Reading and plotting points, the four quadrants and midpoints — one clear step at a time.
6 lessons · 35 questions (A–G) · 20 calculation + 15 word problems · no time pressure
Coordinates fix a position on a grid with two numbers, written (x, y): the first says how far across, the second how far up or down. Pupils most often slip by reading the pair in the wrong order — up before across — or by losing the minus sign on negative coordinates left of or below zero.
At Improve Tuition, a qualified teacher shows a pupil exactly how to track across, then up, with clear grids, in small worked steps — with read-aloud, comfort spacing and a reading tint for pupils who take in maths more easily that way.
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Lesson 1
Reading coordinates
Aim: Read the coordinates of a point: the x-coordinate first, then the y-coordinate.
A point’s position is given by two numbers in brackets, like (3, 5). The first number is the x-coordinate — how far across. The second is the y-coordinate — how far up. A helpful saying is ‘along the corridor, then up the stairs’.
Read across first, then up: (3, 5) is 3 along the x-axis and 5 up the y-axis.
Worked example
Point A is at (3, 5). Read off its x-coordinate and y-coordinate.
Coordinates are always written (x, y) — across first, then up.
The first number is the x-coordinate: that is 3, so 3 across.
The second number is the y-coordinate: that is 5, so 5 up.
So A is 3 across and 5 up. Its x-coordinate is 3 and its y-coordinate is 5.
Check: ‘along the corridor’ (3) comes before ‘up the stairs’ (5). ✓
Watch out
x always comes first. (3, 5) is not the same point as (5, 3).
Try one
Point B is at (6, 2). What is its x-coordinate?
Show answer
The x-coordinate is the first number in the brackets.
Aim: Find the distance between two points that share a row or a column.
If two points have the same y-coordinate they sit on a horizontal line, and the distance between them is the difference of their x-coordinates. If they share the same x-coordinate they sit on a vertical line, and the distance is the difference of their y-coordinates. Distance is counted as a positive number of units.
A(2, 3) and B(7, 3) share the line y = 3, so the gap is 7 − 2 = 5 units across.
Worked example
How far apart are A(2, 3) and B(7, 3)?
Look at the coordinates: both have y = 3, so the points are on the same horizontal line.
On a horizontal line, the distance is the difference of the x-coordinates.
Subtract the smaller from the larger: 7 − 2 = 5.
So A and B are 5 units apart.
Check: counting the squares from x = 2 to x = 7 is 5 steps. ✓
Watch out
Subtract the matching coordinates. Same y → compare the x’s; same x → compare the y’s.
Try one
How far apart are C(4, 1) and D(4, 8)?
Show answer
Both have x = 4, so they are on the same vertical line.
Distance = difference of the y-coordinates: 8 − 1 = 7.
Aim: Use negative coordinates and name the four quadrants.
The axes cross at the origin and split the grid into four quadrants, numbered 1 to 4 anticlockwise from the top right. Left of the y-axis, x is negative; below the x-axis, y is negative. So Quadrant 1 is (+, +), Quadrant 2 is (−, +), Quadrant 3 is (−, −) and Quadrant 4 is (+, −).
The axes split the grid into four quadrants, numbered 1 to 4 anticlockwise from the top right.
Worked example
Which quadrant is the point (−5, 2) in?
Look at the signs of the two coordinates.
x = −5 is negative, so the point is to the left of the y-axis.
y = 2 is positive, so the point is above the x-axis.
Left and above is the top-left region — that is Quadrant 2.
Aim: Find the midpoint of two points by averaging the coordinates.
The midpoint is exactly halfway between two points. Find it by taking the average of the two x-coordinates and the average of the two y-coordinates: add each pair and halve it.
The midpoint M of A(2, 4) and B(6, 4) sits exactly halfway between them, at (4, 4).
Worked example
Find the midpoint of A(2, 4) and B(6, 4).
Average the x-coordinates: add them and halve. (2 + 6) ÷ 2 = 8 ÷ 2 = 4.
Average the y-coordinates the same way: (4 + 4) ÷ 2 = 8 ÷ 2 = 4.
Put the two averages together as a coordinate.
So the midpoint is (4, 4).
Check: 4 is exactly halfway between 2 and 6. ✓
Watch out
Add then halve each coordinate, doing the x’s and y’s separately.
Try one
Find the x-coordinate of the midpoint of (1, 3) and (7, 9).
Aim: Use coordinates to describe positions and journeys on a grid map.
Maps, games and screens all use coordinates to pin down a position. Read a place the same way — across, then up — and find how far something travels along a row or column by subtracting the matching coordinates, just like distance on any grid.
A journey straight up the grid (north) changes only the y-coordinate — here 5 units up.
Worked example
A ship is at (4, 7) and sails north to (4, 12). How far does it sail?
