Problem solving – finding all the combinations
We have all given children the type of problem that requires them to
record all the possible combinations. These present many challenges:

As a teacher you need to decide whether to tell the class how many
combinations can be found or leave it open ended,

Children who do not work systematically loose track of the
combinations they have recorded and so are unsure what to record
next. This often results in combinations being repeated and
frustration!

It is difficult to help children find the combinations they have
omitted without looking in detail at what they have recorded. It
therefore becomes time-consuming to mark.
An important strategy children need to develop is the ability to work
systematically. The following is a suggestion that will help children in this
process and allow meaningful marking to take place alongside the children.
To introduce and strengthen the concept of working systematically the
children can be given a set of cards showing most, but not all, of the
possible combinations. Once given these cards their task is to sort and
classify them to enable them to spot the missing combinations. There are
usually a number of different ways to sort the cards and it is helpful to
share this with the children e.g. they could go round the room and
compare their method of sorting to that of others. All the children then
need to do is record the missing combinations. The marking is simple!
Once children have experienced sorting and classifying alongside finding
combinations they can be shown how to apply this to other questions.
Initially cut some paper into small pieces so they can create their own
combinations on cards and sort them to ensure there are none missing.
The marking can take place alongside the children as they explain the
combinations they have found and how they have sorted them.
When children have begun to develop the skill of recording combinations
systematically they can be encouraged to record them straight away
without needed the physical sorting process.
Some cards for children to sort and identify the missing combinations
have been produced for you to use.
Leicestershire Numeracy Team 2003
Rudolph put four stars
on a tree. He coloured
each star either red or
yellow. In how many
different ways can
Rudolph colour the four
stars?
R
Y
Y
Y
R
R
R
Y
Y
Y
R
Y
R
R
Y
R
Y
R
R
Y
R
Y
R
R
R
R
Y
R
Y
R
Y
Y
R
Y
R
R
Y
R
R
Y
R
R
R
R
R
Y
Y
Y
Y
Y
Y
R
R
Y
Y
Y
Cut out the cards and give them to the children to sort and classify.
There are two missing cards. What are they?
King Arnold has invited some alien
friends for tea (he hopes they
won’t eat him!). If the King sits
on his throne at the head of the
table in how many different
combinations can the aliens sit?
Here are 14 of the 24 possible
combinations. Sort the cards to
help you find the 10 missing
possibilities.
Jed
Jed
Jake
Jake
Jed
Jed
Jake
Jake
Jed
Jed
Jake
Jake
Jed and Jake are pirates. Between
them they have stolen some gold, some
money and some jewels. Can you work
out all the different combinations to
show what each pirate could have taken?
Six have been done for you. Cut them
out and sort them to help you find the
two missing combinations?