### 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?

### How Far Does it Move?

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

# Speeding Up, Slowing Down

### Why do this problem?

This problem provides a visual context in which to consider how speed / 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. This problem could be used together with How far does it move?

### Possible approach

With the interactivity in its initial settings (i.e. dot in the centre) ask group to predict what the path of the red dot will be and what the speed-time graph will look like. Students 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 graph related to what students expected. Confirm understanding by asking what would happen if you changed the polygon to a triangle, a square, ... or a circle.

At this point you might choose to do this card sorting activity which looks at all the possible positions for the starting point for the dot on a triangle. Can the students match the triangle with its corresponding graph and locus?

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 speed-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 is placed on a vertex of a different polygon?
What happens if the dot is in the middle of a side of a polygon?

### Key questions

• Why does the graph jump up or down (i.e. why are there discontinuities)?
• 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 speed-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).

Students could create card sorting activities of their own.

The problem Up and Across develops this work further.

### Possible support

There are a number of variables affecting the speed /time graphs. Fix all but one of the variables and spend time making sense of the impact of that variable. For example, keeping the dot fixed at a vertex and examining what happens for the same polygon but for different sizes, then for different vertices before moving onto different polygons.

The card sorting activity mentioned above keeps the polygon fixed as a triangle. Other card sorting activities could be produced.