Ganita Prakash: Class 7 Mathematics, Part 2 (NCERT)

Sprint 1

Copies You Can Trust

What congruent means, how to test it by turning and flipping, and which measurements you need to copy a shape.

Sprint 2

Three Sides Are Enough

Why three side lengths fix a triangle completely: the SSS rule.

Sprint 3

Naming Congruent Triangles

Why the order of letters in a congruence statement matters, and how to write one correctly.

Sprint 4

Two Sides And The Angle Between

Equal angles only fix the shape. Two sides and the angle between them fix the whole triangle: the SAS rule.

Sprint 5

Why SSA Is Not Enough

Build two different triangles from the same side-side-angle measurements, and see why SSA can't be trusted.

Sprint 6

Two Angles And One Side

Two angles and the side between them (ASA), and how the 180° angle sum turns other cases into ASA (AAS).

Sprint 7

Right Triangles And RHS

The one case where two sides and a non-included angle do work, right triangles (RHS), and all five rules together.

Sprint 8

Angles Of Isosceles And Equilateral Triangles

Use congruence to prove that angles opposite equal sides are equal, and that each angle of an equilateral triangle is 60°.

Sprint 9

Choosing The Right Congruence Test

Look at the measurements given, and decide which rule applies, or that none does.

Sprint 10

Review: Geometric Twins

All five congruence rules, the two cases that fail, and the isosceles results, on the chapter's hardest questions.

Sprint 11

Rakesh's Puzzle: Two Numbers From A Sum And A Difference

Find two numbers when you know only their sum and their difference, and see why the order of the numbers decides the sign.

Sprint 12

Carrom Coin Moves On A Number Line

Turn each carrom strike into a positive or negative number, find where the coin lands with P = a + b, and read jump diagrams.

Sprint 13

Subtracting With Tokens And Zero Pairs

Take away a bigger number from a smaller one by adding zero pairs, and see why subtracting is the same as adding the opposite.

Sprint 14

Multiplying With A Token Bag

See multiplication as putting the same group of tokens into a bag again and again, and find out why 4 × (−2) = −8.

Sprint 15

When The Multiplier Is Negative

A negative multiplier means taking tokens out of the bag. Use zero pairs to see why (−4) × (−2) = +8.

Sprint 16

Same Number, Different Tokens

Different-looking token sets can stand for the same number. Find out whether the product changes when the tokens look different.

Sprint 17

Sign Rules From Patterns

Continue a steady pattern of products below zero and see the four sign rules for multiplying integers appear.

Sprint 18

Brahmagupta's Rules: Fortune And Debt

How Brahmagupta wrote the sign rules for multiplying and dividing in 628 CE, using fortunes and debts.

Sprint 19

Times 1, Times −1, And Swapping The Order

Multiplying by 1 keeps a number, multiplying by −1 gives its opposite, and swapping the two numbers never changes a product.

Sprint 20

Exam Marks And Profit And Loss

Use positive and negative integers to total exam marks with negative marking, and to find a company's profit or loss.

Sprint 21

Mine Lifts And Falling Temperatures

Write a speed with a direction as an integer, and find where a mine lift or a temperature ends up with start + rate × time.

Sprint 22

The Magic Grid Of Integers

Play the grid game from the start of the chapter, multiply four integers with the right sign, and find out why everyone gets −30240.

Sprint 23

Division Of Integers

Divide integers by turning each division into a multiplication question, and use the same sign rules as multiplication.

Sprint 24

Multiplying Three Or More Integers

Multiply three or more integers in any order, and predict the sign of a long product by counting the negative numbers.

Sprint 25

The Distributive Property With Integers

See with tokens why a × (b + c) = a × b + a × c works for negative numbers, and use it to find products quickly.

Sprint 26

Pick The Pattern: Number Machines

Find the hidden rule of a number machine by testing your guess on every row, then use it on the missing row.

Sprint 27

Integer Puzzles: Collatz And Consecutive Numbers

Follow a Collatz-style rule with negative numbers until it loops, and find three consecutive integers with a given product.

