Many different-looking expressions for one growing pattern

What to assume they know

• Expanding a product of two two-term brackets, and collecting like terms (Multiplying two two-term expressions, and where the four terms come from)
• Identity 1A and Identity 1B ((a + b)² and (a − b)²: why there is a middle term at all)
• Identity 1C (Why the product of a sum and the matching difference is a² − b², and the patterns that follow)
• Area of a square and a rectangle, and finding an area by subtracting one region from another
• Substituting numbers into an expression, and evaluating
• The idea of a step number in a growing sequence, and of an expression in that step number

What they should be able to do

• Continue a drawn pattern by one step, and count the units in the drawn steps
• Decompose one pattern in several ways, and write an expression for the step number from each decomposition
• Expand each expression and show that all of them reduce to a single common form
• Explain why agreement at one or two steps does not establish that two expressions are equal, and why expansion does
• Use a derived expression to find the count at a step too large to draw
• Write an expression for a shaded area by subtraction and again by direct measurement, and verify the two agree
• Evaluate competing expressions at given values and confirm they give one number
• Read a figure whose parts are described in letters, and identify the dimensions of each part
• Produce a second method for a pattern or an area, deliberately unlike the first

Where it usually goes wrong

• "Every pattern has one formula, and my job is to find it." It has one value at each step. The chapter prints four expressions for one pattern and invites a fifth, and the point of section 6 is that this is not a problem to be resolved but the normal state of affairs.
• "Different-looking expressions are different expressions." The four printed ones look nothing like each other and are the same. The only way to know is to expand.
• "If two expressions agree at Step 1, they agree." All four agree at Step 1 by construction — they were all read off the same drawing. Agreement at a step is guaranteed and therefore proves nothing about any other step. Make this explicit; it is the transferable idea in the section.
• "My method is wrong because it is not the book's." The chapter prints four and asks whether the reader's matches any of them or is different, with no suggestion that being different is a fault.
• "You cannot subtract to find an area." Two of the printed methods in this section do exactly that, and one of them subtracts four rectangles from a square. Subtraction is a way of counting.
• "The shaded area depends on which student you ask." Three named methods, three different expressions, one number when the letters are replaced. That is the whole design of the Fig. 1 item.
• "Drawing the next figure answers the question." It answers one question. Step 10 is drawable with patience and Step 100 is not, which is what forces the expression.
• "You can read the expression off the picture and stop." The picture tells you how you counted. Whether your count and someone else's always agree is an algebraic question, and expanding is how it is answered.

Questions to check understanding

• Draw the next step of a given pattern and state its count
• Write an expression for the count at the general step, from a stated decomposition
• Given two expressions for one pattern, decide by expanding whether they agree
• Use an expression to find the count at a step beyond the drawn ones
• Write an expression for a shaded region by two different methods and show they agree
• Evaluate competing expressions at given values and confirm a single answer
• Match a described quantity to one or more of several candidate expressions, and say why the others fail
• Produce a second method for a pattern already solved one way

