Why two points are enough to draw the line

What to assume they know

• Two axes, an origin, and why the order of the pair matters — the two axes, the origin, and why the order of a coordinate pair matters
• The four quadrants, and reading a point's signs off its position — the four quadrants and the signs of coordinates in each
• Recovering y = ax + b from two observations — the form y = ax + b and what its two constants do
• A polynomial as an input–output machine — substituting a value into a rule to obtain the output
• Plotting a point from its coordinates on squared paper
• Arithmetic with negative numbers, for the points below the axis

What they should be able to do

• Choose two convenient inputs for a linear equation and compute the matching outputs
• Plot the resulting pair of points, join them with a ruler and extend the line both ways
• State the criterion for a point lying on a line, and apply it by substitution
• Verify a further point against the equation rather than by eye
• Complete a table of paired values for a given linear equation
• Recover the equation of a line from a set of plotted points by comparing each y with its x
• Explain why a linear rule cannot produce a bend, using the constant-step property from §2.3
• Plot a line whose coefficient is a fraction, and one whose coefficient is negative

Where it usually goes wrong

• "I need many points before I can draw the line." Two suffice to fix it. More points are useful as checks, not as construction — and that difference is the point of the chapter's table on p. 28.
• "Two points are enough, so I need not check anything." Any two points give a line. Whether that line is the graph of your equation is tested by a third pair, by substitution. The chapter checks (7, 15) for exactly this reason.
• "The line stops at the two points I plotted." The chapter says to extend it in both directions, and Fig. 2.5 carries arrowheads at both ends. The rule has an output for inputs far outside the plotted window.
• "The straight drawing proves the relationship is linear." A ruler cannot help drawing straight. Compute a third pair before believing the picture.
• "Every line passes through the origin." y = 3x and y = –2x do; y = 2x + 1 does not, and Fig. 2.5 shows it crossing the vertical axis one unit up. The chapter has both kinds on facing pages.
• "(3, 7) and (7, 3) are the same point." They are not. The order of the pair is fixed, and Chapter 1 established it. Plot both to make the difference visible.
• "Choose x = 1 and x = 2 — they are easiest." They are close together, so a small slip in either badly rotates the line. The chapter chooses 0 and 3 for y = 2x + 1 and 0 and 4 for y = ½x; both choices avoid fractions and give a long span.
• "For y = ½x I should take x = 1." That gives y = ½, which has to be plotted between two grid lines. Take x = 4, as the chapter's hint does.
• "Points below the axis need a different method." Example 13's points and Example 12's (–1, –3) use the same substitution and the same plotting; only the signs change.

Questions to check understanding

• Draw the graph of a given linear equation by finding two suitable points
• Decide whether a stated point lies on a given line, and justify it by substitution
• Complete a table of paired values for a given equation
• Recover the equation from a set of plotted points or from a completed table
• Find where a linear polynomial's graph crosses each axis — the form of End-of-Chapter item 10(ii) on p. 38
• Verify a plotted answer against the equation, as End-of-Chapter items 9 and 10 on pp. 37–38 both explicitly require

Examples worth working on the board

Inputs below. Values marked verified are worked out here; where the chapter prints a value I say so. No answers are printed in this chapter and the book has no appended key.

