Vaara Classroom

NCERT · Ganita Manjari · Volume 1 · Chapter 1

Orienting Yourself: The Use of Coordinates

Chapter 1Why Do We Need Coordinates?4:04Why Do We Need Coordinates?You'll learn: Describe an exact location using a reference system and two numbers.Chapter 1How Grid Thinking Developed into Coordinate Geometry4:12How Grid Thinking Developed into Coordinate GeometryYou'll learn: Trace the introduction’s progression from grids and navigation to zero, negative numbers and algebra with geometry.Chapter 1From a Number Line to a Coordinate Plane3:46From a Number Line to a Coordinate PlaneYou'll learn: Explain one-dimensional and two-dimensional location. Show why a plane needs two coordinates.Chapter 1Meet the x-Axis, y-Axis and Origin3:40Meet the x-Axis, y-Axis and OriginYou'll learn: Identify horizontal and vertical axes, perpendicular intersection, origin (0,0) and equal unit marks.Chapter 1Positive and Negative Directions on the Plane3:50Positive and Negative Directions on the PlaneYou'll learn: Connect right/up to positive values and left/down to negative values.Chapter 1What Happens When One Coordinate Is Zero?3:32What Happens When One Coordinate Is Zero?You'll learn: Locate points of the form (x,0), (0,y) and (0,0), including decimal coordinates.Chapter 1What Do the Two Numbers in (x,y) Mean?3:54What Do the Two Numbers in (x,y) Mean?You'll learn: Explain horizontal and vertical positions and perpendicular guides to the axes.Chapter 1Four Quadrants: Finding a Point from Its Signs3:32Four Quadrants: Finding a Point from Its SignsYou'll learn: Explain quadrant numbering and the four sign combinations.Chapter 1Read a Point, Plot a Point—and Check It3:55Read a Point, Plot a Point—and Check ItYou'll learn: Locate points from coordinates and read coordinates from a graph.Chapter 1Does Swapping the Coordinates Change the Point?3:53Does Swapping the Coordinates Change the Point?You'll learn: Explain why ordered pairs depend on order and coincide after swapping only when x=y.Chapter 1Turn a Room into a Two-Dimensional Map4:02Turn a Room into a Two-Dimensional MapYou'll learn: Explain floor plans, scale and object corners. Identify information a floor plan cannot specify.Chapter 1Use Coordinates to Measure and Compare Doorways4:00Use Coordinates to Measure and Compare DoorwaysYou'll learn: Read endpoints, find doorway widths and compare the room and bathroom doors.Chapter 1Three Table Feet Are Given—Where Is the Fourth?3:48Three Table Feet Are Given—Where Is the Fourth?You'll learn: Complete an axis-aligned rectangle, find length and width and discuss placement.Chapter 1Map the Bathroom Using Negative Coordinates4:16Map the Bathroom Using Negative CoordinatesYou'll learn: Read bathroom and shower corners, identify the shower shape and allocate spaces.Chapter 1Will the Bathroom Door Hit the Wardrobe?4:00Will the Bathroom Door Hit the Wardrobe?You'll learn: Reason about a door’s hinged movement and what changes when it is widened.Chapter 1Place a Dining Table Exactly in the Centre4:10Place a Dining Table Exactly in the CentreYou'll learn: Extend the room plan, identify dining-room corners and use equal margins for a centred table.Chapter 1Find Distances Along Horizontal and Vertical Lines3:54Find Distances Along Horizontal and Vertical LinesYou'll learn: Recognise equal x- or y-coordinates and use the absolute difference of the other coordinates.Chapter 1Find a Slanting Distance by Building a Right Triangle3:45Find a Slanting Distance by Building a Right TriangleYou'll learn: Turn horizontal and vertical shifts into perpendicular legs and find direct distance.Chapter 1Build the Distance Formula from the Picture4:04Build the Distance Formula from the PictureYou'll learn: Generalise coordinate differences and derive the square-root distance expression.Chapter 1Use the Distance Formula to Measure a Triangle3:50Use the Distance Formula to Measure a TriangleYou'll learn: Apply the formula to all three sides and interpret exact square-root lengths.Chapter 1Why the Distance Formula Works with Negative Coordinates4:02Why the Distance Formula Works with Negative CoordinatesYou'll learn: Subtract negative coordinates and explain why reversing endpoint order preserves distance.Chapter 1Reflect a Triangle: What Changes and What Stays the Same?3:50Reflect a Triangle: What Changes and What Stays the Same?You'll learn: Track coordinate changes under reflection in either axis and compare side lengths.Chapter 1Predict a Shape Before Plotting Its Points3:41Predict a Shape Before Plotting Its PointsYou'll learn: Predict axis-parallel sides, perpendicular sides and mirror points before checking the graph.Chapter 1Build a Right Triangle Around a Given Point3:50Build a Right Triangle Around a Given PointYou'll learn: Construct a valid right triangle using a given vertex and calculate its three sides.Chapter 1Can a Street Address Identify More Than One Crossing?4:14Can a Street Address Identify More Than One Crossing?You'll learn: Test whether a street-index convention identifies a unique crossing and why direction information matters.Chapter 1Do Two Circular Icons Fit and Overlap on a Screen?4:40Do Two Circular Icons Fit and Overlap on a Screen?You'll learn: Check circle extents against screen boundaries and compare centre distance with the radii.Chapter 1Is This Tilted Quadrilateral Really a Square?4:12Is This Tilted Quadrilateral Really a Square?You'll learn: Use distances to justify the shape and determine its area.Chapter 1Are Three Points on One Straight Line?4:20Are Three Points on One Straight Line?You'll learn: Use distance addition to test collinearity, then plot both sets to verify.Chapter 1Build Isosceles Triangles Using Coordinates4:06Build Isosceles Triangles Using CoordinatesYou'll learn: Construct the two requested triangle types and verify their properties.Chapter 1What Makes a Point the Midpoint?4:28What Makes a Point the Midpoint?You'll learn: Check that a midpoint lies on the segment and divides it equally. Discover coordinate averages.Chapter 1Find a Missing Endpoint from the Midpoint4:12Find a Missing Endpoint from the MidpointYou'll learn: Reverse the coordinate-average relationship to find an unknown endpoint.Chapter 1Divide a Segment into Three Equal Parts4:00Divide a Segment into Three Equal PartsYou'll learn: Extend equal division from halves to thirds and find both internal division points.Chapter 1Is a Point Inside, On or Outside a Circle?4:14Is a Point Inside, On or Outside a Circle?You'll learn: Find a common radius from distances to the origin and classify additional points.Chapter 1Rebuild a Triangle from Its Side Midpoints4:28Rebuild a Triangle from Its Side MidpointsYou'll learn: Combine three midpoint relationships to recover the original triangle vertices.

