The Square Hole
If you take the side length of the yellow equilateral triangle as the unit for length, what would be the size of the hole ?
Alternatively, if the area of the yellow equilateral triangle is taken as the unit for area, what size is the hole then ?
Can you think of a reason why the yellow and purple triangles are equal in area ?
If the side length of the yellow triangle is taken as the unit for length, what length is the third side of the purple isosceles triangles ?
Incidentally, did you notice that the yellow and purple triangles have the same area ? This doesn't require the particular case of one triangle being equilateral, any rectangle split into 4 areas by its diagonals will do.
More obvious now ?
Anyway back to the area of the Square Hole :
Thank-you to Clem, and to Marta & Brittany from MaST Community Charter School, and others who sent in solutions.
Seeing the image as a 'hole' surrounded by four rectangles, with each rectangle made from $2$ yellow (equilateral) and $2$ purple triangles.
The 'height' of the equilateral triangles is $\sqrt{3}$ divided by 2
So the dimensions of each rectangle are $1$ and $\sqrt{3}$
The side of the square hole is therefore $\sqrt{3} - 1$
Why do this problem?
The bisected equilateral triangle ($30-60-90$) is an important shape for students to become familiar with. Each of the $4$ rectangles is made from $2$ equilateral triangles and $2$ isosceles triangles. Together one of each of those two, makes the $30-60$ right-angled triangle.
This problem forces consideration of the side lengths in surd form, and provides an opportunity to become familiar with manipulating surd forms.The switch from one quantity as the given unit to another, emphasises that the choice of unit is a choice for the problem solver and can be considered so as to make the relationships within a problem as clear as possible.
Possible approach
Preceding these questions with 'playtime' using cut out triangles to form patterns may be a very useful preliminary for many students.
Include shapes that have holes, and suggest/invite challenges with respect to that 'hole'.
And maybe include the challenge to lose the 'hole' but keep the square.
Key questions
-
What is the area of the yellow equilateral triangle in terms of its side length ?
- What is the relationship between the area of the yellow triangle and the area of the purple triangle ? [they are equal]
Possible extension
Possible support
There are a number of activities which can provide valuable auxiliary experiences for students working on this problem :
Some children 'play' a long time with cut out triangles arranged on a table, motivated by the aesthetic appeal of emerging pattern possibilities, this is excellent grounding for other mathematical ideas to be built on.