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NCERT · Ganita Manjari · Volume 1 · Chapter 4

Exploring Algebraic Identities

51 videos in this chapter

Understanding and proving identities

7 videos

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.

Perfect-square factorisation

6 videos

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 Substitution

Three-term squares and difference of squares

6 videos

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 Signs3:36

Efficient numerical calculations

3 videos

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 Easier4:02

Algebra tiles and quadratic factors

8 videos

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?3:19

Cubic identities

9 videos

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 Signs

Rational algebraic expressions

3 videos

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 Powers4:03

Dimensions and area applications

5 videos

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 Dimensions3:52

Chapter challenge problems

4 videos

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?3:26

You'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.

Expand and Factor Three-Term Squares with Signs

You'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.

Choose the Identity That Makes the Calculation Easier

You'll learn: Compare two-term and three-term decompositions for a given square or product.

Why 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.

You'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.

Simplify Products of Quadratics and Fourth Powers

You'll learn: Combine several identities and cancel only verified non-zero factors.

Factor Areas and Volumes to Suggest Dimensions

You'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.

Why 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.