CBSE Class 9 · Mathematics
Exploring Algebraic Identities
Official NCERT chapter from Ganita Manjari Part I (book code iemh1). ExamMaster notes are original teaching at CBSE Class 9 depth.
This lesson follows the official chapter “Exploring Algebraic Identities” in Ganita Manjari Part I. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.
- CBSE Class 9
- Medium level
- 8 concepts
1Visualising Identities
An identity is an equality that holds for all allowed values: (a+b)² = a² + 2ab + b². Visualising is a square of side a+b split into a², b², and two ab rectangles. The picture is a count of tiles, not a decoration.
A pretty square that misses one ab is a lying visualise.
Figure. The big square has side a+b. Its area is the a^2 block, the b^2 block, and two ab rectangles — which is (a+b)^2 = a^2+2ab+b^2. Sides are to scale with a:b = 3:2; labels name the pieces, not numerical areas.
How it works
- Draw side a+bThe big square.
- See a², b², and two abThe parts.
- Read a² + 2ab + b²The identity.
2Factorisation Using Algebra Tiles
Algebra tiles: a square tile for a², a rectangle for ab, a small square for b² (or unit tiles as taught). Factorisation is rearranging those tiles into a rectangle whose sides are the factors. If the tiles do not make a rectangle, that grouping is not a factorisation.
A pile of tiles with no sides named is not a factorisation.
Figure. Algebra tiles tile (x+1) by (x+2): one x^2, three x-tiles, two units. That rectangle is x^2+3x+2, so the factors are the side lengths. x:1 = 2:1 is to scale; the labels are the tile names, not areas.
How it works
- Lay the tiles for the given writingThe given.
- Rearrange into a rectangleThe factors are the sides.
- Read the side-lengths as the bracketsThe factorisation.
3Factorisation Without Using Algebra Tiles
Without tiles you use the same identities as rewrites: a² − b² = (a−b)(a+b); x² + 5x + 6 = (x+2)(x+3) by hunting two numbers that multiply to 6 and add to 5. The hunt is the method; a guess that fails the expand-check is not done.
Writing any two brackets that “look busy” is not the hunt.
Figure. Without tiles, split the middle term: find a pair whose sum is the middle coefficient and whose product is the constant — here 2+3=5 and 2\times3=6 — then write (x+2)(x+3). The pair is the test; guessing the brackets first is the long way.
How it works
- Name the identity or the pair-huntThe method.
- Write the bracketsThe candidate.
- Expand to checkMust return the given.
x² + 5x + 6
Factor x² + 5x + 6.
- Pair2 and 3 — product 6, sum 5
- Brackets(x+2)(x+3)
- Expandx²+5x+6
Pro tip. The expand-check is the opposite test.
4Finding New Identities
New identities come from expanding a new writing: (a−b)², (x+a)(x+b), or a taught triple. Finding is expand-and-name, not inventing a law that fails a substitute.
A pattern you saw twice is not an identity until it holds for a third unused pair.
Figure. A new identity from the same area idea: the square of side a+b+c is nine blocks a^2, b^2, c^2 and the three pairs of rectangles 2ab, 2bc, 2ca. Sides are to scale with a:b:c = 2:1:1.
How it works
- Expand the new writingThe left or the right.
- Name the collected formThe new identity.
- Substitute one unused pair as a checkFinding earned a test.
5Simplifying Rational Expressions
A rational expression here is a quotient of polynomials you may cancel a common factor after you factor, not by cancelling terms across a plus. (x²−1)/(x−1) = x+1 when x ≠ 1. Cancelling the x in (x+1)/x is the miss.
Strike-through of two x’s that are not a common factor is a lie.
Figure. (x^2-1)/(x-1) is (x-1)(x+1) over (x-1). The terracotta (x-1) factors cancel when x\neq 1, leaving sage x+1. The dashed strokes mark the cancel; they are not extra factors. Cancelling at x=1 is illegal because the original is undefined.
How it works
- Factor numerator and denominatorThe prepare.
- Cancel a common factor; note the forbidden valueThe simplify.
- Refuse term-cancel across +Honest.
6A definition is a test you can run
An identity is a test: substitute two different allowed pairs; both must hold. If one fails, it was an equation for a special a, not an identity.
Checking only a = 1, b = 0 can hide a fake.
Figure. An identity is a sentence that stays true for every allowed value — (a+b)^2 = a^2+2ab+b^2 on all a,b. An equation asks which values work: x^2=4 only at \pm 2. The test is 'every' versus 'some', not how the letters look.
How it works
- Pick two unused pairsThe object.
- Substitute both sidesThe test.
- If either fails, drop the identity-wordIt was a special case.
7Name the given before the unknown
The given is the writing to expand or factor. The unknown is the other side. Copy every term before you drop an ab.
Forgetting 2ab because the square-picture looked like only two squares is a silent given.
Figure. Name the given before the unknown: the paper gave a+b and ab, not a and b. The identity a^2+b^2=(a+b)^2-2ab uses those two directly: 81-40=41. Solving a quadratic for a and b first is the miss.
How it works
- Copy the given writingThe given.
- Name expand or factorThe unknown.
- Then rewrite and checkGiven first.
8One worked case is enough at this class
One identity-check is enough. A page of (a+b)² clones does not add a new 2ab.
Ten twins of a²+2ab+b² are still one picture.
Figure. One numerical check is enough at this class: 3^2 + 2\times3\times2 + 2^2 = 9+12+4 = 25 = (3+2)^2. Sides are to scale 3:2; the in-cell numbers are those integer areas, not normalised canvas area.
How it works
- Expand or factor one caseThe case.
- A second only as a checkOptional.
- StopThis class.
(a+b)² equals
- a² + 2ab + b²
- a² + b²
- a² + ab + b²
The two rectangles are the 2ab.
Notes
- The official chapter title is “Exploring Algebraic Identities”. Teach the school test for that title, not a contest shortcut.
- If a step needs a later class, stop. The next official chapter will pick it up.
Formulas
- (a+b)² = a² + 2ab + b²
- (a−b)² = a² − 2ab + b²
- a² − b² = (a−b)(a+b)
Recap
Hold these pegs from the official chapter “Exploring Algebraic Identities”. The wording is ExamMaster’s teaching, not a textbook recap.
- Visualising Identities
- An identity is an equality that holds for all allowed values: (a+b)² = a² + 2ab + b².
- Factorisation Using Algebra Tiles
- Algebra tiles: a square tile for a², a rectangle for ab, a small square for b² (or unit tiles as taught).
- Factorisation Without Using Algebra Tiles
- Without tiles you use the same identities as rewrites: a² − b² = (a−b)(a+b); x² + 5x + 6 = (x+2)(x+3) by hunting two numbers that multiply to 6 and add to 5.
- Finding New Identities
- New identities come from expanding a new writing: (a−b)², (x+a)(x+b), or a taught triple.
- Simplifying Rational Expressions
- A rational expression here is a quotient of polynomials you may cancel a common factor after you factor, not by cancelling terms across a plus.
- A definition is a test you can run
- An identity is a test: substitute two different allowed pairs; both must hold.
Practise Exploring Algebraic Identities
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- A 4-question practice set that ends the chapter
- 1 quick check with worked explanations
- Timed mocks scored with the real marking scheme
- Readiness tracked per topic, kept on your device