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'Of All the Areas' printed from http://nrich.maths.org/
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.
Without speaking, draw equilateral triangles on the board, starting as shown in the problem. Write the areas of the first two or three triangles and place question marks next to the rest.
Allow time for reflection and discussion, drawing out ideas such as the use of non-standard units and the interesting result of square numbers.
Present the idea of tilted triangles, discussing how this might be defined before setting the challenge posed in the second part of the problem. A good point to discuss is how we know the triangles are equilateral - those who are convinced that the triangles are equilateral could explain their reasoning to those who aren't.
It is worthwhile giving the class some time to draw out the diagrams and try to come up with their own methods for finding the areas of the tilted triangles, and to share the methods that they find, but if they are struggling to find an efficient way, the pictures in the hints section might be useful as prompts.
Once a few areas have been found, encourage the learners to make conjectures about the areas of much larger triangles with tilt 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
How do you know the tilted triangles are equilateral?
How can you find the area of a tilted triangle in terms of the unit equilateral triangle?
Can you find a generalisation for the area of a tilted triangle?
Can you find a general rule for finding the area of any sized triangle with any tilt?
Are there any areas that it's impossible to make with a tilted triangle?
Focus on the justification of equilateral triangles and the calculation of areas rather than seeking generalisations.