Abscissa and ordinate, and what a coordinate pair actually records

What to assume they know

• Plotting a point on squared paper from an ordered pair, and reading a pair back off a plotted point (Class IX coordinate geometry, which §7.1 names as the starting point)
• Negative numbers on a number line, and subtracting a larger value from a smaller one
• A linear equation in two variables has a graph, and that graph is a straight line
• Square roots of small whole numbers, and the idea that a square root of a non-negative quantity can be taken as non-negative
• Perpendicular lines, and dropping a perpendicular from a point to a line

What they should be able to do

• Say which axis each of the two coordinates is measured from, and use the printed names abscissa and ordinate correctly
• Explain why a point sitting on an axis has one coordinate equal to zero, and write the two families (x, 0) and (0, y) from that reasoning rather than from memory
• Plot a listed set of pairs in order, join them as directed, and describe the figure that results
• Argue that an ordered pair fixes exactly one point of the plane, and that swapping the two entries generally names a different point
• Read a separation directly off the coordinates when both points sit on one axis, and justify the subtraction by treating that axis as a number line
• Identify which quadrant a pair falls in from the signs of its two entries
• State what coordinate geometry is for, in the two directions the chapter claims — geometry handled by algebra, and algebra pictured as geometry

Where it usually goes wrong

• "The x-coordinate is the distance from the x-axis." This is the single most common slip and the chapter's own sentence is built to prevent it. The first entry is measured across to the vertical axis. Teach the crossing explicitly rather than hoping the naming carries it.
• "(3, 5) and (5, 3) are the same point." They are not, and the play in §7.1 is a cheap way to show it: swap any single pair in the list and the drawing breaks. A pair is ordered.
• "A coordinate is a distance, so it cannot be negative." The chapter introduces the idea with the word distance and then, two pages later, plots a point with two negative entries. Say plainly that the word is being used loosely at the start and that the working object is signed.
• "A point sitting on the x-axis has no y-coordinate." It has one; it is zero. The difference matters as soon as a formula asks you to substitute.
• "Coordinate geometry is a way of drawing graphs." The chapter claims traffic in both directions — figures analysed by algebra, and algebra understood through figures.
• "You can always subtract coordinates to get a separation." True only when the two points share an axis or a gridline. AC and BD in Fig. 7.2 are the counter-examples the chapter plants immediately.

Questions to check understanding

• Name the abscissa and the ordinate of a stated pair, and say which axis each was measured from
• Write down the general form of a point lying on a named axis, and give an instance
• Given a plotted point, read off its pair; given a pair, plot it
• Say which quadrant a pair falls in from its signs, and give a pair in a stated quadrant
• Plot and join a listed set of pairs and identify the resulting figure
• Compute a separation between two points that share an axis, by subtraction, and say why the method does not extend
• Decide whether two given pairs name the same point, and justify

Examples worth working on the board

Values marked verified are worked out here on data printed inside pp. 99–112; the chapter prints no answers for its exercises.

