### Why do this problem?

This
problem is the third of three related problems. The first
problem is

Mirror,
Mirror... , and the second is

...on
the Wall . All three problems ask students to consider the
effect of combining two transformations, and then challenge them to
describe the single transformations that produce the same
results.

This problem follows on from the other two

but works equally well on its own. It could be used as extension
work for students learning to draw enlargements from a centre of
enlargement, using fractional scale factors.

Alternatively, this problem/series of problems could provide
suitable task(s) for a unit of work on combined transformations.
Some knowledge of vector algebra might be useful for students
attempting a full proof.

### Possible approach

As an extension task, all that is needed is to provide the
problem as a worksheet to pairs of students who could then make
sense of it together. When they have established the combined
transformation for one specific example, a teacher intervention may
be appropriate, to move the focus to the general case, asking the
key questions below.

With a full class, encourage different students to start with
different flag positions. The teacher intervention above could
become a full class discussion. In theory, all students will have
the same combined transformation, which should be a perfect moment
for a comment on evidence versus proof.

### Key questions

What if the flag was in a different place?

What if you used the other point first?

What if the points were moved?

What, precisely, does the final position of the flag depend
on?

### Possible extension

Ask students to summarise their findings in exactly 20 words
(!) then ask if there is anything further that might be
varied.

Ask (suitably experienced) students to create a dynamic
geometry file that demonstrates their findings.

### Possible support

Spend time drawing accurate enlargements. In
theclass/groupevery enlargement could be drawn on paper with full
details written on and signed; then every student could alternate
between doing an enlargement and checking one off the pile.