NCERT · Ganita Manjari · Volume 1 · Chapter 2

Introduction to Linear Polynomials

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Chapter 2Build an Algebraic Expression from a Shopping Story3:48Build an Algebraic Expression from a Shopping StoryYou'll learn: Translate quantities, repeated groups and a fixed extra into an expression. Identify terms, variables and constants.Chapter 2Why Do Fencing and Planting Produce Different Terms?3:50Why Do Fencing and Planting Produce Different Terms?You'll learn: Model boundary costs and area costs and compare their algebraic terms.Chapter 2One Wire, Many Rectangles: Express the Area3:48One Wire, Many Rectangles: Express the AreaYou'll learn: Use a fixed perimeter to express width and area using one variable.Chapter 2What Is a Polynomial in One Variable?4:10What Is a Polynomial in One Variable?You'll learn: Distinguish expressions involving one variable from the opening two-variable examples.Chapter 2Find Coefficients, Constant Terms and Missing Terms3:49Find Coefficients, Constant Terms and Missing TermsYou'll learn: Read signed coefficients and recognise an absent power as having coefficient zero.Chapter 2Degree: The Highest Power Tells the Type3:53Degree: The Highest Power Tells the TypeYou'll learn: Identify constant, linear, quadratic and cubic polynomials and construct examples.Chapter 2What Makes a Polynomial Linear?3:50What Makes a Polynomial Linear?You'll learn: Recognise degree-one expressions and distinguish them from quadratic and cubic forms.Chapter 2A Square’s Perimeter Grows at a Constant Rate3:53A Square’s Perimeter Grows at a Constant RateYou'll learn: Connect a change in side length to a constant change in perimeter.Chapter 2Joining Fee Plus Match Fee: Build a Linear Rule3:48Joining Fee Plus Match Fee: Build a Linear RuleYou'll learn: Separate a starting amount from an amount added per match and work backwards from a total.Chapter 2When Does an Expression Become an Equation?3:57When Does an Expression Become an Equation?You'll learn: Distinguish a polynomial from a statement equating it to a known total.Chapter 2A Polynomial as an Input-Output Machine3:54A Polynomial as an Input-Output MachineYou'll learn: Explain substitution as a sequence of operations producing an output.Chapter 2Evaluate Polynomials with Positive, Zero and Negative Inputs3:56Evaluate Polynomials with Positive, Zero and Negative InputsYou'll learn: Substitute carefully and distinguish squaring a negative value from multiplying a signed coefficient.Chapter 2How Old Are Salil and His Mother?3:55How Old Are Salil and His Mother?You'll learn: Translate an age relationship and a future total into one equation.Chapter 2Use a Ratio to Split a Total or Difference3:53Use a Ratio to Split a Total or DifferenceYou'll learn: Represent related quantities using one variable and solve.Chapter 2Coin Counts and Coin Values Are Different3:45Coin Counts and Coin Values Are DifferentYou'll learn: Represent the number of each coin and build the total-money equation.Chapter 2Recover a Rectangle’s Dimensions from Its Perimeter3:53Recover a Rectangle’s Dimensions from Its PerimeterYou'll learn: Express length through width and solve the perimeter condition.Chapter 2See Why the Tile Pattern Has 2n−1 Tiles3:45See Why the Tile Pattern Has 2n−1 TilesYou'll learn: Connect the growing picture, a value table and a general rule.Chapter 2Find a Distant Stage—or Work Back from the Tile Count3:50Find a Distant Stage—or Work Back from the Tile CountYou'll learn: Use a stage rule in both directions.Chapter 2Pocket Money Can Follow a Decreasing Linear Pattern3:21Pocket Money Can Follow a Decreasing Linear PatternYou'll learn: Build and interpret a decreasing rule and solve for a target balance.Chapter 2Why the Auto Fare Rule Starts After the First Two Kilometres3:50Why the Auto Fare Rule Starts After the First Two KilometresYou'll learn: Separate the initial flat fare from later per-kilometre charges.Chapter 2Can Area and Volume Have Linear Rules?3:40Can Area and Volume Have Linear Rules?You'll learn: Show how fixing other dimensions makes area or volume linear in one changing dimension.Chapter 2Recognise Linear Growth from Equal Increases3:31Recognise