Checkpoint 6 Maths · Geometry and Measure · 6Gp.01 · 6Gp.02

Coordinates in Four Quadrants

Read and plot coordinates with whole numbers, fractions and decimals in all four quadrants, and use shape properties to plot lines and complete shapes.

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In the curriculum

Read and plot coordinates, including integers, fractions and decimals, in all four quadrants (using a grid). (6Gp.01)
Use what you know about 2D shapes and coordinates to plot points to make lines and shapes in all four quadrants. (6Gp.02)
This means: read and write coordinates like (−3, 2) or (1.5, −0.5), plot them accurately, know the four quadrants, and use the properties of squares, rectangles, parallelograms and triangles to find missing vertices and points on lines.

1. Key words

Word / phrase	Meaning / use
coordinates	الإحداثيات — a pair of numbers (x, y) that gives the position of a point
x-axis	المحور السيني (الأفقي) — the horizontal number line
y-axis	المحور الصادي (الرأسي) — the vertical number line
origin	نقطة الأصل — the point (0, 0) where the axes cross
quadrant	ربع — one of the four parts the axes divide the grid into
plot	يحدد / يرسم نقطة — mark a point accurately on the grid (with a small cross)
vertex (vertices)	رأس (رؤوس) — a corner of a 2D shape
horizontal / vertical	أفقي / رأسي
parallel	متوازي — lines that stay the same distance apart and never meet
midpoint	نقطة المنتصف — the point exactly halfway between two points

2. Reading and writing coordinates

A coordinate grid has two number lines: the x-axis (across) and the y-axis (up and down). They cross at the origin (0, 0). Both axes have negative numbers: on the x-axis, negative numbers are to the left of 0; on the y-axis, negative numbers are below 0.

Coordinates are written in brackets with a comma: (x, y). The x-coordinate (across) always comes first, the y-coordinate (up or down) second. A good way to remember: "along the corridor, then up (or down) the stairs".

To reach A, go 3 along to the right, then 2 up: A is (3, 2). B is (−4, 3), C is (−2, −5), D is (5, −3), E is (0, 4) and F is (−3, 0).

Start at the origin (0, 0).
Go across to the point: right is positive, left is negative. This is the x-coordinate.
Then go up or down: up is positive, down is negative. This is the y-coordinate.
Write (x, y) with brackets and a comma, for example (−2, −5).

Points on the axes. If a point is on the y-axis, you do not move across, so its x-coordinate is 0: E is (0, 4). If a point is on the x-axis, its y-coordinate is 0: F is (−3, 0).

افتكرها كده

الإحداثيات بتتكتب (x, y): الأول x (يمين أو شمال) وبعدين y (فوق أو تحت).
يمين وفوق = موجب. شمال وتحت = سالب.
"امشي في الطرقة الأول، وبعدين اطلع السلم" — يعني أفقي الأول وبعدين رأسي.

3. The four quadrants

The axes divide the grid into four quadrants. They are numbered anticlockwise, starting at the top right. The signs of the coordinates tell you which quadrant a point is in, without drawing it.

The four quadrants and the signs of (x, y) in each one.

Quadrant	x is…	y is…	Example
1st (top right)	positive	positive	(4, 2)
2nd (top left)	negative	positive	(−4, 2)
3rd (bottom left)	negative	negative	(−4, −2)
4th (bottom right)	positive	negative	(4, −2)
on an axis	0 (on the y-axis)	0 (on the x-axis)	(0, −3) or (5, 0): not in any quadrant

4. Fractions and decimals

Points do not have to be where grid lines cross at whole numbers. A point halfway between 1 and 2 has the coordinate 1.5 (or 1½). Halfway between 0 and −1 is −0.5 (or −½). Work out what each small square is worth before you read or plot: count the squares between two labelled numbers.

Here each small square is 0.5 (two squares make 1). P is (1.5, 0.5), Q is (−2.5, 1), R is (−0.5, −1.5) and S is (2, −1.5).

Be careful with negative decimals: −2.5 is between −2 and −3 (further left than −2), not between −2 and −1. Count from 0: −0.5, −1, −1.5, −2, −2.5.

Written as a fraction	Written as a decimal	Where it is
(½, 3)	(0.5, 3)	halfway between x = 0 and x = 1, at height 3
(−1½, 2)	(−1.5, 2)	halfway between x = −1 and x = −2
(2¼, −1)	(2.25, −1)	a quarter of the way from 2 to 3

5. Plotting points to make lines and shapes

To plot a point, start at the origin, move across by the x-coordinate, then up or down by the y-coordinate, and mark it with a small neat cross (×). Then join the points with a ruler if the question asks.

Lines. Points with the same y-coordinate lie on a horizontal line, for example (−3, 2), (0, 2), (5, 2). Points with the same x-coordinate lie on a vertical line, for example (4, −1), (4, 3). The length of a horizontal line is the difference between the x-coordinates: from (−3, 2) to (5, 2) is 3 + 5 = 8 units (count the squares across the y-axis).

Patterns along a sloping line. If points lie on a straight line and are equally spaced, x and y go up (or down) by the same amounts each step. Use this to find more points.

From (−3, −5) to (−1, −1) to (1, 3): each step x goes up 2 and y goes up 4. The next point is (3, 7).

Shapes. Use the properties of the shape to find a missing vertex:

Rectangle / square: sides are horizontal and vertical (when drawn "straight"), so the missing vertex shares its x-coordinate with one neighbour and its y-coordinate with the other.
Square: all four sides equal. A square can also be tilted: its diagonals are equal, cross at right angles and cut each other in half.
Parallelogram: opposite sides are parallel and equal, so the same "move" (across and up) takes you along both opposite sides.
Isosceles triangle: the top vertex is directly above (or below) the midpoint of the base when the base is horizontal.

