Fast mental multiplication, powered by distribution

What to assume they know

• The distributive property, and expanding a single bracket (What happens to a product when you nudge one factor)
• Place value in a multi-digit number, and the effect of multiplying by 10, 100 or 1000
• Column addition with carrying
• Writing a general number by giving each of its digits its own letter (The digit-sum test for 9, and the algebra underneath it)
• Subtracting a number from a round number

What they should be able to do

• 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
• Set out the column addition for a general four-digit number times 11 and say which two digits meet in each column
• Multiply a number of any length by 11 in one line, managing carries correctly
• State the general rule for multiplication by 11 in words
• Adapt the method to 101, and then to 1001 and 10001, saying how the gap between the two copies changes
• Adapt the method to 99 and 999, where the second copy is subtracted
• Explain why the neighbour-sum description is incomplete without the carry, using a case where a digit pair exceeds nine
• Place these methods in their historical setting as iṣṭa-guṇana and name the works that discuss them

Where it usually goes wrong

• "To multiply by 11, write the digit sums between the first and last digits." This is the rule students carry away, and it breaks on the chapter's own example: in 3874 the pair 8 and 7 totals 15, which cannot be one digit of the answer. The chapter's five steps exist to handle exactly that. Demonstrate the failure before giving the repair.
• "Eleven is special." Nothing about 11 is doing the work; what does the work is that 11 is one past ten. That is why 101, 1001 and 10001 all behave the same way with a wider gap, and why 9 and 99 behave the same way in the other direction.
• "The rule is for four-digit numbers." The chapter asks for the rule for a number of any length and then hands over a seven-digit input. Any statement of the rule that mentions four digits is not the rule.
• "Multiplying by 99 needs a new method." It needs the same method with a minus. Splitting 99 as one hundred less one turns the second copy from an addition into a subtraction; the shift is unchanged.
• "A shortcut is less trustworthy than long multiplication." The column layout is the justification: it shows every partial product and where it lands. Nothing is being skipped, only arranged better.
• "The zero in the shifted row is a digit of the answer." It is the units place of the number multiplied by ten. In the printed layout it is set in the same style as the letters, and it is the single most likely thing to be misread.
• "These methods are a recent invention for competitive exams." The chapter names three Indian mathematicians and two verses of one seventh-century work.

Questions to check understanding

• Multiply a given number of four or more digits by 11 in one line
• Multiply by 101, 1001 or 10001 in one line, and state which digits of the answer are copies of the original
• Multiply by 99 or 999 by splitting the multiplier downwards
• State the general rule for multiplication by 11 in words, for a number of any length
• Given an incorrect one-line answer, identify the carry that was dropped
• Rewrite a stated shortcut as a split of the multiplier, and justify it by distributivity
• Name the split that makes a given multiplication easy — for example a multiplier just above or just below a round number

Examples worth working on the board

Inputs. Items marked "printed" are the chapter's own working; the rest the chapter leaves open.

