Transformations on a Grid

Activities to try

1. 1

   Build a reflection vertex by vertex using the perpendicular-equidistant rule

   Why it works: A reflection is produced point by point: every line joining an object vertex to its image is perpendicular to the mirror line, and the image vertex is the same distance beyond the mirror as the object is in front. Because this reverses the vertex order, the shape's sense flips — distinguishing a genuine reflection from a shape merely slid across.

   1. Start with a scalene triangle on a squared grid and a vertical mirror line two squares to its right (choose scalene so a reversal of sense is visible).
   2. For each vertex in turn, draw the line from the vertex to the mirror line that meets the mirror at a right angle (perpendicular to the mirror line); with this vertical mirror it runs along a horizontal grid line. Check the right angle rather than assuming it.
   3. Count the squares from the vertex to the mirror along that perpendicular, then count the SAME number of squares beyond the mirror and mark the image vertex there.
   4. Repeat for every vertex, then join the image vertices to complete the reflected shape.
   5. Compare the vertex order: check that the image reads the mirror-image way round (a clockwise order of corners becomes anticlockwise) — this reversed 'sense' is what makes it a true reflection rather than a slide.
2. 2

   Translate every point by the same vector and contrast it with reflection

   Why it works: A translation moves every point the same distance in the same direction, so orientation and sense are unchanged — the opposite of reflection, which reverses sense. Both leave the shape congruent to the original.

   1. Take the original triangle and set a single translation rule, e.g. '5 squares right, 2 squares up', shown as a staircase arrow.
   2. Apply that identical move to ONE vertex first, then check that every other vertex has to follow the exact same move.
   3. Complete the translated shape and confirm its orientation and sense are unchanged (it reads the same way round as the original).
   4. Place the reflected shape and the translated shape side by side and describe the felt difference: the slide keeps sense, the flip reverses it.
   5. Report the translated image coordinates as (across, up).
3. 3

   Test congruence by tracing and overlaying

   Why it works: Reflection and translation change where a shape is (and reflection changes which way round it reads) but never its size or shape; the technical word for this sameness is 'congruent'.

   1. Trace the original triangle onto tracing paper (or cut out a card copy, noting that real paper/card has slight thickness and only approximates the ideal zero-thickness flat figure it represents).
   2. Lay the tracing directly over the translated image and confirm an exact match in size and shape; for the reflected image, turn the tracing paper OVER to make it match — this flip is itself the reversed sense, and the size and shape are still identical.
   3. Say aloud that the moved shape is congruent to the original: only its position (and, for the reflection, its sense) has changed, not its size or shape.
   4. Answer the challenge 'Is it a new, different triangle now?' — about the mathematical shape represented — by pointing to the exact overlay match.