We Distribute, Yet Things Multiply – Class 8 Maths Chapter 6 Notes & NCERT PDF

Class 8MathematicsChapter 6NCERT book: Ganita Prakash - Mathematics Class 8

Notes and a simple summary of Chapter 6 of the NCERT Class 8 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 6: We Distribute, Yet Things Multiply

Introduction

We Distribute, Yet Things Multiply is about the distributive property — the rule connecting multiplication and addition — and how it produces useful identities, fast mental-multiplication tricks, and different-looking formulas that describe the same pattern.

Distributive property: a (b + c) = ab + ac

It can be pictured as a rectangle of a rows split into b columns and c columns: area ab + area ac.

Key Ideas

6.1 Some Properties of Multiplication

Increments in products. Start with 23 × 27.

Increase 27 by 1: 23 (27 + 1) = 23 × 27 + 23 — the product grows by 23.

Increase both by 1: (a + 1)(b + 1) = ab + (a + b + 1) — the product grows by a + b + 1.

Increase one and decrease the other: (a + 1)(b – 1) = ab + b – a – 1 — the product does not always increase.

These results also hold for negative integers, because integers satisfy the distributive property too.

Identities are equalities of algebraic expressions that are true for all values of the letter-numbers, e.g. a (b + 8) = ab + 8a.

Identity 1: (a + m)(b + n) = ab + mb + an + mn

Each term of the first bracket multiplies each term of the second; signs follow integer rules, so (a – u)(b – v) = ab – ub – av + uv.

Like terms have exactly the same letter-numbers and can be added: a²b + 2a²b = 3a²b. Expanding (a + b)(a² + 2ab + b²) gives a³ + 3a²b + 3ab² + b³.

A pinch of history. The first explicit statement of the distributive property is by Brahmagupta in Brāhmasphuṭasiddhānta (Verse 12.55), who called it khaṇḍa-guṇanam (multiplication by parts). Euclid (geometrically) and Āryabhaṭa (algebraically) used it implicitly.

Fast multiplication. Multiplying by 11 = 10 + 1: 3874 × 11 = 38740 + 3874 = 42614 — each new digit is a sum of neighbouring digits. Similarly for 101 = 100 + 1 and 1001, 10001, … Brahmagupta called such methods iṣṭa-guṇana (Verse 12.56); Sridharacharya and Bhaskaracharya (Līlāvatī) also discussed them.

6.2 Special Cases of the Distributive Property

Identity 1A: (a + b)² = a² + 2ab + b²

Identity 1B: (a – b)² = a² – 2ab + b²

Identity 1C: (a + b)(a – b) = a² – b²

Geometry of 65²: a square of side 65 splits into 60², 5² and two 60 × 5 rectangles: 3600 + 25 + 600 = 4225.

55² = (60 – 5)²: remove two 60 × 5 strips from 60², then add back 5² (removed twice): 3600 – 300 – 300 + 25 = 3025.

Pattern 1: 2(a² + b²) = (a + b)² + (a – b)², e.g. 2(5² + 6²) = 11² + 1².

Pattern 2: 9 × 9 – 1 × 1 = 10 × 8 is a² – b² = (a + b)(a – b).

Sridharacharya's squaring trick (750 CE): a² = (a + b)(a – b) + b². So 197² = 200 × 194 + 9 = 38809.

6.3 Mind the Mistake, Mend the Mistake

Common errors the book asks you to fix:

(5m + 6n)² ≠ 25m² + 36n² — the middle term 60mn is missing.

–3p(–5p + 2q) ≠ –3p + 5p – 2q; correct: 15p² – 6pq.

2(x – 1) + 3(x + 4) ≠ 2x – 1 + 3x + 4; correct: 5x + 10.

5w² + 6w ≠ 11w² — these are not like terms.

(a + 2)(b + 4) ≠ ab + 8; correct: ab + 4a + 2b + 8.

6.4 This Way or That Way, All Ways Lead to the Bay

One pattern of circles can be counted in four ways: (k + 1)² – 1, k² + 2k, k(k + 1) + k, and k(k + 2). Expanding shows they are all k² + 2k, so Step 15 has 15² + 30 = 255 circles. Similarly, Tadang's (m + n)² – 4mn and Yusuf's (n – m)² describe the same shaded square.