Linear Growth from Equal IncreasesYou'll learn: Read starting value and constant increase from a context and a table.Chapter 2Recognise Linear Decay from Equal Decreases3:22Recognise Linear Decay from Equal DecreasesYou'll learn: Model a quantity falling by the same amount per time interval.Chapter 2Model a Growing Plant and a Growing Village3:39Model a Growing Plant and a Growing VillageYou'll learn: Form increasing rules, make tables and predict future values.Chapter 2Model a Phone’s Falling Value and a Prepaid Balance3:35Model a Phone’s Falling Value and a Prepaid BalanceYou'll learn: Form decreasing rules and interpret an exhaustion or depreciation calculation.Chapter 2Find a Fixed Fee and a Per-GB Charge from Two Bills3:36Find a Fixed Fee and a Per-GB Charge from Two BillsYou'll learn: Build two conditions and recover the parameters in y=ax+b.Chapter 2Recover Pricing Rules for Learning Modules and Court Time3:32Recover Pricing Rules for Learning Modules and Court TimeYou'll learn: Transfer the two-observation method to other fixed-fee situations.Chapter 2Build the Celsius–Fahrenheit Relationship3:43Build the Celsius–Fahrenheit RelationshipYou'll learn: Use two known temperature pairs to determine a and b.Chapter 2Draw a Linear Relationship Using Two Points3:32Draw a Linear Relationship Using Two PointsYou'll learn: Turn an equation into coordinate pairs and draw the corresponding straight line.Chapter 2Does a Point Belong to the Line?3:21Does a Point Belong to the Line?You'll learn: Complete a value table and verify point membership by substitution.Chapter 2Read a Linear Rule from Its Plotted Points3:31Read a Linear Rule from Its Plotted PointsYou'll learn: Identify a coordinate relationship from positive and negative input-output pairs.Chapter 2What Does a Positive Slope Change?3:28What Does a Positive Slope Change?You'll learn: Compare rising lines through the origin and relate coefficient size to steepness.Chapter 2What Does a Negative Slope Change?3:25What Does a Negative Slope Change?You'll learn: Compare falling lines and explain how coefficient magnitude affects steepness.Chapter 2Where Does a Line Cross the y-Axis?3:17Where Does a Line Cross the y-Axis?You'll learn: Set x=0 and interpret the signed y-intercept b in y=ax+b.Chapter 2Change the Constant: Why Do the Lines Stay Parallel?3:19Change the Constant: Why Do the Lines Stay Parallel?You'll learn: Hold a fixed and vary b to compare translated lines.Chapter 2Read What Changes When a and b Change4:24Read What Changes When a and b ChangeYou'll learn: Compare the five exercise families using graphs and coefficients.Chapter 2Reverse the Digits and Find the Number3:17Reverse the Digits and Find the NumberYou'll learn: Use place value and two digit conditions to recover a two-digit number.Chapter 2Rearrange an Equation Before Reading Its Slope3:39Rearrange an Equation Before Reading Its SlopeYou'll learn: Rewrite equations in y=ax+b, graph them and identify parallel pairs.Chapter 2Convert Between Kelvin and Fahrenheit3:35Convert Between Kelvin and FahrenheitYou'll learn: Use a linear temperature formula forwards and backwards.Chapter 2Work Done as a Linear Graph3:32Work Done as a Linear GraphYou'll learn: Express work as force times distance and interpret the graph.Chapter 2Recover a Linear Polynomial from Two Points3:19Recover a Linear Polynomial from Two PointsYou'll learn: Find a and b from two graph points, then locate both axis crossings.Chapter 2Find Two Polynomials from Their Sum and a Graph Condition3:22Find Two Polynomials from Their Sum and a Graph ConditionYou'll learn: Combine p(0), a zero of p−q and the sum identity to recover both rules.Chapter 2Count Matchsticks in a Growing Chain of Hexagons3:25Count Matchsticks in a Growing Chain of HexagonsYou'll learn: Derive a stage rule using shared sides and test whether a total is possible.Chapter 2Find a Parallel Line from a Single Point3:29Find a Parallel Line from a Single PointYou'll learn: Find one linear rule from two points, then use its slope to recover a parallel rule.Chapter 2What Do All Lines y=ax+a Have in Common?3:30What Do All Lines y=ax+a Have in Common?You'll learn: Factor a(x+1) and discover common graphical features.

