving puzzles. Once you learn to think in variables, a whole new world of mathematics opens up.'
