Why do this problem?
allows students to explore new ways to see
(visualise) the 'tilted square' figure. It combines work on
coordinates and area with possibel extensions into Pythagoras'
theorem. There is the potential for different insights
that can be utilised to invite discussion about different
A visualisation can be grasped quite easily when someone points
it out, but it is more satisfying and much better for the students'
development if they gradually feel their way around the structure
with moments of revelation.
Invite learners to create tilted squares of their own, identify
coordinates of diagonally opposite corners. Can they usethese to
help to find areas? Share ideas and generalisations as they
Connections may take time to emerge and different insights might
result in different approaches. For example the area of the tilted
square might be found through considering one of the squares and a
rectangle or seen as half way between the areas of the smaller and
larger squares. Give space for learners to find their own
visualisation and share different ideas and approaches.
One important configuration to watch for is this one:
- Can you work form some specific cases to the general?
- How do the areas of the squares and rectangles relate?
- How do the coordinates of opposite coordinates relate to the
dimensions of the inner and outer squares?
Students may be familiar already with a proof of Pythagoras'
Theorem based on this. If not, this is a good moment to include it
and connect the ideas associated with this form.
A problem which focuses on finding the areas of tilted squares