Showing posts with label symmetry. Show all posts
Showing posts with label symmetry. Show all posts

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...

Saturday, March 10, 2007

Parabolas, SATs, Quadratic Functions, Symmetry, Oh My!

The new SAT and other state math assessments are or will be including more Algebra 2 types of questions, particularly those involving quadratic functions. The following was inspired by a recent SAT math problem. As usual, my goal here is not to give conundrums and 'puzzlers'. I'll leave that to the expertise of Jonathan over at jd2718! My intent is to provide enrichment and extensions of questions that students are doing in class. More time is required for these than is normally given for an example presented by the teacher. Hopefully these can be used in the classroom.
The original question on the SAT gave a particular length for segment PQ (see below) and that may be a more reasonable start for most Algebra 2 students. The objective here is to have students apply and extend their knowledge of quadratic functions, graphs, coordinates, symmetry, etc. There are several approaches to this question. If instruction enables students to investigate this problem for 10-15 minutes, students may discover alternate methods that will deepen their understanding of the material. The teacher's role is to gauge the ability level and background of the group to determine how much structure/guidance is needed. This is not obvious at all and requires considerable pedagogical skill and experience.

Consider the graph of the quadratic function f(x) = x2. Assume P, Q are points on the graph so that segment PQ is parallel to the x-axis and let the length of segment PQ be denoted by 2k.
If the graph of g(x) = b - x2, intersects the graph of f(x) at P and Q, express the value of b in terms of k.

Notes:
Encourage several methods, i.e., pair students and require that they find at least two different methods. This is critical to develop that quick thinker who always has the answer before anyone else and does not want to deepen his/her insight. Many students will need to start with a numerical value for the length of segment PQ, say 4. Symmetry is a key idea in this problem, not only with respect to the y-axis, but also with respect to segment PQ! Some will see this quickly, others won't. It is our obligation to think this through in advance and be prepared to guide the investigation. Those who believe this kind of activity is a waste of precious time (so much more content could be covered) will never understand why I believe 'less is more' when it comes to learning math. Profound understanding can never be rushed. Short-cuts, IMO, are PART of a discussion, not the objective. Try it! Can you find at least THREE ways?

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