• The two opening products (Part I, §6.1, p.143). Printed as the challenge: 3874 × 11 and 5678 × 11, with the instruction that the answer be obtained in a single step. Only the first is then worked on the page.
• The split, and the column sum (Part I, §6.1, p.143). Printed: 3874 × 11 becomes 3874 × (10 + 1), which is 38740 + 3874. The page then sets those two numbers out as a column addition with a rule underneath and the total left blank — the reader is meant to look at which digits sit above one another, not at the answer.
• The general four-digit layout (Part I, §6.1, p.143). Printed: a four-digit number whose digits are named, from the thousands place down, d, c, b, a. Multiplying by ten and adding the original gives a column layout in which the top row reads d c b a 0 and the row beneath, indented by one place, reads d c b a. The stated column totals, left to right, are d, then c + d, then b + c, then a + b, then a. Note that the zero in the shifted copy's units place is set in the same italic style as the letters, which makes it easy to misread as a further letter — it is the digit zero.
• The five steps (Part I, §6.1, p.144). Printed: the same product built up one digit at a time, the running answer reading 4, then 14, then 614, then 2614, then 42614, with carried digits drawn as small ringed numerals accumulating above the multiplicand and a plus sign beside them. 42614 is the chapter's own stated result. Two features of the figure matter and are artwork, not text: the ringed carries, and the way the digits of 3874 are picked off from the right.
• The rule to be stated (Part I, §6.1, p.144). The page asks for a general rule for multiplying a number of any length by 11 in one line. Left open.
• Practice inputs for 11 (Part I, §6.1, p.144, a Math Talk item): 94 × 11, 495 × 11, 3279 × 11, 4791256 × 11. The last is seven digits — deliberately, since it defeats anyone who has memorised a four-digit shape rather than the rule.
• The 101 layout (Part I, §6.1, p.144). Printed: for the same general four-digit number, multiplying by 101 gives 100 times it plus itself; the column layout has the top row d c b a 0 0 and the indented row d c b a, with stated totals, left to right, d, c, b + d, a + c, b, a. Notice the column totals at the outer two places at each end are now single digits of the original, because the gap between the copies is two places instead of one; a carry from the middle can still change the left-hand pair (3874 × 101 = 391274).
• The 101 tasks (Part I, §6.1, p.144). Printed: multiply 3874 by 101 in one line; state a general rule; extend the rule to 1001 and 10001.
• Practice inputs for the wider gaps (Part I, §6.1, p.144, a Math Talk item): 89 × 101, 949 × 101, 265831 × 1001, 1111 × 1001, 9734 × 99, 23478 × 999. The last two are the turn of the section — the chapter never sets out a subtraction layout, so the student must see that 99 is a hundred less one and adapt.
• The historical note (Part I, §6.1, p.144). Printed content, paraphrased: such applications of distributivity to easy multiplication were treated at length by Brahmagupta (628 CE), Sridharacharya (750 CE) and Bhaskaracharya (in Lilavati, 1150 CE); in the Brahmasphuṭasiddhānta, verse 12.56, Brahmagupta's name for these fast methods is set in the chapter as ista-gunana. The chapter's earlier history note, on Part I p.142, is what makes verse 12.56 the sequel to verse 12.55.
• The chapter prints no answers for any of the practice inputs. Its own worked values are the split 38740 + 3874, the five partial answers, and 42614.

Figures to have open

• The column-addition schematic for a lettered four-digit number times 11: the number, the same number shifted one place, and each column labelled with which two letters meet. This is the chapter's own device (Part I, §6.1, p.143) and sections 3 to 5 rest on it. Redraw as a schematic.
• The same schematic for 101 (Part I, §6.1, p.144), so the widened gap and the untouched end digits can be seen against the previous figure.
• A carry ladder: the five printed steps of 3874 × 11 with the running answer and the accumulating ringed carries (Part I, §6.1, p.144). The chapter's own figure, and the only place the carry is visible.
• A subtraction layout for a number times 99, built by the explanation to match the two addition layouts. Not in the chapter — flag it as an added construction.
• A timeline carrying Brahmagupta 628 CE, Sridharacharya 750 CE and Bhaskaracharya 1150 CE with the two named works. Standard schematic.
• No photograph is needed.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 6, "We Distribute, Yet Things Multiply", §6.1 "Some Properties of Multiplication", Part I pp.143–144. The printed subheading is "Fast Multiplications Using the Distributive Property", beginning on Part I p.143, with a further printed subheading beneath it naming the multipliers 11, 101 and 1001.
• The lettered layouts are on Part I pp.143 and 144; the five-step figure is on Part I p.144.
• The historical paragraph closing the subsection is on Part I p.144; it is the sequel to the "A Pinch of History" note on Part I p.142, which is carried in Multiplying two two-term expressions, and where the four terms come from.
• The chapter's SUMMARY, Part I p.157, does not mention these methods; its three bullets cover the four-term expansion, its three special cases, and the idea that one problem often has several solution methods and that finding them is a creative process — the last of these being §6.4's theme. Worth knowing before assuming the summary is a revision checklist.