NCERT · Ganita Manjari · Volume 1 · Chapter 3

The World of Numbers

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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.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 NumbersYou'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.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 Products3:12Check Fraction Equality with Cross ProductsYou'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.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 Two3:34There Is Always Another Rational Number Between TwoYou'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.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.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 Roots4:00Extend the Irrationality Proof to Other Square RootsYou'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.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 Digits3:39Repeated Remainders Make Repeated DigitsYou'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.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 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 Numerators3:56Make Enough Space Between Fraction NumeratorsYou'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.

NCERT · Ganita Manjari · Volume 1 · Chapter 4

Exploring Algebraic Identities

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Chapter 4Three Consecutive Squares Always Give Two3:21Three Consecutive Squares Always Give TwoYou'll learn: Explore the opening square-number calculation and formulate the observed pattern.Chapter 4An Identity Is an Equation True for Every Allowed Value3:27An Identity Is an Equation True for Every Allowed ValueYou'll learn: Distinguish an identity from an equation holding only for certain inputs.Chapter 4Build the Square of a Sum from Four Pieces3:13Build the Square of a Sum from Four PiecesYou'll learn: Partition a square of side a+b into a², b² and two ab rectangles.Chapter 4Why the Square Identity Also Works for Negative Numbers3:20Why the Square Identity Also Works for Negative NumbersYou'll learn: Move from area pictures to a distributive proof valid for signed and rational values.Chapter 4When Is the Square of a Sum Bigger Than the Sum of Squares?3:10When Is the Square of a Sum Bigger Than the Sum of Squares?You'll learn: Use the sign of 2ab to compare (a+b)² with a²+b².Chapter 4Square a Binomial with Coefficients and Fractions3:30Square a Binomial with Coefficients and FractionsYou'll learn: Identify the complete expressions playing the roles of a and b and expand correctly.Chapter 4Use Nearby Round Numbers to Calculate Squares3:25Use Nearby Round Numbers to Calculate SquaresYou'll learn: Choose a convenient additive split and evaluate with (a+b)².Chapter 4Read a Perfect Square Backwards3:27Read a Perfect Square BackwardsYou'll learn: Recognise a²+2ab+b² and write its repeated factor.Chapter 4Take Out a Common Factor Before Looking for a Square3:23Take Out a Common Factor Before Looking for a SquareYou'll learn: Extract a common factor to expose a simpler perfect-square expression.Chapter 4Derive the Square of a Difference by Substitution3:19Derive the Square of a Difference by SubstitutionYou'll learn: Replace b with −b in the square-of-a-sum identity.Chapter 4See the Square of a Difference as an Area3:25See the Square of a Difference as an AreaYou'll learn: Remove the two appropriate rectangles from the larger square.Chapter 4Square Numbers Just Below a Round Number3:40Square Numbers Just Below a Round NumberYou'll learn: Use (a−b)² to make subtraction-based calculations efficient.Chapter 4Prove the Consecutive-Squares Pattern3:20Prove the Consecutive-Squares PatternYou'll learn: Express the three squares using n−1,n,n+1 and simplify.Chapter 4Square Three Terms by Grouping Two of Them3:14Square Three Terms by Grouping Two of ThemYou'll learn: Derive (a+b+c)² from the two-term square identity.Chapter 4Find All Nine Regions in a Three-Part Square3:18Find All Nine Regions in a Three-Part SquareYou'll learn: Label the geometrical model and connect paired rectangles to cross terms.Chapter 4Expand and Factor Three-Term Squares with Signs3:36Expand and Factor Three-Term Squares with SignsYou'll learn: Handle a negative term and reverse a six-term perfect-square expansion.Chapter 4Test a Proposed Identity Before Accepting It3:26Test a Proposed Identity Before Accepting ItYou'll learn: Use a counterexample or full expansion to test a claimed universal equality.Chapter 4Read a Four-Square Identity from a Rearrangement3:33Read a Four-Square Identity from a RearrangementYou'll learn: Translate the two rows of shapes into an algebraic relationship.Chapter 4Rearrange a Square to Reveal a Difference of Squares3:31Rearrange a Square to Reveal a Difference of SquaresYou'll learn: Relate a²−b² to a rectangle of sides a+b and a−b.Chapter 4Multiply Numbers Equally Spaced Around a Convenient Centre3:23Multiply Numbers Equally Spaced Around a Convenient CentreYou'll learn: Use the difference-of-squares identity for products.Chapter 4Why Squares of Numbers Ending in Five Have a Shortcut3:42Why Squares of Numbers Ending in Five Have a ShortcutYou'll learn: Use the chapter’s rearranged square identity to explain the calculation pattern.Chapter 4Choose the Identity That Makes the Calculation Easier4:02Choose the Identity That Makes the Calculation EasierYou'll learn: Compare two-term and three-term decompositions