Why do this
problem?
This problem helps learners to come to terms with the
concept of area and also to explore the properties of many
straight-sided shapes. It requires some quite difficult
visualisation when more complex shapes are used. However, the
beginning is very accessible so everyone can become involved
quickly.
Possible approach
You could start by showing the group the beginning of the
problem on the interactive whiteboard by creating multiple
copies of one square and then another square with sides
double the length. Alternatively, you could cut out squares
to attach to a board. Ask children how many of the small
squares will be needed to cover the large square and
encourage them to picture how they would do this, then to
describe it. Depending on the group, it might be a good idea
to continue by looking at the equilateral triangle all
together, which is the first shape that is not entirely
simple because one triangle has to be inverted. Again, ask
the learners to visualise how they think the four triangles
will fit before doing any sketching or manipulating.
Learners can then work in pairs on the problem from these
two sheets so that
they are able to talk through their ideas with a partner.
Scissors are essential and it also helps to have both squared
and isometric paper available for recording. This should be
'dotty paper' if possible, as it is easier to see the shapes
if only the necessary lines are drawn. (Click
here for dotty isometric paper.) Encourage learners to
visualise what they are going to do before they embark on
each new shape.
Some of the later shapes require considerable manipulation to
fit in the four smaller shapes. The hexagons are likely to
prove the first real difficulty and children will reach
different solutions. They may need to count little triangles on
the isometric paper to be convinced that they have actually
fitted four hexagons into the larger one. Possibly the simplest
solution is to cut one hexagon into three rhombi and fit all
together like this:
If many learners are having difficulties, the diagram above
could be drawn on the board. Isometric paper can be very useful
at this point.
At the end, with the whole group together, some solutions and
sketches could be discussed. You may want to draw out the fact
that doubling the sides of the shape has resulted in four times
the area, not double the area. With very visual work such as
this some children show unexpected talent and others, who
usually succeed at once, have blocks. This means that it is
specially helpful if later the children have a few minutes to
jot down their feelings.
How did you feel if you got stuck?
What or who helped you?
How did you feel when you cracked the problem?
Key questions
What do you need to do to fit in this shape?
What have you tried so far?
How will you remember how the shapes fitted together?
Possible extension
Learners could carry on using this sheet which
contains further shapes to explore. They could also draw some
shapes of their own on squared or isometric dotty paper.
Alternatively, or as well, they could go on to making the
bigger shapes three times the size of the small ones.
Possible support
Some children would benefit from having a go with differently
sized squares and rectangles until they have built up their
confidence.
