circles circumference factors pattern tasks

Bike Trail Task

There is two hour parking all around University of Saskatchewan. I once went to move my car (to avoid a ticket) and found that the parking attendant had marked–in chalk–the top of my tire. I wanted to erase the mark so began driving through as many puddles as possible.
I then convinced myself to find a puddle longer than the circumference of my tire–to guarantee a clean slate and a fresh two hours.
As I walked back to campus, I got thinking about the pattern left behind by my tires. For simplicity, let’s take the case of a smaller vehicle–a bike.

If you were to ride a bike through a puddle of a certain width, the trail would look like this:
Is this model correct? Evenly spaced iterations of puddle-width splotches.
Assume that:
width(puddle) < circumference(tire)
and consider the following bike-ish contraptions. Can you predict the pattern? Better yet, can you draw an accurate prediction on graph paper? Assume a six-inch puddle (why not?)
That is the task I present to the students. The emerging patterns are interesting.
Unicycles–one wheel; one pattern.
But now combine them. (Of course, the bike goes in a perfectly straight line…)
A standard bicycle– two wheels; same size.
Alter it slightly. (You may want to encourage colour coding for overlapping paths…)
Old school–two wheels; different sizes.
Exaggerate the difference.
Crazy old school–two wheels; way different sizes.
How does the pattern change? Is it important to know how far apart the wheels are? (Experiment…)
 Just for fun–4 wheels; 3 tracks; 2 sizes.
What do you notice about certain radii? What causes certain patterns to “line-up”?
An interesting task to give a class working on circles, algebraic manipulation, factors, etc.

2 replies on “Bike Trail Task”

I like this idea a lot as an opener/review for circles and circumference, and I also love the pattern-making aspect. You could stretch this idea as far as you wanted – how would it look for a big rig? A tractor? A square tire?

I thought about doing something similar to this the last time I painted my living room (with a roller), but I couldn't quite piece it together in my head – so thanks for putting the idea in a useable form. Speaking of painting…could you drive a bike through paint and use it to mark the center lines? The side lines? How big would the paint "puddle" need to be? What would a vehicle look like that could paint all of the lines at once? OK, now I'm just spitballing, so I'll stop. Thanks again.

Thanks for reading.
I was thinking about the road lines as well. Often I see where a car has driven over them and left a pattern repeating on the road.
The machine may be a chance to incorporate rates into the discussion.
I think "spitballing" is very important; it accesses our creativity.
Thanks again for reading

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