Worked Examples

(x + 2)(x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10.

(6x + 5)² = (6x)² + 2(6x)(5) + 5² = 36x² + 60x + 25.

98 × 102 = (100 – 2)(100 + 2) = 10000 – 4 = 9996.

99² = (100 – 1)² = 10000 – 200 + 1 = 9801.

Calendar square with numbers a, a + 1, a + 7, a + 8: the diagonal products differ by (a + 1)(a + 7) – a(a + 8) = 7, always.

Which is larger, 14 × 26 or 16 × 24? 14 × 26 = (16 – 2)(24 + 2) = 16 × 24 + 32 – 48 – 4 = 16 × 24 – 20. So 16 × 24 is larger.

Figure it Out — Hints

Expand by multiplying every term of one bracket by every term of the other, then combine like terms.

Choose a split that makes squares easy: 104 = 100 + 4, 58 = 60 – 2.

For 'is it always 2 / always a multiple of 4?', expand fully and inspect: (k + 1)(k + 2) – (k + 3) = k² + 2k – 1, which is not always 2.

(a – b)² and (b – a)² are equal.

The pattern (a – b)(a + b) = a² – b², (a – b)(a² + ab + b²) = a³ – b³, … continues to a⁴ – b⁴ and a⁵ – b⁵.

Maths Words

Distributive property (वितरण गुण)

Identity (सर्वसमिका)

Expand (प्रसार करना)

Like terms (सजातीय पद)

Term (पद)

Product (गुणनफल)

Square of a sum (योग का वर्ग)

Difference of squares (वर्गों का अंतर)

Letter-number / variable (चर)

Coefficient (गुणांक)

Commutative property (क्रमविनिमेय गुण)

Historical and Indian Connections

Brahmagupta (628 CE) — khaṇḍa-guṇanam and iṣṭa-guṇana; Sridharacharya (750 CE) — squaring using a² = (a + b)(a – b) + b²; Bhaskaracharya (Līlāvatī, 1150 CE) — fast multiplication methods.

Common Mistakes

Dropping the middle term: (a + b)² is not a² + b².

Distributing to only the first term inside a bracket.

Adding unlike terms (5w² + 6w).

Sign slips when a negative multiplies a bracket.

Maths in Daily Life

Splitting numbers (e.g. 102 = 100 + 2) makes mental multiplication fast. Area problems — like the tiled path in the Dhauli park plan — become simple algebra once you use the distributive property.

💡 Key Learning Points

✓Use the distributive property a(b + c) = ab + ac and Identity 1, (a + m)(b + n) = ab + mb + an + mn, to expand products.
✓Know and apply the identities (a + b)² = a² + 2ab + b², (a – b)² = a² – 2ab + b² and (a + b)(a – b) = a² – b², including their area pictures.
✓Use identities for fast mental arithmetic such as 98 × 102, 65², 197² and multiplying by 11 or 101.
✓Spot and correct common algebra mistakes such as a missing middle term or adding unlike terms.
✓Show that different-looking expressions for one pattern, like (k + 1)² – 1 and k(k + 2), are equal.

👨‍🏫 Teaching Tips

→Model (a + b)² with paper squares and rectangles cut from a 65 × 65 grid (60 + 5) before writing the identity.
→Start with the 23 × 27 'what if one number goes up by 1?' question and let students predict before calculating.
→Run a 'Mind the Mistake' gallery walk: post the book's wrong simplifications and let groups diagnose and fix them.
→Race: mental 98 × 102 or 45 × 55 using Identity 1C versus long multiplication.
→Let groups count the circle pattern their own way, then expand every group's formula to show they match.
→Tell the story of Brahmagupta's khaṇḍa-guṇanam and have students write the verse in modern algebra.
→Use real calendars for the 2 × 2 diagonal-product activity and ask students to prove the difference is always 7.

📋 Assessment Questions

Can the student expand products like (3 – x)(x – 6) and simplify by combining like terms?
Can the student choose and apply the right identity to compute squares and products mentally?
Can the student explain an identity with an area diagram?
Can the student find and correct errors in given simplifications?
Can the student write and check an algebraic formula for a growing pattern?

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