6. We Distribute, Yet Things Multiply

Part I, printed Chapter 66 topics59 min

Chapter 6 of NCERT Mathematics for Class 8 (Part I, printed Chapter 6): We Distribute, Yet Things Multiply. Three sections — The distributive property, The special identities and Getting it right, and getting there differently. Six videos, 59 min in all.

Worked answers to the 20 exercise questions in this chapter →

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Part 1

The distributive property

10 minWhat happens to a product when you nudge one factorMove one factor by one and the product does not move by one. It moves by the other factor.10 min · 12 partsहिंदी10 minMultiplying two two-term expressions, and where the four terms come fromMultiply two brackets and four things come out — not because of a rule with a name, but because one rectangle cut twice has four pieces.10 min · 12 partsहिंदी10 minFast mental multiplication, powered by distributionThe one-line methods for multiplying by 11, 99 and 999 are not five tricks. They are one move, used five times.10 min · 12 partsहिंदी

Part 2

The special identities

9 min(a + b)² and (a − b)²: why there is a middle term at allSquaring a sum is not squaring its parts, and the amount you lose by is not an abstraction. It is two rectangles you can point at.9 min · 12 partsहिंदी10 minWhy the product of a sum and the matching difference is a² − b², and the patterns that followA sum times the matching difference gives four products, like any two brackets. It only looks like two because two of them cancel.10 min · 12 partsहिंदी

Part 3

Getting it right, and getting there differently

9 minMany different-looking expressions for one growing patternA growing pattern does not come with a formula. It comes with a value at every step and any number of honest ways of counting to it.9 min · 12 partsहिंदी

What you will be able to do

  • Predict how a product changes when one factor rises by one, and justify the prediction from the distributive property rather than from two computations
  • Expand a product of two two-term brackets and account for each of the four terms by naming the pair it came from
  • Split a multiplier such as 11, 101 or 999 as a power of ten plus or minus one, and write the product as a sum or difference of two shifted copies
  • Compute the square of a two-digit number by splitting it into a round part and a small part and adding four areas
  • Observe a numerical pattern in doubled sums of squares and identify the two numbers hidden in each result
  • Continue a drawn pattern by one step, and count the units in the drawn steps

What this chapter assumes you already know

  • Multiplication as a rectangular array or as repeated addition, and its commutativity
  • Adding and subtracting integers
  • That two expressions are the same expression when every substitution gives them the same value
  • Multiplying and simplifying fractions
  • Integer sign rules, including the product of two negatives
  • Place value for two-digit numbers, for the long-multiplication comparison
  • Place value in a multi-digit number, and the effect of multiplying by 10, 100 or 1000
  • Column addition with carrying

It builds directly on The carpenter's problem: how to be sure a frame really is rectangular, Body parts and tally marks: counting before numerals, Why the angles of any quadrilateral add to 360° and The digit-sum test for 9, and the algebra underneath it.

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "Add one to a factor, add one to the product."
  • "Up by one and down by one must cancel."
  • "(a + 1)(b + 1) is ab + 1."
  • "Decreases need their own rule."
  • "Distributivity is an arithmetic habit that letters make dubious."
  • "Substituting three values proves an identity."
  • "The increase depends on which bracket you open first."
  • "(a + b)(c + d) is ac + bd."