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# How Far Does it Move?

### Why do this problem?

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### Key questions

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Age 11 to 14

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

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 dot in the centre, ask the group to predict what the path of the 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 number of sides of the polygon.

When the group feel confident, move them on to more challenging situations by moving the dot to a vertex of a pentagon. Ask similar questions about the path of the 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?

Spend time on the path of the 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.

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

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

A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?

N people visit their friends staying N kilometres along the coast. Some walk along the cliff path at N km an hour, the rest go by car. How long is the road?