Reading the pattern out of the first few expansions

What to assume they know

• Squaring and cubing a two-term bracket, from earlier algebra
• Multiplying a polynomial by a two-term bracket, and collecting like terms
• Laws of exponents for a product and for a power of a power, since a bracket such as 2x has to be raised whole
• That an index counts repeated multiplication, and that raising anything to the power 1 leaves it alone
• Every selection was counted once per arrangement of itself is not needed here; this topic deliberately stops short of selection counts, which the next topic introduces

What they should be able to do

• Write out the expansions of a two-term bracket for indices 0 to 4 and say what the page prints beside the zeroth one, and why that condition is there
• State the three regularities §7.2 records about term count, about the two indices moving in opposite directions, and about their sum
• Explain why the two indices in every term must add to the index of the bracket, by reference to how the product is formed
• Build the coefficient array of Fig 7.1 by stripping the letters out of those expansions
• Apply the addition rule to extend the array by one further row, and justify the rule by multiplying the previous expansion by the bracket
• Say why the outermost entry of every row is 1, without appealing to the pattern
• Name the array as Pascal's triangle and give the older Indian name the chapter records for the same arrangement
• Use the row for index 5 to expand a bracket whose two terms are compound, and check the result against the sum-of-indices rule
• State the cost of the triangle as a method, and say what would have to change to reach a high index directly

Where it usually goes wrong

• "The coefficients are just a sequence to memorise." They are the record of a multiplication. Multiplying the row for one index by the bracket splits it into an a-copy and a b-copy, the second offset by one place; the sum of those two copies is the next row. A student who can do that never needs to remember a single row.
• "Row 5 is printed in Fig 7.1 or Fig 7.2." Neither figure goes past index 4. Checked on the printed page and on a close-up. The row for index 5 appears only as loose numbers in the text before it is used.
• "Raising 2x to the fourth gives 2x⁴." The whole term is raised, so it gives 16x⁴. This single slip destroys four of the six coefficients in the worked fifth power on p. 127.
• "The condition beside the zeroth power is a typographical leftover." It is the one place in the list where the identity can fail, and it fails for a reason a student can state.
• "1 sits at the ends because the pattern looks nicer that way." The outermost terms are the ones where every bracket contributes the same letter, so there is nothing to add anything to; the addition rule is adding a neighbour that is not there.
• "The rows are symmetric by coincidence." Swapping the roles of the two terms turns the bracket into itself, so the expansion has to survive being read backwards. The symmetry is forced.
• "The triangle solves the problem." It solves it slowly. The page's own twelfth-power example exists to show that a recurrence is not enough.

Questions to check understanding

• Write the next row of the array given the previous one, and say which additions produced which entries
• Expand a bracket to the fourth, fifth or sixth power using the appropriate row
• Expand a bracket whose terms are compound, such as one built from 2x and 3y, and state the coefficient of a nominated term
• Give the number of terms in an expansion from the index alone
• Check a printed expansion for errors using the sum-of-indices rule
• Short-answer items asking why the outer entries are 1, or why each row reads the same in both directions

Examples worth working on the board

Items marked verified are worked out here from the chapter's printed data; this chapter prints no answers on these pages.

• The five printed identities (§7.2, p. 126). Indices 0 to 4, in a column. The zeroth power is given as 1 and carries a condition printed to its right on the same line: the two terms must not sum to zero. The first power returns the bracket unchanged. The square has coefficients 1, 2, 1; the cube has 1, 3, 3, 1. The fourth is not stated flat — the page obtains it as the cube multiplied by the bracket, and only then gives 1, 4, 6, 4, 1. That printed multiplication is the hinge of this whole topic.
• Why the condition on the zeroth line matters. Verified by inspection: if the two terms cancel, the bracket is 0 and the zeroth power of 0 is not defined, so the identity would be asserting something meaningless. Every other line on the page holds with no condition at all. This is the chapter's first exclusion and it is content, not fine print.
• The three regularities (§7.2, p. 126, numbered (i), (ii) and (iii)). One: the count of terms exceeds the index by one, illustrated on the page with the square, which has three terms. Two: across successive terms the index on the first quantity falls by one each time while the index on the second rises by one. Three: in every single term the two indices add to the index of the whole bracket.
• Fig 7.1 (p. 127). A two-column table headed with the words Index and Coefficients. Five rows, labelled 0 to 4, holding 1; 1 1; 1 2 1; 1 3 3 1; 1 4 6 4 1, staggered so that each row sits centred under the one above. Read off the page image, since the numbers are set as artwork.
• Fig 7.2 (p. 127). The same five rows and the same fifteen numbers, with six small shaded downward-pointing triangles laid over them: one between the pair in the row for index 1, two in the row for index 2, three in the row for index 3, none in the row for index 4. Each triangle sits above the entry that the two numbers flanking it add up to. Verified on the printed page: Fig 7.2 adds no row beyond index 4, so the two figures cover exactly the same five indices — the second one annotates the first rather than extending it.
• The row for index 5, printed as six bare numbers in the running text of §7.2 on p. 127: 1, 5, 10, 10, 5, 1 — six, because a row at index n carries n + 1 entries, which is the observation this brief teaches. It is drawn in neither Fig 7.1 nor Fig 7.2, though Fig 7.3 on the next page does draw it. The page produces it by the addition rule and then uses it immediately.
• A compound binomial raised to the fifth power (§7.2, p. 127). Input: the bracket whose first term is 2x and whose second is 3y, raised to the fifth. The page pairs the row 1, 5, 10, 10, 5, 1 with the three regularities and gets, term by term, 32x⁵, then 240x⁴y, then 720x³y², then 1080x²y³, then 810xy⁴, then 243y⁵. Verified: the third coefficient is 10 × 8 × 9, and the fourth is 10 × 4 × 27, which is where students who write 2x⁵ instead of 32x⁵ come apart.
• The cost statement (§7.2, p. 128). To use the triangle for the twelfth power you must first write every row up to index 12, and the page says plainly that this grows worse as the index grows. That sentence is the motive for the next topic and should close the explanation, not open it.

Figures to have open

• Fig 7.1 redrawn as a labelled schematic: an index column on the left, the staggered coefficient rows on the right, headings kept. The chapter's own figure (p. 127); its numbers are artwork and do not extract, so redraw rather than reproduce.
• Fig 7.2 redrawn: the same five rows with six downward markers placed exactly where the chapter places them, each pointing at the entry its two neighbours produce. The chapter's own figure (p. 127).
• A step-by-step shift-and-add panel: one row, duplicated, the lower copy displaced by one position, the two summed column by column into the row beneath. Standard schematic, and the core image of this topic — the chapter draws nothing like it.
• A term-anatomy callout for a single term of an expansion, naming the coefficient and the two indices. Standard schematic.
• No photograph is required. The portrait of Pascal printed on p. 126 with his dates beneath it is decorative here and the timeline can carry the same information.

Where this sits in the book

• NCERT Class XI Mathematics, Chapter 7 Binomial Theorem, §7.1, p. 126, for the framing problem and the numerical cases it names.
• §7.2, p. 126, for the five printed identities, the condition attached to the zeroth power, and the three numbered regularities.
• §7.2, p. 127, for Fig 7.1, Fig 7.2, the bold subheading naming the triangle, the attribution to Pingala, the row for index 5 and the worked fifth power.
• §7.2, p. 128, first paragraph, for the twelfth-power cost statement.
• Historical Note, p. 134, for the dates and the sequence of names used in section 9.
• Forward pointer inside the same chapter: the rewriting that removes the recurrence begins on p. 128 and is the subject of Rewriting the triangle with selection counts, so any row is reachable directly.