### Walk and Ride

How far have these students walked by the time the teacher's car reaches them after their bus broke down?

### Fence It

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

### Speeding Up, Slowing Down

Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the red dot affects its speed at each stage.

# How Far Does it Move?

### Why do this problem?

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.

### Possible approach

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?

### Key questions

• 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?

### Possible extension

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

### Possible support

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.