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?
Notes:
(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...
Monday, August 6, 2007
Two Geometry Problems - Dozens of Methods to Ponder...
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11 comments:
After a quick glance, I cast my voted for
1) 4/3 (approx. 1.3333...)
2) 24/13 (approx. 1.846)
I did them both by a ratio method, using variables for actual lengths of the sides of the rectangles.
The value depends on the students. In some cases this would make nice problem solving. In others, the problem solving might be rough, but the discussion after might have more value.
I want my students to know that shooting in numbers would work. And I think my students would be a bit gunshy about assuming that either figure were a square; I strongly emphasize using what I call "non-special" diagrams so that they don't accidentally suggest false conclusions to themselves.
I agree with Justin's answers.
On the first, I just "eyeballed" it (it is easy to see how to break the rectangle into 4 triangles equal in area to ABE). For the second, I plugged in variables for the ratios, dropped a perpendicular from E, and got areas for different parts of the diagram in terms of xy and then got a value for xy by knowing the area of the smaller quadrilateral, which I plugged into the total area formula.
As to Dave's questions:
I don't know enough HS students to know how they would approach these. I think my middle schoolers could do the first one. Maybe I'll try it out this fall.
I would not encourage students to assume the figures are squares. I think it takes a good deal of sophistication to see when you can make such an assumption "without loss of generality" and I think that would be dangerous ground at that level, for most kids.
(g) is the biggest question for me as a newish and untrained math teaching volunteer. What I've settled on over the past few years is a combination of two things. I generally see them on a Monday and the following Wednesday (then not again for 2 weeks) so what I tend to do is introduce a topic/ trick/ approach/ etc. on Monday, and give them problems to work on with me available to help during that class period. I have trouble knowing how to give hints without giving away too much, though. I suspect that comes with experience, more than any particular training.
Then I give them problems to "struggle with" for homework, with instructions to work on them for up to 30 minutes (I give enough work to keep the top students busy for 30 minutes, but tell the class not to expect to finish all the problems in that time, and to try to find problems they are most comfortable with to try first.) Then on Wednesday, the students who had success with some of the problems share their approaches. I'll also highlight if another student started down a good path but didn't quite get to the right answer, and we'll work out together as a class how we might finish the problem using that approach. If I did the problem a different way than they did, I'll show that too, and if I'm working from problems where I have solutions, and that's different, then I'll show that too. I try to show as many different ways of doing something as I can, because you never know what is going to "click" and stick with an individual student. And also, just seeing lots of different ways to approach things helps "prime the pump" for when they are stuck not knowing where to start with a new problem.
justin--
you nailed it! Any thoughts on the other issues I've raised? Also, it's nice to hear from a good friend.
jonathan--
The discussion afterward is where some of the most important learning takes place!
mathmom--
I agree with your approach. I would probably do things similarly. The issue of how to give hints and how to guide the investigation is so crucial. Yes, you learn from experience (which means, "from one's mistakes!"). Generally speaking, 'less is more' as usual. If you give away too much, there's no going back. If you give away too little, that's correctable. Once you know your students well, you'll know how much to push and pull each one.
I got 24/11 for the second one... I can't figure out my mistake. I'm thinking it might be that I assumed that the "1" part of the given ratios were the same for each side of the rectangle (that EB and FC are the same length). Is that an assumption I can make?
UGH! I just found my mistake. I broke the second rectangle into 12 square units, where DEFC is part of that whole. Subtracting out the smaller triangle I pulled the end-all-be-all of careless errors: I forgot to divide by two when calculating the area! After that I correctly set up and solved a proportion that gave me my answer.
That's one of my better qualities of being a teacher- I can anticipate most common errors (since I also make them).
I too struggle with how much to "help" my students (especially at the beginning of the year when we are all new to each other).
I don't want their frustration with a task to become so overwhelming that they give up - yet I don't want to immediately give the answer or a hint either. I want each of my students to develop the ability to know when their answer is "correct" or on the right track, without having to get input from me on every problem.
I too value the discussions after a problem that required effort. Often I wonder though, no matter how much I try to include each student - how many are actually processing what is being said?
Dave,
I think that main difficulty for students would be trying to figure out the values for the sides. They very often just want to put numbers in where there are none. I think in these cases, it would actually be more difficult if you tried to have numbers instead of variables.
Also, I left out the part where I broke them into smaller triangles, like mathmom. I think that's a vital piece to try to convey to students.
many many points to ponder here...
jackie--
As a dedicated, thoughtful and competent educator, you plan a lesson carefully and thoroughly and implement it the best you can. In the process of guiding and leading students toward enlightenment (after all, 'educate' means to lead forth), you ask many questions, walk among the students (rather than exclusively stand in front of the room), watching, listening, making suggestions. There will always be a few who are not invested but you goad and prod them lovingly anyway using humor (even sarcasm if not overdone or hurtful). You recognize that some students need a quiet place to reflect on the lesson and process the ideas/concepts in the lesson by themselves (I think best when no one is on top of me or staring at me or putting me under pressure). And then... We assess the learning. If time permits, we give the group another similar problem immediately or to try at home or the following day or several days later or on a traditional assessment. If the problem was very difficult, we can make it a bonus problem on a test and later make it count. The strategies go on forever, but there's no book that will teach you this. It's part of the art and science of teaching. But art isn't all inborn or instinctual. Much of it comes from taking risks, experimenting, reflecting, learning from our mistakes and after a few decades pretending you actually know what you're doing! The fact is that, from reading your comments, I already know what an outstanding teacher you will make. I hope that helps a little bit...
I will have more to say about Justin's insightful comments (and others too) as well as issues like plugging in numbers vs. an algebraic approach vs. making special assumptions about diagrams, but I wanted this comment to focus on the last paragraph of jackie's comment.
As always, I am deeply indebted to each of my readers and especially those who take the time to write such thoughtful statements. From the recent links to some of my posts (particularly the percent posts), it appears that other bloggers appreciate the dialogue that is taking place here and are recommending their readers to visit. However, they are making it clear that it is the comments (and, in particular, the intelligence and eloquence of the commenters) to my posts that make it all worthwhile - thank you!!
Dave,
It helps more than "just a little". Thank you for the eloquent reminder of the practices I have to continue to practice!
I realized reading comments that my own work is not represented.
I assume (without even thinking about it) that the area of the entire figure is unity.
One triangle is one-half of one-quarter of two-thirds or one-twelfth. The other is one-half of a whole times three-quarters, or three-eighths. Their sum is eleven-twenty-fourths. The quad'al is unity - the sum or thirteen twenty-fourths of unity. And then unity in this case must be the twenty-four thirteenths.
I don't know where I picked up this idea of 'unity,' but I use it in place of a variable. By burying the concept of variable, I make my work entirely into arithmetic. And, I hadn't realized, but I assumed that each rectangle was a unit x unit square.
And I like doing this stuff without pencil, when possible. You want to know what made me pause? Adding the fractions!
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