for a given square or product.Chapter 4Multiply Two Linear Expressions with Algebra Tiles3:13Multiply Two Linear Expressions with Algebra TilesYou'll learn: Build a rectangle from one x² tile, x tiles and unit tiles.Chapter 4Arrange Tiles to Discover Two Factors3:23Arrange Tiles to Discover Two FactorsYou'll learn: Reverse the area model to read dimensions from a quadratic.Chapter 4Why Must the Two Numbers Have the Right Sum and Product?3:19Why Must the Two Numbers Have the Right Sum and Product?You'll learn: Explain why splitting 7x as 3x+4x fits twelve unit tiles but 2x+5x does not.Chapter 4Generalise the Product (x+a)(x+b)3:15Generalise the Product (x+a)(x+b)You'll learn: Use tiles and distributivity to derive x²+(a+b)x+ab.Chapter 4Multiply Linear Factors with Different x-Coefficients3:26Multiply Linear Factors with Different x-CoefficientsYou'll learn: Generalise the rectangle model to non-monic factors.Chapter 4Factor Quadratics with Positive and Negative Terms3:27Factor Quadratics with Positive and Negative TermsYou'll learn: Find signed number pairs and express monic quadratics as two linear factors.Chapter 4Recover Missing Factors When the Leading Coefficient Is Not One3:22Recover Missing Factors When the Leading Coefficient Is Not OneYou'll learn: Use coefficient matching or middle-term splitting on the chapter’s non-monic examples.Chapter 4Two Ways to Combine Known Identities3:19Two Ways to Combine Known IdentitiesYou'll learn: Compare expansion first with factoring a difference of squares first.Chapter 4Derive the Cube of a Sum by Multiplication3:15Derive the Cube of a Sum by MultiplicationYou'll learn: Multiply (a+b) by its squared expansion and collect like terms.Chapter 4Build a Cube from Two Cubes and Six Cuboids3:14Build a Cube from Two Cubes and Six CuboidsYou'll learn: Visualise the cubic identity as a volume decomposition.Chapter 4Derive the Cube of a Difference and Track Its Signs3:32Derive the Cube of a Difference and Track Its SignsYou'll learn: Substitute −b into the cube-of-a-sum identity.Chapter 4Recognise a Perfect Cube in an Expanded Expression3:31Recognise a Perfect Cube in an Expanded ExpressionYou'll learn: Recover a repeated cubic factor from four terms.Chapter 4Calculate Cubes Using Nearby Convenient Numbers3:49Calculate Cubes Using Nearby Convenient NumbersYou'll learn: Choose a sum or difference and apply a cubic identity.Chapter 4Factor a Sum or Difference of Two Cubes3:24Factor a Sum or Difference of Two CubesYou'll learn: Multiply the proposed factors and observe cancellation of mixed terms.Chapter 4Does x−y Also Divide Fourth and Fifth Power Differences?3:21Does x−y Also Divide Fourth and Fifth Power Differences?You'll learn: Extend the chapter’s factor pattern using known identities and multiplication.Chapter 4Derive the Three-Variable Cubic Identity3:20Derive the Three-Variable Cubic IdentityYou'll learn: Expand and cancel to link a sum factor to x³+y³+z³−3xyz.Chapter 4Find a Sum of Cubes Without Finding the Three Numbers3:32Find a Sum of Cubes Without Finding the Three NumbersYou'll learn: Combine the square-of-three-terms identity with the cubic identity.Chapter 4Factor First, Then Cancel Common Factors3:31Factor First, Then Cancel Common FactorsYou'll learn: Simplify a rational expression by complete factorisation.Chapter 4Use Square and Cube Identities in Rational Expressions3:59Use Square and Cube Identities in Rational ExpressionsYou'll learn: Recognise powers and factor denominator quadratics before simplifying.Chapter 4Simplify Products of Quadratics and Fourth Powers4:03Simplify Products of Quadratics and Fourth PowersYou'll learn: Combine several identities and cancel only verified non-zero factors.Chapter 4Recover a Rectangle’s Dimensions from Its Tiles3:32Recover a Rectangle’s Dimensions from Its TilesYou'll learn: Turn a square, strips and unit squares into a factored area expression.Chapter 4Find Pool Dimensions by Factoring an Equation3:30Find Pool Dimensions by Factoring an EquationYou'll learn: Model a length difference and known area, then apply the zero-product property.Chapter 4Factor Areas and Volumes to Suggest Dimensions3:52Factor Areas and Volumes to Suggest DimensionsYou'll learn: Interpret factors as possible positive dimensions and divide an area by a given width.Chapter 4How Much Area Does a Path Around a Square Add?3:29How Much Area Does a Path Around a Square Add?You'll learn: Subtract the original square area from the larger outside square.Chapter 4A Number Plus Its Reciprocal: Use Factorisation3:17A Number Plus Its Reciprocal: Use FactorisationYou'll learn: Clear a non-zero denominator, factor the quadratic and check both solutions.Chapter 4Two Given Linear Factors Constrain a Quadratic3:26Two Given Linear Factors Constrain a QuadraticYou'll learn: Use the stated factors to compare coefficients and prove a relationship.Chapter 4Use a Sum and Pairwise Products in a Cubic Identity3:38Use a Sum and Pairwise Products in a Cubic IdentityYou'll learn: Express the second factor using the square-of-a-sum identity.Chapter 4Why Is n³−n Always Divisible by Six?3:26Why Is n³−n Always Divisible by Six?You'll learn: Factor into three consecutive integers and reason about divisibility.Chapter 4Evaluate Cubic Expressions from a Given Relationship3:36Evaluate Cubic Expressions from a Given RelationshipYou'll learn: Use a three-variable cubic identity to avoid finding individual variable values.

