Steepness as the tangent of an angle, and the one line that has none

What to assume they know

• Plotting points in the plane, and the four coordinate results §9.1 recalls: distance between two points, the section formula, the mid-point as its equal-ratio case, and the coordinate expression for the area of a triangle
• The tangent function on the interval from zero to a straight angle, from Chapter 3 — in particular that it is positive on acute angles, negative on obtuse ones, zero at both ends, and has no value at a right angle
• Right triangles and the tangent of an acute angle as a ratio of two sides
• Signed differences: that a coordinate difference carries direction as well as size
• Measuring an angle anticlockwise from a fixed ray, and the convention that fixes its sign

What they should be able to do

• State what the inclination of a line is, and give the closed range it lies in
• Explain why a line meets the x-axis in two angles and say which one the chapter keeps as the inclination
• State the chapter's definition of slope in your own words, and name the single inclination for which it delivers nothing
• Give the slope of the x-axis and explain why the y-axis has none
• Rebuild the acute-inclination argument from the right triangle drawn in Fig 9.3(i)
• Rebuild the obtuse-inclination argument and say precisely where the minus sign is introduced and where it is absorbed
• Show that exchanging the two points leaves the quotient unaltered, and say why
• Compute a slope from two given points, recognising both degenerate outcomes
• Convert an inclination given in degrees into a slope, and read an inclination back off a slope
• Decide, from the sign of a slope alone, whether the inclination is acute or obtuse

Where it usually goes wrong

• "Slope is defined as rise over run." Not here. Rise over run is a theorem in this chapter, proved twice over in §9.2.1. If a student cannot say what slope is without drawing two points, they have the by-product and not the definition.
• "A vertical line has infinite slope." It has none. Infinity is not a value of the tangent function; the tangent simply has no value at a right angle. Saying "infinite" invites students to compare it with other slopes, and nothing follows from that comparison.
• "The inclination is whichever angle looks smaller." Fig 9.2 marks both angles precisely so this can be refused. The inclination is the anticlockwise one from the positive x-direction; for a falling line that is the obtuse one, and its tangent is correspondingly negative.
• "A negative slope means the line is below the axis." It means the inclination is obtuse — the line falls as you read left to right. Where it sits relative to the axes is a separate question the slope does not answer.
• "You must take the points in left-to-right order." You need not. Both differences flip together, and the quotient survives. Making students verify this on Example 1(a) is cheaper than policing their subtraction.
• "The obtuse case needs a different formula." It needs a different argument — that is why §9.2.1 splits into two cases — and produces the same formula. Skipping case 2 leaves students with a rule proved only for lines that rise.
• "Zero slope and no slope are two ways of saying the same thing." Example 1(b) and 1(c) are printed adjacently to stop exactly this. One has a numerator of zero, the other a denominator of zero, and only one of them is a number.
• "Area zero just means the triangle is small." The Remark on p. 152 reads it the other way: a vanishing area means no triangle was ever there, so the three points lie on one line. That reading is what makes the area formula useful later in the chapter.

Questions to check understanding

• Given two points, compute the slope, and state which of the two degenerate cases applies when it does
• Given an inclination in degrees, write the slope; given a slope, name the inclination
• Given an angle measured from the positive y-direction, convert it to an inclination before computing the slope
• Decide whether three given points are collinear, by slopes or by the area result, and say which method you used
• State the range of the inclination and explain why the endpoints behave differently from the interior
• Explain in words why a vertical line has no slope, without using the word "infinity"
• Reproduce either case of the §9.2.1 argument from a labelled figure

Examples worth working on the board

Inputs only. Values marked verified are worked out here on data printed inside pp. 151–156.