Sprint 28

Integer Puzzles: Alien Coins And Expressions

Pay exact amounts with +13 and −9 coins, then use brackets and signs to make expressions as big or as small as possible.

Sprint 29

Review: Operations With Integers

Check what you can now do with integers, revise every rule, and solve mixed problems from the whole chapter.

Sprint 30

The Biggest Tile That Fits

Find the biggest square tile that covers a 12 ft by 16 ft floor exactly, and meet the Highest Common Factor (HCF).

Sprint 31

Rice Bags And Jump Jackpot

Pack rice from two farms in equal bags using as few bags as possible, and find the longest jump that lands on two treasures.

Sprint 32

Prime Factorisation

Break any number into primes with a factor tree or the division method, and see why you always get the same primes.

Sprint 33

Every Factor From Prime Factors

Build every factor of a number from its prime factors, and learn how one counterexample can break a claim.

Sprint 34

Common Factors From Prime Factors

Find all the common factors of two numbers by spotting the subparts their prime factorisations share, then pick the biggest.

Sprint 35

The HCF Shortcut

Find the HCF directly: take each prime the numbers share, the fewest times it appears. Works for three numbers too.

Sprint 36

Torans, Gajak And The LCM

Find the shortest toran two people can both make, and the next day for free gajak, and meet the Lowest Common Multiple (LCM).

Sprint 37

The LCM From Prime Factors

A multiple contains the whole prime factorisation of the number. Build the LCM by taking each prime the most times it appears.

Sprint 38

Patterns And General Statements

Turn examples into rules that always hold, such as 'if one number divides the other, it is their HCF', and write them with algebra.

Sprint 39

What Happens When Numbers Are Doubled

Double both numbers and the HCF doubles. Find out when the HCF of two multiples of the same number is exactly that number.

Sprint 40

One Ladder For HCF And LCM

Divide two numbers by common factors in one ladder and read off both the HCF and the LCM, then speed it up with bigger divisors.

Sprint 41

HCF × LCM = Product

Discover that the HCF times the LCM of two numbers is their product, see why, and use it to find a missing value.

Sprint 42

HCF Word Problems: Cubes In A Box

Decide which cube sizes pack a box with no gaps, and find the biggest number that divides two numbers.

Sprint 43

LCM Word Problems: Stars, Cows And A Circle Game

Find when repeating patterns line up, how many cows the cowherd had, and how many children were playing Fire in the Mountain.

Sprint 44

Two Challenges: The Dog And The Rabbit, And Mahaviracharya's Fractions

Work out when a dog catches a rabbit with a head start, and add five fractions from Mahaviracharya using the LCM.

Sprint 45

Review: Finding Common Ground

Check what you can now do with factors and multiples, revise the HCF and LCM rules, and decode the book's mystery colours.

Sprint 46

Decimals Recap: Tenths, Hundredths And Thousandths

Read decimals as tenths, hundredths and thousandths, and divide any number by 10, 100 or 1000 by moving the point.

Sprint 47

Multiplying Decimals With Fractions

Multiply decimals by writing them as fractions over 10, 100 or 1000: the cost of pens, petrol mileage, a week's walk and an area.

Sprint 48

The Decimal Places Rule

Multiply decimals fast: multiply as whole numbers, then place the point by adding up the decimal places.

Sprint 49

Is The Product Always Bigger?

Predict whether a decimal product is bigger or smaller than the numbers you multiply by comparing each with 1.

Sprint 50

Decimal Multiplication In Real Life

Turn shopping, measurement and profit problems into decimal multiplications: shirts, coins, oranges and notebooks.

Sprint 51

Dividing Decimals By 10, 100 And 1000

Cut a 3.9 m ribbon into 10 and 100 pieces, and divide any decimal by 10, 100 or 1000 by moving the point left.

Sprint 52

Division That Gives A Decimal

Share 29 m of ribbon between 2 and then 4 friends, by turning the division into a fraction over 10 or 100.

