Stretching fractions

Imagine a strip with a mark somewhere along it. Fold it in the middle so that the bottom reaches back to the top. Stetch it out to match the original length. Now where's the mark?
Age
14 to 16
Challenge level
filled star filled star filled star
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem



This problem is about an iterative process.

To iterate means to repeat, so an iterative process involves repeating something many times.

Imagine some dough, Plasticine or Blu-Tack, something that can be made into a strip then stretched.

We are going to take a length of this material, which we'll regard as the unit length, and put a mark at some fraction distance along it.

Now we are going to follow a procedure and see where our mark ends up.

  1. The material is folded in the middle so that the bottom reaches back to the top.

  2. The material is now only half a unit in length and twice as fat, so it is rolled out or stretched uniformly to become one unit in length again.

  3. Finally we'll note the new position of our mark.



And that's the process we'll be repeating.

Now let's try with an actual fraction.

Starting for example at $\frac{1}{5}$

Stretching Fractions


First Iteration : We fold to get

Stretching Fractions
then roll out, back to unit length


Stretching Fractions
The new position is $\frac{2}{5}$


Second Iteration : fold again


Stretching Fractions
and roll back out to unit length


Stretching Fractions
The position is now $\frac{4}{5}$


Third Iteration : fold again


Stretching Fractions
and roll back out to unit length


Stretching Fractions
The position is now $\frac{2}{5}$ for the second time


But we know what happens after $\frac{2}{5}$, it goes to $\frac{4}{5}$, then $\frac{2}{5}$ again, and so on for ever.


So what are you invited to investigate?

  1. What happens for other start fractions?

  2. Does everything go to a loop?

  3. What size loops appear and for what fractions?

In fact you'll want to describe those loops very carefully.

This problem came to our attention via an ATM workshop led by Dave Hewitt, from the School of Education, University of Birmingham.

We appreciate his permission to pass it on.