Examples worth working on the board

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

• The circle pattern (Part I, §6.4, p.150, artwork, with a Math Talk marker). Three drawn steps of filled black circles, followed by a dotted continuation. Counted off the printed page: Step 1 holds 3 circles, Step 2 holds 8, Step 3 holds 15. Printed questions: draw the next figure; how many circles it holds; how many there are at Step 10; and an expression in the step number.
• Method 1 (Part I, §6.4, p.151, artwork). Each step drawn as a square of circles, one row and one column longer than the step number, with the bottom-right circle drawn as a dashed empty outline to show it is absent; the whole square is framed in red. The four printed expressions run 2 squared less 1, 3 squared less 1, 4 squared less 1, 5 squared less 1, each rewritten with the step number visible inside the bracket, and the general form is the square of one more than the step number, less one.
• Method 2 (Part I, §6.4, p.151, artwork). Each step drawn as a square whose side is the step number, framed in red, with one extra column marked in green at the right and one extra row marked in green underneath. The printed expressions are the square of the step number plus twice the step number.
• Method 3 (Part I, §6.4, p.151, artwork). Each step drawn as a rectangle whose sides are the step number and one more than it, framed in red, with an extra row marked in green beneath. The printed expressions run 1 times 2 plus 1, 2 times 3 plus 2, 3 times 4 plus 3, 4 times 5 plus 4, each rewritten to show the step number in both places, and the general form is the step number times one more than itself, plus the step number.
• Method 4 (Part I, §6.4, p.152, artwork). Each step drawn as a single rectangle with no remainder, framed in red. The printed expressions run 1 times 3, 2 times 4, 3 times 5, 4 times 6, each rewritten with the step number in both factors, and the general form is the step number times two more than itself.
• The four simplifications (Part I, §6.4, p.152). Printed side by side in four columns; each of the four expressions is expanded and every one of them ends at the square of the step number plus twice the step number. The page states plainly that all four routes reach the same answer when carried out correctly, and names that expression as the count at the general step.
• The owl box (Part I, §6.4, p.152). Printed: mathematics often admits several ways of looking at a pattern and several ways of approaching one problem; finding them takes creativity, and exploring the ones that are not your favourite is worth doing. This is the section's moral and the chapter states it explicitly.
• The step-15 question (Part I, §6.4, p.152). Printed: use the formula to find the count at Step 15. This is the payoff of section 9 — the drawing route is now hopeless and the expression is not.
• The square-tile ring (Part I, §6.4, p.152, artwork). Three drawn steps of pink square tiles forming a hollow square ring. Read off the printed page: Step 1 is a three-by-three block with the single middle tile absent; Step 2 is a four-by-four block with a two-by-two hole; Step 3 is a five-by-five block with a three-by-three hole. Printed questions (Part I p.153, with a Math Talk marker): how many tiles are in each figure; how many at Step 4 and at Step 10; an expression for the general step; and whether more than one method can be found. The tile counts of the three drawn steps are data; everything else is the exercise.
• The four-rectangle square (Part I, §6.4, p.153, artwork). A square whose side is the total of two letters, holding four congruent rectangles arranged around a central region drawn with diagonal hatching. The shorter dimension is marked with a small double-headed arrow across the top; the longer is marked down the right side. Printed: two named methods. One takes the whole square's area and subtracts the four rectangles. The other observes that the central region is itself a square whose side is the difference of the two letters. The page then asks that both be expanded and checked against each other.
• Fig. 1 and its three methods (Part I, §6.4, pp.153–154, artwork). Printed figure: three congruent rectangles arranged as an upright I inside a taller rectangle, with the region outside the I drawn with diagonal hatching; the page states that all three rectangles share the same dimensions. Three hand-drawn sketches beside the text label the parts. Printed methods, each named for a student: the first takes a square whose side is the larger letter and subtracts a rectangle of the two letters; the second takes the outer rectangle, whose height is the larger letter plus twice the smaller, and subtracts all three rectangles; the third takes twice one flanking piece, whose width is half the difference of the letters and whose height is the larger letter. The page asks that all three be expanded and shown equivalent, and then gives substitution values: the larger letter 8 and the smaller 3.
• The dashed region in the L figure (Part I, §6.4, p.154, artwork, with a Math Talk marker). Printed figure: one overall rectangle split into two regions — an L-shaped band tinted pink running down the right side and along the bottom, drawn with a solid outline on every edge, and the remaining upper-left rectangle filled with diagonal hatching. The only dotted lines in the figure are that hatched rectangle's top and left edges. So "the dashed region" the instruction asks for is the hatched upper-left rectangle, not the pink L: its sides are the full width less the band width, by the full height less the band depth. The pink L is its complement within the same overall rectangle. Four labels: the band's width marked at the top, the full height marked at the right, the band's depth marked at the left, and the full width marked along the bottom — and the top and left labels are the same letter, so a single uniform band width is what the drawing states, not an inference. Printed substitution values: the height 6, the band width 3.5, the full width 9. Printed instruction: use more than one method. No answer, and the expression stays off screen — it is the exercise.
• The chapter-end set, Part I pp.154–156. Item 3 offers candidate expressions to be matched against descriptions; part (i) is about two more than a square number and part (ii) about the total of the squares of two consecutive numbers, with seven candidates offered for the second. Item 4 is the calendar item, carried in What happens to a product when you nudge one factor. Item 5 lists four claims to be verified. Item 7 is the three-consecutive-numbers item, carried in Why the product of a sum and the matching difference is a² − b², and the patterns that follow. Item 8 asks for the expression behind a described procedure and a proof about it. Item 10 is the Dhauli park, carried in (a + b)² and (a − b)²: why there is a middle term at all. Item 11 is the two step sequences below.
• Item 11's two sequences (Part I p.156, artwork). Printed questions for each: draw the next figure, give the count at Step 10, and write an expression for the general step. First sequence (yellow tiles), read off high-magnification the printed page: each step is a Z — a horizontal block whose width is two more than the step number and whose height is the step number, with one arm rising from the block's right-hand column and another descending from its left-hand column. As drawn, Step 1 has a three-by-one block with arms of 3 above and 3 below, 9 tiles in all; Step 2 has a four-by-two block with arms of 4 and 4, 16 tiles; Step 3 has a five-by-three block with arms of 6 and 6, 27 tiles. See the Notes — the drawn arm lengths do not continue a single rule, and a reviewer should look before this item is used. Second sequence (blue squares): each step is a solid rectangle with one square missing from its bottom row — the rectangle has one more column than the step number and two more rows. As drawn, Step 1 has 5 squares, Step 2 has 11 and Step 3 has 19.
• The chapter prints no answers to any exercise item. Its own printed working in this section is the four methods' expressions, their four expansions, and the common form they all reach.

