Symmetric Trace
Problem
Before we begin we need to check something - it's about symmetry.
A pattern continues forever in both directions.
Imagine it's on a roll of paper and two strips are torn off, one of which is turned upside-down and placed underneath the other.
It is not possible to shift the lower strip horizontally so that it lines up and matches the upper strip.
On the other hand for the next pattern. . .
Even with the second piece upside-down the two pieces can still be made to line up and match.
Now to start the real problem.
This problem is about that kind of symmetry.
The pattern is a trace from a point on a rolling wheel.
Before starting, you may find it useful to explore How far does it move? .
Point 1 is on the circumference of the wheel and its trace looks like this:
Trace One
Forget the wheel for a moment and just concentrate on the trace pattern.
If this trace was turned upside-down you would certainly not be able to line it up with itself.
Point 2 is somewhere inside the wheel and its trace looks like this :Trace Two
Would "Trace Two" line up with itself upside-down?
Justify your answer, if you can.
The third trace is made where a horizontal line from Point 1 intersects with a vertical line through the centre of the wheel. It looks like this :
Trace Three
Can "Trace Three" line up with itself upside-down?
Justify your answer this time.Getting Started
This is a Three Star Challenge so you won't be expecting a quick fix.
But as a hint : split the trace into chunks that mean something.Firstly an obvious chunk would be the trace for one revolution of the wheel because that's clearly going to repeat (the period).
You might notice that the first half and the second half of the period match each other in a particular way - how would you describe that kind of mathematical matching and, most importantly, can you account for it?
All three traces have this property but the "same when upside-down" quality isn't there for all of them.
What makes that "same when upside-down" work?Student Solutions
Also if you copy into Word, and rotate a copy, you could use one of the drawing tool to trace over one of the curves you want to compare and then move that new line around to see if it fits over the rotated copy.
Trace 2 does not line up with itself upside down. This can be spotted by looking at the lowest point on the line, the curve is sharper than at the highest point on the line, and so when rotated these two can not line up.
Trace 3 on the other hand does line up with itself upside down. This can be spotted by the fact that when you rotate the graph and slide it along it lines up. Another method is to draw a line at the top:
You can now notice that when the wheel has turned 180 degrees, then the rotated diagram is the same as the one above, and so the rotated graph does line up with itself.
Teachers' Resources
- To get a sense of how the wheel motion gives each trace.
- To identify and accurately describe the geometric properties possessed by these traces.
- And then to bring those two together in an account which connects what is observed in the trace with the characteristics of the motion.