Position and Direction
Section: Geometry | Syllabus: Cambridge Primary Mathematics (0845)
Coordinates in All Four Quadrants
Coordinates describe the exact position of a point on a grid, using two numbers that can be positive, negative, or even fractions and decimals. When both positive and negative coordinates are used, the grid extends in all four directions from a central point called the origin.
- A coordinate is written as (x, y): the x-coordinate gives the horizontal position, the y-coordinate gives the vertical position
- The grid is divided into 4 quadrants by two number lines crossing at the origin, (0, 0)
- The signs of x and y follow a pattern in each quadrant: (+,+) top right, (−,+) top left, (−,−) bottom left, (+,−) bottom right
- Coordinates can include fractions and decimals, not just whole numbers, e.g. (2.5, −3) or (−1.5, 4)
Each quadrant has its own pattern of positive and negative x and y values
Worked Example: Plotting a Point with Negative Coordinates
- Question: Plot the point (−3, 2).
- Step 1: Start at the origin (0, 0)
- Step 2: Move 3 places left, since x is negative
- Step 3: Move 2 places up, since y is positive
- Answer: The point lies in the top-left quadrant
Common Mistakes
MistakeReading the coordinates in the wrong order, e.g. plotting (−3, 2) by moving up 3 then left 2
Fixalways move horizontally (x) first, then vertically (y) - x always comes before y in (x, y)
Plotting Lines and Shapes from Coordinates
Once several coordinates are plotted, joining them in order can form a line or a complete 2D shape, across all four quadrants.
- Plot each coordinate as a point, then join the points in order with straight lines
- To check a shape is correct, count how many vertices it should have and confirm every one has been plotted and joined
- A shape can span more than one quadrant, crossing the x-axis or the y-axis
Worked Example: Plotting a Shape from Coordinates
- Question: Plot the points (−2, 1), (2, 1), (2, −2) and (−2, −2), then join them in order. What shape is formed?
- Step 1: Plot each point, then join (−2, 1) to (2, 1) - a horizontal side of length 4
- Step 2: Join (2, 1) to (2, −2) - a vertical side of length 3
- Step 3: Continue joining the remaining points in order, then join the last point back to the first
- Answer: A rectangle, 4 units by 3 units, spanning all four quadrants
Common Mistakes
MistakeJoining the points in the wrong order, creating a crossed or incorrect shape
Fixjoin the coordinates in the order they are given - joining points out of order can make a shape that crosses over itself
Translating Shapes
A translation slides a shape to a new position without rotating, reflecting, or changing its size. Every point on the shape moves the same distance in the same direction.
- A translation is described by how far to move horizontally and how far to move vertically, e.g. "3 right and 2 down"
- Every point on the original shape has a corresponding point on the translated image, found by applying the same horizontal and vertical movement to each one
- The shape and size stay exactly the same - only the position changes
Every vertex moves by the same amount - 3 right and 2 down - to reach its corresponding point
Worked Example: Translating a Point
- Question: A vertex is at (1, 4). Translate it 3 right and 2 down. What are the new coordinates?
- Step 1: Moving right increases the x-coordinate: 1 + 3 = 4
- Step 2: Moving down decreases the y-coordinate: 4 − 2 = 2
- Answer: (4, 2)
Common Mistakes
MistakeAdding when moving left or down, instead of subtracting
Fixmoving right or up adds to the coordinate; moving left or down subtracts from it
Reflecting Shapes
A reflection creates a mirror image of a shape across a mirror line. Every point on the shape and its reflection are the same distance from the mirror line, but on opposite sides.
- The mirror line can be vertical, horizontal, or diagonal
- Each point on the reflected image is the same perpendicular distance from the mirror line as the matching point on the original shape
- The shape and size stay the same - only its position and orientation change, becoming a mirror image
Each point and its reflection are the same perpendicular distance from the mirror line
Worked Example: Reflecting a Point in a Vertical Line
- Question: A point is at (2, 3). Reflect it in the vertical line x = 0 (the y-axis). What are the new coordinates?
- Step 1: The y-coordinate does not change, since the mirror line is vertical
- Step 2: The x-coordinate becomes the same distance on the opposite side of the line: 2 becomes −2
- Answer: (−2, 3)
Common Mistakes
MistakeMeasuring the reflected distance from the edge of the grid instead of the mirror line
Fixalways measure the distance from the mirror line itself, not from the edge of the grid
Rotating Shapes
A rotation turns a shape 90° around a fixed point called the vertex of rotation, either clockwise or anticlockwise. The shape's size stays the same, but its orientation changes.
- A rotation needs 3 things to describe it fully: the angle of turn (90° here), the direction (clockwise or anticlockwise), and the vertex it turns around
- For a 90° clockwise turn: a point directly right of the vertex moves to directly below it; below moves to the left; left moves above; above moves to the right
- The distance from the vertex to each point stays exactly the same before and after the rotation
Every point keeps the same distance from the vertex, but turns 90° clockwise around it
Worked Example: Rotating a Shape 90° Clockwise
- Question: A shape has a vertex 3 squares directly above the centre of rotation. Where does this vertex end up after a 90° clockwise rotation?
- Step 1: For a 90° clockwise turn, a point above the centre moves to directly to the right of it
- Step 2: The distance from the centre stays the same: 3 squares
- Answer: The vertex ends up 3 squares directly to the right of the centre of rotation
Common Mistakes
MistakeRotating around the wrong point, such as the centre of the shape instead of the given vertex
Fixalways rotate around the specific vertex stated in the question - the shape's own centre is only correct if that happens to be the given vertex
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