Ganita Prakash

Chapter 1 of Ganita Prakash (Class 7 Maths) is about large numbers — lakhs, crores, arabs, millions, and billions — exploring how to read, write, compare, round, and work with them through real-life contexts and multiplication patterns.

• 11 lakh (1,00,000) is 1 followed by 5 zeroes; in the Indian system commas are placed in a 3-2-2-2 pattern from right to left.
• 21 crore (1,00,00,000) is 1 followed by 7 zeroes and equals 100 lakhs; 1 arab equals 100 crores (also written 1,000,000,000 = 1 billion in the American system).
• 3The Indian naming system is also used in Bhutan, Nepal, Sri Lanka, Pakistan, Bangladesh, Maldives, Afghanistan, and Myanmar; the words 'lakh' and 'crore' come from the Sanskrit lakṣha and koṭi.
• 4The American/International system groups digits in a 3-3-3-3 pattern: thousand, million, billion.
• 5Rounding up means the approximated number is more than the actual number (e.g., ordering 750 sweets for 732 people); rounding down means it is less (e.g., quoting ₹450 for an item costing ₹470).

Chapter 2 of Ganita Prakash (Grade 7) covers Arithmetic Expressions — teaching students how to read, write, compare, and evaluate mathematical phrases using the four operations, brackets, and the concept of terms.

• 1An arithmetic expression is a mathematical phrase formed using numbers and operations (+, –, ×, ÷); its value is the number it evaluates to, written with '=' (e.g., 13 + 2 = 15).
• 2Expressions can be compared using '=', '<', or '>' based on their values; comparisons can often be made by reasoning rather than full computation (e.g., 1023 + 125 < 1022 + 128 because Joy gains 3 more than he starts with fewer).
• 3Brackets resolve ambiguity: evaluate the expression inside brackets first before applying outer operations (e.g., 30 + (5 × 4) = 50, not 140).
• 4Terms are the parts of an expression separated by a '+' sign after all subtractions are converted to additions of negative numbers (e.g., 83 – 14 has terms 83 and –14).
• 5The commutative property of addition: swapping the order of terms does not change the sum (Term 1 + Term 2 = Term 2 + Term 1), and this holds even when terms are negative numbers.

Chapter 3 of Ganita Prakash Class 7 — 'A Peek Beyond the Point' — introduces decimal numbers by extending the Indian place value system to tenths, hundredths, and thousandths, and teaches students to read, write, compare, locate on a number line, and add or subtract decimal numbers.

• 1A unit can be split into 10 equal one-tenth parts; each one-tenth can further be split into 10 one-hundredth parts, giving 100 one-hundredths in a unit.
• 2The decimal system is based on the number 10 ('decem' in Latin, cognate to Sanskrit 'daśha'), extending the Indian place value system so each place is 10 times smaller than the one immediately to its left.
• 3A decimal point ('.') is used to separate the whole number part from the fractional part; the digits to the right of the point represent tenths, hundredths, thousandths, etc.
• 4Reading decimals: 70.5 is read as 'seventy point five' (or seventy and five-tenths); 7.05 is 'seven point zero five' (or seven and five-hundredths); 0.274 is 'zero point two seven four' — not 'two hundred and seventy four'.
• 5Adding zeros to the right of a decimal does not change its value: 0.2 = 0.20 = 0.200, all equal 2 tenths. But 0.2, 0.02, and 0.002 are different quantities.

Chapter 4 of Ganita Prakash Class 7, 'Expressions Using Letter-Numbers', introduces algebraic expressions — mathematical expressions that use letters (called letter-numbers) to represent numbers, allowing general relationships and patterns to be written concisely as formulas.

• 1Letters (called letter-numbers) are used to represent unknown or varying numbers; expressions containing letter-numbers are called algebraic expressions.
• 2The multiplication sign between a number and a letter-number is omitted by convention — 4 × n is written as 4n, with the number written first.
• 3Algebraic expressions take a number value when the letter-numbers are replaced by specific numbers (e.g., if a = 23 in a + 3, the value is 26).
• 4Mathematical relations expressed using algebraic expressions are called formulas (e.g., the perimeter of a square with side q is 4q).
• 5Like terms are terms that involve the same letter-numbers (e.g., 5c, 3c, 10c); unlike terms have different letter-numbers (e.g., 18c and 11d).

Chapter 5 of Ganita Prakash Grade 7 covers parallel and intersecting lines, teaching students how to identify, draw, and reason about the angles formed when lines meet on a plane surface or are cut by a transversal.

• 1When two lines intersect on a plane they form four angles; vertically opposite angles (e.g. ∠a and ∠c) are always equal to each other.
• 2Adjacent angles formed by two intersecting lines are called linear pairs and always add up to 180°.
• 3Perpendicular lines are a pair of intersecting lines where all four angles are equal to 90°; a square symbol marks the right angle in diagrams.
• 4Parallel lines lie on the same plane and never meet however far they are extended; a set of parallel lines is marked with a single arrow (>) in diagrams, a second set with two arrows.
• 5A transversal is a line that intersects two other lines; it creates eight angles with a maximum of four distinct angle measures (due to vertically opposite pairs).

