The four quadrants, and reading a point's signs off its position

What to assume they know

• Two axes, an origin, and why the order of the pair matters — axes, origin, units, and why the pair is ordered
• Signed numbers, and what changing a sign does to a position on a line
• Perpendicular distance from a line
• Reading a plan to a stated scale, and area of a rectangle and of a trapezium
• Roman numerals I to IV

What they should be able to do

• Name the plane and its axes using the chapter's three printed names for it
• State the numbering of the four quadrants and the sign pattern that holds in each, and derive each pattern rather than recalling it
• Explain each coordinate as a perpendicular distance from the other axis, and say why that phrasing makes the sign pattern automatic
• Given a point's coordinates, name its quadrant; given a quadrant, write down a point in it
• Explain why the axes themselves belong to no quadrant
• Set up axes with a stated range and scale before plotting, and say why the range has to be chosen before the first point is marked
• Read coordinates off a scaled floor plan spanning more than one quadrant, and compute lengths, widths and a fourth corner from them
• Decide a physical question — whether a swinging door meets a wardrobe — by comparing coordinates

Where it usually goes wrong

• "There are four quadrants because someone drew four boxes." There are four because there are two independent sides to choose — left or right of one axis, above or below the other — and two choices of two options give four cases.
• "The sign pattern for each quadrant has to be learnt by heart." Read it off the position instead: is the point right of the y-axis, and is it above the x-axis? Two answers give the two signs, in that order, every time.
• "Quadrant numbering starts at the bottom left, like a table." It starts where both coordinates are positive and runs anticlockwise, as Fig. 1.4 letters it.
• "A point on an axis is in the nearest quadrant." The axes are the boundaries; a point on one is in no quadrant. This is why every quadrant statement in the chapter concerns points with both coordinates non-zero.
• "Negative coordinates only turn up in made-up textbook examples." They turn up in a bathroom. The origin was put in the corner of one room, and everything on the other side of that corner has to be described with negatives — which is why the exercise's y-axis has to run down to −15.
• "y is the distance from the y-axis." It is the other way round: the y-coordinate is measured from the x-axis. The name records which axis you read the number against, not which axis you measure from, and the chapter's perpendicular-distance sentence at p. 6 is where to fix this.
• "A larger coordinate means a point further from the origin." (−6, 9) is further from the origin than (8, 0) despite the smaller first number. Distance is the next module's business, and the pair alone does not order points by it.

Questions to check understanding

• Name the quadrant of a given point, and write a point in a named quadrant
• Given a point's coordinates, state its perpendicular distance from each axis
• Predict which sides of a quadrilateral are parallel or perpendicular from the coordinates of its vertices, then verify by plotting (end-of-chapter item 3, p. 12)
• Given a point's x-coordinate only, say which quadrants it could lie in and what else is needed to fix it (end-of-chapter item 2, p. 12)
• Complete a rectangle from three given vertices
• Read a floor plan spanning more than one quadrant and report a room's corners
• Decide a clearance question by comparing a computed length with a coordinate
• Plot vertices meeting stated conditions, one vertex per named quadrant (end-of-chapter item 8, p. 12)