• Fig 9.1 (§9.1, p. 151). A recall diagram: the points (6, −4) and (3, 0) plotted on one pair of axes, with dashed guide lines dropped from (6, −4) to both axes. Read off the printed page: (3, 0) is drawn sitting on the positive x-axis, and the label for (6, −4) sits to the right of the point in the fourth quadrant. Use it to make the point that one of the two coordinates being zero is a statement about distance from an axis, not a defect.
• The four recalled results (§9.1, pp. 151–152), each with the chapter's own illustration attached:
  • distance between (6, −4) and (3, 0). Verified: the two differences are −3 and 4, so the distance is 5.
  • the section formula, illustrated on A(1, −3) and B(−3, 9) cut internally in the ratio 1 : 3. Verified: the dividing point is the origin — both coordinates come out zero, which makes it a memorable check.
  • the mid-point rule, presented as the section formula with the two parts equal.
  • the area of the triangle on (4, 4), (3, −2) and (−3, 16). Verified: the bracket evaluates to −54, and the area is 27. Since 27 is not zero, these three points are not collinear — worth saying aloud, because the Remark immediately afterwards turns a zero area into a collinearity test and students read the example and the Remark as one thing.
• Fig 9.2 (§9.2, pp. 152–153). One line through the origin region crossing the x-axis, with both angles marked at the crossing: θ on the right of the line and the supplementary angle on the left. This is the figure the whole section hangs on — because it shows there are two candidate angles and the anticlockwise rule is what makes "the" angle well defined.
• The two ends of the range (§9.2, p. 153). A line parallel to the x-axis, or the x-axis itself, has inclination zero; a line parallel to the y-axis has a right angle. Verified consequence: the slope runs over every real number as the inclination sweeps the open interval below a right angle and the open interval above it, and it takes no value at the one angle in between — so "every real number is some line's slope, and one line's slope is no real number" is exactly true.
• Fig 9.3(i) (§9.2.1, p. 153). P(x₁, y₁) and Q(x₂, y₂) on a line of acute inclination, with Q above and right of P. QR is dropped perpendicular to the x-axis at R, and PM is drawn perpendicular to RQ, so M is the corner of a right triangle MPQ. Read off the printed page: the angle at P inside that triangle is marked θ, the same letter as the inclination marked at the crossing with the x-axis, and both right angles are marked with squares. Verified: MQ is y₂ − y₁ and MP is x₂ − x₁, and the equality of the two marked angles is the corresponding-angles fact for the x-axis and the parallel line PM.
• Fig 9.3(ii) (§9.2.1, p. 154). The same construction with the line now falling to the right: Q sits up and to the left of P, M is on the left, R is below M on the axis, and a right-angle square sits at each of M and R. Only one arc is drawn at the x-axis crossing and it carries θ. The supplementary quantity is lettered up at P, where it names the interior angle of the right triangle — the angle the page then works with. A redraw must keep it on that vertex; sliding it down beside the crossing severs the very correspondence the proof runs on. Verified: here MQ is y₂ − y₁ and MP is x₁ − x₂, the tangent of the angle at P is their quotient, and the minus sign carried in from the supplement cancels against the reversed denominator, so the printed expression is the same one as in case 1.
• The order-swap check. Verified: replacing (x₁, y₁) and (x₂, y₂) by each other negates numerator and denominator together, so the slope is unchanged. The chapter's Summary on p. 174 prints both orderings side by side, which is the same observation.
• Example 1 (§9.2.2 running on to §9.2.3, pp. 155–156), four parts, all sharing the point (3, −2):
  • (a) with (−1, 4). Verified: differences 6 and −4, slope −3/2.
  • (b) with (7, −2). Verified: numerator 0, denominator 4, slope 0 — a line parallel to the x-axis.
  • (c) with (3, 4). Verified: numerator 6, denominator 0 — no slope. Note the two points differ in y and agree in x, which is the vertical case.
  • (d) an inclination of 60°. Verified: the slope is √3, about 1.732. Handing all four over together is the point: one anchor point, four outcomes, including both degenerate ones.
• Exercise 9.1 items that belong to this topic, with their data intact:
  • q3 asks how far apart P(x₁, y₁) and Q(x₂, y₂) are, given that the segment PQ runs parallel to each axis in turn. Verified: the size of the y-difference, then the size of the x-difference.
  • q5: the line joining the origin to the mid-point of the segment on P(0, −4) and B(8, 0). Verified: the mid-point is (4, −2) and the slope is −1/2.
  • q7: a line making 30° with the positive y-direction, measured anticlockwise. Verified: the inclination is 120° and the slope is −√3. This is the item that tests whether a student has understood that the inclination is measured from one named ray and not from whichever axis is nearer.
  • q9: the angle made with the x-axis by the line through (3, −1) and (4, −2). Verified: the slope is −1, so the inclination is 135° and the acute angle at the crossing is 45°.
  • q11: a line through (x₁, y₁) and (h, k) with slope m — a one-line consequence of §9.2.1 that is really the point-slope form arriving early.
  • q1 (vertices (−4, 5), (0, 7), (5, −5), (−4, −2)) and q2 (an equilateral triangle of side 2a with its base along the y-axis, centred at the origin) exercise the §9.1 recall rather than slope. Verified for q2: the base runs from (0, a) to (0, −a) and the apex is at (±a√3, 0), so there are two admissible answers.

Figures to have open

• Fig 9.2 redrawn, with both angles at the x-axis crossing marked and only one of them named as the inclination. The chapter's own figure; section 3 cannot be taught without both marks present.
• Fig 9.3(i) and Fig 9.3(ii) redrawn as a matched pair sharing one layout, so the only visible difference is the direction the line runs and the position of M. These are the chapter's own figures and carry the whole of §9.2.1.
• A tangent graph over the range from zero to a straight angle, with the vertical break at the right angle drawn and labelled. Standard schematic; it is what makes "no slope" look inevitable rather than arbitrary.
• Example 1's four lines on one pair of axes, all through (3, −2). Standard schematic built from the chapter's data.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 9 "Straight Lines", §9.1 Introduction, pp. 151–152 — Descartes, the name analytical geometry, Fig 9.1, the four recalled results and the collinearity Remark
• §9.2 Slope of a Line, pp. 152–153 — the two supplementary angles, the inclination and its range, Fig 9.2, Definition 1, the refused right angle, and the slopes of the two axes
• §9.2.1 Slope of a line when coordinates of any two points on the line are given, pp. 153–154 — Figs 9.3(i) and 9.3(ii) and the two-case proof
• Example 1, pp. 155–156, sits inside the run of §9.2.2 and is the first worked use of both the definition and the quotient
• Exercise 9.1, pp. 158–159, questions 1, 2, 3, 5, 7, 9 and 11
• The chapter Summary, p. 174, restates the quotient in both point orders and repeats the two axis cases