I haven't read any comments yet about this problem, but I promised an algebraic solution...
Represent the legs as 6 + x and 6 - x. It takes experience for students to think of something like this, but once they see it, they'll use it again.
Then the square of the hypotenuse will be (6+x)^2 + (6-x)^2 = 72 + 2x^2, which is minimized when x = 0, i.e., when the legs are both 6 and 6. Yes, the algebra is challenging here for most, but isn't it wonderful to see how this innocent problem connects to quadratic functions! BTW, most students will buy into the idea that minimizing the square of the hypotenuse leads to the desired result.
Using the same representation, the area = (1/2)(6+x)(6-x) = (1/2)(36-x^2). This is maximized when x = 0. Ah, the power of algebra. Of course, this derivation is not appropriate for most middle schoolers, but many will internalize this when substituting values for the legs. This can also be facilitated by choosing values like 6.1 and 5.9 and using the calculator to show that the hypotenuse will be slightly longer than 6radical2 and the area slightly less than 18.
I'm not suggesting that problems such as these 'a curriculum make', nor can one do these frequently, nor are they appropriate for all! I am suggesting that our students need to be exposed to some of these with time provided for discovery and dialogue at the high school level as well as middle school. Do you find many examples like these in our textbooks? If you do, pls send me the ISBN #. If not, I will try to provide a few more of my own!!
Stay tuned for Monday's Warm Up Problem. It's much more of a 5-minute opener and is designed for grades 5-8.
I'm risking losing a good part of my audience here by not ranting about inequities in education and the need for a national math curriculum. Never fear, I will return to that as well. The 'inner math child' in me is dominant right now however!
Saturday, January 6, 2007
Legs of a right triangle revisited...
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2 comments:
It might be easier for students, initially, to see that one leg is x and the other is 12-x. You can pursue the same Pythagorean Theorem analysis, and exercise their ability to find the vertex of a parabola by completing the square to put it in standard form.
The algebra feels manageable for high school students, but getting them to generate representations or models for the situation is very difficult. Almost all of my high school Geometry students (10th grade-ish) shut down when I ask a question like this.
However, I love the challenge that you're throwing out here. We should absolutely be able to ask questions like this, and have students rise to at least try to solve them. How do we create a classroom culture that supports that?
Thanks for the support mrc! I’ve taught this both ways, with students generally coming up with x and 12-x on their own. After they see the advantage of using the 6 + x, 6 – x approach most seem to prefer it. I can recall one very sharp young lady commenting, “Aren’t you just writing the sides in terms of how far they are away from the average of the 2 numbers, namely 6. After they revived me, I concurred!
As far as creating a classroom culture, most of the math problems we assign that are not simple mechanical exercises can lead to fruitful discussion if we ask probing questions that require students to go beyond the explicit curriculum. NCTM’s Curriculum Focal Points addresses the need at least up to 8th grade for a distillation of our math curriculum to insure that students are exposed to and can demonstrate understanding of essential math ideas. This is exactly what Bill Schmidt, the chair of our TIMSS committee meant when he argued for a more coherent vision with more depth and less breadth — in English, LESS IS MORE! As long as high school math teachers feel compelled to ‘cover the text’ there will never be enough time to allow students to gain a deeper understanding of topics. In the meantime I have a few suggestions for enriching what we currently teach. That was one the main purposes of this blog. These warmups were really intended to provoke reflection about HOW we develop conceptual understanding. The real meat lies in my comments following the problems. Some readers may choose to just put these problems up for the quick opener, others may decide to go further. That’s the prerogative of each educator out there. I’m only suggesting that one could go further. Now that we have the technologies available to us, students can explore outside the 40+ minutes of class time and submit their ideas for group reflection (wikis, threaded discussions) or for teacher comment. We’re just scratching the surface. It’s a really exciting time and, naturally, I’m winding down...
Dave
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