Ganita Prakash

Chapter 1 of Class 8 maths, 'A Square and a Cube', teaches perfect squares and perfect cubes through their definitions, properties, patterns, and methods to find square roots and cube roots using factors and prime factorisation.

• 1Perfect squares are numbers like 1, 4, 9, 16, 25 obtained by multiplying a number by itself (n²)
• 2All perfect squares end with 0, 1, 4, 5, 6, or 9; never with 2, 3, 7, or 8
• 3Only numbers with an odd number of factors are perfect squares (e.g., 1, 4, 9, 16, 25, 36)
• 4Perfect cubes are numbers like 1, 8, 27, 64, 125 obtained by multiplying a number by itself three times (n³)
• 5Sum of the first n odd numbers equals n² (e.g., 1+3+5+7+9=25=5²)

Chapter 2 of Class 8 maths, "Power Play", introduces exponential notation, the laws of exponents, and applications of powers including exponential growth, scientific notation, and large number comparisons.

• 1Exponential notation: n^a means n multiplied by itself a times (e.g., 2^3 = 8)
• 2Laws of exponents: n^a × n^b = n^(a+b), (n^a)^b = n^(a×b), n^a ÷ n^b = n^(a−b), and n^0 = 1 for any n ≠ 0
• 3Exponential growth is multiplicative (doubling every step) versus linear growth (fixed addition per step); paper folded 46 times reaches the Moon, but 1.92 billion steps of 20 cm each are needed on a ladder
• 4Negative exponents: n^(−a) = 1 / n^a (e.g., 2^(−3) = 1/8)
• 5Scientific notation: express very large numbers as x × 10^y where 1 ≤ x < 10 and y is an integer (e.g., 5,900 = 5.9 × 10^3)

Chapter 3 of Class 8 Maths, "A Story of Numbers", traces the evolution of number systems across human history, from early counting methods using body parts and tally marks to sophisticated place-value systems like the Egyptian, Mesopotamian, Mayan, and ultimately the Hindu number system that we use today.

• 1One-to-one mapping is a foundational counting principle: assigning each object to a unique symbol or physical marker (like sticks or body parts) to determine quantity
• 2Landmark numbers are reference points used in a number system—the Egyptian system uses 1, 10, 100, 1000; the Roman system uses I (1), V (5), X (10), L (50), C (100), D (500), M (1000)
• 3A base-n number system has landmark numbers that are powers of a fixed number n (for example, Egyptian is base-10 with powers 1, 10, 100, 1000…; Mesopotamian is base-60 with powers 1, 60, 3600…)
• 4Place-value (positional) notation represents numbers using the position of symbols to indicate which power of the base they represent, dramatically reducing the need for new symbols
• 5Zero (0) as both a placeholder digit and a number in its own right was a breakthrough invented in ancient India around 200 BCE, enabling unambiguous number representation and efficient arithmetic

Chapter 4 of Class 8 maths, "Quadrilaterals", teaches four-sided figures and their properties through geometric reasoning. It covers rectangles, squares, parallelograms, rhombuses, kites, and trapeziums, with emphasis on proofs using congruence, angle sums, and diagonal properties.

• 1The sum of all angles in any quadrilateral is always 360°
• 2A rectangle is a quadrilateral with all 90° angles; its diagonals are equal in length and bisect each other
• 3A square is a special rectangle where all sides are equal and diagonals bisect each other at right angles (90°)
• 4A parallelogram has opposite sides parallel and equal, opposite angles equal, and diagonals that bisect each other
• 5A rhombus has all four sides equal, opposite angles equal, and diagonals that bisect each other at 90° and bisect the angles

Chapter 5 of Class 8 maths, "Number Play", explores patterns and properties of numbers, focusing on divisibility rules, parity (even and odd numbers), properties of multiples and factors, and number puzzles like cryptarithms and digital roots.