NCERT · Ganita Manjari · Volume 1 · Chapter 7

The Mathematics of Maybe: Introduction to Probability

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Chapter 7Probability Measures How Likely Something Is3:50Probability Measures How Likely Something IsYou'll learn: Introduce uncertainty and likelihood using the chapter’s everyday questions.Chapter 7Know What Could Happen Without Knowing What Will Happen3:50Know What Could Happen Without Knowing What Will HappenYou'll learn: Explain random experiments, individual trials and possible outcomes.Chapter 7Why Can a Coin Toss Be a Fair Way to Decide?3:39Why Can a Coin Toss Be a Fair Way to Decide?You'll learn: Connect fairness to a symmetric coin and an unbiased random toss.Chapter 7Read the Probability Scale from Zero to One3:47Read the Probability Scale from Zero to OneYou'll learn: Place impossible, less likely, even-chance, more likely and certain events on a scale.Chapter 7Fractions, Decimals and Percentages Describe the Same Chance4:01Fractions, Decimals and Percentages Describe the Same ChanceYou'll learn: Translate between numerical forms and interpret probability statements.Chapter 7Explain a Likelihood Ranking Instead of Guessing a Number4:00Explain a Likelihood Ranking Instead of Guessing a NumberYou'll learn: Rank the chapter’s everyday events and explain the assumptions behind each judgement.Chapter 7Estimate Probability from What Actually Happened4:00Estimate Probability from What Actually HappenedYou'll learn: Define experimental probability as event frequency divided by trials.Chapter 7Run a Coin Experiment and Record the Results3:49Run a Coin Experiment and Record the ResultsYou'll learn: Organise a20-toss experiment and calculate observed relative frequencies.Chapter 7A Paper Cup Has Three Outcomes—Are They Equally Likely?3:53A Paper Cup Has Three Outcomes—Are They Equally Likely?You'll learn: Collect landing-position data and assign empirical probabilities.Chapter 7Reason About a Fair Die Without Running an Experiment3:47Reason About a Fair Die Without Running an ExperimentYou'll learn: Count favourable faces among six equally likely faces.Chapter 7Count Letter Cards, Not Just Different Letters3:52Count Letter Cards, Not Just Different LettersYou'll learn: Use equally likely physical positions when a letter repeats.Chapter 7Two Possible Results Do Not Automatically Mean Fifty-Fifty4:03Two Possible Results Do Not Automatically Mean Fifty-FiftyYou'll learn: Test whether equally likely outcomes are justified.Chapter 7Why Experiment and Theory Can Give Different Answers3:49Why Experiment and Theory Can Give Different AnswersYou'll learn: Compare relative frequency with the fair-model probability and discuss larger experiments.Chapter 7A Coin and Die Have No Memory3:49A Coin and Die Have No MemoryYou'll learn: Explain independence and the gambler’s fallacy.Chapter 7Use a Survey to Estimate a Random Student’s Preference4:03Use a Survey to Estimate a Random Student’s PreferenceYou'll learn: Read a frequency table and find a probability for a randomly selected member of the observed group.Chapter 7A Sample Can Suggest What a Larger Population Prefers3:56A Sample Can Suggest What a Larger Population PrefersYou'll learn: Scale sample proportions to estimate quantities for a whole school.Chapter 7Estimate Sweet Colours and School Club Choices3:47Estimate Sweet Colours and School Club ChoicesYou'll learn: Calculate sample probabilities and scale proportions to population estimates.Chapter 7Read a Tyre-Lifetime Table as Probability Evidence3:43Read a Tyre-Lifetime Table as Probability EvidenceYou'll learn: Add relevant category frequencies and divide by the observed total.Chapter 7List Every Possible Outcome Exactly Once3:40List Every Possible Outcome Exactly OnceYou'll learn: Define a sample space S and its number of elements n(S).Chapter 7The Question Determines the Detail in a Sample Space3:52The Question Determines the Detail in a Sample SpaceYou'll learn: Refine an outcome description to match what is recorded.Chapter 7An Event Is a Selection of Outcomes3:44An Event Is a Selection of OutcomesYou'll learn: Represent an event as a subset of the sample space.Chapter 7One Sample Space Can Record Numbers or Colours3:57One Sample Space Can Record Numbers or ColoursYou'll learn: List the requested integer outcomes and separate physical balls from colour categories.Chapter 7Combine a Die Roll with a Coin Toss3:44Combine a Die Roll with a Coin TossYou'll learn: Use ordered pairs to list the outcomes of two different random actions.Chapter 7Use a Table to Organise Choices Without Missing Any3:54Use a Table to Organise Choices Without Missing AnyYou'll learn: List combinations systematically and identify a selected subset.Chapter 7Build a Two-Toss Tree One Branch at a Time3:46Build a Two-Toss Tree One Branch at a TimeYou'll learn: List paths from the first toss through the second toss and label complete outcomes.Chapter 7Exactly One Head and At Least One Head Are Different Events3:34Exactly One Head and At Least One Head Are Different EventsYou'll learn: Select complete paths that match the event wording.Chapter 7Fruit Colours Can Give Unequal Tree Branches3:46Fruit Colours Can Give Unequal Tree BranchesYou'll learn: Draw a weighted tree for one selection from each basket.Chapter 7Find the Chance of Two Pens Having the Same Colour3:55Find the Chance of Two Pens Having the Same ColourYou'll learn: Use replacement to keep second-draw probabilities unchanged and combine matching-colour paths.Chapter 7Count Even Cards and Balls That Are Not Red3:39Count Even Cards and Balls That Are Not RedYou'll learn: Identify favourable outcomes in simple uniform selections and interpret not-red.Chapter 7Find Probabilities from the Repeated Letters of PEACE3:41Find Probabilities from the Repeated Letters of PEACEYou'll learn: Calculate events involving several target letters or excluding one letter.Chapter 7Read an Eight-Section Spinner3:50Read an Eight-Section SpinnerYou'll learn: Count single and grouped favourable sectors.Chapter 7Two Dice Need Thirty-Six Ordered Outcomes3:49Two Dice Need Thirty-Six Ordered OutcomesYou'll learn: Build a6×6 outcome grid and find a prime sum greater than 5.Chapter 7Impossible and Certain Events with Two Dice3:47Impossible and Certain Events with Two DiceYou'll learn: Construct events with probability 0 and 1 from the full sample space.Chapter 7List Three-Coin Outcomes and Count Exactly Two Heads3:43List Three-Coin Outcomes and Count Exactly Two HeadsYou'll learn: Extend the tree to eight equally likely complete sequences.Chapter 7Add a First-Coin Condition to a Three-Coin Event3:48Add a First-Coin Condition to a Three-Coin EventYou'll learn: Select only paths satisfying both conditions.Chapter 7Without Replacement, the Second Draw Changes3:50Without Replacement, the Second Draw ChangesYou'll learn: Update branch counts when the first selected ball is laid aside.Chapter 7Find the Probability of Two Different Colours3:45Find the Probability of Two Different ColoursYou'll learn: Combine all ordered different-colour paths or subtract same-colour probability from 1.Chapter 7Compare Sample Spaces With and Without Replacement3:47Compare Sample Spaces With and Without ReplacementYou'll learn: List numbered pairs and explain the presence or absence of repeated-number pairs.Chapter 7Even Four-Digit Numbers Formed Without Repeating Digits3:52Even Four-Digit Numbers Formed Without Repeating DigitsYou'll learn: List or organise equally likely arrangements and count those with an even last digit.Chapter 7Guessing Exactly Two Correct Answers on Three Questions3:59Guessing Exactly Two Correct Answers on Three QuestionsYou'll learn: Organise correct/incorrect path patterns while retaining unequal branch probabilities.Chapter 7Recording Only the Number of Heads Changes the Sample Space3:41Recording Only the Number of Heads Changes the Sample SpaceYou'll learn: Choose the valid set of recorded head counts and explain why competing lists fail.Chapter 7Use Area to Model Where a Random Drop Lands3:39Use Area to Model Where a Random Drop LandsYou'll learn: Compare the circle’s area with the containing rectangle’s area.

