Symmetry

A symmetry is a transformation that makes a figure match itself. In line symmetry, a line acts like a mirror: folding along it makes the two parts land exactly on each other. Check the corners and edges. A balanced-looking picture or two equal areas alone do not prove a match.

The opening image shows two tree drawings. The left drawing matches across its vertical line; the right drawing does not. Real trees need not match like an ideal drawing.

Find the fold lines

A square has four lines of symmetry: two through the midpoints of opposite sides and two through opposite corners. A rectangle that is not a square has two, through the midpoints of opposite sides. Its corner diagonals divide it into equal areas, but are not fold lines that make the parts coincide.

A triangle with exactly two equal sides has one line of symmetry. A triangle with three unequal sides has none. A circle has infinitely many: every line through its center works. Turning the whole figure does not change how many symmetry lines it has.

Construct a matching part

Choose a crease and mark a point on one side. Its reflected partner is the same distance straight across the crease, measured at a right angle. A point already on the crease stays put. Mark partner corners and join them in the same boundary order. You can check by folding or tracing.

A turn is a different test

Rotational symmetry means a figure matches itself after a turn greater than zero and less than a full turn. A square matches after a quarter turn. A non-square rectangle matches after a half turn. A full turn by itself proves nothing: every figure returns to its starting position then.

Look carefully at real examples

Leaves, faces and butterflies may be approximately symmetric, with small differences between their sides. Do not assume every natural example has exact matching parts. The symmetry of a printed letter also depends on its font and design. In mathematics, state which figure and which transformation you are testing.

Try the Grade 4 workshop

Test folds, construct reflected parts and explain an original design in the Fold-and-Match Studio. Keep your paper work for a grown-up to review.