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Why do this problem?
offers opportunties to consider different methods
of listing systematically. It can be used to introduce or revisit
sample space diagrams, and with some students, tree diagrams.
Play the game a few times for real.
"Is this a fair game? How can we be sure?"
Class work in pairs trying to decide and to develop an argument to
justify their conjectures.
After about ten minutes, stop to discuss the merits of different
arguments and representations. This may be an appropriate point to
highlight the benefits of different systematic methods for listing
all possibilities, using sample space diagrams and, if pupils have
met them before, tree diagrams.
Finding a fair game can become a class activity:
Students help to create a class list of all distinct starting
points for the game (for example, four ribbons can be either $1R$
and $3B$ or $2R$ and $2B$). These are written on the board for $3,
4, 5, \ldots$ribbons.
Distribute the task of checking which combinations are fair and
record them on the board as pairs of pupils decide.
There are not many solutions that work and if pupils are to notice
a pattern amongst the combinations that are fair they may need to
consider up to a total of $16$ ribbons.
Spend some time conjecturing about more than $16$ ribbons and
How can you decide if the game is fair?
How many goes do you think we need to be confident of the
likelihood of winning?
Are there efficient systems for recording the different possible
Can you justify your conclusions?
Justifying the general rule
When the work is being shared out amongst pupils in the class the
smaller number of ribbons can be given to pupils who struggle most.
They can also use physical objects, such as coloured counters, to
check their findings.