Sprint 53

Long Division Beyond The Ones

See long division as fair sharing, place by place, and keep going past the ones into tenths, hundredths and thousandths.

Sprint 54

Dividing A Decimal By A Counting Number

Share 9.5 kg of sugar into 4 bags and find 0.06 ÷ 5 with long division, placing the point before the tenths.

Sprint 55

Dividing By A Decimal

Make a decimal divisor whole by multiplying both numbers by 10, 100 or 1000, then divide as usual. Ravi's scooter trip and more.

Sprint 56

Does This Ever End? Never-Ending Decimals

Find out why 10 ÷ 3 and 1 ÷ 7 never end, meet the magic number 142857, and see which fractions give ending decimals.

Sprint 57

When Division Makes Numbers Bigger

Predict whether a quotient is bigger or smaller than the dividend by comparing the divisor with 1, and order expressions without calculating.

Sprint 58

Look Before You Leap: Decimals And The Calendar

See how 0.2422 of a day adds up over centuries, and why leap years follow the rules for 4, 100 and 400.

Sprint 59

Everyday Decimals

Choose the cheaper snack, work out profit and how many books fit on a shelf, convert metric units, and read fine number lines.

Sprint 60

Decimal Puzzles

Solve Sridharacharya's division problem from the Patiganita with decimals, and place digits to build the biggest and smallest products.

Sprint 61

Review: Another Peek Beyond The Point

Check what you can now do with decimals, revise the multiplication and division rules, and solve a Hidato puzzle.

Sprint 62

Statistical Questions And Statements

Tell a statistical question, one answered by collecting data that varies, from a question with one fixed answer.

Sprint 63

The Average: One Number For A Whole Group

Use the mean to compare groups of different sizes, and see it as the fair share when everything is shared equally.

Sprint 64

Know Your Onions: Comparing Data And Dot Plots

Compare two towns' onion prices using minimum, maximum, total, range and mean, and see the spread on a dot plot.

Sprint 65

Outliers And The Median

Find the median for odd and even counts, spot outliers, and see how an outlier pulls the mean away from the median.

Sprint 66

How Tall Is Your Class? Spread And Centre

Answer 'How tall is the class?' using both the spread (minimum and maximum) and the centre (mean and median) of dot plots.

Sprint 67

Zero Median Runs: Zero Is Data, 'Did Not Play' Is Not

Treat 0 as a real value but leave out missing ones, and choose sensible data to average: England's 407 and Sita's mango tree.

Sprint 68

Mean And Median Workshop

Practise choosing and finding the mean and median for real data: date palms, pets, tap water, newborn babies and sumo wrestlers.

Sprint 69

Clubbing The Columns: Double Bar Graphs

Read a double bar graph using its scale, and draw one from raw data with a sensible scale and a legend.

Sprint 70

Read A Graph Like A Detective: Rockets And Sunshine

Read complex graphs in two steps, identify what is shown and then infer, and check which statements the data can support.

Sprint 71

Graphs In The Wild: Cricket, Animal Speeds And Electric Vehicles

Read three real-world graphs, a T20 runs-per-over chart, animal speeds and EV registrations, and spot claims they can't support.

Sprint 72

Data Detective: Telling Tall Tales About Heights

Investigate 30 years of Indian children's average heights, test claims against the table, and estimate from the trend.

Sprint 73

Data Detective Casebook

Solve mixed data cases: compare pockets, quiz scores and Sudoku times, split a class at the median, and check claims about skyscrapers.

Sprint 74

Be A Data Collector: Projects With Your Own Data

Plan, collect, show and summarise your own data: estimating lengths, sentence lengths, names, and how long a minute feels.

Sprint 75

Review: Connecting The Dots

Check what you can now do with data, revise mean, median, outliers and graphs, and crack the chapter's lock puzzle.

Sprint 76

The Perfect Eye: Perpendicular Bisectors

Find the exact centres for a symmetrical eye, prove with congruence that they give the perpendicular bisector, and construct it in three steps.