• 1Parity rules: odd ± odd = even; even ± even = even; odd ± even = odd; switching +/- signs between numbers changes the result by an even amount
• 2Divisibility by 10, 5, 2: determined by the units digit (0 for 10; 0 or 5 for 5; even for 2)
• 3Divisibility by 9: a number is divisible by 9 if and only if the sum of its digits is divisible by 9; digital root (repeated digit sum) equals the remainder when divided by 9
• 4Divisibility by 3: a number is divisible by 3 if the sum of its digits is divisible by 3
• 5Divisibility by 11: alternating digit sums — add digits in odd places, subtract digits in even places; if difference is 0 or multiple of 11, the number is divisible by 11

Chapter 6 of Class 8 Maths, "We Distribute, Yet Things Multiply", explores the distributive property of multiplication over addition and uses it to solve multiplication patterns, expand algebraic expressions, and develop fast multiplication techniques for special numbers.

• 1The distributive property: a(b + c) = ab + ac, and its extension (a + m)(b + n) = ab + mb + an + mn
• 2When product ab is increased by 1 on one factor, the product increases by the other factor; when both increase by 1, the increase is a + b + 1
• 3Square of sum identity: (a + b)² = a² + 2ab + b²
• 4Square of difference identity: (a − b)² = a² − 2ab + b²
• 5Difference of squares identity: (a + b)(a − b) = a² − b²

Chapter 7 of Class 8 Maths, "Proportional Reasoning-1", teaches how to identify when quantities change proportionally and use ratios to solve real-world problems. It covers observing similarity in change, working with ratios in simplest form, proportional reasoning with the Rule of Three (Trairasika), and dividing quantities into specified ratios.

• 1Ratios a : b express that for every 'a' units of the first quantity, there are 'b' units of the second quantity, where 'a' and 'b' are the terms
• 2Two ratios are proportional (a : b :: c : d) when their terms change by the same factor, verified by reducing both to simplest form or using cross multiplication (ad = bc)
• 3Ratios are reduced to simplest form by dividing both terms by their HCF (Highest Common Factor)
• 4The Rule of Three (Trairasika) solves for unknown quantities: if a : b :: c : d, then d = (b × c) / a
• 5To divide a quantity x into ratio m : n, the parts are m × (x / (m+n)) and n × (x / (m+n))

Chapter 8 of Class 8 maths, "Fractions in Disguise", introduces percentages as fractions with denominator 100, and teaches how to convert between fractions, decimals, and percentages, then apply percentages to real-world scenarios like profit-loss, discounts, and compound interest.

• 1Percentages are fractions with denominator 100: x% means x out of every 100
• 2Conversions: express any fraction as a percentage by multiplying by 100, or convert a percentage to a fraction by writing it over 100
• 3Finding a percentage of a quantity: y% of n = (y/100) × n, also equals the fraction multiplied by the number
• 4Percentage change: (change/original) × 100 for increases or decreases; compounding means interest is added back to principal each period
• 5Profit and loss: profit % = (profit/cost price) × 100; loss % = (loss/cost price) × 100; discount reduces marked price

Chapter 9 of Class 8 maths, "The Baudhayana-Pythagoras Theorem", teaches the fundamental relationship between the sides of a right-angled triangle, where the square of the hypotenuse equals the sum of squares of the other two sides (a² + b² = c²), along with applications in geometry and number theory.

• 1The square constructed on the diagonal of a square has double the area of the original square
• 2For a right-angled triangle with sides a, b and hypotenuse c: a² + b² = c²
• 3The hypotenuse of an isosceles right triangle with equal sides of length a is c = a√2
• 4√2 is approximately 1.41421356 and cannot be expressed as a terminating decimal or fraction
• 5Baudhayana triples are sets of three positive integers (a, b, c) satisfying a² + b² = c², such as (3, 4, 5), (5, 12, 13), and (8, 15, 17)

Chapter 10 of Class 8 maths, "Proportional Reasoning-2", extends proportional reasoning concepts by teaching ratios with multiple terms, how to divide quantities in given ratios, pie chart construction using proportional angles, and inverse proportions where quantities change in opposite directions.

