The following is an open-ended problem for Algebra 2 students...
Enjoy it but I'd really like to hear how you might implement this in the classroom. Part of homework? A bonus? An open-ended activity in class? Students working independently or in pairs? Part of an assessment? At what point would you use this? At the end of the chapter on quadratic functions?
(a) Consider the function f(x) = 6x - x2.
If P and Q are the points of intersection of the graph of f with the x-axis and R is a point on the portion of the graph above the x-axis, what is the maximum area of triangle PQR?
(b) Consider the quadratic function whose x-intercepts are the nonzero numbers p and q, p > q, and whose y-intercept is -pq.
(i) Explain carefully why the graph of this function has a maximum point no matter what the signs of p and q are.
(ii) Write an expression for the y-coordinate of the vertex of the graph of this function in terms of p and q (simplified).
Note: This appears to be a standard problem using the formula -b/2a, but there are other approaches and the result may surprise you.
Wednesday, February 14, 2007
Algebra 2 Challenge for 2-15-07
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12 comments:
When you get your students responses, could you give us a breakdown of how many perople used each technique? I'd be interested in how many people recognize the importance of the base-altitude formulation of area in part (a), and how many compute the quadratic in (bi) as opposed to how many reason about the signs of p and q, and how many argue from symmetry in (bii).
insightful questions, eric! this is exactly why i carefully developed this question - to deepen student understanding by inviting a variety of approaches...
unfortunately, i do not teach this course, however, i will share this with our Algebra 2 and Precalculus teachers. They may want to try this on Friday, the day before the Feb vacation. I'm hoping other readers will want to do the same with their students but there's not a high probability of an algebra 2 teacher googling this topic or is there? actually, it's amazing how many teachers are looking for problems and/or activities on a variety of topics...pretty random...
I really liked b:i; what a neat relationship!
thanks, darmok!
i was trying to develop another expression for the maximum 'height' of an inverted parabola in terms of its x-intercepts; i started with the special case y= -x^2+c in which the x-intercepts were
p and -p which led to c = -p^2 (which is independent of the sign of p!) and generalized it. Our textbooks rarely ask students to delve beneath the surface and questions like these usually show up only on math contests. The non-honors student therefore does not get to see these kinds of challenges unless the teacher finds them and modifies them appropriately. i gave part (a) to allow most students to begin thinking about the ideas in a simpler context, although some may struggle with that one too. Part (a) is an SAT-type question.
What also gets me excited about math (and which i try to convey to my students) is how everything (well, almost?) in math is connected somehow.
Thus, in the quad eqn, -x^2+bx+c=0, 'b' is the opposite of the sum of the x-intercepts (roots), and '-c' is the product of the x-intercepts. Students might learn a formula like -b/a for the sum of the roots of a quadratic, but would they think of applying it in this problem? Some might, but it's one of the roles of the educator to think about these ideas so they can be communicated to the student. One of my purposes is to present some non-routine problems that require students to explore ideas rather than simply apply them mechanically as they are often asked to do. I'd like to see ALL students at least try a few more challenging problems! If presented properly, they may come to realize some of the beauty of our subject. C'mon, at least WE have to appreciate its beauty!!
oops, change 'c' in previous comment to c = p^2 not
-p^2; sorry...
Silly me. I actually wrote out a general quadratic function for (b) and found the coefficients, rather than recognizing that I should start with a(x-p)(x-q).
I like the graphical reasoning of (bi) that eric mentions.
TC,
Insecure students always try to compute too much. I do hope that students know what a parabola looks like, and that they can reason from it. I'd add this problem:
(biii) Why does the conclusion of (bi) fail if p = 0 or if q = 0? Why does it fail if p = q?
nice additions, Eric! I avoided these by requiring p>q and nonzero but it makes the question richer to let students analyze ALL cases...
tc, putting the equation into standard quad form is a natural thing to do! I think most of us would do that. I'm hoping students will see how graphical properties of parabolas and graphical interpretations in general are a powerful way of thinking
Math can definitely have beauty—certainly it can have intrinsic beauty, but especially when coupled to science.
Oops. I was wrong with part of my last post. The colclusion of (bi) still holds if p = q ≠ 0. However, the geometry is a bit different.
Oh, well. You know what we call it when one emphasizes coordinate algebra over geometrical thinking? Putting Descartes before the horse.
Just thought I'd let you know that I am an Algebra 2 teacher who stumbled across your page here via google, and intend to pose this question to my students sometime in the near future (we're just now reviewing factoring, haven't tied it to graphing quadratics, yet)
Just thought I'd let you know that I am an Algebra 2 teacher who stumbled across your page here via google, and intend to pose this question to my students sometime in the near future (we're just now reviewing factoring, haven't tied it to graphing quadratics, yet)
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