Another Quadratic Function Problem 2-16-07 through 2-20-07

You may want to read the comments for this post. Answers and possible solutions are discussed. There is also considerable discussion about teaching techniques for f(x-h).

The parabola problem from 2-15-07 generated some interesting discussion. I haven't had a chance to see it implemented with our Algebra 2 classes yet but I'll let you know if and when...
Today's problem is along the same lines. I'm trying to provide some problems that are exclusively high school math content for this time of year. There are dozens of outstanding problem-solving sites for MathCounts and similar middle-school competitions but there appears to be a dearth of secondary math problem-solving sites (or I haven't found them yet!). Again, how might one use the problem below? As a bonus or an extended in-class activity or a performance assessment or ??? How many would regard this question as suitable only for honors or accelerated students? My take is that if students are exposed to higher levels of thinking and know they are expected to learn how to do these and held accountable on an assessment, they will adjust. Not all will experience equal success but that's ok too! Many should be able to do part (a) or are my expectations way too high?

(a) Consider the quadratic function f(x) = 4(x+4)².
The graph intersects the line y = k, k>0, in 2 distinct points B and C.
The rectangle whose base is on the x-axis and 2 of whose vertices are B and C
has area 64. Determine the value of k. Show method clearly.

(b) Now let's generalize the result of (a).
Consider the quadratic function f(x) = a(x-h)², a>0.
The line y = k, k>0, intersects the graph of f in two distinct points B and C. The rectangle whose base is on the x-axis and two of whose vertices are B and C has area R.
(i) Explain graphically (not algebraically) why the area, R, of this rectangle is independent of h.
(ii) Express k in terms of a and R. Check that your formula for k gives the value you obtained from part (a).

Algebra 2 Challenge for 2-15-07

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.

An Open Letter to Susan Ohanian

A comment I sent to Susan Ohanian:

Susan,
I plan on expounding on the following comments in greater detail in my soon to be created blog and I invite you to visit it and comment often when it's ready...
I've only read some of your thoughts and feelings regarding standards, testing, NCLB, etc. I admire your passion and courage to take on the establishment and challenge the current 'de-constructive' movement as you might perceive it. I agree wholeheartedly that there is now a testing mania in our country that can potentially do more harm than good. Note that if I weren't a centrist I would have phrased that differently! [I bet you’re reacting negatively to my self-characterization since you may see only 2 camps here in this 'holy war’.] You're probably trying to read between my lines and predict whether I am your friend or foe. I hope we can be friends and enjoy our similarities and differences. I know we will disagree on my next few comments but I need to say them to you with the same emotion and passion you exhibit. Ok, here goes...
First, I do not see it as inconsistent that one can have reasonable clear goals for students and still nurture and allow children to develop in their own unique fashion. Since math is my specialty, I will use ladders with its rungs as my metaphor for the acquisition of mathematical skills and concepts. There are different math ladders to climb for the different parts of the whole of mathematics and these ladders are in fact dependent upon each other, but each ladder must be climbed rung by rung. You can;t get to the 5th rung from the bottom if the 2nd, 3rd, and 4th rungs are missing, This is the nature of mathematical knowledge as I see it. Now each child can make it to the next rung in a myriad of ways but she still needs to get there if she wants to continue climbing and eventually move on to more sophisticated math ladders that are even steeper and with more rungs. A child can climb math ladders at her own pace, stopping along the way or even needing to return to lower rungs or starting all over again to regain her strength.
This is how I view the need for math standards for bands of grade levels and ultimately for specific math courses at the high school. From my perspective, the mathematical exposure of a student sitting in classroom X in district Y should be approximately the same if that same student moved to classroom Z in district W. To me, this is a no-brainer because it's about content. No one tells me HOW I must teach my Advanced Placement students in Calculus. The College Board does insist however that my students must be exposed to the same core content -- the main ideas and principles of Calculus -- as every other student of AP Calculus. Never once in over 3 decades have I ever felt constrained by this or by the test. My creativity is not restricted, nor do I expect all my students to solve problems the same way. The dialogue in the class is fruitful and thought-provoking. I don't race through the content to finish ahead of everyone else because I understand the nature of how children learn. I believe with conviction that less is truly more when it comes to helping students develop conceptual understanding, but they still need to know their trig identities and the unit circle 'cold'!

In summary, for math at least, the journeys may be different but there are certain destinations one must have in K-8. After that, there can be many different destinations!

Again, good luck with your mission...
Sincerely,
Dave Marain