Distance when the segment is parallel to an axis

What to assume they know

• Two axes, an origin, and why the order of the pair matters — axes, origin, ordered pairs
• The four quadrants, and reading a point's signs off its position — quadrants, and reading coordinates off a plan
• Subtraction across zero and with negative numbers
• The idea that a length is a non-negative quantity
• Squares and square roots of small whole numbers

What they should be able to do

• Recognise from a pair of coordinates alone whether a segment is parallel to an axis, lies on an axis, or is slanted
• Compute the length of a segment parallel to the x-axis as the size of the difference of the x-coordinates, and the y-parallel case correspondingly
• Explain why the order of subtraction does not change the length, and why the result is written with absolute-value bars
• Read the four sides of a rectangle straight off its corner coordinates, and compute its perimeter and area
• Compute widths and gaps from figures in the chapter — a doorway, a wardrobe, a room wall — and check the results against the dimensions printed in the plan
• State how this case sits inside the general distance formula, and what happens to the formula when one difference is zero

Where it usually goes wrong

• "A distance always needs the square-root formula." When one difference is zero the square root undoes the square and leaves a single subtraction. Using the full formula is not wrong, just slower — and running it without noticing the zero is how students end up computing √16 instead of writing 4.
• "You can subtract in whichever order and keep the sign you get." You can subtract in either order; you may not keep a negative answer as a length. Both subtractions have their place — the signed shift matters when direction is the point, the size matters when a length is.
• "A length can come out negative." Nothing measured by a ruler can. The bars in the summary are there to say so in symbols.
• "Parallel to the x-axis means lying on the x-axis." The bed's front edge sits four feet above it and behaves identically. What makes the rule work is the shared coordinate, not the location.
• "You have to count the squares on the grid." Counting is a check, not the method, and it fails as soon as a coordinate is 1.5 or −2.9. Subtract instead.
• "If both coordinates are negative the subtraction changes." It does not: 0 − (−6) is 6 in exactly the way 7 − 3 is 4. The chapter has you do one of these in the bathroom and another when it reflects a triangle into negative territory at p. 11.
• "Because the axes have different names, horizontal and vertical distances are different kinds of thing." Both are read off the same unit, which was marked equally on both axes back at §1.3 — that is exactly what makes a single formula possible.

Questions to check understanding

• Given two coordinate pairs, decide whether the segment is horizontal, vertical or slanted, and give its length when it is one of the first two
• Compute the perimeter and area of a rectangle from its four vertices
• Find the fourth vertex of a rectangle from three, then give all four side lengths
• Given one endpoint and a length, find the possible positions of the other endpoint on a line parallel to an axis (two answers)
• Identify from a list of vertex pairs which sides of a figure are parallel to an axis (end-of-chapter item 3(ii), p. 12)
• Explain why a computed difference of −4 and a length of 4 are not in conflict
• Given a point's x-coordinate and a line through it parallel to the y-axis, say what all the points on that line have in common (end-of-chapter item 2, p. 12)

Examples worth working on the board

Values marked verified are worked out here on the printed inputs. The chapter prints no answers.

