KS3 · Maths

Plotting coordinates in all four quadrants

A treasure map says "3 across, 2 up". Easy. The next says "3 across the other way, 2 down". Coordinates are just that: directions from one starting point.

Maths · Coordinates

Read the corners

A: squares across from the origin 2. B: squares up or down from the origin 3. Relationship: Corner P is 2 across and 3 up from the origin, so P is (2, 3): x first, then y. Now slide A and B to read Q, R and S.

A: squares across from the origin2B: squares up or down from the origin3Line A up with a corner, then B level with it.

Corner P is 2 across and 3 up from the origin, so P is (2, 3): x first, then y. Now slide A and B to read Q, R and S.

Four corners, one in each quadrant. Slide A along the x-axis to read how far across, then B along the y-axis to read how far up or down.

Watch out: The readouts are distances, so they never show a minus sign. You decide the sign from the side of the axis you are on: left of the origin or below it means negative.

Maths · Order matters

Same numbers, same point?

Two friends are plotting. One plots (1, 4). The other plots (4, 1).

Are they plotting the same point? Pick the idea closest to yours.

Yes — it is the same two numbers, so it is the same point.No — but the y coordinate comes first, so you go up first.Yes — they are both in the first quadrant, so they are the same point.No — (1, 4) is 1 across and 4 up, but (4, 1) is 4 across and 1 up.

How sure are you?

I'm sureI think soI'm guessing

Yes — it is the same two numbers, so it is the same point.

This is a common misconception

The numbers matter, but so does their order. In (1, 4) you go 1 across and 4 up. In (4, 1) you go 4 across and 1 up. Those two journeys finish in different places.

No — but the y coordinate comes first, so you go up first.

This is a common misconception

You have spotted that they differ, but the order is the other way round. The x coordinate is written first and you travel in the x direction first: across, then up or down. So (1, 4) is 1 across and 4 up.

Yes — they are both in the first quadrant, so they are the same point.

This is a common misconception

A quadrant is a whole area, not a single place. Plenty of points share a quadrant. (1, 4) and (4, 1) are both in the first quadrant, but they are different points.

No — (1, 4) is 1 across and 4 up, but (4, 1) is 4 across and 1 up.

This matches the evidence

Exactly. The x coordinate is always written first and travelled first, so swapping the two numbers gives a different point.

x first, y second, every time. Swap the numbers and you have moved to a different point.

Maths · Quadrants

Which quadrant?

Sort each point into its quadrant using only its signs. No plotting allowed.

Still to sort

First quadrant (0)

Top right of the origin

Where the line is: The quadrants are numbered anticlockwise, starting at the top right and going round to the top left, then bottom left, then bottom right.

Second quadrant (0)

Top left of the origin

Third quadrant (0)

Bottom left of the origin

Fourth quadrant (0)

Bottom right of the origin

5 of 5 still to sort.

Maths · Between the grid lines

Halfway points

Coordinates do not have to sit on grid lines. Drag each tag to where you think the value belongs on this one axis, then check.

A — Halfway between 1 and 2B — Midpoint of 1 and 5C — Midpoint of −3 and 5D — Halfway between −2 and −1

Predict, then check

Look closely at the numbers along each axis.

On one grid, the x-axis is labelled 0, 2, 4, 6, 8 and the y-axis is labelled 0, 1, 2, 3, 4. Is that a sensible way to label it?

Slide or change?

Same rule for every vertexvsDifferent change for different vertices

The same triangle, A(1, 1), B(4, 1), C(1, 3), moved two different ways.

Focus

Rule applied to the vertices

Same rule for every vertex

Add 3 to the x coordinate of every vertex

Different change for different vertices

Add 3 to x for A and B, but only 1 to x for C

The insight

This is the deciding difference: was every vertex treated the same way?

New vertices

Same rule for every vertex

A(4, 1), B(7, 1), C(4, 3)

Different change for different vertices

A(4, 1), B(7, 1), C(2, 3)

How far each point moves

Same rule for every vertex

Every point moves 3 in the positive x direction

Different change for different vertices

Points move different distances in the x direction

What happens to the shape

Same rule for every vertex

It slides across, unchanged

Different change for different vertices

It changes: C no longer sits above A

Is it a translation?

