Why the product of a sum and the matching difference is a² − b², and the patterns that follow

What to assume they know

• Expanding a product of two two-term brackets (Multiplying two two-term expressions, and where the four terms come from)
• Identity 1A and Identity 1B, the squares of a sum and of a difference ((a + b)² and (a − b)²: why there is a middle term at all)
• Like terms, and that a term and its negative cancel
• Area of a square and of a rectangle, and that cutting and rearranging a region leaves its area alone
• Squares of the round numbers up to a few hundred, and of the single digits
• That verifying a claim on several examples is not the same as proving it

What they should be able to do

• Observe a numerical pattern in doubled sums of squares and identify the two numbers hidden in each result
• Prove that twice a sum of two squares equals the square of the sum plus the square of the difference, by adding Identities 1A and 1B
• Read a list of numerical instances as a conjecture, state the conjecture in letters, and test it by expanding
• Expand (a + b)(a − b) and explain which two terms cancel and why
• Give an area argument for the same identity by cutting a strip from a square and moving it
• Use Identity 1C forwards, to compute a product of two numbers straddling a round number
• Use Sridharacharya's rearrangement of the identity to square a number by multiplying two easier numbers
• Choose the shift that makes one of the two factors round
• Decide whether the two patterns depend on the numbers being counting numbers, and justify the decision
• Derive consequences: the square of the middle of three consecutive numbers against the product of its neighbours, and writing a given number as a difference of two squares

Where it usually goes wrong

• "a² − b² is (a − b)²." The commonest confusion in the chapter, and the reason the previous topic comes first. One is a product of two different brackets, the other a square. Substitute 5 and 3 and get 16 against 4.
• "A sum times a difference should have four terms." It does have four, before collecting. Two of them are one rectangle added and the same rectangle removed. Show them cancelling rather than asserting that they do.
• "Four instances make a proof." Pattern 2 is printed as four lines that all work, and the chapter's very next move is to ask whether it is a true identity — because four lines cannot tell you. This is the most transferable idea in the section.
• "(a − b)² and (b − a)² are different." Squaring destroys the sign, so neither is greater. Students expect a size question and the answer is that the question dissolves. Part I p.149 no. 1 is built on this.
• "45 times 55 has to be multiplied out." Both sit five away from 50. The skill is spotting the middle, not remembering the identity.
• "43 times 45 is not a sum-and-difference pair." It is, around 44 — and unlike 98 and 102 nothing on the page tells you so.
• "The patterns are facts about whole numbers." The proof is algebra and never mentions what kind of numbers the letters are, which is exactly what Part I p.149 no. 4 is asking the student to notice.
• "Sridharacharya's method needs a shift of one." Choose the shift so that one factor lands on a round number — the chapter's own second example takes a shift of 3 to reach 200.
• "Cutting and moving a piece could change the area." It cannot, and saying so out loud is worth a sentence: the area argument and the algebra are two accounts of one fact, not two facts that happen to agree.

Questions to check understanding

• Compute a product of two numbers straddling a round number, without long multiplication
• Square a number by choosing a shift that makes one factor round, and state the shift chosen
• Expand a sum times the matching difference, and identify the two terms that cancel
• Given a list of numerical instances, state the pattern in letters and decide by expanding whether it always holds
• Write a given number as a difference of two squares, and say how many ways there are
• Compare two products without computing them, by writing each as a difference of squares
• Verify or refute a claim about a difference of two squared expressions
• Justify whether a proved identity continues to hold for negative numbers and for fractions
• Prove the relation between the square of a number and the product of its two neighbours

