Monday, July 30, 2012

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.



  1. 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.

  2. @Jeff
    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