Mathematics Content (Class VI-VIII) — PSTET Paper II Study Notes

Overview

Mathematics Content for Classes VI-VIII forms a substantial portion of PSTET Paper II, testing your command of upper-primary mathematical concepts. This section evaluates not just your procedural skills but also your conceptual clarity — the kind needed to teach these topics effectively to students aged 11-14.

The syllabus spans seven major areas: Number System, Algebra, Geometry, Mensuration, Commercial Mathematics, Data Handling, and Practical Geometry. Questions typically test fundamental understanding rather than complex problem-solving, but they often include traps based on common student misconceptions. Mastering this content is essential because pedagogical questions frequently reference these mathematical ideas.

Expect 15-20 content-based questions. Focus on building rock-solid basics — definitions, properties, formulas and their applications in straightforward problems.

Key Concepts

**Integers extend whole numbers** to include negatives. Operations on integers follow specific sign rules: negative × negative = positive; negative × positive = negative.

**Rational numbers** are numbers expressible as p/q where q ≠ 0. Every integer is rational (denominator = 1). Rational numbers are dense — between any two rationals, infinitely many rationals exist.

**Algebraic expressions** use variables to represent unknown quantities. An equation states that two expressions are equal; an identity is true for all values of variables.

**Triangle properties** are foundational: angle sum = 180°; exterior angle = sum of two interior opposite angles; sum of any two sides > third side.

**Area and perimeter** differ fundamentally — perimeter is boundary length (linear units), area is surface covered (square units). Volume measures space occupied (cubic units).

**Percentage is "per hundred"** — a standardised way to compare quantities. Profit/Loss percentage is always calculated on Cost Price, not Selling Price.

**Mean, median and mode** are measures of central tendency. Mean uses all values; median is the middle value; mode is most frequent. Each has specific use cases.

**Probability** measures likelihood of an event, ranging from 0 (impossible) to 1 (certain). Probability = Favourable outcomes ÷ Total outcomes.

Formulas / Key Facts

### Number System

a^m × a^n = a^(m+n)
a^m ÷ a^n = a^(m-n)
(a^m)^n = a^(mn)
a^0 = 1 (for a ≠ 0)
a^(-n) = 1/a^n

### Algebra

(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
(a + b)(a - b) = a² - b²
(a + b)³ = a³ + 3a²b + 3ab² + b³
Linear equation in one variable: ax + b = 0, solution x = -b/a

### Geometry

Sum of interior angles of polygon = (n - 2) × 180°
Each angle of regular polygon = (n - 2) × 180° / n
Pythagoras theorem: In right triangle, hypotenuse² = base² + perpendicular²

### Mensuration

Area of triangle = ½ × base × height
Area of circle = πr²; Circumference = 2πr
Surface area of cuboid = 2(lb + bh + hl)
Volume of cuboid = l × b × h
Volume of cylinder = πr²h; Curved surface area = 2πrh

### Commercial Mathematics

Profit % = (Profit/CP) × 100
SP = CP × (100 + Profit%)/100
Simple Interest = (P × R × T)/100
Compound Interest: Amount = P(1 + R/100)^T

### Data Handling

Mean = Sum of observations / Number of observations
Probability of event E = n(E)/n(S)

Worked Examples

**Example 1: Rational Numbers** Simplify: (-3/4) + (5/6) - (1/2)

Solution: LCM of 4, 6, 2 = 12 = (-9/12) + (10/12) - (6/12) = (-9 + 10 - 6)/12 = -5/12

**Example 2: Algebraic Identity** Evaluate 103² using identity.

Solution: 103² = (100 + 3)² Using (a + b)² = a² + 2ab + b² = 100² + 2(100)(3) + 3² = 10000 + 600 + 9 = 10609

**Example 3: Profit and Loss** A shopkeeper buys an article for ₹800 and sells it for ₹920. Find profit percentage.

Solution: Profit = SP - CP = 920 - 800 = ₹120 Profit % = (120/800) × 100 = 15%

**Example 4: Mensuration** Find the volume and total surface area of a cylinder with radius 7 cm and height 10 cm. (Use π = 22/7)

Solution: Volume = πr²h = (22/7) × 7 × 7 × 10 = 1540 cm³ Total surface area = 2πr(r + h) = 2 × (22/7) × 7 × (7 + 10) = 44 × 17 = 748 cm²

Common Mistakes

**Confusing (-a)² with -a²** → (-3)² = 9, but -3² = -9. Brackets matter — the negative is squared only when inside brackets.

**Adding/subtracting fractions without common denominator** → Students write 1/2 + 1/3 = 2/5. Always find LCM first: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

**Calculating percentage on wrong base** → Profit % is on CP, discount % is on Marked Price, not SP. Mixing these bases gives wrong answers.

**Forgetting units conversion in mensuration** → When length is in metres and breadth in centimetres, convert to same unit before calculating area.

**Applying Pythagoras theorem to non-right triangles** → The theorem works only for right-angled triangles. Check for the 90° angle first.

**Confusing surface area with volume** → Surface area needs material to cover; volume needs material to fill. Read questions carefully for what is asked.

Quick Reference

Integers: ...−3, −2, −1, 0, 1, 2, 3... (whole numbers + negatives)

Rational number between a and b: (a + b)/2

To solve linear equation: isolate variable by inverse operations

Triangle inequality: any side < sum of other two sides

Discount = Marked Price − Selling Price

Mode can be found without calculation — just identify most frequent value

Probability of sure event = 1; probability of impossible event = 0

Mathematics Content (Class VI-VIII)