Of All the Areas
Why do this problem?
In this problem, students are encouraged to measure areas using a triangular unit and come up with a general formula for finding the area of a tilted triangle. It brings together geometrical thinking, algebra, and sequences, and offers a different perspective on Pythagoras' Theorem.
Possible approach
For a similar problem using the more familiar context of a square dotty grid and leading to Pythagoras's Theorem, see Tilted Squares. For introductory work on using a triangular unit to measure area, see Isometric Areas and More Isometric Areas.
Allow time for reflection and discussion, drawing out ideas such as the use of non-standard units and the interesting result of square numbers.
Once a few areas have been found, encourage the students to make conjectures about the areas of much larger triangles with a tilt of 1, and to justify their ideas. The lesson could be structured in a similar way to the lesson in these videos of the task Tilted Squares.
Key Questions
Possible Support
You could start with the problems Isometric Areas and More Isometric Areas.
Focus on the justification that the tilted triangles are equilateral, and on calculating the areas rather than seeking generalisations.
Possible Extension
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