VaaraVaara Classroom

Videos Practice Progress

Videos
Practice
Progress

←All chaptersBack to videos

NCERT · Ganita Manjari · Volume 1 · Chapter 3

The World of Numbers

51 videos in this chapter

Counting and the origins of numbers

5 videos

Chapter 3Counting Before Numerals: Match One Object to Another3:26Counting Before Numerals: Match One Object to AnotherYou'll learn: Explain one-to-one correspondence and why counting numbers were needed.Chapter 3Tally Marks, Grouping and the History of Counting3:36Tally Marks, Grouping and the History of CountingYou'll learn: Explore the chapter’s bone-tally examples and the need for larger-number notation.Chapter 3Trade Ratios and Prime-Number Groups3:26Trade Ratios and Prime-Number GroupsYou'll learn: Apply counting to a trade ratio and identify the property of a numerical group.Chapter 3Why Counting Numbers Are Not Enough for Subtraction3:22Why Counting Numbers Are Not Enough for SubtractionYou'll learn: Test closure under addition and subtraction and motivate larger number systems.Chapter 3Count to Twelve on One Hand3:30Count to Twelve on One HandYou'll learn: Explain the finger-joint counting method and its connection with base twelve.

Zero and integers

6 videos

Chapter 3Zero as a Placeholder and as a Number3:35Zero as a Placeholder and as a NumberYou'll learn: Distinguish an empty place in numeral notation from zero as a quantity used in arithmetic.Chapter 3What Happens When We Calculate with Zero?3:22What Happens When We Calculate with Zero?You'll learn: Explain a−a=0, a+0=a, a−0=a and a×0=0.Chapter 3Debts and Fortunes: Why We Need Negative Numbers3:21Debts and Fortunes: Why We Need Negative Numbers

Rational numbers and arithmetic

7 videos

Chapter 3From Parts of a Whole to Rational Numbers3:24From Parts of a Whole to Rational NumbersYou'll learn: Define p/q with integer numerator and non-zero integer denominator. Include negative fractions and integers.Chapter 3Different Fractions Can Name the Same Number3:32Different Fractions Can Name the Same NumberYou'll learn: Generate equivalent signed fractions and reduce to lowest terms.Chapter 3Check Fraction Equality with Cross Products3:12

Rational numbers on the number line

5 videos

Chapter 3Place Positive, Negative and Improper Fractions on a Line3:23Place Positive, Negative and Improper Fractions on a LineYou'll learn: Divide unit intervals equally and locate fractions on either side of zero.Chapter 3Absolute Value Measures Distance, Not Direction3:29Absolute Value Measures Distance, Not DirectionYou'll learn: Explain absolute value and distance between two rational points.Chapter 3There Is Always Another Rational Number Between Two3:34

Rational-number applications

2 videos

Chapter 3How Many Complete Kurtas Can the Tailor Make?3:31How Many Complete Kurtas Can the Tailor Make?You'll learn: Divide mixed-number measurements and interpret a whole-object result.Chapter 3Solve a Rational-Number Equation and Reason About Its Sign3:18Solve a Rational-Number Equation and Reason About Its SignYou'll learn: Solve a fractional-coefficient equation and infer a product’s sign from a relation.

Irrational lengths and proofs

7 videos

Chapter 3A Unit Square Reveals a New Kind of Number3:25A Unit Square Reveals a New Kind of NumberYou'll learn: Use the diagonal of a unit square to motivate √2 and irrational numbers.Chapter 3Why √2 Cannot Be a Fraction3:25Why √2 Cannot Be a FractionYou'll learn: Follow the contradiction proof from a lowest-terms fraction to both numerator and denominator being even.Chapter 3Extend the Irrationality Proof to Other Square Roots4:00

Real numbers and decimal expansions

13 videos

Chapter 3Rational and Irrational Numbers Share One Real Number Line3:44Rational and Irrational Numbers Share One Real Number LineYou'll learn: Connect natural numbers, integers, rationals, irrationals and reals in a nested classification.Chapter 3Why Rational Decimals Either Stop or Repeat3:30Why Rational Decimals Either Stop or RepeatYou'll learn: Distinguish terminating decimals from repeating ones and use long division.Chapter 3Repeated Remainders Make Repeated Digits3:39

Closing perspective

1 video

Chapter 3Why √−1 Is Not on the Real Number Line2:59Why √−1 Is Not on the Real Number LineYou'll learn: Explain why squaring any real number cannot produce −1 and introduce the chapter’s final question.

Chapter challenge problems

5 videos

Chapter 3Find More Reciprocals with Cyclic Repeating Blocks3:46Find More Reciprocals with Cyclic Repeating BlocksYou'll learn: Test reciprocal decimals and compare rotations of their repeating blocks.Chapter 3What Does the Fourth Decimal Place Tell Us?3:38What Does the Fourth Decimal Place Tell Us?You'll learn: Relate a last non-zero fourth decimal digit to a numerator over 10⁴ and examine reduction.Chapter 3Make Enough Space Between Fraction Numerators3:56

You'll learn: Introduce negative values, zero and positive integers on one number line.

Chapter 3Add and Subtract Integers with Meaning3:33Add and Subtract Integers with MeaningYou'll learn: Interpret signed changes and subtraction of a negative amount.

