This
problem helps students to consolidate their understanding of
how equations of the form $y=mx+c$ describe the gradient and
position of lines. Students explore the effect of combining
reflections and translations and justify why a pair of
transformations is (or isn't) commutative.

This is good preparation for future work on transforming the
graphs of more complicated functions.

This builds on Translating
Lines and Reflecting
Lines.

Start by introducing the problem. Hand out this
set of transformation cards and suggest that students work in
pairs, recording the order of transformations they have tried, and
the eventual outcomes. Encourage students to work in rough,
sketching just the essential properties of the lines at each stage.
Allow some working time before bringing the class together.

Suggest to the class that findings could be shared, and
organise the board so that all the results can be displayed:
eventual outcomes and the corresponding transformations. Invite
students to record the results they have so far. In order to avoid
mistakes creeping in, suggest students tick any results they can
confirm.

Allow some more working time, with students adding to the
record of results. The class may wish to organise themselves so
that different groups are allocated the job of investigating
particular combinations.

Questions that may emerge:

How do we know that we have considered all the possible
combinations?

Why are all the eventual outcomes parallel to each
other?

How many different outcomes are possible?

Could we have predicted the number of different outcomes in
advance?

Why do certain combinations produce the same outcome as
others?

This can lead to a discussion about commutativity: why does
the order matter for some pairs of transformations but not for
others? Once students have established which pairs of
transformations are commutative, it is worth returning to the
results on the board to appreciate why some combinations led to the
same eventual outcome.

How does each transformation affect the gradient and the
intercept?

Why do some pairs of transformations produce the same outcome
regardless of the order?

Students could consider what happens when the same four
transformations are applied to different starting graphs. What
similarities are there between results? Can they be
explained?

Students could also consider the effect on a graph of rotation
through multiples of 90 degrees and investigate what happens when
these are combined with translations and reflections.

Ensure that students are secure about the relationship between
a line's properties and its equation. Encourage students to sketch
the graphs of different equations and then use the interactivity
below to check their work.

This text is usually replaced by the Flash movie.