Same rule for every vertex

Yes

Different change for different vertices

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Directions from a starting point: across first, then up or down.

What you need to know

The x-axis runs across and the y-axis runs up and down. They cross at the origin, which has coordinates (0, 0).
A coordinate pair is written (x, y): x first, y second. (1, 4) and (4, 1) are different points.
To plot a point, start at the origin, travel the x distance in the x direction, then the y distance in the y direction. A negative number means travel the other way.
The axes split the grid into four quadrants, numbered anticlockwise. The signs of x and y tell you which quadrant a point is in.
Coordinates can be fractions or decimals, and each axis needs a scale that stays consistent along its whole length.

The big picture

A coordinate pair is a set of directions from the origin: the first number says how far across (x), the second how far up or down (y), and a minus sign means go the other way. The signs also tell you which of the four quadrants the point is in.

Key points

1Coordinates are directions from the origin: across by x first, then up or down by y.

2A minus sign reverses the direction: negative x goes the negative x way, negative y goes down from the origin.

3Signs fix the quadrant: (+, +) first, (−, +) second, (−, −) third, (+, −) fourth.

4The midpoint of two values is found by adding them and halving the result.

5A shape is described by the coordinates of its vertices. Apply the same rule to every vertex and the shape is translated.

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Worked example

Problem

Plot the point (−4, 2.5) and say which quadrant it is in.

⚠ Watch out

Swapping the numbers: reading (1, 4) as "1 up, 4 across". The first number is always x, so (1, 4) is 1 across and 4 up, and (4, 1) is a different point.

🧠

Memory hook

Along the corridor, then up the stairs: x first, then y (and down the stairs if y is negative). The quadrants then run round anticlockwise: 1 top right, 2 top left, 3 bottom left, 4 bottom right.

✓

Check yourself

Without drawing: which quadrant is (−6, −2) in? Is (−2, −6) the same point? (Third quadrant. No: it is 2 left, 6 down, not 6 left, 2 down.)

Flashcards

(12)

What is the origin?

The point where the x-axis and y-axis cross. Its coordinates are (0, 0).

Which way does each axis run?

The x-axis is horizontal (across) and the y-axis is vertical (up and down).

In a coordinate pair (x, y), which number comes first, and which way do you move first?

x comes first, and you move first in the x direction (across), then in the y direction (up or down).

Are (1, 4) and (4, 1) the same point?

No. (1, 4) is 1 across and 4 up. (4, 1) is 4 across and 1 up.

What do negative coordinates tell you to do?

Travel from the origin in the negative direction: a negative x is a move in the negative x direction, and a negative y is a move down.

How are the four quadrants numbered, and what is a quadrant?

A quadrant is one of the four areas the axes divide the grid into. They are numbered anticlockwise: first (+, +), second (−, +), third (−, −), fourth (+, −).

Can coordinates be non-integers?

Yes. An integer is any positive or negative whole number or zero, but between any two grid values there are infinitely more points, such as halfway between 1 and 2. They can be fractions or decimals, in any quadrant.

How do you find the midpoint of two values?

Add them and halve the result. For 1 and 5: 1 + 5 = 6, and half of 6 is 3.

What rule must the scale on a coordinate axis follow?

It must be consistent all the way along that axis. The two axes may use different scales, such as steps of two on x and steps of one on y.

How do you describe a shape on a coordinate grid?

By the coordinates of its vertices (corners). The shape can sit in one quadrant or across several.

What makes a movement of a shape a translation?

Every point moves the same distance in the same direction. Applying the same rule to every vertex does this: adding 3 to every x coordinate slides the shape 3 in the positive x direction.

If the vertices of a shape are changed by different amounts, is that a translation?

No. If the vertices are not all changed by the same rule, the shape changes and the result is not a translation.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Same numbers, same point?

Which quadrant?

Halfway points

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