Examples worth working on the board

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

• Fig. 1.4, the four quadrants (p. 6). Fine green ruling; the x-axis ticked from −8 to 7, the y-axis from −5 to 4, arrowheads on all four ends. The four regions are lettered Quadrant I (upper right), Quadrant II (upper left), Quadrant III (lower left) and Quadrant IV (lower right), so the numbering runs anticlockwise. Two points are marked with their coordinates printed: Q (−5, 3) and S (3, −5). The origin is labelled O (0, 0). Read on the printed page: only those two points and the origin are marked, and neither Q nor S is on an axis.
• The four sign cases as the chapter lists them (p. 6), stated so an explanation can derive rather than recite: the x-coordinate is positive exactly when the point is to the right of the y-axis, and the y-coordinate is positive exactly when the point is above the x-axis. Two independent yes/no decisions give 2 × 2 = four combinations, which is why there are four regions and not three or five. Verified: the count of quadrants and the count of sign patterns are the same number for that reason.
• The chapter's own worked reading (p. 7). S (3, −5) is placed in Quadrant IV with x-coordinate 3 and y-coordinate −5; Q (−5, 3) is placed in the second quadrant. The student is then told to redraw it, put those two points on, and add a point of their own in Quadrant I and another in Quadrant III, writing down the coordinates.
• The pair the chapter supplies without comment. Q (−5, 3) and S (3, −5) are the same two numbers in the two possible orders, and they sit in opposite quadrants. Use them for the sign-pattern demonstration; the observation is added here, not the chapter's claim.
• Think and Reflect (p. 7), four prompts: what value the first coordinate must take once a point is known to sit on the y-axis; whether the matching statement holds for the other axis; whether the reversed pair can ever land on the original point; and whether the equality condition on x and y is correctly stated. Verified: anything sitting on the y-axis has 0 in its first slot, anything on the x-axis has 0 in its second, and a reversed pair coincides with the original exactly when the two coordinates are equal.
• Exercise Set 1.2's set-up (p. 7). On a graph sheet, axes drawn with 1 cm = 1 unit, the x-axis marked from (−7, 0) to (13, 0) and the y-axis from (0, −15) to (0, 12). Verified: that is a sheet 20 cm wide and 27 cm tall — the exercise is choosing a range large enough to hold everything Fig. 1.5 shows, which is the practical reason to plan a range before plotting.
• Fig. 1.5, the home on the axes (p. 7). Printed coordinates on the figure: O (0, 0), A (12, 0), B (12, 10), C (0, 10), R₁ (11.5, 0). Lettered without coordinates: the bathroom corners P, R, F; the shower corners S, H, W, R; the wardrobe corners W₁–W₄; the bed corners S₁–S₄; the room door end D₁; the bathroom door ends B₁, B₂. Read off the figure's own ticks on the printed page: P (−6, 0), R (−6, 9), F (0, 9), S (−6, 6), H (−3, 6), W (−2, 9), D₁ (8, 0), W₁ (3, 0), W₂ (7, 0), W₃ (7, 2), W₄ (3, 2). The exercise text itself prints B₁ (0, 1.5) and B₂ (0, 4) (p. 5).
• Which quadrant each room is in. Verified from the readings above: the bedroom occupies the first quadrant (x from 0 to 12, y from 0 to 10); the bathroom lies to the left of the y-axis in the second quadrant (x from −6 to 0, y from 0 to 9); the dining room of item 4 lies below the x-axis, straddling the third and fourth quadrants. That is the answer to "why does this exercise need negative numbers at all", and it is worth stating.
• The bathroom's dimensions confirm the readings. Verified: 0 − (−6) = 6 across and 9 − 0 = 9 up, which is the 6 ft × 9 ft printed inside Fig. 1.1 (p. 3). Independently, item 4 states the dining room's length runs from P to A; with P (−6, 0) and A (12, 0) that is 18 ft, exactly the length the item prints. Two separate printed facts agree with the coordinates I read off, which is why those readings can be trusted.
• Item 1, the study table (p. 8). Three feet at (8, 9), (11, 9) and (11, 7). Verified: the fourth foot is at (8, 7); the table measures 11 − 8 = 3 by 9 − 7 = 2, so 3 ft × 2 ft, footprint 6 sq ft. The item also asks whether the spot is any good — it sits in the top right of the bedroom, clear of the bed and the wardrobe, and near the corner where Fig. 1.1 draws the potted plant — and then asks whether the table's height can be made out at all. It cannot: the plan records two measurements only.
• Item 2, the door swing (p. 8). The bathroom door is hinged at B₁ (0, 1.5) and swings inwards, into the bedroom; its leaf is 2.5 ft long, from the width computed in Exercise Set 1.1. The wardrobe occupies x from 3 to 7 and y from 0 to 2. Verified: the swinging tip stays within 2.5 ft of the hinge, so it never reaches x = 3 and the door clears the wardrobe by half a foot. The item then asks what changes if the door is made wider — verified: a leaf of 3 ft would just touch the wardrobe's near corner region and anything beyond 3 ft would sweep into it, so widening the door means moving the wardrobe or rehanging the door on its other edge.
• Item 3, the bathroom (p. 8). (i) The four corners, in the item's own order O, F, R, P — verified as (0, 0), (0, 9), (−6, 9), (−6, 0). (ii) What shape the showering area SHWR is, and the coordinates of its four corners — verified from the readings: S (−6, 6), H (−3, 6), W (−2, 9), R (−6, 9); SH and RW are both horizontal, of lengths 3 ft and 4 ft, and 3 ft apart, while SR is vertical and HW is slanted, so the region is a right trapezium of area ½ × (3 + 4) × 3 = 10.5 sq ft. (iii) The student must then mark off a 3 ft × 2 ft washbasin space and a 2 ft × 3 ft toilet space and give the corner coordinates. This part reads as open but is not: Fig. 1.5 already draws both fixtures inside the bathroom, the toilet against the left wall and the washbasin below it, so a student is placing the given rectangles over positions the figure has already fixed. The bathroom has 54 sq ft, of which the shower already takes 10.5.
• Item 4, the other rooms (p. 8). The dining room is 18 ft long and 15 ft wide, its length running from P to A. Verified: with P (−6, 0) and A (12, 0) the corners are (−6, 0), (12, 0), (12, −15) and (−6, −15), and its centre is at (3, −7.5). A 5 ft × 3 ft table placed exactly at that centre has feet at (0.5, −6), (5.5, −6), (5.5, −9) and (0.5, −9) if the 5 ft side runs along the x-axis — and at (1.5, −5), (4.5, −5), (4.5, −10), (1.5, −10) if it runs the other way. Both are correct; the item does not fix the orientation, and an explanation should show that the answer is a family, not a single quadruple.

Figures to have open

• Fig. 1.4 redrawn (p. 6): axes ticked −8 to 7 and −5 to 4, the four regions numbered, Q (−5, 3) and S (3, −5) marked. The chapter's own figure.
• Fig. 1.5 redrawn (p. 7): the bedroom in the first quadrant, the bathroom in the second with the shower's slanted boundary, the wardrobe, the bed, both doorways, and every letter the exercise refers to — O, A, B, C, P, R, F, S, H, W, W₁–W₄, B₁, B₂, D₁, R₁. The chapter's own figure, and the whole exercise set is unanswerable without it. Redraw to scale on a visible grid.
• An extension of the same plan downwards to show the dining room in the third and fourth quadrants. This must be drawn: the chapter asks the student to sketch it and prints no such figure.
• A 2 × 2 decision table for the sign patterns. Standard schematic.
• A quarter-circle swing overlay for the door, with the wardrobe edge marked at x = 3. Standard schematic.

Where this sits in the book

• NCERT Ganita Manjari Class 9 (Part I), printed Chapter 1, §1.3, pp. 6–7, with Fig. 1.4 (p. 6) and the Think and Reflect box (p. 7)
• Exercise Set 1.2, pp. 7–8, with Fig. 1.5 (p. 7); items 1–4 on p. 8
• The printed dimensions the coordinate readings are checked against are inside Fig. 1.1 (p. 3); B₁ and B₂ are printed in Exercise Set 1.1 (p. 5)
• End-of-Chapter Exercises items 2, 3 and 8 (p. 12) test this material
• The Chapter Summary restates the four sign patterns and the axis forms (p. 15)