A GP plots as a curve, and what that curve tells you

What to assume they know

• The common ratio and arⁿ⁻¹ (Common ratio, and why the nth term is ar^(n−1))
• Why an AP plots as points on a straight line (An AP plots as points on a straight line)
• Plotting ordered pairs and reading coordinates off a grid (Two axes, an origin, and why the order of the pair matters)
• Multiplying and comparing decimals, including repeated multiplication by 0.75
• Fractions of a quantity, for the one-sixth condition
• The fractal count and area sequences (Fractals: self-similarity generates a GP)

What they should be able to do

• Build a stage-and-value table for a GP and read ordered pairs off it
• Plot the pairs from a GP and describe the shape, contrasting it with an AP's plot
• Explain why a constant ratio other than 1 cannot produce collinear points, and why the ratio 1 is the exception that proves the rule
• Interpret the increasing steepness as the rise being proportional to the current value
• Describe what a plot with ratio below 1 does, and state that it approaches zero without reaching it
• Generate a bounce-height sequence from a stated drop height and ratio
• Decide from the sequence how many bounces are needed to fall below a stated fraction of the original height
• Read values off a printed chart and check them against the arithmetic that produced them

Where it usually goes wrong

• "The points are not straight because the drawing is rough." They cannot be straight. For 3, 6, 12, 24 the successive rises are 3, 6 and 12 — the rise triples along with the value. Straightness requires equal rises, by definition.
• "Bending upward means the sequence is growing fast." Bending means the rise is growing. Fig. 8.10 B also bends and its values fall. What the bend reports is a constant ratio, and the direction depends on whether that ratio is above or below 1.
• "The falling plot reaches zero at about stage 6." It never reaches zero; every term of (3/4)ⁿ is positive. It gets as close as you like, which is what the chapter says on p. 191, and a chart with a coarse vertical scale will make it look like arrival.
• "The line drawn through the points is the graph." As with an AP, the sequence exists only at whole stage numbers. There is no bounce number 3½.
• "24 feet is the first term of the GP." It is the drop height. The GP of bounce heights starts at 18, which is what p. 193 states, even though Fig. 8.11 plots 24.00 ft at bounce number 0.
• "The chart's labels are the exact values." 10.12 is a truncation of 10.125, the fractal areas on Fig. 8.10 B are rounded to two decimals, and the 7.9 on Fig. 8.11 does not match the working at all. Reading a chart is not the same as computing.
• "Because 4.27 rounds to 4, the sixth bounce satisfies the condition." It does not: 4.27 feet is above 4 feet. Rounding before comparing is how this question gets answered wrongly.
• "A graph gives you the total." It shows each term. The total of the bounce heights is a separate computation, and this chapter supplies no shortcut for it.

Questions to check understanding

• Complete a stage-and-value table for a GP and write the nth-term entry
• Plot the pairs from a given GP and state what the shape shows
• Given two plots, one known to be arithmetic and the other known to be geometric, say which is which and give the reason. (State that pairing in the question. A bend on its own does not establish a geometric progression, so an unpaired plot cannot be classified from its shape alone.)
• Describe what happens to a GP with ratio below 1 as the positions continue
• Generate a bounce-height sequence from a drop height and a percentage
• Find how many steps are needed for a decaying GP to fall below a stated fraction of its starting value
• Compute a total distance for a bouncing ball over a stated number of bounces
• Read a value off a printed chart and check it against the rule that generated it

Examples worth working on the board

Values marked verified are worked out here on the chapter's stated data; this book prints no answer key.

