This problem provides a visual context in which to consider how
distance / time graphs represent movement over time. It allows
opportunity for learners to discuss and refine their ideas. Asking
learners to predict, to justify their predictions and to consider
modifying their views can help address misconceptions and improve
understanding.

With the interactivity in its initial settings (i.e. dot in
the centre) ask the group to predict what the path of the red dot
will be and what the distance-time graph will look like. Learners
could sketch the path and graph in advance, before seeing the
polygon roll. Their suggestions could be compared and discussed
before making a final joint judgement on the shapes of the path and
the graph.

Run the interactivity. Discuss how the graphs related to what
learners expected. Confirm understanding by asking what would
happen if you changed the polygon to a triangle, a square, ... or a
circle.

When the group feel confident, move them on to more
challenging situations by moving the red point to a vertex of a
pentagon. Ask similar questions about the path of the red dot and
the distance-time graph.

Allow plenty of time for discussing/comparing different ideas
before running the interactivity. The pause button is useful to
focus on the different stages of the journey and to ask for
conjectures about what will follow.

Ask pairs or groups to work on new questions, agreeing and
drawing the graph and path together before using the interactivity
to confirm their ideas.

Suitable questions are:

What happens if the dot is moved to a different vertex?

What happens if the dot isplaced on a vertex of a different
polygon?

What happens if the dot is in the middle of a side of a
polygon?

- What does the gradient of the graph relate to?
- Why does the dot speed up and slow down at different stages of the "journeys"?
- If we change - (the polygon/position of dot) - what will be the same about the graph and what will be different?

Pairs could draw out a distance-time graph they have generated
and post it as a challenge for others to establish which initial
settings were used (in the least number of guesses).

Imagine a rectangle, a semicircle or some other shape rolling
along.

Follow-up problems concentrating on speed-time graphs and
$(x,y)$ position against time are
Speeding Up, Slowing Down and
Up and Across

Spend time on the path of the red dot for different polygons
and positions so that learners become confident with predicting its
locus.

Talk with students about the time the polygon took to turn in
one part of its rotation, and how far the dot would travel if
placed in various positions for that rotation. Draw conclusions
about the speeds it would be travelling, in each of those
positions. Talk through speed-while-rotating in relation to radius,
give real examples - like being in different positions on a
children's roundabout, or attempting to run to keep up with a long
rotating arm/beam etc.