ABCD is a rectangle in both figures. Figures are not drawn to scale.
1. In Figure 1, E is the midpoint of side BC. If the area of AECD is 1, what is the area of ABCD?
2. In Figure 2, E is a point on side AB such that AE:EB = 3:1. F is a point on side BC such that
BF:FC = 2:1. If the area of DEFC = 1, what is the area of ABCD?
(a) Although the first question is very similar to an actual SAT problem, both questions admit multiple solution paths and strategies that can help students develop both their spatial reasoning and analytical skills.
(b) The first question is similar to some textbook questions and would be rated as above-average difficulty on the SAT's, although most visiting this site would not consider it difficult. There are many variations on these kinds of problems. The most famous one is to consider the figure formed by connecting the midpoints of the sides of a rectangle and showing its area is half of the whole.
(c) The 2nd question is more discriminating and requires more than intuition.
(d) How many of today's students have a well-developed sense of fractional parts of the whole?
(e) Try to imagine how a variety of learners, say 9th or 10th graders, would approach these. Do you think the majority would attempt a visual approach - cutting up each irregular shape into common figures or perhaps dissecting the entire rectangle into equal parts? Would any students consider plugging in arbitrary lengths for the sides of the original rectangles even though specific values are given (one could then use similarity to finish it)? Is it a good strategy to encourage students to assume each rectangle is in fact a square? How many students would attempt an algebraic setup?
(f) What is the point of spending so much time discussing various approaches to a problem? Is it really worth the time expended when there is so much more material that one must complete? You all know my "less is more" mentality and that there are no shortcuts to developing problem-solving facility.
(g) Which is more important for learning to take place: Allowing the student to struggle but arrive at a solution with some strategic guidance from the instructor OR Allowing dialogue to occur enabling students to see how their peers are approaching it? What kinds of questions might the instructor ask in facilitating the activity? What exactly is our role when students are engaged? Is there a course one can take or a book from which educators can learn such pedagogy or does one learn from experience and/or from their peers?
(h) Is this more about solving geometry problems or developing problem-solving skills? Will I ever stop asking such inane questions...