An AP plots as points on a straight line

What to assume they know

• The common difference and the nth-term formula (Common difference, and why the nth term is a + (n − 1)d)
• Plotting an ordered pair on a pair of axes (Two axes, an origin, and why the order of the pair matters)
• What the two coefficients of a linear rule do to its line — steepness and where it cuts the vertical axis (What a and b do to the line: slope, y-intercept, and parallel families)
• A constant difference as the signature of a linear pattern (A constant difference is the signature of a linear pattern)
• Degree of a polynomial in one letter (Univariate polynomials and what degree names)

What they should be able to do

• Build a stage-and-value table from a growing pattern and read ordered pairs off it
• Plot the pairs coming from an AP and describe what the plot shows
• Explain why a constant common difference forces the points to be collinear
• Rearrange a + (n − 1)d into the form dn + (a − d) and identify each coefficient in the picture
• Read the common difference off a plotted AP as its steepness
• Say what the value at position zero means, and why it is not the first term
• Explain why the graph of a sequence is a set of points and not a continuous line
• Use the shape of a plot as a test for whether a sequence is an AP

Where it usually goes wrong

• "The points happen to line up in this example." They cannot fail to. Equal horizontal steps with equal vertical rises is the definition of a straight line; the constant common difference is the constant rise.
• "The graph is a line." It is five dots. The line is drawn to make the pattern visible and to let you read off the steepness. Nothing lives at (2.5, 7), even though the drawn line passes through it.
• "The line crosses the vertical axis at the first term." For 4n − 3 it crosses at −3, and the first term is 1. The crossing point is a − d, one step before the sequence starts.
• "Steeper means bigger terms." Steepness reports d, the rise per step. A progression can start high and climb gently or start low and climb hard, and the two lines cross.
• "A straight-line plot proves the sequence is an AP." It proves it for the positions you plotted. Five collinear dots are strong evidence and not a proof; the constant-difference check is the proof.
• "If the plot is not straight, there is no rule." 1, 4, 9, 16 bends and has the rule n². Bending rules out an AP, not a rule.
• "Position can go on the vertical axis, it makes no difference." Swap them and the steepness becomes 1/d and every reading in the section changes. The chapter's convention — position across, value up — has to be stated before the first point is plotted.

Questions to check understanding

• Complete a stage-and-value table for a growing pattern and write the nth-term entry
• Plot the pairs from a given AP and state what the plot shows
• Given a plotted line through the points of an AP, read off the common difference
• Rewrite a + (n − 1)d in the dn + c form and say what c equals
• Decide from a plot whether a sequence can be an AP, and justify it by the differences
• Explain why a sequence's graph is a set of separated points
• Predict, before plotting, how two given APs will compare on the same axes

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 (§8.4.1, p. 181, printed as a two-row table with a shaded label column). Stage numbers 1, 2, 3, 4, 5, then an ellipsis column, then n. Tile counts 1, 5, 9, 13, 17, then the ellipsis, then 4n − 3. The table is the bridge the section is built on: it puts position and value into two rows so that pairing them is the obvious next move.
• The five ordered pairs (p. 182): (1, 1), (2, 5), (3, 9), (4, 13), (5, 17), with the stage number taken as the first coordinate. The page states that they fall on one straight line.
• Fig. 8.4 (p. 182). Drawn on a green square-ruled grid. The vertical axis is labelled in twos and runs up to 22; the horizontal axis is marked in twos from −2 to 10; both carry printed axis names. The five points are marked and labelled with their coordinates, and a straight line with an arrowhead at each end is drawn through them. All five points sit in the first quadrant, while the grid itself extends a little to the left of the vertical axis.
• The rearrangement — the argument of section 5, worked from the chapter's own formula. a + (n − 1)d expands to a + dn − d, which is dn + (a − d). Verified on the chapter's own case: a = 1 and d = 4 give 4n − 3, and the chapter prints exactly that simplification on p. 181. So the steepness is 4, the common difference, and the constant is −3, which is 1 − 4 = a − d.
• The zero-position value. For 4n − 3 substituting n = 0 gives −3. Verified. There is no stage 0 in Fig. 8.3 and no term in position 0, so −3 is where the drawn line would cross the vertical axis and nothing more. This is the single most useful place to distinguish the line from the sequence.
• A half-position test. For 4n − 3 substituting n = 2.5 gives 7. Verified. The point (2.5, 7) lies on the drawn line, and there is no such term — the tile pattern has no stage two-and-a-half. Show that point in a different colour.
• The two prediction exercises (§8.4.1, p. 182). Verify that 2, 5, 8, 11, … and −5, −1, 3, 7, … are APs, write their nth terms, and say what the plots look like. Verified: 3n − 1 and 4n − 9, so the first plot rises 3 per step and the second rises 4 per step. Ask for the prediction before plotting; the chapter's phrasing invites exactly that.
• The recursive rule, restated on p. 183 as t₁ = a with tₙ = tₙ₋₁ + d for n ≥ 2, and worked for the tile pattern as t₁ = 1, tₙ = tₙ₋₁ + 4. Useful here because the recursion is the step-by-step instruction the staircase in the picture draws.
• A contrast to hold back for section 10 (a deliberate forward reference to §8.6.2, p. 191): the geometric pattern 3, 6, 12, 24, 48 gives the pairs (1, 3), (2, 6), (3, 12), (4, 24), (5, 48), and the chapter says outright that these do not fall on a straight line. Showing the two plots side by side is what makes section 10 a diagnosis rather than a description; the full treatment is A GP plots as a curve, and what that curve tells you.
• A non-AP with a rule, for the same section: 1, 4, 9, 16, 25 from §8.1 (p. 174). Verified: gaps 3, 5, 7, 9, so the plot bends. A sequence can have a perfectly good explicit rule and still not be an AP.

Figures to have open

• Fig. 8.4 redrawn: five labelled points on a square grid with a straight line through them, axes named, vertical scale to about 22. The chapter's own figure (p. 182) is the model; redraw it as a clean schematic rather than reproducing the printed grid art. The five coordinate labels must stay, since section 3 reads them aloud.
• A staircase overlay for section 4 — treads of 1 and risers of d between consecutive points. Standard schematic, and the figure that carries the argument.
• The same plot extended left to show the crossing at −3, with the crossing point marked as not-a-term. Standard schematic.
• A side-by-side of the AP plot and the geometric plot of 3, 6, 12, 24, 48 for section 10. Both come from the chapter (Figs. 8.4 and 8.9); the second is treated in full in A GP plots as a curve, and what that curve tells you.

Where this sits in the book

• Chapter 8, §8.4.1 Visualising an AP, pp. 181–183. The stage-and-value table closes p. 181; the ordered pairs, the collinearity claim and Fig. 8.4 are on p. 182; the recursive rule and Example 5 open p. 183.
• Fig. 8.4 and its caption, p. 182 — the caption is where the chapter uses the phrase linear pattern.
• Two inline Exercise prompts on p. 182, both of which ask what the plot looks like as well as what the nth term is.
• Deliberate cross-references outside this topic: the corresponding table and plot for a geometric progression are §8.6.2, pp. 190–191, treated in A GP plots as a curve, and what that curve tells you; the vocabulary for reading a straight line off its rule — steepness and the crossing on the vertical axis — is Chapter 2 of this book, not Chapter 8.
• The summary, p. 196, restates the nth-term formula but says nothing about plotting.