This extended investigation problem will encourage creative
hypothesis making and testing. Hypotheses can be tested out on the
pattern maker giving instant feedback on whether a hypothesis needs
to be rejected.The code system will encourage translation of vague
visual statements into precise mathematical statements.
Get a feel for the interactivity and the resulting codes and
grids. What patterns can we see? What differences or similarities?
Stress that, whilst the computer uses a deterministic algorithm to
determine the chance of a pattern being selected randomly this
cannot ever be sure that a pattern was random or not in the way
described. However, students can create firm hypotheses to test out
the computer's algorithm. This can then be tested against the
computer. Whilst it is not possible to accept a hypothesis with
certainty, it is possible to reject a hypothesis with certainty if
a run of the computer provides contradictory evidence.
Students will need to grapple with these ideas and should be
prepared to mull over the problem over a period of time. Perhaps
the patterns could be printed out and displayed for students to
consider at their leisure.
- In what way are the patterns random?
- What features do codes share and how do they differ?
- How does changing a pattern slightly affect the code?
- How might you group different patterns together? Would certain
groupings of patterns be considered more or less likely than
- Can you think of any patterns which would be very, very
unlikely to be generated at radom?
Students might consider different ways in which they might choose
to decide on the likelihood of a given pattern. Could they produce
an algorithm? An example might be to declare a pattern unlikely if
it contains less then 5 connected regions.
Try the simpler problem Random