As posted on twitter.com/dmarain ...
Diagonal of a rectangle has length 6 and makes a 30° angle with a side.
(a) Area of rectangle=?
(b) If diagonal has length d, area=?
(1) A moderate difficulty problem for SATs? Appropriate or too hard for a PARCC assessment with both parts?
(2) Should diagram be given or is drawing part of what's being assessed?
(3) Will some students recognize that the expression in terms of 'd' is the formula for the area of an equilateral triangle of side length d? If no one does then is it our responsibility to model and facilitate "connection-making"? Uh, yes ...
More interestingly, will some students realize that the rectangle divides into two 30-60-90 triangles which can be rearranged to form an equilateral triangle? Hence, the title of this post!
If we create this kind of environment in our classes it may happen. I think we're all conditioned to thinking that's only for the top honors groups, and only for a few students. But, for me, helping ALL children discover and uncover the beauty of mathematics was my raison d'être for teaching. Idealistic perhaps but when that is lost, what's left?