Sunday, 22 February 2009
Two Nice Questions
I have talked before about Dave Marain's web site at Math Notations. He has a new contest for grade ten and under that has a wide enough span of problems that it even attracks some middle schoolers. If you teach in the grade eight to eleven range, you might want to drop in, and perhaps participate. He covers some nice stuff.
In particular, he posts a Math Problem of the Day (with a solution) and all the ones I have seen have been really interesting challenges. Good for bright kids who need a challenge that textbooks often don't offer. One of the problems from his last contest illustrates a nice property that too few teacherx and students know about quadratics.
The question was:
" The graphs of y = 2x+3 and y = -x2 + bx + c intersect in 2 distinct points P and Q, where P is on the y-axis. Let V denote the vertex of the graph of the parabola.
(a) Determine all values of b for which the points Q and V coincide."
Most students who are at all clever know that the constant term, c, of a quadratic will give the y-intercept, and if you ask them what happens when you change "c", they will tell you it just moves the whole graph up or down without changing the shape. So if the two points intersect on the y-axis, they must have the same constant term.. ie, that the c term of the quadratic must be 3.
When asked what the A term of a quadratic does the usual answer is that it makes the quadratic "narrower." I really don't like this answer for several reasons. I hope students, even those who say the curve is "narrower", realize that it has a domain from negative to positive infinity, and therefore it never gets "narrow". What they mean, I hope, is that it goes up more quickly for larger values of A. and less quickly for A values closer to zero.
When you ask them about B, they are often less certain. They may tell you it moves the vertex (or the graph) left or right, and maybe they can even give a specific distance (someting about h==b/2A) but in fact there is a little more happening than that. In the image above, the red line is the locus being traced by the vertex as the variable b is animated and a is held as -1, while c=3. It looks quite clearly as if it follows a parabolic path with a leading coefficient that is the negative of the original function, and a vertex at the y-intercept. The confirmation of this should not be beyond a good Alg II, pre-calc student.
A couple of days ago Dave also posted a "Problem of the Day" about a regular dodecagon that got me thinking about a problem I haven't worked out yet. A regular Dodecagon for the Greek Impaired student is a twelve sided polygon. Think of a clock face with all the hour marks connected in sequence. What was given in the problem, is that if you draw a chord from any vertex, to the vertex four hours away, the length of this segment squared is the area of the dodecagon. Ok, I didn't know that. The proof was a pretty easy use of pre-calc tools, primarily the law of cosines and areas of isosceles triangles... but ... I wondered, how often does that happen? How often does a chord between two vertices of a regular polygon represent a quadrature of the figure (the square root of its area)... and what if we allowed easy extensions of this...(ie from a vertex to a midpoint of some edge).
Anyway, a big point, Dave's Problems of the Day are pretty good for provoking mathematical thought in students and teachers, so check them out.