Chapter 6 of Ganita Prakash (Class 7) is 'Number Play', a chapter that explores parity (odd and even numbers), magic squares, the Virahāṅka–Fibonacci sequence, and cryptarithms through puzzles and activities.

• 1Parity rule: the sum of any number of even numbers is always even; the sum of an odd count of odd numbers is odd, while an even count of odd numbers gives an even sum.
• 2Two consecutive numbers always have opposite parity (one even, one odd), so their sum is always odd — used to reason about age puzzles and similar problems.
• 3The word 'parity' denotes the property of being even or odd; parity rules apply to sums, differences, and products, and can be used to check whether a puzzle has any solution at all.
• 4In a 3×3 grid filled with numbers 1–9 (no repetition), the sum of all numbers is 45, so every row sum and every column sum in a magic square must equal 15 (the magic sum).
• 5Observation 2: In a 3×3 magic square using 1–9, the centre must always be 5; Observation 3: the numbers 1 and 9 cannot occupy corner positions.

Chapter 7 of Ganita Prakash Grade 7, titled 'A Tale of Three Intersecting Lines', teaches students how to construct triangles using a compass and ruler, how to determine whether a triangle can exist for given side lengths using the triangle inequality, and how to prove that the angles of any triangle always sum to 180°.

• 1An equilateral triangle has all three sides of equal length; it is constructed by drawing two arcs of equal radius from the endpoints of the base and joining the intersection point to both endpoints.
• 2Any triangle with given side lengths can be constructed efficiently using a compass: draw arcs of the required radii from the base vertices and let their intersection be the third vertex.
• 3The triangle inequality states that each side length must be less than the sum of the other two; sets such as 10, 15, 30 fail this check (30 > 10 + 15) and cannot form a triangle.
• 4When the longest side is taken as the base and circles of the two smaller radii are drawn from its endpoints, the circles intersect internally if and only if the triangle inequality is satisfied — confirming a triangle exists.
• 5A triangle can be constructed given two sides and their included angle (SAS): draw the base, construct the given angle at one endpoint, mark the second side along the angle arm, and join the endpoints.

Chapter 8 of Ganita Prakash (Class 7) is about Working with Fractions — covering multiplication and division of fractions, including fraction-by-whole-number, fraction-by-fraction, the concept of reciprocal, and comparing the product or quotient to the original numbers.

• 1Multiplying a fraction by a whole number: multiply the whole number by the numerator and keep the denominator (e.g., 3 × (1/4) = 3/4).
• 2To multiply a fraction by a whole number, first divide the multiplicand by the denominator of the multiplier, then multiply the result by the numerator of the multiplier.
• 3Multiplying two fractions follows Brahmagupta's formula: (a/b) × (c/d) = (a×c)/(b×d), first stated in the Brahmasphutasiddhanta (628 CE).
• 4The area of a rectangle whose sides are fractions equals the product of those fractional sides, giving a geometric interpretation of fraction multiplication.
• 5When cancelling common factors before multiplying fractions (apavartana), divide numerators and denominators by their common factors to reach the lowest form directly — a process so well known by 150 CE that the Jain scholar Umasvati mentioned it in a philosophical work.

Chapter 1 'Geometric Twins' of Ganita Prakash Grade 7 introduces congruence — the property of figures that have the same shape and size and can be superimposed exactly on each other. It develops five conditions for triangle congruence (SSS, SAS, ASA, AAS, RHS) and uses congruence to prove properties of isosceles and equilateral triangles.

• 1Congruent figures have the same shape and size; they can be superimposed exactly, one over the other, and a figure may be rotated or flipped before superimposing.
• 2SSS (Side Side Side): if two triangles have all three sidelengths equal, they are congruent.
• 3SAS (Side Angle Side): if two sides and the included angle of one triangle equal those of another, the triangles are congruent.
• 4ASA (Angle Side Angle): if two angles and the included side of one triangle equal those of another, the triangles are congruent.
• 5AAS (Angle Angle Side): congruence still holds when the equal side is not included between the two equal angles.

Class 7 Maths Chapter 10 'Operations with Integers' covers multiplication and division of positive and negative integers using the token model and number line, and establishes the commutative, associative, and distributive properties for integer multiplication.

• 1Token model: green tokens represent +1 and red tokens represent −1; a zero pair (one green + one red) cancels to zero.
• 2A positive multiplier means placing tokens into an empty bag; a negative multiplier means removing tokens from the bag.
• 3Sign rules for multiplication: positive × positive = positive; negative × negative = positive; positive × negative = negative; negative × positive = negative.
• 4Division of integers follows the same sign rules as multiplication: (−a) ÷ b = −(a ÷ b), a ÷ (−b) = −(a ÷ b), and (−a) ÷ (−b) = a ÷ b.
• 51 × a = a for all integers; −1 × a = −a (the additive inverse) for all integers.