• The chapter's opening instruction (p. 27, foot). To draw a linear equation it directs the reader to locate two points the line passes through, and takes y = 2x + 1 as the worked case.
• The two points (p. 28). At x = 0, y = 1, giving the point A (0, 1). At x = 3, y = 7, giving B (3, 7). The chapter names the coordinates of A explicitly — 0 for the x-coordinate and 1 for the y-coordinate — then plots both, joins them, and extends the line in both directions.
• Fig. 2.5 (p. 28, captioned as the straight line y = 2x + 1). Read off the printed page: it is drawn over printed green squared paper. The vertical axis is marked 1 to 8 and labelled y-axis; the horizontal axis is marked –2, –1, 0 and 1 to 5 and labelled x-axis. Both axes carry arrowheads at both ends. The line is drawn in green with arrowheads at both ends, and A (0, 1) and B (3, 7) are labelled beside their points.
• Why x = 0 and x = 3, and not x = 1 and x = 2. Verified, an added point — both chosen inputs give whole-number outputs, and they sit three apart so the ruler has a long span to work with. Two points close together amplify any plotting error into a large error of direction. The chapter simply makes the choice; the reasoning is worth saying out loud, because the reader has to make the same choice unaided in Exercise Set 2.6.
• Think and Reflect, p. 28. A table to complete for points on the same line. The x row reads 1, 2, 5, 7, 9, 12, 20. The y row has just two entries printed: 3 under x = 1, and 15 under x = 7. Verified — the missing outputs are 5, 11, 19, 25 and 41.
• The verification the chapter performs (p. 28). It states that (1, 3) and (7, 15) lie on the line, that membership of a line is exactly the condition that the coordinates satisfy its equation, and that (7, 15) can be confirmed by putting x = 7 and y = 15 into y = 2x + 1. Verified — 2 × 7 + 1 = 15, so the pair agrees.
• Why a linear rule cannot bend. Verified, and an added argument — take equal steps along x and the outputs step by equal amounts, which is the §2.3 property. Successive plotted points therefore rise by the same amount each time you move the same distance across. So the hop from one plotted point to the next is the same displacement wherever you are on the plot, which means the direction never changes — and a path that never changes direction is straight. Put the emphasis there, on the direction staying fixed. A path that turned by the same amount at each stage would not be straight at all; steady turning is what produces a circular arc or a polygon. Concretely on y = 2x + 1: at x = 0, 1, 2, 3 the outputs are 1, 3, 5, 7, so every unit right is two units up, everywhere. The chapter asserts that the plot is a straight line and does not argue it. This is the topic's best content and must be presented as reasoning offered, not quoted.
• Example 12 (pp. 28–29). Plot (–1, –3), (0, 0), (1, 3), (3, 9) and (4, 12) on graph paper, join (–1, –3) to (4, 12) with a ruler, and observe that all of them fall on one straight line. The chapter then asks the reader to guess the equation from the relation between the coordinates, and answers it: each y-coordinate is three times its x-coordinate, so y = 3x.
• Fig. 2.6 (p. 29). Read off the printed page: printed squared paper with the horizontal axis marked in twos from –12 to 20 and the vertical axis marked in twos from –4 to 12. All five points are labelled with their coordinate pairs. Note that the drawn line is not visible as a separate stroke in this figure — it is the plotted points that carry the example.
• Example 13 (p. 29). Plot (–3, 6), (–2, 4), (0, 0), (1, –2), (2, –4) and (3, –6), and join (–3, 6) to (3, –6) with a ruler. The chapter again asks the reader to guess the equation from the relation between the coordinates.
• Two things about Example 13 the teacher must know. First, the chapter does not print the equation for this one. Verified — the line is y = –2x, since each output is twice its input with the sign reversed. Second, the item lists and Fig. 2.7 labels six coordinate pairs, while the sentence that follows counts only five as lying on the line.
• Fig. 2.7 (p. 29). Read off the printed page: printed squared paper, the horizontal axis marked in twos from –12 to 12 and the vertical axis from –6 to 6. All six points are labelled — and, as in Fig. 2.6, no stroke is drawn joining them, although Example 13 tells the reader to lay a ruler from (−3, 6) to (3, −6). Both figures on this page give the points only; the line is the reader's to add.
• Example 12 and 13 as a matched pair. Verified, an added pairing — y = 3x climbs left to right and y = –2x falls; both pass through (0, 0); both are read off by comparing each y with its x. Presenting them together makes the sign the only thing that changed, which sets up the next topic. The chapter prints them consecutively without drawing the comparison.
• Example 14 (p. 29). Draw y = ½x, y = x and y = 2x by choosing suitable points. The printed hint suggests (0, 0) and (4, 2) for y = ½x, and asks the reader to verify that both lie on it. Verified — ½ × 4 = 2, so (4, 2) agrees. The choice of 4 rather than 1 or 3 is the whole lesson of the hint: with a fractional coefficient, pick an input the denominator divides.
• What the drawing has established, and what it has not. Verified — joining two points always produces a straight line, whatever rule they came from. So the straightness of the drawing is not evidence that the relationship is linear. The evidence is that further pairs, computed independently, also land on it — which is exactly what the p. 28 table and the (7, 15) substitution do.

Figures to have open

• A coordinate grid, drawn as a clean schematic rather than reproduced from the book's green printed graph paper, on which points can be plotted from computed coordinates and a line drawn and extended with arrowheads. This replaces Figs. 2.5, 2.6 and 2.7 (pp. 28–29) and carries almost the whole topic.
• A staircase overlay for section 4: equal horizontal steps with equal vertical rises, drawn against the line. Standard schematic, and it is the visual form of the argument the chapter does not make.
• A completable table of paired values that can be filled cell by cell and then cross-checked against plotted points. Standard schematic; the chapter's own table on p. 28 is the model.
• Example 13 must be drawn with six labelled points, matching the coordinate pairs the chapter lists rather than the count of five its sentence gives.

Where this sits in the book

• NCERT Ganita Manjari, Class 9, printed Chapter 2, "Introduction to Linear Polynomials", §2.6 "Visualising linear relationships", pp. 27–29. The section opens at the foot of p. 27; the two points, Fig. 2.5, the table and the on-the-line criterion are on p. 28; Example 12 begins at the foot of p. 28; Fig. 2.6, Example 13, Fig. 2.7 and Example 14 with its hint are all on p. 29.
• Chapter 1 of the same book supplies the plane, the axes and the ordered pair; see Two axes, an origin, and why the order of the pair matters and The four quadrants, and reading a point's signs off its position.
• Backward link inside the chapter: the constant-step property used in section 4 is established on p. 19 and developed on pp. 21–23.
• Forward link inside the chapter: Example 14's three lines are drawn in Fig. 2.8 on p. 30 and brought onto one set of axes in Fig. 2.9 on p. 31, where the chapter draws its conclusions about the coefficient. That material is What a and b do to the line: slope, y-intercept, and parallel families.
• Exercise Set 2.6, p. 36, is the drawing practice; End-of-Chapter starred items 7, 9, 10 and 13 on pp. 37–39 are the assessment forms.