Both positions have x = 4, so the ship moves straight up a vertical line.
Going north means the y-coordinate increases; the distance is the difference of the y-coordinates.
Subtract: 12 − 7 = 5.
So the ship sails 5 units north.
Check: from y = 7 up to y = 12 is 5 squares. ✓
Watch out
‘North’ means up (y increases); ‘east’ means right (x increases). Match the coordinate to the direction.
Try one
A robot at (2, 3) moves to (9, 3). How far does it travel?
Show answer
Both positions have y = 3, so it moves along a horizontal line.
Distance = difference of the x-coordinates: 9 − 2 = 7.
Aim: Use coordinates with shapes, translations (slides) and reflections (mirror lines).
You can find a shape’s missing corner by matching the coordinates it must share with the other corners. A translation slides a point: add to the x-coordinate to go right, to the y-coordinate to go up. A reflection in the x-axis flips the sign of the y-coordinate; a reflection in the y-axis flips the sign of the x-coordinate.
A slide of 5 squares right adds 5 to the x-coordinate: (3, 4) moves to (8, 4).
Worked example
A counter at (3, 4) is moved 5 squares to the right. What is its new position?
Moving right changes only the x-coordinate; up or down would change the y-coordinate.
Add 5 to the x-coordinate: 3 + 5 = 8.
The y-coordinate stays the same: 4.
So the new position is (8, 4).
Check: the counter moved 5 squares right and none up, so only x changed. ✓
Watch out
Right/left changes x; up/down changes y. A reflection flips the sign of just one coordinate.
Try one
A point at (5, 2) is reflected in the x-axis. What is its new y-coordinate?
Show answer
Reflecting in the x-axis flips the point to the other side of that axis, so the y-coordinate changes sign.
The y-coordinate 2 becomes −2.
So the new y-coordinate is −2.
Check: (5, 2) and (5, −2) are the same distance from the x-axis. ✓
31.A counter at (3, 4) is moved 5 squares right. What is its new x-coordinate?
Answer:
Show working
Moving right changes only the x-coordinate.
Add 5 to the x-coordinate: 3 + 5 = 8.
So the new x-coordinate is 8. ✓
32.A counter at (3, 4) is moved 2 squares up. What is its new y-coordinate?
Answer:
Show working
Moving up changes only the y-coordinate.
Add 2 to the y-coordinate: 4 + 2 = 6.
So the new y-coordinate is 6. ✓
33.A point at (5, 2) is reflected in the x-axis. What is its new y-coordinate?
Answer:
Show working
Reflecting in the x-axis flips the sign of the y-coordinate.
y = 2 becomes −2.
So the new y-coordinate is −2. ✓
34.A point at (5, 2) is reflected in the y-axis. What is its new x-coordinate?
Answer:
Show working
Reflecting in the y-axis flips the sign of the x-coordinate.
x = 5 becomes −5.
So the new x-coordinate is −5. ✓
35.A drone at (−3, 4) moves 7 squares right. What is its new x-coordinate?
Answer:
Show working
Moving right changes only the x-coordinate.
Add 7 to the x-coordinate: −3 + 7 = 4.
So the new x-coordinate is 4. ✓
0 of 35 answered
Guide
How do coordinates work?
Coordinates describe a point on a grid using two numbers in brackets, (x, y). The first is the x-coordinate — how far across from zero — and the second is the y-coordinate — how far up or down. The point (3, 2) means three across and two up.
The axes split the grid into four quadrants. To the left of zero the x-values are negative; below zero the y-values are negative. The midpoint of two points is found by averaging their x-values and their y-values.
Common mistakes. Reading the pair the wrong way round (up before across), dropping the minus sign on negative coordinates, or miscounting the squares from zero.
Common questionsWhich coordinate comes first, x or y?
The x-coordinate (across) always comes first, then the y-coordinate (up or down): (across, up). A common reminder is “along the corridor, then up the stairs.”
What are the four quadrants?
The two axes split the grid into four regions. Each quadrant has a different combination of positive and negative x and y values.
How do you find the midpoint of two points?
Add the two x-coordinates and halve the result, then do the same with the two y-coordinates. That gives the point exactly halfway between them.
What does a negative coordinate mean?
A negative x-value is to the left of zero and a negative y-value is below zero, so (−2, −3) is two left and three down.
What are coordinates used for?
They locate points for graphs, maps, shapes, translations and reflections — anywhere a position needs to be described with numbers.
Stuck on this topic?
A teacher can find the exact gap
Practising coordinates on your own is a strong start. If the same marks keep slipping, a qualified teacher can pinpoint the precise gap and fix it. Improve Tuition offers one-to-one maths tuition with our maths tutors in Batley and online — the first assessment is free.