This chapter, 'Finding Common Ground,' teaches Class 7 students how to find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of numbers using prime factorisation, and explores their properties, patterns, and an efficient combined procedure for computing both at once.

• 1The Highest Common Factor (HCF) of two or more numbers is the highest of their common factors; it is also known as the Greatest Common Divisor (GCD).
• 2The Lowest Common Multiple (LCM) of two or more numbers is the lowest of their common multiples.
• 3Prime factorisation — writing a number as a product of primes using the division method — simplifies finding both HCF and LCM and makes the process more reliable than listing all factors or multiples.
• 4To find the HCF using prime factorisation, identify the common prime factors and take the minimum number of times each occurs across all the given numbers' factorisations.
• 5To find the LCM using prime factorisation, take all prime factors that appear in any of the numbers and include the maximum number of times each occurs across any factorisation.

This chapter, 'Another Peek Beyond the Point' from Ganita Prakash Grade 7, teaches students how to multiply and divide decimals by extending the place-value procedures used for counting numbers. It also explores an application of decimal arithmetic to explain how leap years are designed to keep calendars aligned with Earth's actual revolution around the Sun.

• 1Decimals are extensions of the Indian place-value system representing decimal fractions (tenths, hundredths, thousandths, etc.) and their sums.
• 2To divide by 10, 100, 1000 etc., move the decimal point to the left by as many places as there are zeroes in the divisor.
• 3To multiply two decimals: remove the decimal points, multiply the resulting whole numbers, then place the decimal point so that the product has as many decimal digits as the sum of decimal digits in the multiplier and multiplicand.
• 4Whether the product of two decimals is greater or less than the numbers multiplied depends on whether the numbers are greater than 1, between 0 and 1, or a mix of both.
• 5Dividing a decimal by 10, 100, 1000 etc. means moving the decimal point to the left; dividing by a decimal divisor means multiplying both dividend and divisor by 10, 100, etc. to convert the divisor to a whole number.

Chapter 13 of Ganita Prakash Grade 7 (Connecting the Dots…) introduces students to statistics — covering how to frame statistical questions, calculate the arithmetic mean and median, identify outliers, and read or draw dot plots and clustered bar graphs to compare data.

• 1A statistical statement is a claim about a phenomenon expressed in terms of numerical values, proportions, probabilities, or predictions.
• 2A statistical question is one that can be answered by collecting data, where variability in the data is expected — for example, 'How tall are Grade 7 students in our school?'
• 3The Arithmetic Mean (AM) = Sum of all values ÷ Number of values; it can also be understood as the 'fair-share' or equal-share value across the data.
• 4Ancient Indian mathematicians referred to the Arithmetic Mean by terms such as samamiti (Bhāskarācārya, 1150 CE), samarajju (Brahmagupta, 628 CE), samīkaraṇa (Mahāvīrācārya, 850 CE), and sāmya (Śrīpati, 1039 CE) — all reflecting the idea of an 'equalising' value.
• 5When group sizes differ, the total (sum) is not a fair comparison measure; the mean per unit (e.g., runs per match) should be used instead.

Chapter 14 of Ganita Prakash Class 7 covers Constructions and Tilings, teaching students how to use only an unmarked ruler and compass to construct perpendicular bisectors, angle bisectors, 60°/90°/45° angles, parallel lines, and regular hexagons, and then explores tiling — covering a plane region without gaps or overlaps.

• 1A perpendicular bisector of a line segment XY is constructed using a compass by drawing arcs from X and Y with the same radius above and below XY; the line joining the two intersection points is the perpendicular bisector.
• 2Any point that is at equal distance from both endpoints X and Y of a line segment lies on the perpendicular bisector of XY.
• 3A 90° angle at a point O on a line is constructed by first marking two points X and Y equidistant from O, then drawing the perpendicular bisector of XY through O.
• 4An angle bisector is constructed using the SSS congruence condition: mark equal lengths OA = OB on the two arms, then find point C such that AC = BC; OC bisects the angle.
• 5An angle can be copied using a ruler and compass by constructing congruent triangles (SSS condition), ensuring equal arm lengths and equal arcs.

Chapter 7 'Finding the Unknown' from Class 7 Ganita Prakash (Part II) introduces algebraic equations — teaching students how to frame an equation using a letter-number for an unknown quantity and solve it systematically by performing the same operation on both sides.

• 1An algebraic equation is a statement of equality between two algebraic expressions, written with an '=' sign, e.g. 2n + 1 = 99 or 6y + 7 = 4y + 21.
• 2The process of finding the value(s) of the letter-number for which LHS equals RHS is called solving the equation.
• 3Trial and error is one method: substitute different values until LHS = RHS, but it can be inefficient.
• 4The systematic method uses inverse operations — when the same operation is performed on both sides of an equation, equality is maintained.
• 5Three key rules for isolating the unknown: (a) removing an added/subtracted term requires placing its additive inverse on the other side; (b) removing a multiplied factor requires dividing the other side by that factor; (c) removing a divisor requires multiplying the other side by that divisor.