Lines of symmetry

Activities to try

1. 1

   Build the butterfly's missing wing, point by point

   Why it works: Reflection sends each point perpendicular to the mirror line to an equal distance on the far side; building partner-by-partner (counting equal perpendicular distances) and folding to check turns a 'looks about right' sketch into a construction you can justify and check.

   1. Start from half a butterfly drawn on squared paper beside a clearly marked vertical mirror line.
   2. For each key point of the given half (e.g. a wingtip), count the squares perpendicular from the mirror line to that point.
   3. Count the SAME number of squares straight out on the far side, perpendicular to the mirror, and mark the partner point.
   4. Repeat for every corner/key point, then join the partners in the SAME order as the original half.
   5. Fold the paper along the mirror line (holding it up to the light if needed), or stand a mirror/Mira on the line and compare the reflection of the given half with your drawn half. Check that the two halves line up, allowing for small inaccuracies in drawing and folding.
2. 2

   Reflect, don't slide: spot the difference

   Why it works: Reflection reverses 'sense' — a true mirror image faces the opposite way, unlike a shape merely slid (translated) across, which keeps the same orientation and, for a lopsided half like a butterfly wing, will not fold onto the original half.

   1. Look at two completions of the same half-butterfly: one where the second wing points the opposite way, one where a copy of the left wing has been slid across so both wings point the same way.
   2. Decide which is truly symmetric and explain how you can tell.
   3. Fold each along the mirror line and watch which one's halves coincide.
   4. Say what folding reveals about the slid copy (the wingtips miss).
3. 3

   Points on the line, and new mirror directions

   Why it works: The 'perpendicular and equidistant' rule is constant whatever the mirror's orientation; points on the line are fixed, and the method generalises from vertical to horizontal and sloping lines.

   1. Work a figure that has a key point sitting right ON the mirror line; decide where its partner goes (it stays put — distance zero, not duplicated or shifted).
   2. Complete a figure whose mirror line is HORIZONTAL, counting squares perpendicular to that line (now up/down).
   3. Later, try a figure with a SLOPING mirror line (on squared paper, a diagonal line at 45°), keeping the same rule: each partner the same perpendicular distance from the line, counted in corner-to-corner diagonal steps across the squares.
   4. Fold (or imagine the fold) along each mirror line to check the halves coincide.