Geometry, Measurement & Data
Geometry & Coordinates: Cartesian Grid and Transformations
Grade 4 · Grade 5 · Grade 6
- ✓Identify and plot points on a Cartesian grid using ordered pairs.
- ✓Describe the location of points on a Cartesian grid using coordinates.
- ✓Perform and describe translations (slides) of 2D shapes.
- ✓Perform and describe reflections (flips) of 2D shapes across a line of reflection.
- ✓Perform and describe rotations (turns) of 2D shapes about a point.
Key concepts
A Cartesian grid is a system used to locate points using two perpendicular number lines. The horizontal line is called the x-axis, and the vertical line is called the y-axis. They meet at a point called the origin (0,0). For Junior grades, we usually focus on the first quadrant where both x and y values are positive.
An ordered pair (x, y) that tells you the exact location of a point on a Cartesian grid. The first number (x) tells you how many units to move horizontally from the origin (right for positive, left for negative). The second number (y) tells you how many units to move vertically (up for positive, down for negative). Remember: 'run before you jump' – move along the x-axis first, then the y-axis.
A translation is a transformation that moves a shape from one location to another without turning it or flipping it. Every point in the shape moves the same distance in the same direction. You describe a translation by stating the direction (e.g., right, left, up, down) and the number of units moved.
A reflection is a transformation that creates a mirror image of a shape across a line called the line of reflection. The reflected shape is the same size and shape as the original, but it is flipped. Each point in the original shape is the same distance from the line of reflection as its corresponding point in the reflected shape.
A rotation is a transformation that turns a shape around a fixed point called the centre of rotation. The shape keeps its size and shape, but its orientation changes. You describe a rotation by stating the centre of rotation (e.g., the origin), the angle of rotation (e.g., 90 degrees, 180 degrees), and the direction (e.g., clockwise, counter-clockwise).
Key facts to remember
- 1A Cartesian grid uses two number lines, the x-axis (horizontal) and y-axis (vertical), to locate points.
- 2The point where the x-axis and y-axis meet is called the origin, with coordinates (0,0).
- 3Coordinates are always written as an ordered pair (x, y), where x is the horizontal position and y is the vertical position.
- 4A transformation changes the position or orientation of a shape without changing its size or shape.
- 5Translation (slide) moves a shape without turning or flipping it.
- 6Reflection (flip) creates a mirror image of a shape across a line of reflection.
- 7Rotation (turn) turns a shape around a fixed point called the centre of rotation.
Worked examples
Example 1
Plot the points A(2,3), B(5,1), C(1,5) on a Cartesian grid. Then identify the coordinates of point D if it is 3 units right and 2 units down from point C.
Answer
A(2,3), B(5,1), C(1,5) are plotted on the grid. Point D is (4,3).
Always start at the origin (0,0) when plotting points unless you are moving from an existing point.
Example 2
A triangle has vertices at P(1,1), Q(3,1), and R(2,4). Translate the triangle 4 units right and 1 unit up. Give the new coordinates of the vertices.
Answer
The new vertices of the translated triangle are P'(5,2), Q'(7,2), and R'(6,5).
When translating, every point in the shape moves the exact same way.
Example 3
Reflect a rectangle with vertices S(1,2), T(4,2), U(4,4), V(1,4) across the line x=5. Give the new coordinates of the reflected rectangle.
Answer
The new vertices of the reflected rectangle are S'(9,2), T'(6,2), U'(6,4), and V'(9,4).
When reflecting, the distance from the original point to the line of reflection is equal to the distance from the reflected point to the line of reflection.
Common mistakes
- ✗Mixing up the x and y coordinates (e.g., plotting (3,2) instead of (2,3)).
- ✗Counting grid lines instead of spaces/units when plotting points or performing transformations.
- ✗Forgetting to label the axes or the origin on a grid.
- ✗Confusing the different types of transformations (slide, flip, turn) and their effects.
- ✗Incorrectly identifying the line of reflection or centre of rotation, leading to an incorrect transformation.
Exam tips
- ★Always start at the origin (0,0) when plotting points, unless you are moving from an existing point.
- ★Remember the phrase 'run before you jump' to help you remember to move along the x-axis first, then the y-axis.
- ★Use a ruler and pencil to draw clear grids and shapes. This helps prevent errors and makes your work easy to read.
- ★For transformations, apply the rule to each vertex of the shape to find its new position, then connect the new vertices to draw the transformed shape.
- ★Double-check your coordinates and transformations by visually inspecting your work on the grid to ensure it makes sense.
Ready to practise?
Try a problem on this topic
Snap a photo or type a question — get step-by-step working instantly.