Up and across
Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the dot affects its vertical and horizontal movement at each stage.
Problem
Take a look at the interactivities below which show regular polygons "rolling" along the horizontal surface, leaving a trace of the path of the dot.
In the first interactivity, the graph records the horizontal distance travelled by the dot.
In the second interactivity, the graph records the height of the dot over time.
Experiment by positioning the dot at the centre of the polygons, at one of the vertices or at the centre of one of the sides of the polygons and explore how this affects the two graphs.
Challenge:
Can you now work out what produced the following two graphs?
Can you work out how many sides the polygon had and where the dot was placed?
Try to explain how you worked it out.
Getting Started
Try to approach the problem systematically.
As you go along try to understand why the graph takes the shape that it does:
- by relating it to the rolling polygon and the journey of the red dot
- by trying to predict what will happen before you set the polygon rolling
Could the dot have been on the centre of a polygon?
Try for each of the polygons.
Could the dot have been on the centre of the base of a polygon?
Try for each of the polygons.
Could the dot have been on the centre of one of the sloping sides of a polygon?
Try for each of the polygons.
Could the dot have been on the centre of a side opposite the base of a polygon?
Try for each of the polygons.
Could the dot have been on a vertex opposite the base of a polygon?
Try for each of the polygons.
Could the dot have been on a vertex on the base of a polygon?
Try for each of the polygons...
Alternatively...
- try all possible positions of the dot in a triangle,
- and then in a square,
- and then in a pentagon,
- and then in a hexagon...
Student Solutions
Congratulations to those of you who deduced the correct answer to this problem.
Cameron from Boroughbridge Primary was one of the first to send in a solution. He used a combination of observation and trial and error to begin the problem:
If you put the dot in the centre of a side and look at the height chart, it goes down and then immediately up again. This also happens on the height chart that you are trying to make. If you try this with different shapes, a hexagon makes the right pattern.
He then considered on which side the dot should be placed:
Then I knew that as the highest points were at the start, the dot must go on the centre of the side that is 1 anti-clockwise of the top side on a hexagon.
Robert from Cowbridge Comprehensive sent in a solution with slightly different reasoning:
The height did reach zero, but it did not stay at zero, meaning that the dot could not be on a corner, but had to be in the middle of a side.
The height graph had four of the curves before hitting zero, so the dot had to be on the fourth side to touch the ground.
Therefore the shape was a hexagon, with the dot in the middle of the upper left side.
Hi to 'The Nrich Team'
My 'top' S4 set were this month inspired by your various 'Rolling Polygon' problems.
The class divided into groups of 2 or 3 students. Exact values were the order of the day. The Equilateral triangle and Square were fine though much discussion was needed regarding the final form of the answers. The pentagon proved a lot more challenging with the Golden ratio eventually surfacing.
The groups spent over a week working on this investigation with excitement mounting as the sequence developed. Predictions were made at the stage when the hexagon revealed the number 12 as the 4th term. For the Septagon exact values were not possible so conjectures for the 5th term of the sequence were tested using very accurate calculator work (Sine Rule & Cosine Rule etc).
I feel that the challenge and sheer range of technique required for this investigation has benefited my students immensely. They are aged 14 to 15 and produced work of impressive depth and quality. Thanks for the stimulation that your questions have provided ... keep up the great work. I have attached the write-up produced by David, Nicholas and Robert as it was a superb exposition.... I'm sure you will agree! All the best Ken Nisbet
Teachers' Resources
Why do this problem?
This problem provides a visual context in which to consider how graphs can represent horizontal and vertical movement over time. It allows opportunities 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 is the third of three related problems. The first two problems are How far does it move? and Speeding up, slowing down .
Possible approach
With the dot in the centre, ask students to predict what the graphs 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 decision about the graphs.
Run the interactivity. Discuss how each graph relates to what students expected. Confirm understanding by asking what would happen if you changed the number of sides on the polygon. The pause button could be used to focus discussion on the different stages of the graphs and to make conjectures about what will follow.
At this point you might choose to do this card sorting activity which includes possible positions for the starting point for the dot on a triangle and a square. Can the students match these with the corresponding graphs?
When the group feel confident, move them on to more challenging situations by moving the point to a vertex of a pentagon. Ask similar questions about the path of the dot and the corresponding graphs. Allow plenty of time for discussing/comparing different ideas before running the interactivity, which you could also run with different polygon and point positions.
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?
Key questions
- Some graphs start at the origin and others don't. Can you explain why?
- Some graphs have sections that are horizontal and others don't. Can you explain why?
- Why is the horizontal speed not constant during each stage of the "journey"?
- If we change - (the polygon/position of dot) - what will be the same about the graphs and what will be different?
Possible support
There are a number of variables affecting the horizontal distance and and height graphs. Fix all but one of the variables and spend time making sense of the impact of that variable. For example, trying the dot at different vertices of one polygon, then looking at edges and interior points, before moving onto different polygons.
Possible extension
Pairs could draw out a pair of horizontal or height graphs they have generated and post them 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.