Examples worth working on the board

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

• Pattern 1's four printed instances (Part I, §6.2, p.147, under the printed subheading "Investigating Patterns"). Printed exactly as four lines: twice the total of 2 squared and 1 squared equals 3 squared plus 1 squared; twice the total of 3 squared and 1 squared equals 4 squared plus 2 squared; twice the total of 6 squared and 5 squared equals 11 squared plus 1 squared; twice the total of 5 squared and 3 squared equals 8 squared plus 2 squared. Hand all four over as data — the second and third are the ones that make the pattern findable, since 6 and 5 give 11 and 1 immediately.
• The prompt and the printed clue (Part I, §6.2, p.148). Printed: take a pair of natural numbers, total their squares, and try to write twice that total as a total of two squares; try more pairs; then the page supplies one further instance written suggestively, with 5 and 6 producing the square of their sum and the square of their difference.
• The proof of Pattern 1 (Part I, §6.2, p.148). Printed in full: Identity 1A and Identity 1B set one above the other, then added; the like terms named — the two squares of the first letter, the two squares of the second, and the two middle terms totalling zero — and the conclusion stated as twice the total of the two squares equalling the square of the sum plus the square of the difference.
• Pattern 2's four printed instances (Part I, §6.2, p.148). Printed as four lines, each with the two squares written as products: 9 times 9 less 1 times 1 equals 10 times 8; 8 times 8 less 6 times 6 equals 14 times 2; 7 times 7 less 2 times 2 equals 9 times 5; 10 times 10 less 4 times 4 equals 14 times 6. The page asks that the pattern be described in letters so that its generality can be decided.
• The conjecture, the test, and Identity 1C (Part I, §6.2, p.148). Printed: the conjectured identity, then the expansion by the distributive property, then the observation that the two middle terms total zero, then Identity 1C boxed. The page also notes that the same product already appeared as item 5(i) of the §6.1 "Figure it Out" set on Part I p.143.
• A printed slip in the expansion line (Part I, §6.2, p.148). In the display line that expands the product, the fourth term is set as the square of the first letter where the argument needs the square of the second. Both the printed page and the extracted text agree, so this is a printed error and not an extraction artefact. The boxed Identity 1C immediately beneath it is correct. Anyone showing that page must correct the line or the derivation ends in zero.
• The geometric task and its hint figure (Part I, §6.2, pp.148–149). The task, flagged with a "Try This" marker, is to show the identity geometrically. The hint figure on Part I p.149 is artwork: a large block tinted blue-grey with a narrower block tinted red standing against its right edge, the two together making one rectangle; the top edge is marked a across the blue-grey part and b across the red; the left side carries a long arrow marked a for the whole height and a short arrow marked b for a band at the bottom; that bottom band is drawn in dashed outline only, with a dashed vertical line inside it continuing the blue-grey/red boundary; and a curved arrow runs from the red block down to the dashed band beside the printed question of what the shape becomes when that piece is moved.
• An added reading of that figure — flag it as a reconstruction, not a quotation. The square of side a is the blue-grey block together with the dashed band beneath it. Removing a small square of side b from the corner of the band leaves a piece of the band whose sides are b and a − b; rotate that piece and stand it against the right edge, which is where the red block is drawn, and the shape becomes a rectangle a + b across by a − b down. The area never changed, so the difference of the two squares is that rectangle.
• Forward inputs for Identity 1C (Part I, §6.2, p.148): 98 times 102, and 45 times 55. Both straddle a round number, and finding the middle is the skill.
• Sridharacharya's rearrangement (Part I, §6.2, p.149). Printed: the identity restated so that a square equals the product of the shifted pair plus the square of the shift, credited to Sridharacharya (750 CE), followed by the printed question of why the restatement is true. Two printed workings: 31 squared taken with a shift of 1, giving 32 times 30 plus 1, and stated as 961; and 197 squared taken with a shift of 3, giving 200 times 194 plus 9, and stated as 38809. Both are the chapter's own results.
• §6.2 "Figure it Out" (Part I p.149). No. 1: which is greater, the square of a − b or the square of b − a, with a justification asked for. No. 2: write 100 as a difference of two squares. No. 3 (a Math Talk item): find 406 squared, 72 squared, 145 squared, 1097 squared and 124 squared using the identities so far. No. 4: whether the two patterns are restricted to the counting numbers, whether they survive negative integers, and whether they survive fractions, with a justification asked for in each case. All four are open on the page. No. 3 is the natural place to make the choice of shift explicit — 145 and 1097 both sit near a round number, 72 and 124 do not.
• Chapter-end inputs. Part I p.154 no. 1(ii), 397 times 403 using Identity 1C, and no. 1(iv), 43 times 45 using Identity 1C — the second of those is the interesting one, because the middle number is not written down anywhere and the student has to find it. Part I p.154 no. 2(ii), the product of 3a − 9b and 3a + 9b. Part I p.155 no. 5(iv): the claim that the square of 6n + 2 less the square of 4n + 3 is five short of a square number. Part I p.155 no. 7: take any three numbers in a row, square the middle one, take away whatever the outer two multiply to, do the same with further triples, write down what is observed as an equation, and expand both sides to confirm it is an identity. Part I p.156 no. 9: which is larger, 14 times 26 or 16 times 24, and which is larger, 25 times 75 or 26 times 74, without computing the products fully.
• The chapter prints no answers to any exercise item. Its own computed values in this section are the four instances of each pattern, 961 and 38809.

Figures to have open

• The cut-and-slide area figure: a square of side a, a corner square of side b marked for removal, and the remaining strip rotated into place to form a rectangle a + b by a − b. The chapter supplies a hint version (Part I, §6.2, p.149) and section 6 is built on it. Redraw as a step-by-step schematic in three or four frames rather than reproducing the printed art — the printed figure shows only the end state and a question.
• Two stacked bars showing twice the total of two squares splitting into the square of the sum and the square of the difference. Standard schematic; the chapter proves Pattern 1 only algebraically.
• A number line with a marked middle and the two factors placed symmetrically about it, reusable for 98 and 102, for 45 and 55, and for 43 and 45. Standard schematic.
• A Notation panel for Sridharacharya's rearrangement, with the shift shown as a choice rather than a constant. 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", §6.2 "Special Cases of the Distributive Property", Part I pp.147–149. The printed subheading is "Investigating Patterns", beginning on Part I p.147, with the two patterns labelled "Pattern 1" and "Pattern 2".
• Identity 1C is boxed on Part I p.148; the geometric hint and the Sridharacharya material are on Part I p.149.
• §6.2 "Figure it Out", Part I p.149, items 1 to 4.
• Backward pointer, printed on Part I p.148: the same product was set as item 5(i) of the §6.1 "Figure it Out" set on Part I p.143, carried in Multiplying two two-term expressions, and where the four terms come from.
• Chapter-end "Figure it Out", Part I p.154 nos. 1(ii), 1(iv), 2(ii); Part I p.155 nos. 5(iv), 7; Part I p.156 no. 9.
• The chapter's SUMMARY, Part I p.157, lists this identity third among the special cases.