Not that long ago I was talking to a true great in education research, Zalman Usiskin. When we were talking about length and perimeter he questioned why we used perimeter in the title for this particular subset of waypoints in the framework. Zalman felt that in fact we should use the term paths, as a perimeter is a special type of path. I’ll be honest – this stopped me in my tracks and really made me think about what length, paths, perimeters are all about.
Not long after this discussion I was discussing fundamental ideas in maths with Martin Hyland. We were thinking about what some of the basic concepts are that we may take for granted but that younger children need to develop explicitly. Very often these ideas are crucial for later mathematical progression. The specific example applicable here is the fact that the shortest path between two points is the straight line between them (assuming they lie on a flat plain) and this is normally defined as the distance between the two points.
My mind wanders in this role. It’s a mix of beautiful fractals (I’ve been looking at similarity recently), carefully constructed Euclidean proofs (which I’ve never really completed in their full complexity) and arguing voices regarding the best approach.
Most recently whilst reading about the Erlanger Programme, which emphasises transformational geometry over Euclidean, I thought about how geometric shapes relate to time-distance graphs. This was seeded by a comment that motions (in the mathematical sense) and movement or displacement are not the same thing; which is a whole other argument and blog!
Imagine walking along the path that is the perimeter of a shape at a steady rate. Your time distance graph would look something like this:
But, what if the time distance graph was with respect to the ‘centre’ (I’ll leave you to decide how you define that one!) of the shape?
In the simplest case, a circle…
Why does this shape form this graph?
What about a square with side length m?
Why does this shape form this graph? What position on the square do the peaks and troughs represent? Now this is just my sketch, but what is the nature of the curves? Are they sections of quadratics?
What about a rectangle with side lengths n and m (n<m)?
Again why does this shape form this graph? What position on the rectangle do the peaks and troughs represent? As before, this is just my sketch, but what is the nature of the curves? Are they sections of quadratics? Are each of the curves translations of each other? Why are the values ½n, ½m and √(1/2n)2+(1/2m)2 significant?
In my mind this kind of activity is fascinating and would lead beautifully into looking at individual components such as the height above the centre of a shape. Consider this in relation to the centre of a circle and experience another beautiful entrance into the fascinating world of trigonometry.
Another approach would be that of Texas A&M Society of Physics students. Rather than travelling around the perimeter of the shape they designed a floor that would allow you to ride a bike with square wheels without bumping along. (https://www.youtube.com/watch?v=LgbWu8zJubo) Same problem different situation, still captivating, and now I want to ride a bike with rectangular wheels.
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SOMETHING TO TRY:
KS1: Having marked a spot in the classroom ask the pupils all to stand exactly three paces away from the spot. What shape have they made? What’s special about this shape? How could they use string, a drawing pin and a pencil to draw one?
KS2: Choose a selection of shapes each with a point identified somewhere on their interior. Ask pupils to find the places on the shape’s perimeters that are the furthest away from the point and those that are closest. Can there be more than one place? Ask them to explain their reason/method.
KS3: The red dot travels around the square at a constant speed. A graph is drawn of the time (x-axis) against height (y-axis) from the shown starting position. The two possible versions are shown below. Why are there two? What’s happening in each? What are the significant features of the graphs?
Draw the equivalent graphs for the following rectangle:
KS4: Draw the time distance graph, with respect to the centre, as you travel around a regular hexagon at a constant rate. Identify key positions along this journey on your graph.
KS5: Explain how you would draw the time distance graph, with respect to the centre, as you travel around a regular n-sided polygon at a constant rate. Identify key positions along this journey on your graph.
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