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...

Another Quadratic Function SAT Problem

Have you noticed the SAT Tips of the Week in the sidebar? These are intended both for math teachers and students.

The "new" SAT has a few 2nd year algebra questions and, typically, there is at least one 'parabola' problem usually expressed in function form. Here are two different versions - one multiple choice and one "grid-in" (student-generated response).
Can you predict which one might give most students more difficulty?
It might be interesting to list all of the skills, knowledge and concepts being tested here. Are all of these typically included in your Algebra 2 course? Do students get enough exposure to these kinds of problems?


Version I
For some constant r, the graph of the quadratic function f(x) = -x² + 2rx is a parabola with x-intercepts at P and Q and vertex V. What is the area of ΔPQV, in terms of r?

(A) r² (B) r³ (C) 2r²

(D) 2r³ (E) 4r³


Version II

For some constant r, the graph of the quadratic function f(x) = -x² + 2rx is a parabola with x-intercepts at P and Q and vertex V. If the area of ΔPQV equals 27, what is the value of r?

Answers, solutions, strategies and comments will appear below the Read more...



Answers/Solutions/Comments/...

Version I
Answer: (B) r³

Possible Solution (no frills):
Factoring, we have f(x) = -x(x-2r); x-intercepts are 0 and 2r. Therefore base of triangle has length 2r.
The x-coordinate of the vertex is r (why?), so y-coord = f(r) = -r(r-2r) = r².
Area of triangle = (1/2)(2r)(r²) = r³.

Version II
Answer: r = 3

Possible Solution:
From Version I, we obtain r³ = 27, so r = 3.

Comments

• These kinds of questions typically appear among the last 3-4 problems on a section, meaning they are of above-average difficulty. Students who are in Algebra 2 or beyond should definitely attempt it. After reviewing it, most students may conclude it's not very hard at all!
  
• Testmakers are more frequently using a parameter like 'r' to make it more difficult to merely punch it into the graphing calculator and read off the intercepts and vertex.
• This kind of question could also appear on standardized tests like the Algebra 2 End of Course Exam from Achieve/ADP or other state tests.
• Pick up a copy of 10 Real SATs from the College Board to find several other practice problems like this.
• One could modify and extend these problems in many ways. For example inscribe the parabolic region (bounded by the cruve and the x-axis) in a rectangle and determine its area, a simple variation. More interestingly is to note that the ratio of the area enclosed by the parabola to the area of the rectangle is 2:3, a famous result proved in Calculus.
• Skills, knowledge required for this question? Worth enumerating in my opinion...
  
• I also believe strongly that our students should be tackling these kinds of problems on a regular basis to deepen their understanding of the relationship among the function, the coordinates of key points and the geometry. This used to be known as Analytic Geometry.
  

...Read more

Another Quadratic Function Challenge

The line y = k intersects the graph of y = 3x² in points A and B. Points C and D are on the x-axis and ABCD is a rectangle. If the area of ABCD = 128/9, what is the value of k? Calculators not allowed. Show your method clearly!

The above question is designed for students in their second year of algebra or precalculus. It can also be used for practice for the Algebra 2 questions on the SAT, although it is somewhat above that level. Many students, including the more advanced, tend to struggle with problems like these because they don't have that much experience with coordinate geometry questions. These types of problems are critical for their later development. Once the students have done a few of these they do not find them so formidable. A useful pedagogical tool is to let them try it, review the method clearly, then erase the board and call on students to recall each step. Tell them you will do this, encouraging them to take good notes and pay careful attention! When using it on an assessment, make it a bonus the first time, then make it count. Some of my readers may recall a similar parabola problem a few months ago.

What goes up...Applying Quadratic Functions

This question was inspired by a released SAT question from a couple of years ago. Some of you may recall this question that appeared on the first released Sample Test for the 'new' SAT. Can you think of some reasons why this question was used by ETS as a Sample problem?

Because the height function given was not exactly in 'standard' form (using a,h, and k), even the strongest students I administered this problem to resorted to complicated algebra or used a physics formula (s = 0.5at²+...). They missed the point that when given the vertex of a parabola you're given more than just an ordered pair! I believe our students need more experience with this type of 'free-response' application. We see these in some textbooks but is enough time devoted to them or is that left to the physics teacher?

As usual, I've modified the question and developed it into an open-ended problem with several parts. Pls don't get exercised about the lack of reality of the physical model!

A model rocket is projected vertically upward from a point 877.5 ft above the ground and after 2.5 seconds reaches its maximum height of 1440 ft. We are given its height above the ground as a function of t:

h(t) = p(q-3t)² + r, where t is in sec, h is in ft; p, q, r are constants.


(a) Determine the values of p, q and r.

(b) Rewrite the given function in 'standard' form: h(t) = a(t-h)² + k.

(c) Determine, algebraically, all values of t for which h(t) = 1080. Explain, in terms of the motion of the rocket, why there are exactly 2 such values.

(d) After how many seconds did the object hit the ground? Use algebra.

(e) Verify your results by analyzing the function using graphing calculator technology.

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?