Mathematics • Year 8 • Unit 2 • Lesson 1
Cartesian Plane — Mixed Challenge
Pull together everything from Lesson 1: ordered pairs, axes, the origin, sign patterns and quadrants. Six mixed problems, one "find the mistake", and one open-ended puzzle about rectangles on the plane.
1. Mixed problems — choose the right move
Each question uses a different combination of ideas from Lesson 1. Decide what's being asked BEFORE you write. Show your reasoning. 3 marks each
1.1 State the quadrant (or axis) for each point: A(7, 2), B(−4, −1), C(0, −6), D(−3, 5).
1.2 A point R has coordinates (−6, 0). Describe in words exactly where R sits relative to the origin, and state which axis it lies on.
1.3 Points P(2, 5) and Q(2, −5) are reflections of each other across one axis. State which axis, and explain in one sentence why.
1.4 A point S(a, b) has a < 0 and b > 0. State its quadrant, and give one example of possible whole-number coordinates for S.
1.5 A point T is 5 units below the origin and 3 units to the right. Write T's coordinates, state its quadrant, and write the coordinates of the point you get if you reflect T across the x-axis.
1.6 The four points (3, 0), (0, 3), (−3, 0) and (0, −3) form a square whose corners all sit on the axes. Write down (i) which axis each point is on, and (ii) the coordinates of the centre of the square.
2. Find the mistake
Another student tried to identify the quadrant of the point W(−4, −7). Their working is below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — name the quadrant of W(−4, −7):
Line 1: x = −4, y = −7.
Line 2: x is negative, y is negative, so sign pattern is (−, −).
Line 3: Quadrants go CLOCKWISE from the top right: I, II, III, IV.
Line 4: So (−, −) is the bottom right → Quadrant IV.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? Revisit lesson § "Quadrant numbering". The numbering direction is the key thing to check.3. Open-ended challenge — find the missing corner
This question has more than one valid answer. 4 marks
3.1 Three corners of a rectangle on the Cartesian plane are A(2, 3), B(−2, 3) and C(2, −3). A fourth point D is to be added so that ABCD is a rectangle.
(a) Find the coordinates of D. Explain how you used the coordinates of A, B and C to work it out.
(b) State the quadrant of each of the four corners A, B, C and D.
(c) The rectangle is "centred on the origin" — explain in one sentence why this is true. (Hint: think about what's halfway between the corners.)
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Quadrants
A(7, 2): (+, +) → Quadrant I. B(−4, −1): (−, −) → Quadrant III. C(0, −6): x = 0 → y-axis (no quadrant). D(−3, 5): (−, +) → Quadrant II.
1.2 — R(−6, 0)
R sits 6 units to the left of the origin, with no vertical movement (y = 0). It lies on the x-axis. Because one coordinate is 0, R is not in any quadrant.
1.3 — P(2, 5) and Q(2, −5)
Same x (= 2), opposite y values. That makes them a mirror pair across the x-axis: reflecting P down across y = 0 lands you exactly on Q.
1.4 — S(a, b) with a < 0, b > 0
Pattern (−, +) → Quadrant II. Example: S(−4, 7) (any negative x and positive y is valid).
1.5 — T 5 below, 3 right
3 right → x = 3; 5 below → y = −5. So T(3, −5), pattern (+, −) → Quadrant IV. Reflecting T across the x-axis flips the sign of y only: image is (3, 5), which is in Quadrant I.
1.6 — Square on the axes
(3, 0) is on the x-axis (right). (0, 3) is on the y-axis (up). (−3, 0) is on the x-axis (left). (0, −3) is on the y-axis (down). All four corners are equally distant from the origin, so the centre of the square is at the origin (0, 0).
2 — Find the mistake
(a) The mistake is on Line 3 (and the wrong answer is then carried into Line 4).
(b) Quadrants are numbered anticlockwise, not clockwise. Going anticlockwise from the top right: I (top right) → II (top left) → III (bottom left) → IV (bottom right).
(c) Corrected working:
x = −4, y = −7.
Sign pattern (−, −) puts the point in the bottom-left region.
Bottom-left is Quadrant III (not IV).
So W(−4, −7) is in Quadrant III.
3 — Missing corner of a rectangle
(a) A(2, 3) and B(−2, 3) share y = 3, so AB is a horizontal side. A(2, 3) and C(2, −3) share x = 2, so AC is a vertical side. D must match B's x (= −2) and C's y (= −3), so D = (−2, −3).
(b) A(2, 3) → Quadrant I. B(−2, 3) → Quadrant II. D(−2, −3) → Quadrant III. C(2, −3) → Quadrant IV. (One corner in each quadrant.)
(c) Each pair of opposite corners is the mirror image of the other across the origin (e.g. A(2, 3) ↔ D(−2, −3)), so the centre of the rectangle is exactly halfway between them — the origin (0, 0).
Marking: 2 marks for D = (−2, −3) with a clear reason; 1 mark for naming all four quadrants; 1 mark for explaining the rectangle is centred on the origin.