Rectangle ABCD: D must be in line with A (same x) and in line with C (same y), so D is (−4, −1).

A tilted square PQRS. The diagonals PR and QS are equal, cross at right angles and cut each other in half at (2, 1). So S is 3 below (2, 1): S = (2, −2).

إزاي تكمل الشكل

في المستطيل: الرأس الناقص بياخد الـ x من جاره الأول والـ y من جاره التاني. وفي متوازي الأضلاع: نفس "الحركة" (كام يمين وكام فوق) من رأس للتاني بتتكرر على الضلع المقابل.

6. Worked examples

Worked example 1 Look at the first grid in section 2. Write the coordinates of points B and C. [1]

B: 4 to the left (−4), then 3 up (3): (−4, 3).
C: 2 to the left (−2), then 5 down (−5): (−2, −5).
Mark-scheme style: 1 mark for both correct. Common mistake: writing (3, −4) for B (the numbers swapped).

Worked example 2 (a) In which quadrant is the point (−7, 2)? [1] (b) Rania says the point (0, −3) is in the 4th quadrant. Is she correct? Explain. [1]

(a) x is negative (left) and y is positive (up): the 2nd quadrant.
(b) No. The x-coordinate is 0, so the point is on the y-axis, 3 below the origin. Points on an axis are not in any quadrant.

Worked example 3 Look at the decimal grid in section 4, where each small square is 0.5. Write the coordinates of R and S. [2]

Each small square = 1 ÷ 2 = 0.5.
R: 1 small square left of 0 = −0.5; 3 small squares down = −1.5. R = (−0.5, −1.5).
S: 4 small squares right = 2; 3 small squares down = −1.5. S = (2, −1.5).
Common mistake: counting each small square as 1, giving (−1, −3) for R.

Worked example 4 A(−4, 3), B(2, 3) and C(2, −1) are three vertices of rectangle ABCD. (a) Write the coordinates of D. [1] (b) Work out the area of the rectangle. Each square is 1 cm². [1]

(a) AB is horizontal (both y = 3) and BC is vertical (both x = 2). D is below A (x = −4) and level with C (y = −1): D = (−4, −1).
(b) Length AB: from −4 to 2 is 4 + 2 = 6. Height BC: from 3 down to −1 is 3 + 1 = 4. Area = 6 × 4 = 24 cm².
Common mistake: working out the length as 2 − 4 = −2 by ignoring the minus sign. Count the squares, crossing 0.

Worked example 5 P(−1, 1), Q(2, 4) and R(5, 1) are three vertices of a square PQRS. Find S. [2]

Plot the points: this square is tilted. Its diagonals are PR and QS.
Midpoint of PR: x halfway between −1 and 5 is 2, y = 1. So the diagonals cross at (2, 1).
The diagonals of a square are equal and cut each other in half. PR has length 6, so QS also has length 6, with half (3) above (2, 1) and half below. Q(2, 4) is 3 above, so S is 3 below: S = (2, −2).
Check with "moves": P to Q is 3 right and 3 up; so S to R must also be 3 right and 3 up. From S(2, −2): 3 right, 3 up gives (5, 1) = R ✓.

Worked example 6 The points (−3, −5), (−1, −1) and (1, 3) lie on a straight line. They are equally spaced. Write the coordinates of the next point on the line. [1]

x goes −3 → −1 → 1: up 2 each time. Next x = 3.
y goes −5 → −1 → 3: up 4 each time. Next y = 7.
Next point: (3, 7).

Common mistakes

Swapping the order: (x, y) means across first, then up or down. (2, 5) and (5, 2) are different points.
Counting from 1 instead of from 0, or counting the lines wrongly when crossing an axis.
Going the wrong way for negatives: negative x is left, negative y is down.
Reading decimal grids as if every square were 1. Check the scale first.
Putting −2.5 between −2 and −1 (it is between −2 and −3).
Saying a point like (0, 4) or (−3, 0) is in a quadrant: points on the axes are not in any quadrant.
Forgetting brackets and the comma, or writing two numbers that cannot be read clearly.

How Checkpoint asks this

"Write the coordinates of point A." Answer in the form (x, y).
"Plot the point (−3, 1.5) on the grid. Label it P." Use a small neat cross exactly on the position and write the letter.
"A, B and C are three vertices of a square/rectangle/parallelogram. Plot point D to complete the shape and write its coordinates." Sketch the shape first, then check side lengths and parallel sides.
"Which quadrant…?" or "Draw a ring around the point that is in the 3rd quadrant."
"Tick (✓) the points that lie on the line…" Look for the same x (vertical line) or same y (horizontal line), or follow the pattern.
Grids may go up in 0.5 or 2: always read the scale before counting.
Use a sharp pencil and a ruler for plotting and joining points; examiners check accuracy.

7. Practice

Remember

Coordinates are (x, y): across first, then up or down. Start counting from the origin (0, 0).
Right and up are positive; left and down are negative.
Quadrants go anticlockwise from the top right: 1st (+, +), 2nd (−, +), 3rd (−, −), 4th (+, −). Points on an axis are in no quadrant.
Check the scale: a small square may be 0.5 (or another value), so points like (1.5, −0.5) sit between the grid numbers.
Same y = horizontal line; same x = vertical line. Count squares across 0 to find lengths.
Use shape properties (equal sides, parallel sides, right angles, diagonals) to find a missing vertex, then check it.

All pages in this course

Start here

Checkpoint 6 Maths: The Test and How to Prepare

Number: integers and calculation

Number: place value, decimals, fractions, percentages and ratio

Geometry and measure

Statistics and probability

Practice tests