• How §1.4 opens (p. 8). The section begins by taking this case as already known — points on an axis, or a segment parallel to an axis — and names two examples in Fig. 1.5, W₁W₂ and S₁S₂, before turning to the slanted case. So the exposition of the easy case is a single sentence and two labels; the content of this topic is what that sentence is relying on.
• W₁W₂, the wardrobe's front edge (Fig. 1.5, p. 7). Read off the figure's ticks: W₁ (3, 0) and W₂ (7, 0). Verified: both y-coordinates are 0, so the segment lies along the x-axis; its length is 7 − 3 = 4 ft, which is exactly the 4 ft printed as the wardrobe's width inside Fig. 1.1 (p. 3). The matching back corners W₄ (3, 2) and W₃ (7, 2) give the same 4 ft along the line y = 2, and W₁ to W₄ gives 2 − 0 = 2 ft — together, the 4 ft × 2 ft the plan prints.
• S₁S₂, the bed's front edge (Fig. 1.5, p. 7). Both corners sit at y = 5, so the segment is parallel to the x-axis without lying on it, and its length is the difference of the two x-coordinates. Use this one for the method only. Neither endpoint carries a printed x-coordinate, and neither drawn corner lands on a labelled tick: S₁ stands about half a unit clear of the y-axis, with a hatched headboard strip drawn in the gap, and S₂ lies between +6 and +7. So no number should be spoken for S₁S₂. That is itself the teaching point: the procedure is fixed by the two coordinates agreeing in y, whatever the figure turns out to say.
• Reiaan's room as a rectangle (Fig. 1.3, p. 5; the same corners recur in Fig. 1.5). Printed coordinates: O (0, 0), A (12, 0), B (12, 10), C (0, 10). Verified: OA = 12 − 0 = 12 ft along the x-axis; AB = 10 − 0 = 10 ft along the line x = 12; BC = 12 − 0 = 12 ft along the line y = 10; CO = 10 ft along the y-axis. Perimeter 44 ft, area 120 sq ft, matching the 12 ft × 10 ft printed in Fig. 1.1 (p. 3). Notice which coordinate is shared in each of the four cases — that is the whole calculation.
• The two doorway widths. Verified: the bathroom door's ends are printed as B₁ (0, 1.5) and B₂ (0, 4) (p. 5), sharing x = 0, so the width is 4 − 1.5 = 2.5 ft. The room door runs from D₁ (8, 0) — read off the ticks — to the printed R₁ (11.5, 0), sharing y = 0, so its width is 11.5 − 8 = 3.5 ft. One vertical case and one horizontal case, from the same plan.
• Subtracting the other way round. Verified: 3 − 7 = −4 while 7 − 3 = 4, and the wardrobe is 4 ft wide either way. Show both subtractions and then the bars: the chapter's summary states the horizontal case as the absolute value of the difference of the two x-coordinates, and the vertical case as the absolute value of the difference of the two y-coordinates (p. 15). The running text of §1.4 makes the same point without the bars, saying at p. 11 that the sign of a shift along an axis makes no difference to what is being measured.
• The legs of the coming triangle (Figs. 1.6–1.7, pp. 9–10). The chapter's own next step computes CD = 7 − 3 = 4 and AC = 4 − 1 = 3 for A (3, 4), D (7, 1) and C (3, 1). Verified: C shares its x-coordinate with A and its y-coordinate with D, so AC and CD are precisely the two cases of this topic — which is why the general formula needs no new measuring idea, only this one twice over.
• Crossing zero. Verified from the same plan: the bathroom's bottom wall runs from P (−6, 0) to O (0, 0), a length of 0 − (−6) = 6 ft, while P to A (12, 0) spans 12 − (−6) = 18 ft — that longer span is not a wall of the bathroom at all but the whole southern boundary of bathroom and bedroom together, which is why it doubles as the dining-room length the exercise prints at p. 8. A subtraction across zero is the case students get wrong; the chapter supplies a real one.
• What the general formula does here. Verified: with y shared, the second difference is 0, so the formula of p. 10 returns the square root of the first difference squared — the size of that difference and nothing else. So there is one rule in the chapter, not three.

Figures to have open

• The room rectangle with corners O (0, 0), A (12, 0), B (12, 10), C (0, 10) and all four side lengths derived. Redrawn from Fig. 1.3 (p. 5).
• A detail of the wardrobe with W₁–W₄ and their coordinates, and of the bathroom door with B₁ and B₂. Redrawn from Figs. 1.3 and 1.5.
• A horizontal line drawn above the x-axis with tick marks aligned to the axis below it, to make the "same ruler" claim visible. Standard schematic; the chapter prints nothing like it and section 3 depends on it.
• The A–C–D corner of Fig. 1.7 (p. 10) with the two legs labelled 3 and 4, used only as the closing pointer. The chapter's own figure.
• No photograph is needed.

Where this sits in the book

• NCERT Ganita Manjari Class 9 (Part I), printed Chapter 1, §1.4 "Distance Between Two Points in the 2-D Plane", opening paragraph, p. 8
• The two named examples W₁W₂ and S₁S₂ are lettered in Fig. 1.5 (p. 7); the room corners and the two doorways are in Fig. 1.3 (p. 5) and Exercise Set 1.1 (p. 5)
• The absolute-value statements are the two Chapter Summary bullets on p. 15; the sign-of-the-shift remark is in the running text at p. 11
• The two legs computed as coordinate differences are on p. 9, with Fig. 1.7 on p. 10
• End-of-Chapter Exercises items 2 and 3 (p. 12) test the axis-parallel case directly