Figures to have open

• The circle pattern's first three or four steps, redrawn so that each of the four decompositions can be overlaid on the same drawing in turn. The chapter's own figures (Part I, §6.4, pp.150–152) put each method on its own row; overlaying them on one shape is what makes section 6 land. Redraw as a schematic.
• The four-rectangle square with the central hatched region, both dimensions marked (Part I, §6.4, p.153).
• Fig. 1: three congruent rectangles forming an upright I inside a taller rectangle, with the flanking regions hatched, and enough labelling to support all three named methods (Part I, §6.4, pp.153–154). The chapter's own sketches are hand-drawn and informal; a clean schematic is needed, and each method should get its own copy with only its own construction lines showing.
• The L figure (Part I, §6.4, p.154): the pink L-shaped band solid-outlined on every edge, and the hatched upper-left rectangle whose top and left edges are dotted — that hatched rectangle being the region the item asks for — with the band width, band depth, full width and full height marked, the band width and depth sharing one letter as printed.
• The square-tile ring at three steps, and the two sequences of item 11 at three steps each (Part I pp.152 and 156). These are the exercises, so the figures must be faithful in count as well as shape.
• 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", §6.4 "This Way or That Way, All Ways Lead to the Bay", Part I pp.150–154. The section opens at the foot of Part I p.150, immediately after §6.3.
• Methods 1 to 3 are on Part I p.151; Method 4 and the four simplifications are on Part I p.152; the tile ring is drawn on Part I p.152 and questioned on Part I p.153; the four-rectangle square and Fig. 1 are on Part I p.153; the third method for Fig. 1 and the L-shaped region are on Part I p.154.
• Chapter-end "Figure it Out", Part I pp.154–156. Items 3, 5, 8 and 11 belong to this topic; items 4, 7, 9 and 10 are carried by the m01 and m02 briefs and are cross-referenced rather than repeated.
• The chapter's SUMMARY, Part I p.157, closes with this section's claim: that one problem often admits several routes to the same correct answer, and that finding them is a creative act.
• §6.3 "Mind the Mistake, Mend the Mistake" (Part I p.150) is marked video: no in the spine and has no brief. Its twelve items are error-spotting practice on the expansions this topic and the m01 topics cover, and they make a good class exercise after any of these videos.