• The stage table (§8.6.2, printed at the top of p. 191 as a two-row table with a shaded label column). Stage numbers 1, 2, 3, 4, 5, an ellipsis column, then n. Counts 3, 6, 12, 24, 48, the ellipsis, then 3 × 2ⁿ⁻¹. This is the deliberate twin of the AP table on p. 181.
• The five pairs (p. 191): (1, 3), (2, 6), (3, 12), (4, 24), (5, 48). The page states that these do not fall on one straight line.
• Fig. 8.9 (p. 191). Drawn on a green square-ruled grid, with both axes named. The five points are marked, labelled with their coordinates, and joined by a smooth red bending line. The horizontal axis is marked 0 to 5. Read off the page: the vertical axis carries the labels 5, 10, 15, 20, 25, 30, 35, then 45, then 50, at nine evenly spaced gridlines — the label 40 is missing, so the top of the printed scale does not match its own spacing. A redrawn figure must restore 40, and a teacher must not treat the printed top labels as the real scale.
• Fig. 8.10 A (p. 192). The Sierpiński triangle's piece counts plotted against stage number: (0, 1), (1, 3), (2, 9), (3, 27), (4, 81), joined by a steeply bending line. The vertical axis is labelled in tens up to 80, so the point at 81 sits just above the topmost labelled gridline. Both axis names are printed.
• Fig. 8.10 B (p. 192). The same fractal's black area plotted against stage number: (0, 1), (1, 0.75), (2, 0.56), (3, 0.42), (4, 0.32), joined by a line that falls steeply and then flattens. The vertical axis runs to 1.2 in steps of 0.2. Verified: the printed labels are the exact values (3/4)ⁿ rounded to two decimals — 0.5625 → 0.56, 0.421875 → 0.42, 0.31640625 → 0.32. Worth showing the exact fractions beside the plotted decimals.
• What the chapter says the two plots mean (p. 191): as the stages increase the piece count rises very fast while the black area diminishes, getting closer and closer to 0. That sentence is the whole of section 6.
• Example 10, the bouncing ball (p. 192). A ball is dropped from 24 feet, and each bounce reaches three quarters of the height before it. Two questions are asked: write the heights for five successive bounces, and find how many bounces are needed for the ball to stay below one sixth of the height it was dropped from. The page then prints seven bounces of working: 24 × 0.75 = 18, 18 × 0.75 = 13.5, 13.5 × 0.75 = 10.125, 10.125 × 0.75 = 7.59375, 7.59375 × 0.75 = 5.695, 5.695 × 0.75 = 4.27125, 4.27125 × 0.75 = 3.2034375.
• Where the progression starts (p. 193). The page identifies the bounce heights as a GP with first term 18 and ratio 0.75, listing 18, 13.5, 10.125, 7.594, 5.695, 4.27125, 3.2034375, …. Note: 24 feet is the drop, not a bounce, so it is not a term of this GP. Fig. 8.11 nonetheless plots it at bounce number 0, which is a sensible chart choice and a genuine trap for a student reading the axis label literally.
• The one-sixth condition (pp. 192–193). Verified: one sixth of 24 feet is 4 feet; the sixth bounce reaches about 4.27 feet, still above it, and the seventh reaches about 3.20 feet, below it. The page's conclusion is that the ball stays below one sixth after the seventh bounce, which agrees. This is why the printed working runs to seven bounces when the question asked for five.
• Two printed slips in the bounce data, both confirmed on the printed pages rather than inferred from extracted text. First, the chapter rounds mid-chain: it computes the fifth bounce as 5.695 and then multiplies that rounded value onward, so its 4.27125 and 3.2034375 drift slightly from the exact values. Verified: carried exactly, 24 × 0.75ⁿ gives 5.6953125, 4.271484375 and 3.203613281 for bounces 5, 6 and 7. The drift changes nothing about the answer. Second, in Fig. 8.11 the fourth bounce is labelled 7.9 ft, while the working on p. 192 gives 7.59375 and every other label on the chart is set to two decimals. Read on p. 193, the label is 7.9. Treat it as a typographic slip for 7.59; a redrawn chart should carry 7.59 ft.
• Fig. 8.11 (p. 193), titled on the chart itself. Bounce number along the horizontal axis from 0 to 7, with that axis named beneath; height in feet up the vertical axis, marked in fives to 25, with the axis named as such. Eight points are marked, each with a dashed vertical dropline to the axis and a printed value beside it: 24.00, 18.00, 13.50, 10.12, 7.9, 5.70, 4.27 and 3.20 ft. Note that 10.12 is 10.125 truncated rather than rounded up.
• The transfer exercise (Exercise Set 8.3 item 5, p. 193). A ball dropped from 80 metres rising to 60% of its previous height each time; find the height after the 5th bounce, and the total vertical distance covered up to its sixth landing. Verified: the bounce heights are 48, 28.8, 17.28, 10.368 and 6.2208 metres, so the fifth bounce reaches 6.2208 m. The total distance is the first fall of 80 m plus each of those five heights counted twice, once up and once down: 80 + 2 × 110.6688 = 301.3376 m, about 301.34 m. Part (ii) is the item worth flagging to anyone — the chapter states no formula for the total of a GP anywhere, so the only route it leaves open is adding the five heights directly. That is entirely feasible here, and an explanation should say so plainly rather than implying a missing formula.

Figures to have open

• Fig. 8.9 redrawn: five labelled points with a smooth bending line, axes named, and a vertical scale in fives that includes 40 — the printed scale omits that label and must not be copied. From p. 191.
• Figs. 8.10 A and 8.10 B redrawn as a matched pair on a shared stage axis, one rising to 81 and one falling from 1 toward 0, with the exact fractions shown beside the rounded decimals. From p. 192; the pairing is what makes section 5 work.
• A rise-comparison overlay for sections 3 and 4: the same plot with the vertical rises bracketed and labelled, so that "the rise grows with the value" is read rather than asserted. Standard schematic; the chapter does not draw it.
• Fig. 8.11 redrawn: bounce number 0 to 7 across, height in feet up, eight points with droplines, values 24.00, 18.00, 13.50, 10.125, 7.59, 5.70, 4.27 and 3.20 ft, and a horizontal reference line at 4 feet for the one-sixth condition. That reference line is the explanation's addition and is what turns the chart into an answer. From p. 193.
• A side-by-side of the AP plot (Fig. 8.4, p. 182) and the GP plot (Fig. 8.9, p. 191) for section 1. Both are the chapter's own.

Where this sits in the book

• Chapter 8, §8.6.2 Visualising a GP, pp. 190–193. The section heading is at the foot of p. 190; the table, the pairs and Fig. 8.9 are on p. 191; Figs. 8.10 A and 8.10 B and Example 10's working are on p. 192; the GP identification, the one-sixth conclusion and Fig. 8.11 are on p. 193.
• Figures 8.9 (p. 191), 8.10 A and 8.10 B with their shared caption (p. 192), and 8.11 (p. 193).
• Exercise Set 8.3, p. 193, item 5 — the transfer task. Items 1, 2, 4 and 6 belong to Common ratio, and why the nth term is ar^(n−1), item 3 to A recursive rule builds each term from the ones before it, and item 7 to Fractals: self-similarity generates a GP.
• Deliberate cross-references outside this topic: the AP table and plot are §8.4.1, pp. 181–182, treated in An AP plots as points on a straight line; the fractal sequences plotted in Fig. 8.10 are derived in §8.6.1, treated in Fractals: self-similarity generates a GP.
• The summary, p. 196, restates the GP formula and says nothing about plotting.