NCERT · Ganita Manjari · Volume 1 · Chapter 8

Predicting What Comes Next: Exploring Sequences and Progressions

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Chapter 8A Sequence Is an Ordered List of Terms3:48A Sequence Is an Ordered List of TermsYou'll learn: Explain position, term value, ellipsis and finite versus infinite sequences.Chapter 8Build Triangular Numbers by Adding the Next Row3:35Build Triangular Numbers by Adding the Next RowYou'll learn: Connect dot arrays with cumulative sums of natural numbers.Chapter 8Odd Borders Build the Square Numbers3:38Odd Borders Build the Square NumbersYou'll learn: Connect successive odd additions with enlarging square arrays.Chapter 8A New Sequence Can Come from Running Totals3:50A New Sequence Can Come from Running TotalsYou'll learn: Generate partial sums and compare them with the original terms.Chapter 8Read Subscripts and Include Negative or Fractional Terms3:45Read Subscripts and Include Negative or Fractional TermsYou'll learn: Use tₙ, sₙ, uₙ and identify increasing, decreasing or signed sequences.Chapter 8An Explicit Rule Finds a Term Directly from Its Position3:36An Explicit Rule Finds a Term Directly from Its PositionYou'll learn: Substitute a positive integer index into a term formula.Chapter 8Work Backwards to Find a Term’s Position3:41Work Backwards to Find a Term’s PositionYou'll learn: Solve a term equation and test whether the index is a positive integer.Chapter 8Generate Sequences from Linear and Quadratic Rules3:50Generate Sequences from Linear and Quadratic RulesYou'll learn: Evaluate several explicit rules and compare the resulting sequences.Chapter 8A Recursive Rule Needs a Starting Value3:40A Recursive Rule Needs a Starting ValueYou'll learn: Generate terms from the preceding term and compare with an explicit description.Chapter 8Use a Recursive Machine That Doubles and Adds Three3:43Use a Recursive Machine That Doubles and Adds ThreeYou'll learn: Calculate successive terms and check a proposed value by continuing the rule.Chapter 8A Recursive Rule Can Grow Much Faster Than Addition3:41A Recursive Rule Can Grow Much Faster Than AdditionYou'll learn: Evaluate a rule that multiplies the previous term by one less than itself.Chapter 8The Next Virahanka–Fibonacci Term Uses the Previous Two3:46The Next Virahanka–Fibonacci Term Uses the Previous TwoYou'll learn: Generate the sequence from two initial values and explain why both are needed.Chapter 8A Rule Can Use the Previous Three Terms3:46A Rule Can Use the Previous Three TermsYou'll learn: Apply a three-term recurrence and contrast its initial conditions with the two-term rule.Chapter 8Recognise an Arithmetic Progression by Its Constant Difference3:50Recognise an Arithmetic Progression by Its Constant DifferenceYou'll learn: Compare successive differences and identify the special class of sequences.Chapter 8Why an AP’s nth Term Uses n−1 Steps3:35Why an AP’s nth Term Uses n−1 StepsYou'll learn: Count the jumps from the first term to derive tₙ=a+(n−1)d.Chapter 8APs Can Decrease or Have Fractional Terms3:43APs Can Decrease or Have Fractional TermsYou'll learn: Identify a and d in positive, negative, fractional and decimal examples.Chapter 8Plot an AP as Discrete Points on a Straight Line3:56Plot an AP as Discrete Points on a Straight LineYou'll learn: Turn a term table into ordered pairs(index, value).Chapter 8Write the Same AP Explicitly and Recursively3:48Write the Same AP Explicitly and RecursivelyYou'll learn: Translate between tₙ=a+(n−1)d and tₙ=tₙ₋₁+d.Chapter 8Find a Distant AP Term or Test Membership3:28Find a Distant AP Term or Test MembershipYou'll learn: Use the general term forwards and backwards.Chapter 8Recover an AP from Two Term Conditions3:43Recover an AP from Two Term ConditionsYou'll learn: Use differences between term equations to find a and d.Chapter 8Taxi Fares and Salaries Can Form APs3:43Taxi Fares and Salaries Can Form APsYou'll learn: Identify the first observed term and the fixed increment in a context.Chapter 8Pair a Sum with Its Reverse3:37Pair a Sum with Its ReverseYou'll learn: Add1–10 in opposite directions so every column has the same total.Chapter 8Derive the Sum of the First n Natural Numbers3:36Derive the Sum of the First n Natural NumbersYou'll learn: Generalise the reversed-sum proof and connect it to a rectangular dot array.Chapter 8Find a Sum That Starts Somewhere Other Than One3:50Find a Sum That Starts Somewhere Other Than OneYou'll learn: Subtract two initial sums to retain exactly the intended interval.Chapter 8The nth Triangular Number Is a Sum3:31The nth Triangular Number Is a SumYou'll learn: Connect the original triangle arrays to the closed term formula.Chapter 8Count and Add the Two-Digit Multiples of Three3:35Count and Add the Two-Digit Multiples of ThreeYou'll learn: Identify the first and last terms, count the steps and add the required multiples.Chapter 8A GP Multiplies Where an AP Adds3:54A GP Multiplies Where an AP AddsYou'll learn: Compare equal differences with equal ratios using the two