• The two families of axis points (§7.1, p. 99). The chapter records that a point sitting on the horizontal axis has the shape (x, 0), and one on the vertical axis the shape (0, y). An axis point is at zero remove from its own axis, so the reading taken against the other axis is what survives.
• The join-the-dots play (§7.1, p. 99). The full list, in the order the page gives it. Outline, joined in sequence and closed back to the start: A(4, 8), B(3, 9), C(3, 8), D(1, 6), E(1, 5), F(3, 3), G(6, 3), H(8, 5), I(8, 6), J(6, 8), K(6, 9), L(5, 8), back to A. Then three separate triangles: P(3.5, 7), Q(3, 6), R(4, 6); X(5.5, 7), Y(5, 6), Z(6, 6); and S(4, 5), T(4.5, 4), U(5, 5). Finally four straight strokes: S to (0, 5) and to (0, 6); U to (9, 5) and to (9, 6). Verified, and this is an added reconstruction rather than anything the page states: every one of those points maps onto another point of the list under the reflection x → 9 − x. A pairs with L, B with K, C with J, D with I, E with H, F with G; the two three-point triangles swap with each other; S swaps with U and T sits on the mirror line itself. So the drawing is exactly symmetric about the vertical line x = 4.5. Read against that symmetry it is a face: two spikes rising to height 9 as ears, the two small triangles at height 6–7 as eyes, the downward triangle at height 4–5 as a nose, and the four rays running out to the left and right edges as whiskers. The page asks what the picture is and does not answer.
• Distances read straight off one axis (§7.2, Fig. 7.2, p. 100). Given A(4, 0) and B(6, 0), the page reads OA = 4 and OB = 6 from the drawing and gets AB by subtraction. Given C(0, 3) and D(0, 8) on the vertical axis, the same move gives CD. Verified: AB = 2 units and CD = 5 units. Fig. 7.2 shows all four of these points on one pair of axes, and the two axes are not ruled alike — the vertical one is numbered 1 to 8, the horizontal one only 1 to 6, with the labels (4, 0) and (6, 0) printed under the horizontal axis and C(0, 3) and D(0, 8) printed beside the vertical one.
• The first genuinely two-dimensional case, kept for the next topic (§7.2, p. 100). From the same figure the page also takes A across to C, using OA = 4 and OC = 3, and gets 5; and B across to D, getting 10. Verified: AC = 5 and BD = 10. These two are the hinge — they are the first separations in the chapter that cannot be got by subtracting one coordinate from another.
• Signs and quadrants. The chapter's own later figures supply the counterweight to the word distance: Fig. 7.4 (p. 101) plots Q(−5, −3) with both entries negative, and Exercise 7.1 question 1 (iii) (p. 105) sets the pair (a, b) against (−a, −b). Hand both over as evidence that a coordinate carries a direction.
• What the machinery is for (§7.1, p. 99). Two pointers backwards: a linear equation in two variables, with its two leading coefficients not both zero, draws as a straight line; and the quadratic graph met in Chapter 2, with its leading coefficient non-zero, draws as a parabola. The page also lists fields where the subject gets used — art, navigation, seismology, engineering and physics.

Figures to have open

• A schematic of the chapter's Fig. 7.2 (p. 100): axes ruled to about 8, with A(4, 0) and B(6, 0) on the horizontal axis and C(0, 3) and D(0, 8) on the vertical one, and the segments AC and BD drawn in. This is the chapter's own figure and the whole of sections 9 and 10 hangs on the four points sharing one pair of axes; redraw it rather than reproducing the printed art.
• The plotted result of the §7.1 play. Must be drawn from the printed list of pairs, not sourced elsewhere, because the point of the section is that the picture is a consequence of the coordinates.
• A single-point callout with two perpendicular measurement arrows, each labelled with the axis it terminates on. Standard schematic.
• A four-quadrant frame with the sign pattern of each region marked, so section 8 has somewhere to put Q(−5, −3). Standard schematic.
• No photograph is needed anywhere in this topic.

Where this sits in the book

• NCERT Mathematics, Class X, Chapter 7 "Coordinate Geometry", §7.1 "Introduction", p. 99 — the Class IX recap, the naming of abscissa and ordinate, the two axis families, the join-the-dots play, the backward pointers to the linear graph and to the parabola of Chapter 2, and the list of fields of application.
• §7.2 "Distance Formula", p. 100, Fig. 7.2 — the four axis points and the two cross-axis separations that this topic sets up and hands on.
• Forward pointer inside the same chapter: Fig. 7.4, p. 101, is where negative coordinates first appear, and it is the evidence section 8 needs.
• Forward pointer: Exercise 7.1 question 1 (iii), p. 105, sets (a, b) against (−a, −b) — the algebraic form of the same point about signs.
• Backward pointer the chapter makes itself: the quadratic graph of Chapter 2.