Class 6 Mathematics Notes Chapter 11 (Algebra) – Mathematics Book

Detailed Notes with MCQs of Chapter 11: Algebra from your NCERT Class 6 Mathematics textbook. This chapter is fundamental, not just for your current studies, but it lays the groundwork for higher mathematics and is often tested in basic aptitude sections of government exams. Pay close attention.

Chapter 11: Algebra - Detailed Notes

1. Introduction to Algebra

• What is Algebra? Algebra is a branch of mathematics that uses letters (like x, y, a, b, l, etc.) to represent numbers. These letters are called variables. It helps us generalize arithmetic rules and solve problems where quantities are unknown.
• Why Variables? Instead of working only with specific numbers (like 2, 5, -7), variables allow us to talk about numbers in general or represent quantities whose value can change or is currently unknown.
• Constants: Numbers with fixed values (like 4, 100, 0, -15) are called constants.

2. Matchstick Patterns (Introduction to Rules using Variables)

• The chapter often starts with patterns (like making letters with matchsticks) to illustrate how rules can be expressed using variables.
• Example: To make the letter 'L', you need 2 matchsticks.
  • For 1 'L': 2 matchsticks (2 x 1)
  • For 2 'L's: 4 matchsticks (2 x 2)
  • For 3 'L's: 6 matchsticks (2 x 3)
  • For 'n' 'L's: 2n matchsticks.
• Here, 'n' is the variable representing the number of 'L's, and '2n' is the rule (an algebraic expression) to find the total matchsticks needed.

3. The Idea of a Variable

• A variable can take different numerical values. Its value is not fixed.
• We use letters like x, y, z, a, b, c, l, m, n, p, etc., to denote variables.
• Contrast: A constant has a fixed value (e.g., the number 5 always means five). A variable (e.g., 'x') can represent 5 in one problem and maybe 10 or -2 in another.

4. Use of Variables in Common Rules

• Geometry: Variables make writing formulas concise.
  • Perimeter of a Square: If the side length is 'l', the perimeter is l + l + l + l = 4l. Here 'l' is the variable.
  • Perimeter of a Rectangle: If length is 'l' and breadth is 'b', the perimeter is l + b + l + b = 2l + 2b = 2(l + b). Here 'l' and 'b' are variables.
  • Area of a Square: Side 'l', Area = l × l = l².
  • Area of a Rectangle: Length 'l', Breadth 'b', Area = l × b (or lb).
• Arithmetic Properties: Variables help state general rules.
  • Commutativity of Addition: a + b = b + a
  • Commutativity of Multiplication: a × b = b × a
  • Associativity of Addition: (a + b) + c = a + (b + c)
  • Associativity of Multiplication: (a × b) × c = a × (b × c)
  • Distributivity of Multiplication over Addition: a × (b + c) = a × b + a × c

5. Expressions with Variables

• An algebraic expression is a combination of constants, variables, and arithmetic operations (+, -, ×, ÷).
• Formation:
  • y + 5: 5 added to y
  • t - 7: 7 subtracted from t
  • 10a: 10 multiplied by a
  • x / 3 (or x/3): x divided by 3
  • 2y - 5: First y multiplied by 2, then 5 subtracted from the product.
• Writing Statements as Expressions: This is a key skill.
  • "Sarita has 10 more marbles than Amina. If Amina has x marbles, Sarita has..." -> x + 10 marbles.
  • "Raju's father's age is 2 years more than 3 times Raju's age. If Raju's age is y years, his father's age is..." -> 3y + 2 years.
  • "The price of rice per kg is ₹5 less than the price of wheat per kg. If wheat costs ₹p per kg, rice costs..." -> ₹(p - 5) per kg.

6. Introduction to Equations

• An equation is a statement that two expressions are equal. It always contains an equality sign (=).
• Example: x + 10 = 30 is an equation.
  • x + 10 is the Left Hand Side (LHS).
  • 30 is the Right Hand Side (RHS).
• Difference from Expression: An expression (like x + 10) just represents a value, while an equation (x + 10 = 30) makes a claim of equality.
• Solution of an Equation: A value of the variable that makes the equation true (i.e., makes LHS equal to RHS) is called the solution or root of the equation.
  • In x + 10 = 30, if we try x = 20, LHS = 20 + 10 = 30. Since LHS = RHS (30 = 30), x = 20 is the solution.
  • If we try x = 15, LHS = 15 + 10 = 25. Since 25 ≠ 30, x = 15 is not the solution.
• Solving (Class 6 Method): At this level, solutions are often found by:
  • Trial and Error: Trying different values for the variable until the equation holds true.
  • Inspection: Looking at the equation and reasoning out the value (e.g., in m + 5 = 8, we can see m must be 3).

Key Takeaways for Exams:

• Understand the difference between a variable and a constant.
• Be able to form simple algebraic expressions from word statements.
• Know how variables are used in basic geometric formulas and arithmetic properties.
• Understand what an equation is and what it means to find its solution.
• Be able to check if a given value is a solution to a simple equation.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on Chapter 11 concepts, suitable for practice:

1. Which of the following is a variable?
   (A) 7
   (B) -100
   (C) The number of days in January
   (D) The temperature in a city on a given day

2. The expression for "5 subtracted from p" is:
   (A) 5 - p
   (B) p - 5
   (C) 5 + p
   (D) 5p

3. If the side of a regular pentagon is denoted by 's', its perimeter is:
   (A) 3s
   (B) 4s
   (C) 5s
   (D) 6s

4. Which of the following is an equation?
   (A) x + 3
   (B) 2y - 5 > 10
   (C) 7a + 1 = 15
   (D) 4b - 9

5. Leela is Radha's younger sister. Leela is 4 years younger than Radha. If Radha's age is 'x' years, what is Leela's age?
   (A) 4x
   (B) x + 4
   (C) x - 4
   (D) 4 - x

6. The rule for the number of matchsticks required to make a pattern of 'n' squares (like ☐☐☐...) in a row is:
   (A) 2n + 2
   (B) 4n
   (C) 3n + 1
   (D) 4n - 1

7. What is the value of the expression 2m - 3 if m = 5?
   (A) 7
   (B) 10
   (C) 13
   (D) 4

8. Which value of 'y' is a solution to the equation y + 8 = 12?
   (A) 20
   (B) 8
   (C) 12
   (D) 4

9. The statement "A number x multiplied by 3 and then 2 added to the product" can be written as:
   (A) x + 3 + 2
   (B) 3x + 2
   (C) 3(x + 2)
   (D) 2x + 3

10. In the formula for the area of a rectangle, A = l × b, what are l and b?
    (A) Both are constants
    (B) Both are variables
    (C) l is a variable, b is a constant
    (D) l is a constant, b is a variable

Answer Key for MCQs:

1. (D) - Temperature can vary. Others are fixed numbers or definitions.
2. (B)
3. (C) - A pentagon has 5 sides. Perimeter is the sum of side lengths.
4. (C) - Only this option has an equality sign (=) relating two expressions.
5. (C) - Leela is younger, so subtract 4 from Radha's age.
6. (C) - 1 square: 4 sticks. 2 squares: 7 sticks (4+3). 3 squares: 10 sticks (7+3). Pattern: 4 + 3(n-1) = 4 + 3n - 3 = 3n + 1.
7. (A) - 2(5) - 3 = 10 - 3 = 7.
8. (D) - 4 + 8 = 12.
9. (B)
10. (B) - Length and breadth can vary for different rectangles.

Study these notes carefully. Understanding these basics of Algebra is crucial for your future mathematical journey and competitive exams. Let me know if any part needs further clarification.