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 approaches.
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 arise.
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 is Making Squares