• 1Two ratios a : b and c : d are proportional if a × d = b × c (cross-multiplication test)
• 2Representative Fraction (RF) on maps shows the ratio between map distance and actual geographical distance (e.g., 1 : 60,00,000 means 1 cm on map = 60 km on ground)
• 3Ratios with multiple terms (a : b : c : d) are proportional if all terms scale by the same factor
• 4To divide a quantity x in ratio p : q : r : s, each part is x × (term / sum of all terms)
• 5Pie chart angles are proportional to data: angle = (value / total) × 360°

Chapter 11 of Class 8 maths, "Exploring Some Geometric Themes", teaches students about fractals—self-similar shapes that repeat at smaller scales—and methods for visualizing three-dimensional solids through nets, projections, and isometric drawings.

• 1Fractals are self-similar geometric patterns that repeat at progressively smaller scales, found in nature (ferns, trees, coastlines) and in mathematical constructions
• 2The Sierpinski Carpet is constructed by dividing a square into 9 smaller squares, removing the center, and repeating; at step n, there are 8^n remaining squares
• 3The Sierpinski Gasket (Triangle) is formed by joining midpoints of an equilateral triangle and removing the central triangle; the Koch Snowflake replaces each side with a bump-shaped structure by dividing into thirds
• 4A net is a flat pattern that can be folded into a 3D solid; a cube has 11 possible net structures, a regular tetrahedron has 2, and a dodecahedron has 43,380
• 5Faces are flat surfaces, edges are line segments forming face boundaries, and vertices are points where edges meet (e.g., a cuboid has 6 faces, 12 edges, 8 vertices)

Chapter 12 of Class 8 maths, "Tales by Dots and Lines", teaches data visualization and statistical measures through dot plots and line graphs. Students learn how mean and median represent the 'centre' of data, how they change when values are added or removed, and how to visualize trends over time using line graphs and infographics.

• 1Mean represents the 'centre' of data where total distances from values on the left equal total distances on the right
• 2When a value greater than the mean is added, the mean increases; when a value less than the mean is added, the mean decreases
• 3If all data values are increased by a fixed number, the mean also increases by that same fixed number
• 4If all data values are multiplied by a fixed number, the mean is also multiplied by that same fixed number
• 5Median of a dataset is the middle value; adding a value greater than median increases the median, adding one less than median decreases it

Chapter 13 of Class 8 maths, "Algebra Play", uses algebra to explain and create mathematical tricks, puzzles, and numerical patterns such as 'Think of a Number' tricks, number pyramids, calendar grids, and divisibility tricks.

• 1"Think of a Number" tricks always yield the same final result regardless of starting number (e.g., doubling, adding 4, dividing by 2, subtracting original = 2)
• 2Number pyramids: each number is the sum of the two directly below it; unknown values can be found using algebraic equations
• 3Calendar magic: finding which date was chosen from the sum of a 2×2 grid (sum = 4a + 16, where a is top-left number)
• 4Largest product strategy: to maximize a 2-digit × 1-digit product using 3 given digits, use the largest digit as the multiplier and arrange the other two in decreasing order
• 5Divisibility tricks work via algebra: reversing a 2-digit number and finding the difference gives 9(b – a), always divisible by 9

Chapter 14 of Class 8 maths, "Area", teaches students how to measure and calculate the area of different shapes using unit squares and mathematical formulas, covering rectangles, triangles, parallelograms, rhombuses, trapeziums, and any polygon.

• 1Area of a rectangle = length × width (measured in square units like cm²)
• 2A diagonal divides a rectangle into two congruent triangles, each with area equal to half the rectangle
• 3Area of a triangle = ½ × base × height, regardless of triangle type or position
• 4Regions with the same perimeter can have different areas, and vice versa — perimeter is not indicative of area
• 5Area of a parallelogram = base × height (using the perpendicular height, not the slant side)