growth pictures.Chapter 8Why a GP’s nth Term Has Exponent n−13:35Why a GP’s nth Term Has Exponent n−1You'll learn: Count repeated multiplications to derive tₙ=arⁿ⁻¹.Chapter 8Find the Common Ratio in Growing and Shrinking GPs3:42Find the Common Ratio in Growing and Shrinking GPsYou'll learn: Divide consecutive non-zero terms and derive term formulas.Chapter 8Negative Ratios Can Make the Signs Alternate3:53Negative Ratios Can Make the Signs AlternateYou'll learn: Generate terms and verify a signed ratio.Chapter 8Describe a GP with a Recursive Rule3:51Describe a GP with a Recursive RuleYou'll learn: Translate the explicit expression into a repeated-multiplier rule.Chapter 8Use a Known GP Term to Find a Later Term3:31Use a Known GP Term to Find a Later TermYou'll learn: Move a fixed number of positions by multiplying by the ratio that many times.Chapter 8Which GP Term Equals a Given Number?3:34Which GP Term Equals a Given Number?You'll learn: Match the target with a power of the common ratio.Chapter 8Build the Sierpiński Triangle One Removal at a Time4:00Build the Sierpiński Triangle One Removal at a TimeYou'll learn: Join side midpoints and remove each central small triangle repeatedly.Chapter 8Count the Small Triangles with a Power of Three3:41Count the Small Triangles with a Power of ThreeYou'll learn: Explain why each retained triangle produces three at the next stage.Chapter 8More Triangles Can Occupy Less Total Area3:41More Triangles Can Occupy Less Total AreaYou'll learn: Use the retained fraction to model the shrinking shaded area.Chapter 8Plot AP and GP Behaviour Side by Side3:55Plot AP and GP Behaviour Side by SideYou'll learn: Compare constant additive change with multiplicative growth and decay.Chapter 8Build and Count a Sierpiński Square Carpet3:41Build and Count a Sierpiński Square CarpetYou'll learn: Remove the central ninth in every retained square and count the remaining pieces.Chapter 8Find the Shrinking Area of the Square Carpet3:42Find the Shrinking Area of the Square CarpetYou'll learn: Apply the retained fraction8/9 at every stage.Chapter 8A Bouncing Ball’s Heights Form a Shrinking GP3:50A Bouncing Ball’s Heights Form a Shrinking GPYou'll learn: Generate bounce maxima and find the first bounce below a specified threshold.Chapter 8Count Both Upward and Downward Travel Between Bounces3:31Count Both Upward and Downward Travel Between BouncesYou'll learn: Distinguish rebound height from total vertical distance travelled.Chapter 8A Shifted Recursive Sequence Can Reveal a GP3:28A Shifted Recursive Sequence Can Reveal a GPYou'll learn: Transform the recurrence into a geometric rule and identify a target term.Chapter 8Count Multiples Within a Fixed Number Range3:46Count Multiples Within a Fixed Number RangeYou'll learn: Use the smallest and largest allowable multiples and the AP term formula.Chapter 8Recover an AP from Sums of Two Terms3:41Recover an AP from Sums of Two TermsYou'll learn: Convert term-sum information into two linear conditions on a and d.Chapter 8When Does a Natural-Number Sum First Exceed One Thousand?3:51When Does a Natural-Number Sum First Exceed One Thousand?You'll learn: Use the triangular-sum formula and test neighbouring integer positions.Chapter 8Write One Hundred as Consecutive Natural Numbers3:37Write One Hundred as Consecutive Natural NumbersYou'll learn: Use a starting value and number ofterms to find every valid representation.Chapter 8A Doubling Bacterial Culture Uses Stage Zero3:43A Doubling Bacterial Culture Uses Stage ZeroYou'll learn: Model the initial count and subsequent hourly doublings.Chapter 8Find a GP from an Initial Sum and Later-Term Relation3:49Find a GP from an Initial Sum and Later-Term RelationYou'll learn: Express the two conditions using a and r and find possible progressions.Chapter 8Find Three GP Terms from Their Sum and Product3:51Find Three GP Terms from Their Sum and ProductYou'll learn: Use the middle term and equal ratios to constrain the first and third terms.Chapter 8Equally Spaced GP Positions Also Form a GP3:33Equally Spaced GP Positions Also Form a GPYou'll learn: Compare the ratios between terms separated by the same number of steps.Chapter 8Recover Three GP Terms from a Sum and Sum ofSquares3:39Recover Three GP Terms from a Sum and Sum ofSquaresYou'll learn: Combine algebraic identities with the GP middle-term relation.Chapter 8A Sum-Based Recurrence Can Collapse to Doubling3:38A Sum-Based Recurrence Can Collapse to DoublingYou'll learn: Subtract consecutive recurrence statements to simplify the rule.Chapter 8Find a Familiar Sequence Hidden in Another Recurrence3:44Find a Familiar Sequence Hidden in Another RecurrenceYou'll learn: Compare consecutive cumulative-sum rules and recognise a two-term recurrence.

Coming soon

Volume 1 · Chapter 5 · I’m Up and Down, and Round and Round · Volume 1 · Chapter 6 · Measuring Space: Perimeter and Area

Volume 1 · Chapter 5 · I’m Up and Down, and Round and Round50 videos coming
Volume 1 · Chapter 6 · Measuring Space: Perimeter and Area69 videos coming