Chapter 3Why Does a Negative Times a Negative Become Positive?3:40Why Does a Negative Times a Negative Become Positive?You'll learn: Explain the chapter’s signed multiplication rules, then apply integer division.

Chapter 3Track Temperature Changes and a Trader’s Balance3:40Track Temperature Changes and a Trader’s BalanceYou'll learn: Translate a sequence of changes into an integer calculation.

Check Fraction Equality with Cross Products

You'll learn: Use ad=bc to check equality when denominators are non-zero.

Chapter 3Add and Subtract Rational Numbers3:36Add and Subtract Rational NumbersYou'll learn: Find a common denominator, combine signed numerators and simplify.

Chapter 3Multiply and Divide Rational Numbers3:31Multiply and Divide Rational NumbersYou'll learn: Multiply numerators and denominators and divide by a non-zero fraction using its reciprocal.

Chapter 3Which Arithmetic Laws Still Work for Fractions?3:37Which Arithmetic Laws Still Work for Fractions?You'll learn: Explain closure, commutativity of addition/multiplication and distributivity.

Chapter 3Does This Fraction Equation Have Just One Answer?3:17Does This Fraction Equation Have Just One Answer?You'll learn: Simplify both sides and recognise an identity rather than a unique solution.

There Is Always Another Rational Number Between Two

You'll learn: Use averaging to generate a number strictly between two distinct rational numbers.

Chapter 3Find Several Rational Numbers in One Interval3:40Find Several Rational Numbers in One IntervalYou'll learn: Use common denominators or repeated averaging to list the required number of interior points.

Chapter 3Find Numbers Between Very Close Decimals3:44Find Numbers Between Very Close DecimalsYou'll learn: Refine decimal places to reveal rational values in a narrow interval.

Extend the Irrationality Proof to Other Square Roots

You'll learn: Adapt the contradiction method and identify the divisibility step needed.

Chapter 3Construct √2 and Transfer It to the Number Line3:24Construct √2 and Transfer It to the Number LineYou'll learn: Build a right triangle and use a compass arc to locate the exact irrational length.

Chapter 3Build More Square Roots with a Square Root Spiral3:51Build More Square Roots with a Square Root SpiralYou'll learn: Extend successive unit-perpendicular triangles and identify hypotenuse lengths.

Chapter 3Pi: An Exact Number with Fractional Approximations3:27Pi: An Exact Number with Fractional ApproximationsYou'll learn: Define π as circumference divided by diameter and distinguish exact π from a rational approximation.

Chapter 3How Can Adding More Terms Approach Pi?3:32How Can Adding More Terms Approach Pi?You'll learn: Explore the chapter’s alternating infinite-series example through finite partial sums.

Repeated Remainders Make Repeated Digits

You'll learn: Explain why a long-division process eventually terminates or cycles.

Chapter 3Predict a Terminating Decimal Without Dividing3:41Predict a Terminating Decimal Without DividingYou'll learn: Reduce a fraction first, then inspect prime factors of its denominator.

Chapter 3How Many Decimal Places Will a Fraction Need?3:39How Many Decimal Places Will a Fraction Need?You'll learn: Balance powers of 2 and 5 to turn a reduced denominator into a power of ten.

Chapter 3Turn a Terminating Decimal into a Fraction3:42Turn a Terminating Decimal into a FractionYou'll learn: Use place value to write a power-of-ten denominator and reduce.

Chapter 3Turn a Pure Repeating Decimal into a Fraction3:32Turn a Pure Repeating Decimal into a FractionYou'll learn: Multiply by a power of ten matching the cycle length, subtract and solve.

Chapter 3A Non-Repeating Start Followed by One Repeating Digit3:46A Non-Repeating Start Followed by One Repeating DigitYou'll learn: Align repeating tails after moving the non-repeating prefix.

Chapter 3Convert a Mixed Decimal with a Repeating Block3:50Convert a Mixed Decimal with a Repeating BlockYou'll learn: Extend tail alignment to a multi-digit repeating cycle and select the correct powers of ten.

Chapter 3Why the Digits 142857 Keep Rotating3:34Why the Digits 142857 Keep RotatingYou'll learn: Explore a cyclic number arising from the repeating block of 1/7.

Chapter 3Investigate the Repeating Blocks of Thirteenths3:37Investigate the Repeating Blocks of ThirteenthsYou'll learn: Compute and compare repeating blocks for successive multiples of 1/13.

Chapter 3A Decimal Can Have a Pattern Without Repeating3:55A Decimal Can Have a Pattern Without RepeatingYou'll learn: Classify rational and irrational numbers using exact form and decimal behaviour.

Chapter 3Why 0.999… Is Exactly 13:36Why 0.999… Is Exactly 1You'll learn: Use algebraic cancellation to prove equality and explain two decimal representations of terminating values.

Make Enough Space Between Fraction Numerators

You'll learn: Choose an equivalent common denominator with enough integer numerator positions between two fractions.

Chapter 3Can Three Rational Numbers Satisfy Both Zero Conditions?3:09Can Three Rational Numbers Satisfy Both Zero Conditions?You'll learn: Combine the sum and pairwise-product conditions to obtain a sum of squares equal to zero.

Chapter 3Prove That the Average Really Lies Between Two Numbers3:18Prove That the Average Really Lies Between Two NumbersYou'll learn: Justify both rationality of the average and its strict location between distinct endpoints.