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In Key Stage 2 you work with coordinates across all four quadrants (including negative values), and learn how to transform shapes by translating, reflecting, and rotating them.
A coordinate describes a position on a grid using two numbers in brackets.
The first number is the x-coordinate (how far across). The second number is the y-coordinate (how far up).
Memory tip: "Along the corridor, then up the stairs" — x first, then y.
Example: the point (4, 3) is 4 along and 3 up.
The point where the axes cross (0, 0) is called the origin.
When we extend the x and y axes into negative values, we get four quadrants.
y
|
| Q2 | Q1
-+--------+------- x
| Q3 | Q4
|
Examples:
Use the signs of (x, y) to identify the quadrant before plotting:
flowchart TD
A[Point x, y] --> B{Sign of x?}
B -->|Positive| C{Sign of y?}
B -->|Negative| D{Sign of y?}
C -->|Positive| E["Quadrant 1<br/>+, +<br/>e.g. 3, 5"]
C -->|Negative| F["Quadrant 4<br/>+, -<br/>e.g. 5, -1"]
D -->|Positive| G["Quadrant 2<br/>-, +<br/>e.g. -4, 2"]
D -->|Negative| H["Quadrant 3<br/>-, -<br/>e.g. -3, -6"]
A translation slides a shape from one position to another without rotating or reflecting it. Every point moves the same distance in the same direction.
A translation is described as a movement in x and a movement in y.
Example: Translate triangle A by (+3, -2) (3 right, 2 down).
Key fact: A translated shape is congruent to the original (same size and shape, just in a different position).
A reflection flips a shape over a mirror line. Each point of the reflected shape is the same distance from the mirror line as the original, but on the opposite side.
Reflecting in the x-axis: the x-coordinate stays the same; the y-coordinate changes sign.
Reflecting in the y-axis: the y-coordinate stays the same; the x-coordinate changes sign.
Reflecting in the line y = x: swap the x and y coordinates.
Key fact: A reflected shape is congruent to the original, but it is a mirror image.
A rotation turns a shape around a fixed point called the centre of rotation. You need to state:
Rotating 90 degrees clockwise around the origin:
Rotating 180 degrees around the origin:
Rotating 90 degrees anti-clockwise around the origin:
Key fact: A rotated shape is congruent to the original.
| Transformation | Changes position? | Changes orientation? | Changes size? |
|---|---|---|---|
| Translation | Yes | No | No |
| Reflection | Yes | Yes (mirror flip) | No |
| Rotation | Yes | Yes | No |
| Enlargement | Yes | No | Yes |
All transformations except enlargement produce congruent shapes.
To fully describe a transformation you need to give all the information:
A triangle ABC has vertices at A(2, 3), B(5, 3), and C(2, 6). The class is asked to apply three transformations to it and to record the new vertices each time.
Step 1 — Plot the original triangle. A is 2 right and 3 up. B is 5 right and 3 up. C is 2 right and 6 up. Joining these points gives a right-angled triangle: AB is horizontal (3 squares long), AC is vertical (3 squares long), and BC is the slanted hypotenuse. The right angle is at A.
Step 2 — Translation by (-3, +2). Subtract 3 from every x-coordinate and add 2 to every y-coordinate.
The shape is the same — three squares wide, three squares tall — but it has slid 3 left and 2 up. The new triangle is congruent to the original.
Step 3 — Reflect the original in the y-axis. The y-coordinates stay the same; the x-coordinates change sign.
The reflected triangle is now in Quadrant 2. It is a mirror image — what was on the right of A is now on the left of A''. The right angle is now at A'' but the orientation has flipped.
Step 4 — Rotate the original 90 degrees clockwise about the origin. The rule is (x, y) becomes (y, -x).
The new triangle is in Quadrant 4. Lengths are preserved (still 3 by 3), but the triangle has turned: what was vertical (AC) is now horizontal, and what was horizontal (AB) is now vertical.
Step 5 — Verify congruence. All three image triangles have sides of length 3, 3, and the slant length matching the original. Distances and angles are preserved by translation, reflection, and rotation — only enlargement changes side lengths.
Step 6 — Describe each transformation precisely. A complete description must include enough information to reproduce it: for the translation, the column vector or "3 left, 2 up"; for the reflection, the mirror line; for the rotation, the centre, angle, and direction. Saying simply "rotated" earns no marks at SATs.
Step 7 — Combine two transformations. Reflect the translation result (triangle A'B'C') in the x-axis. Each reflected vertex keeps its x-coordinate and changes the sign of its y-coordinate.
The combined effect of "translate by (-3, 2), then reflect in x-axis" is not the same as "reflect in x-axis, then translate by (-3, 2)" — order of transformations matters, except for two translations (which commute).
Step 8 — Use coordinates to find a midpoint. The midpoint of two points (a, b) and (c, d) is ((a+c)/2, (b+d)/2). Find the midpoint of A(2, 3) and B(5, 3):
Midpoint = ((2+5)/2, (3+3)/2) = (3.5, 3)
This is a useful skill for symmetry problems where the mirror line passes through the midpoint of two corresponding points.
Step 9 — Find a missing vertex of a parallelogram. A parallelogram has three known vertices: P(1, 1), Q(5, 1), and R(7, 4). Find the fourth vertex S so that PQRS forms a parallelogram (in that vertex order).
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