Sprint 77

Right Angles Anywhere: At A Point, From Outside, And With A Rope

Draw a right angle at a point on a line or through a point outside it, and see how the Śulba-Sūtras did it with a rope.

Sprint 78

Bisecting Angles: 45° And The Eight-Petalled Flower

Bisect any angle using SSS congruence, and use repeated bisection to build 45° designs like the eight-petalled flower.

Sprint 79

Copying Angles And Drawing Parallel Lines

Copy any angle exactly using SSS congruence, and use that to construct a line parallel to a given line.

Sprint 80

Arch Designs: Trefoil And Pointed Arches

Plan support lines for symmetrical arches, and construct trefoil and pointed arches with equal angles and equal lengths.

Sprint 81

Regular Hexagons From Six Equilateral Triangles

Check whether angles fit around a point using the 360° total, and see why six equilateral triangles make a regular hexagon.

Sprint 82

60°, 30° And 15° Angles, And The Six-Pointed Star

Construct 60° from an equilateral triangle, halve it to 30° and 15°, and build hexagons, stars and designs.

Sprint 83

Tangram Puzzles: Seven Pieces, Many Shapes

Rearrange the seven tangram pieces into new shapes, reasoning about the pieces' sizes and angles.

Sprint 84

Tiling Grids With Dominoes: Counting And Colouring

Decide whether a grid can be covered with 2 × 1 tiles by counting squares, and use chessboard colouring to prove some regions impossible.

Sprint 85

Tiling The Whole Plane

Explain which regular polygons tile the plane using the angles at a corner, and meet tilings in art and in beehives.

Sprint 86

Review: Constructions And Tilings

Check what you can now construct and explain, revise the key constructions, and solve mixed problems on constructions and tilings.

Sprint 87

Balancing Mobiles And Scales: Find The Hidden Weights

Find unknown weights on mobiles and scales: both sides weigh the same, and taking equal weights off both sides keeps the balance.

Sprint 88

From Patterns And Pictures To Equations

Write an equation for an unknown from a pattern or a balance picture, and learn the words equation, LHS, RHS and solving.

Sprint 89

Guessing Vs Balancing: Two Ways To Solve

Solve by trial and error, see why it can be slow, and learn that doing the same thing to both sides keeps an equation true.

Sprint 90

Solving Equations Step By Step

Get the unknown alone by removing terms and factors from both sides, collect unknowns on one side, and always check your answer.

Sprint 91

Shortcut Moves: Moving Terms Across The Equals Sign

Use the shortcut where a term changes sign as it crosses '=', and know why it works.

Sprint 92

Word Problems Into Equations: Tiles, Parties And Savings

Choose an unknown, write an equation from a real situation, solve it, and answer in words; solve bracketed equations in more than one way.

Sprint 93

Number Tricks And Problems With Two Unknowns

Explain 'think of a number' tricks with algebra, and solve two-unknown problems by writing one unknown using the other.

Sprint 94

Working Backwards: Make Equations With A Given Solution

Build equations with a chosen solution by doing the same thing to both sides, and use one equation to find related values without solving.

Sprint 95

Practice Lab: Solve, Check And Find Your Way Out

Solve many kinds of equations, check every answer, and use your solutions to find a path through the book's equation maze.

Sprint 96

Puzzles And Real Life: Machines, Blobs, Bricks And Giraffes

Turn puzzles, number machines, pictures and stick patterns into equations and solve them.

Sprint 97

Mind The Mistake, Mend The Mistake

Find the exact step where a solution goes wrong, say what the mistake is, and correct it.

Sprint 98

A Pinch Of History: Bījagaṇita And Brahmagupta's Formula

See how algebra grew in India and travelled the world, read equations in old Indian notation, and use Brahmagupta's formula for Ax + B = Cx + D.

Sprint 99

Review: Finding The Unknown

Check what you can now do with equations, revise the key moves, and solve fresh problems on your own.