Saturday, August 1, 2009

Using "SAT-Type" Problems to Develop Understanding of Quadratic Functions in Algebra

f(x) = t-2(x+4)2 where t is a constant.
If f(-8.3) = f(a) and a > 0, what is the value of a?

This type of question is of the Grid-in type (or short constructed response) that now appears on standardized testing like the SAT-I and ADP Algebra 2.

I administered it to a group of strong SAT students recently and the students who completed Alg II struggled with it. As our president might say, this was a "teachable moment!"

A few thoughts...
Should textbooks include more questions of this type both as examples and regular homework exercises? As you might guess, I'm very much opposed to having questions labeled as Standardized Test Practice in texts or appear in a separate section of the text or in ancillaries.

By the way, by including the label "SAT-type problems" in the title of this post I'm trying to engender both positive and negative response. Those of you who have followed this blog for 2- 1/2 years know that what I'm really referring to are "conceptually-based questions." Some of you react adversely to the idea that standardized test questions should influence our curriculum or how we teach. N'est-ce pas?

Your comments...


Totally_clueless said...

Nice one, Dave!

I guess the advanced students start thinking more in terms of detailed solutions rather than looking at the basic aspects (or rather, big picture) like symmetry.

For the non-SAT advanced student, I would make it more interesting by saying f(-7.3)=f(a) where a>0, and see what they come up with.


Pat B said...

Nice problem. I would like to get back and try it with my kids... I focus on the vertex form a lot and really push symmetry, so I hope they would do well, but will shut up before I jinx them...
I think part of the problem is that we almost never encourage students write stuff like y-3 [or f(x)-3]= 2(x-4). Many software programs (TI-8*) for example, won't allow entries of this kind. I think the idea of "centers" and shifts are more difficult when they see a different shift inside the parentheses vrs outside.

Dave Marain said...

tc and Pat==
Thanks for your support!

Yes, I wonder what they would do with that clever variation. If they grasp the symmetry, it's obvious, but...

I agree with you that texts and curriculum should be more consistent with translations, since the point-slope form of a line and the general "h-k" forms for other conics follow your suggestion. Then of course they wouldn't find it so strange or confusing. However, the emphasis in most curricula/texts is on the function aspect for linear and quadratics so the "y =' or "f(x) =" forms have won out for now. This is usually students' introduction to functions so it's believed that the "y = " form is important. Until of course someone writes a text using your approach!

Now do either of you believe that some students would be test-savvy or clever enough to "plug in" a simple value for the parameter 't' and use a graphing calculator to solve this problem? Currently, testmakers believe a parameter discourages use of a calculator but I've seen otherwise!

Dave Marain said...

Another thought, Pat...
A variant of the form you suggested used to show up when studying the focus-directrix form of the parabola:
y-k = (1/4p)(x-h)^2.
I'm assuming this is still taught in most precalculus curricula. I can't imagine not discussing the significance of the focus of a parabola!