CBSE Class 8 Mathematics (Ganita Prakash)

Chapters: 14

1. A Square and a Cube

Square numbers · Cubic numbers · A pinch of history

Square Roots: Build a Square, Find Its Side – The square root of a number n is the number that, multiplied by itself, gives n. √25 = 5 because 5 × 5 = 25. Picture n tiles arranged in a square: the root is the side. Perfect squares (1, 4, 9, 16, …) have whole-number roots. Other roots are irrational and lie between two whole numbers; we estimate them or simplify them, like √50 = 5√2.
Cubes and Cube Roots – The cube of a number n is n × n × n = n³. It is the number of unit blocks in a cube whose edge has n blocks. A perfect cube is a whole number that is the cube of a whole number (1, 8, 27, 64, 125 …). The cube root ∛m undoes cubing: ∛64 = 4 because 4³ = 64. Find it by grouping equal prime factors in threes, or by estimating with the last digit. 1729 is the smallest number that is a sum of two cubes in two ways.

Experiencing growth by powers · Exponential notation and operations · The other side of powers · Powers of 10

Laws of Exponents – An exponent tells how many times a base is multiplied by itself. Same base: multiply → add exponents, divide → subtract, power of a power → multiply. a⁰ = 1, a⁻ⁿ = 1/aⁿ and a^(1/n) is the nth root.
Scientific Notation (Standard Form) – Scientific notation writes any number as a × 10ⁿ, where 1 ≤ a < 10 and n is a whole number (an integer). For big numbers the decimal point moves left and n is positive. For small numbers it moves right and n is negative. To multiply, multiply the a's and add the powers; to divide, divide the a's and subtract the powers; then fix a so it is between 1 and 10. Unit prefixes like kilo (10³), mega (10⁶), milli (10⁻³), micro (10⁻⁶) and nano (10⁻⁹) are powers of ten with names.

Early number systems · The idea of a base · Place value representation

Place Value, Ordering and Negative Numbers – In our number system every place is worth 10 times the place on its right. A digit's value = digit × the value of its place. To compare numbers, line up the places and compare from the left. Negative numbers lie left of zero; the further left, the smaller.
Number Bases: Grouping in Tens, Twos and Sixties – A base is the size of the groups we count in. In base 10 we make bundles of ten, so each place is worth 10 times the place to its right: ones, tens, hundreds. In base b the places are worth 1, b, b², b³ … and only digits 0 to b − 1 are used. Computers use base 2 (binary, digits 0 and 1). Time and angles still use base 60, from ancient Babylon: 60 seconds make a minute and 60 minutes make an hour.

Rectangles and squares · Angles in a quadrilateral · Parallelograms and rhombuses · Kites and trapeziums

Coming soon

Multiples and divisibility · Digits in disguise

Divisibility: Multiples, Divisors, GCD and LCM – a is divisible by b when a = b × k for a whole number k: b is a divisor (factor) of a and a is a multiple of b. Every whole number a can be written as a = b × q + r with 0 ≤ r < b (division with remainder); b divides a exactly when r = 0. Quick tests tell divisibility by 2, 3, 4, 5, 6, 8, 9, 10 and 11 from the digits. The GCD is the greatest common divisor, found by prime factors or by Euclid's algorithm; LCM is the least common multiple, and GCD × LCM = a × b. Two numbers are coprime when their GCD is 1.
Mathematical Reasoning: Puzzles You Can Solve with Rules – Reasoning questions test whether you can spot a rule and use it. Odd one out: find the rule most items share. Syllogism: draw circles for "all", "some" and "no" and accept only what MUST be true. Blood relations: draw a family tree. Coding-decoding: find how letters or numbers shift. Cryptarithms: digits hide behind letters; use place value, carries and odd/even (parity) to find them.

Properties of multiplication · Special products (identities)

Coming soon

Ratios · Sharing in a ratio and unit conversion

Ratio and Proportion – A ratio a : b compares two amounts of the same kind. Divide both parts by the same number to simplify it. To share an amount in a ratio, add the parts, find one part, then multiply. Two quantities are in direct proportion when y = k·x (double one, double the other) and in inverse proportion when x·y = k (double one, halve the other).

Percentages

Percentages – Per cent means "out of 100", so x% = x/100. To find x% of an amount, multiply by x/100. Percentage change = (change ÷ original) × 100. To increase by r%, multiply by (1 + r/100); to decrease by r%, multiply by (1 − r/100). Successive changes multiply their multipliers. For reverse percentage, divide the new value by the multiplier.

Doubling and halving a square · The theorem and triples

Pythagoras Theorem – In any right-angled triangle, the square on the longest side (the hypotenuse) equals the sum of the squares on the other two sides: a² + b² = c².

Ratios in maps and multi-term ratios · Inverse proportion

Ratio and Proportion – A ratio a : b compares two amounts of the same kind. Divide both parts by the same number to simplify it. To share an amount in a ratio, add the parts, find one part, then multiply. Two quantities are in direct proportion when y = k·x (double one, double the other) and in inverse proportion when x·y = k (double one, halve the other).

Fractals · Visualising solids

Fractals and the Sierpinski Triangle – A fractal is a pattern made by repeating the same rule again and again, so a small part looks like the whole. This is called self-similarity. In the Sierpinski triangle, each step removes the middle of every triangle: the number of triangles goes 1, 3, 9, 27 … (3ⁿ) and the area left is multiplied by 3/4 each time.
3D Shapes: Faces, Edges, Vertices, Nets and Views – A 3D shape (solid) has length, width and height, so it takes up space. We describe it by counting its faces (flat or curved surfaces), edges (lines where two faces meet) and vertices (corners). Prisms have two equal ends joined by rectangles; pyramids have one base and triangles that meet at a top point; cylinders, cones and spheres have curved surfaces. A net is a flat pattern that folds into the solid. Drawing a solid from the top, front and side gives its three views. For every solid with flat faces, Faces + Vertices − Edges = 2 (Euler's rule).

Averages · Visualising and interpreting data

Showing Data: Dot Plots, Bar Graphs, Histograms and Line Graphs – Graphs turn a list of numbers into a picture. A dot plot puts one dot per item above a number line. A bar graph compares separate categories with gaps between bars. A histogram shows grouped numerical data in class intervals with bars that touch. A line graph shows change over time. A graph can mislead if its axis does not start at 0, its scale is uneven or its pictures are the wrong size.

Algebra behind puzzles

Linear Equations in One Variable – A linear equation in one variable has one unknown (like x) with power 1, for example 2x + 3 = 11. An equation is like a balance: both sides are equal. To solve it, do the same thing to both sides (add, subtract, multiply or divide by the same non-zero number) until x is alone. Always check by putting the answer back in.

Areas of quadrilaterals and triangles

Coming soon