This Year 6 four-quadrant coordinates and transformations lesson is written for pupils aged 10-11 and follows the National Curriculum. In short, it shows you how to use coordinates in all four quadrants and translate and reflect shapes. Before you start, it helps to revise converting units, area, perimeter and volume and angles, circles and 3D shapes, because this lesson builds on them.
Year 6 four-quadrant coordinates and transformations lesson: what you will learn
A coordinate grid helps us describe position. In Year 6 the grid has four quadrants, so coordinates can be negative.
Starter activity
Zara is playing a treasure hunt game. The treasure is at a secret point, and she is given the instructions "start at the origin, move 4 to the left and 3 up". Where does she end up? How can she describe the position of the treasure so that her friend can find it too? In this chapter you will use coordinates in all four quadrants, and use translations and reflections to move shapes around the grid.
The coordinate grid
Definition: axes and quadrants
The x-axis runs left to right and the y-axis runs up and down. They cross at the origin, (0, 0), and split the grid into four quadrants.
Key fact: signs of coordinates
Right and up are positive. Left and down are negative. So (−3, 2) is 3 left and 2 up.
Reading and writing coordinates
Rule: coordinates
Write coordinates as (x, y). Always give the x-coordinate (along) first and the y-coordinate (up or down) second.
Example 1
Points A(1, 1), B(5, 1) and C(5, 4) are three corners of a rectangle. Find the fourth corner.
The fourth corner is directly above (1, 1) and level with (5, 4). It has x-coordinate 1 and y-coordinate 4. The point is (1, 4).
Distances on the grid
Example 2
How far is (−3, 2) from (4, 2)?
Both points are at y = 2, so count along: from −3 to 4 is 3 + 4 = 7 units.
Translations
Definition: translation
A translation moves every point of a shape the same distance in the same direction, without turning or flipping.
Method: translating a point
- Add or subtract from the x-coordinate for a move right or left.
- Add or subtract from the y-coordinate for a move up or down.
- Plot the new point.
Example 3
Translate the point (2, 3) by 4 to the right and 5 down.
x: 2 + 4 = 6. y: 3 − 5 = −2. The image is (6, −2).
Example 4
A triangle has a corner at (−1, −2). Translate it 3 left and 4 up.
x: −1 − 3 = −4. y: −2 + 4 = 2. The image is (−4, 2).
Reflections in the axes
Rule: reflecting in the axes
Reflecting in the x-axis changes the sign of the y-coordinate: (a, b) becomes (a, −b). Reflecting in the y-axis changes the sign of the x-coordinate: (a, b) becomes (−a, b).
Example 5
Reflect (−4, −1) in the x-axis and then in the y-axis.
In the x-axis the y-coordinate changes sign, so the image is (−4, 1). In the y-axis instead, the image is (4, −1).
Watch out
Do not swap x and y. Count carefully across zero: from −2 to 3 is 5 steps, not 1.
More about the coordinate grid
Method: translating a shape
- Write down the coordinates of each corner.
- Add the horizontal move to every x-coordinate (add if right, subtract if left).
- Add the vertical move to every y-coordinate (add if up, subtract if down).
- Plot the new points and join them in the same order.
Example 7
Triangle ABC has corners A(−2, 1), B(0, 1) and C(−2, 4). It is translated 3 to the right and 5 down. Write the new corners.
Add 3 to each x-coordinate and subtract 5 from each y-coordinate. A′ is (−2 + 3, 1 − 5) = (1, −4). B′ is (0 + 3, 1 − 5) = (3, −4). C′ is (−2 + 3, 4 − 5) = (1, −1). So the new corners are A′(1, −4), B′(3, −4) and C′(1, −1).
Example 8
Reflect the point (−5, 3) in the x-axis and in the y-axis.
For a reflection in the x-axis the y-coordinate changes sign and the x-coordinate stays the same: (−5, −3). For a reflection in the y-axis the x-coordinate changes sign: (5, 3). A point and its image are always the same distance from the mirror line.
Example 9
Find the distance from (−3, −2) to (4, −2).
The y-coordinates are the same, so the points lie on a horizontal line. The distance is the difference between the x-coordinates. From −3 up to 0 is 3 and from 0 to 4 is 4, so the distance is 3 + 4 = 7 units. You can also subtract: 4 − (−3) = 7.
Example 10
Describe the transformation that maps (3, 4) onto (−1, 1).
The x-coordinate goes from 3 to −1, which is 4 to the left. The y-coordinate goes from 4 to 1, which is 3 down. This is a translation of 4 left and 3 down. It is not a reflection, because reflections change signs but do not shift by a fixed amount.
Watch out: common mistakes
Coordinates are always (x, y), never (y, x). When you count across zero, count both parts: from −2 to 3 is 2 + 3 = 5 steps, not 1. When you translate left or down, you must subtract. Also, a translation and a reflection are different: a reflection flips the shape, but a translation only slides it without turning it.
The four quadrants
The quadrants are numbered 1, 2, 3 and 4 going anticlockwise from the top right. In quadrant 1 both coordinates are positive. In quadrant 2, x is negative and y is positive. In quadrant 3 both are negative. In quadrant 4, x is positive and y is negative.
Did you know?
The coordinate grid is named after the French mathematician René Descartes. About 400 years ago he realised that any position on a flat surface can be described by two numbers. The story says that he had the idea while watching a fly walk across the ceiling of his bedroom. His system is now used in maps, computer games and satellite navigation.
Otto’s tip
Along the corridor, then up the stairs! x comes first, y comes second.
How to use this Year 6 four-quadrant coordinates and transformations lesson
First, read the key points so that you know where the Year 6 four-quadrant coordinates and transformations lesson is going. Then work through each definition and worked example, and cover the solution before you read it. After that, practise with the four-quadrant coordinates and transformations worksheet and check every question with the worksheet answers. When you feel ready, test yourself with the multiple choice questions or the quick quiz. Finally, sit the four-quadrant coordinates and transformations test and mark it with the test answers.
Also, the starters are a good warm-up, so you can keep the key facts fresh. Moreover, all the pages of this chapter work together, so you can move between them at any time. For example, many pupils read the lesson on Monday, practise on Wednesday and test themselves on Friday. In short, this Year 6 four-quadrant coordinates and transformations lesson gives you everything you need for the chapter.
To sum up, regular practice works better than one long session. For instance, ten minutes a day is often enough. Besides, mistakes are useful, because they show what to revise next. Then, once a method feels easy, try a harder question. As a result, you build confidence step by step.
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