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Chapter 4 · Period 1 of 15 · 50 min

Consecutive-Square Pattern and the Promise of Algebra

Introduction · Discovery/Inquiry

This opening unit positions the chapter as a journey from numeric surprise to algebraic explanation, so let students linger on the conjecture stage before any algebraic language is introduced. Watch for students who pick the same three integers each time and so miss the generality of the pattern — prompt them to try a set that includes a negative integer. Example 1, p.68 may be assigned as a self-study read before any further work if students want a written verification.

0-12Ask students to pick any three consecutive integers and compute their squares. They record the result of (smallest square) + (largest square) − 2×(middle square) for two or three chosen sets and note the invariant. Pairs share results with neighbours.
12-28Walk through Example 1, p.68 with the class, confirming the invariant algebraically. Pose the think-and-reflect prompt from the section: test four consecutive squares for a similar pattern. Students work individually, compute, and conjecture what the analogous invariant might be.
28-42Class discussion: students share their four-consecutive-square experiments. Draw out the idea that algebra can explain numeric patterns that seem coincidental, and frame the chapter's agenda — developing and proving algebraic identities.
42-50Quick individual write: students record in one or two sentences what they think an 'algebraic identity' might mean, based on the pattern just explored. Collect as an exit check on prior language.
Assessment

Competency (C-3.1). Discover and prove algebraic identities by deriving each identity from a geometric or distributive-property argument and then using it.

Learning outcome. Students can verify and conjecture the invariant produced by adding the smallest and largest of three consecutive square numbers and subtracting twice the middle square.

Priya notices that for the three consecutive integers 5, 6, 7 the expression 5^2 + 7^2 − 2×6^2 equals 2. Her classmate Rohan says this will work for ANY three consecutive integers, not just 5, 6, 7. Write an algebraic argument that decides whether Rohan is right. Let the three consecutive integers be (n−1), n, (n+1). Expand each square and simplify the expression (n−1)^2 + (n+1)^2 − 2n^2. Then state clearly what your working shows.

Look for

  • Correct representation of three consecutive integers as (n−1), n, (n+1).
  • Correct expansion: (n−1)^2 = n^2−2n+1 and (n+1)^2 = n^2+2n+1.
  • Correct simplification: (n^2−2n+1) + (n^2+2n+1) − 2n^2 = 2.
  • Explicit conclusion that the result is 2 for all integers n, confirming Rohan's claim.
  • Recognition that this is an identity — true for all values of n — not merely a numerical coincidence.
Chapter 4 · 15 periods of 50 min

Also saved as 9 periods or 12 periods.

  • 2Building the Square Identity from a Partitioned Square
    Visualising Identities
  • 3Reversing the Identity: Factorising Perfect-Square Trinomials
    Factorisation of Algebraic Expressions Using Identities
  • 4Proving the Consecutive-Square Result and Deriving (a-b)^2
    Factorisation of Algebraic Expressions Using Identities
  • 5The Three-Term Square Identity: Derivation and Application
    More Identities
  • 6Factorising with the Three-Term Identity and Verifying an Identity
    More Identities
  • 7Algebra Tiles: Building Products as Rectangles
    Factorisation Using Algebra Tiles
  • 8Splitting the Middle Term Without Tiles: Factorising Quadratics
    Factorisation Without Using Algebra Tiles
  • 9Middle-Term Splitting Extended: Applications and Harder Quadratics
    Factorisation Without Using Algebra Tiles
  • 10Deriving the Cube Identities: Expansion and Geometric Decomposition
    Finding New Identities
  • 11Sum and Difference of Cubes, and the Three-Variable Cube Identity
    Finding New Identities
  • 12Finding New Identities by Combining Results
    Finding New Identities
  • 13Simplifying Rational Algebraic Expressions by Factorisation
    Simplifying Rational Expressions
  • 14Applied Factorisation: Rectangle and Pool Dimension Problems
    Simplifying Rational Expressions
  • 15Whole-Chapter Synthesis: Connecting Identities, Proof, and Application
    Synthesis
Ganita Manjari · Class 9 · 14 chapters

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