The statement in the prompt is always true. If, for example, the lines are perpendicular, then the reflections can be represented by a 180o rotation with a centre where the lines intersect. Students' initial questions have included:
- What do 'map', 'object', and 'image' mean?
- How many (and what) transformations are there?
- How would you arrange the lines of symmetry?
- Can you do more than two reflections?
- Could we use any shape?
- You can only get back to the object by doing two 'reverse' reflections.
- Which way would the line of symmetry be facing if you could do the two reflections with one reflection only?
- Could the shape be three-dimensional?
- Does it work for any shape?
In the orientation phase, students, with surprising frequency, present a diagram like the one on the right to illustrate the prompt.
The inquiry has proceeded with a discussion of how the two lines might be arranged (and the templates below have proved useful). After that, students might suggest trying different combinations of transformations, which leads to listing all the pairs involving reflection, rotation and translation.
Questions that have arisen at this point include:
- Does the order of the transformations matter?
- Will you be able to use the same transformation for a 90o or 270o rotation combined with a translation or reflection as you do for a rotation of 180o?
- How is the result different if the shape has one or more lines of reflection? (Shall we exclude shapes with lines of symmetry?)
- Is it possible to do the combined transformation in more than one way?
This inquiry has worked well with all year groups in secondary school. Extra challenge can be developed by including an enlargement as one of the transformations:
- Is it possible to map the object onto the image of an enlargement combined with another transformation using a single enlargement that has a negative scale factor?
In this inquiry, students often decide to work in pairs or groups with each one inquiring into specific combinations. Their results can feed into a document of the class's findings (see the examples below).
These are the outcomes of inquiries into the prompt by year 8 all-attainment classes:
Templates Pairs of perpendicular and parallel lines for two reflections.