Shape and Space - Symmetry (Line and Rotational)

Line Symmetry (Reflectional Symmetry): A shape possesses line symmetry if a line, called the axis or line of symmetry, can be drawn through it such that one half is a perfect mirror image of the other. Visually, if you imagine folding the shape along this line, the two parts would match up exactly with no overlapping edges.

Lines of Symmetry in Regular Polygons: For any regular polygon with $n$ sides, there are exactly $n$ lines of symmetry. For instance, a regular pentagon has 5 lines of symmetry, each passing from a vertex to the midpoint of the opposite side, while a square has 4 lines consisting of two lines connecting opposite midpoints and two lines connecting opposite vertices.

Rotational Symmetry: This occurs when a shape can be rotated around a fixed center point by an angle less than $360^\circ$ and still look identical to its original position. Visually, imagine a star shape being spun around its center; if it fits into its own 'outline' multiple times during a full turn, it has rotational symmetry.

Order of Rotational Symmetry: This represents the total number of times a shape looks exactly the same as it performs one full $360^\circ$ rotation. A shape that only looks like itself after a complete $360^\circ$ turn is said to have an order of symmetry of $1$. For example, a standard parallelogram has an order of $2$ because it looks the same at $180^\circ$ and $360^\circ$.

Center of Rotation: This is the specific fixed point around which a shape is rotated to check for rotational symmetry. In regular polygons, this point is the exact geometric center where all lines of symmetry intersect.

Angle of Rotation: The smallest angle a shape must be turned to appear exactly as it did in its starting position. For a shape with rotational symmetry of order $n$, the angle is found by dividing a full circle by the order. For a regular hexagon, the angle is $60^\circ$ because its order is $6$.

Symmetry in 2D Shapes vs. 3D Objects: While Grade 6 focuses on 2D planes, it is helpful to visualize that 2D symmetry is about lines and points, whereas 3D symmetry involves planes of symmetry. A rectangle on paper has 2 lines of symmetry, but a rectangular prism (box) has multiple planes of symmetry that slice the volume into equal mirror halves.