[Update as of 6-2-07: Solutions to most parts of the problem are now posted in the Comments.]
The following problem is typical of a somewhat difficult Algebra 2/Precalculus type of SAT problem. Students who have been shown the 'quick and easy way' to solve this definitely have an advantage when taking the test. This has been developed into an activity for Algebra 2 students who might be taking the SAT on June 2nd. However, the activity explores more than just a strategy for solving the problem efficiently to get an answer for the test. Students will be asked to solve the problem the traditional way as well and analyze why this question should not appear on the test. Two separate graphical interpretations are included to deepen student understanding, one for Algebra 2 students and a more sophisticated one for the Precalculus class. Is it worth spending 25 minutes or more on one problem in class? I'll let you judge...
Consider the system of equations:
1/x + 1/y = 1/4
1/(x+y) = 1/3
(a) (SAT-type of thinking): Show that xy = 12 (without solving for x and y).
Note: I'm giving the 'answer' here so that students will focus on the method; also for part (f).
(b) (Algebra 2): Solve the system (i.e., find all possible solutions for x and y). Write your answer(s) as ordered pairs.
(c) Verify, algebraically, that your solutions satisfy the original equations.
(d) Explain why this exact question should not appear on the SATs (although this type of question has appeared frequently).
(e) (Precalculus extension): Graph the system:
1/x + 1/y = 1/4
1/(x+y) = 1/3
Notes/Hints:
You must graph each equation separately. Do not replace the first equation by xy = 12.
Hint: Solve each equation for y.
(i) Explain clearly why the graph of the first equation has both vertical and horizontal asymptotes. Label these in your graph.
Note: The customary way is to solve for y first. See if you can also determine the asymptotes without changing the form of the equation!
(ii) Does your graph of the first equation contain the origin? Explain why or why not.
(iii) How many solutions of the system are evident from the graph? Explain the reason for this.
(f) (Algebra 2): Now solve the system
x+y = 3
xy = 12
(i) In what way is this system equivalent to the original system?
(ii) How many solutions of this system are evident from the graph? Explain the reason for this.
Saturday, May 26, 2007
Investigating an SAT Algebra Problem - Going Beyond the SAT Strategy for Deeper Meaning
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9 comments:
Here's one more connection you can make, though it is somewhat accidental.
The Arithmetic Mean of two numbers is the simple average:
A(x,y) = (x+y)/2.
The Geometric Mean of two numbers is the square root of the product:
G(x,y) = √(xy)
The Harmonic mean of two numbers is the riciprocal of the arithmetic mean of the reciprocals:
H(x,y) = 1/A(1/x, 1/y)
= 1/{[(1/x)+(1/y)]/2}
= 2xy/(x+y).
Now, and this can be done in Algebra 2 classes, for any positive x and y,
A(x,y) ≥ G(x,y), with equality if and only if x = y.
How? Square both sides.
Does (x+y)^2/4 ≥ xy?
Does (x+y)^2 ≥ 4xy?
Does x^2 + 2xy + y^2 ≥4xy?
Does x^2 −2xy + y^2 ≥ 0?
Yes! The left side is (x − y)^2.
It's easy to work this backwards. You do have to justify each step, but you are never dividing by a variable expression, and so the task is easy.
Now, remember H? H(x,y) = 1/A(1/x, 1/y).
Now, A(x,y) ≥ G(x,y). what happens if we replace x and y with their reciprocals?
A(1/x, 1/y) ≥ G(1/x, 1/y).
But, that's also
1/H(x,y) ≥ 1/G(x,y). Prove the G part. It's a snap.
So, H(x,y) ≤ G(x,y). In fact,
A(x,y) ≥ G(x,y) ≥ H(x,y).
What are A and H for the x and y in your example?
A is 3/2. H is 8. Something is wrong there.
I could tell something was wrong with the problem early, but that just formalized it.
Eric--
As always, you go several steps beyond! Relating the problem to the AM-GM-HM inequalities is nice... You correctly intuited and proved that the variables cannot represent real numbers in this example.
Now, here's my point. Sometimes when these types of problems appear on standardized tests, they may involve non-real values of the variables even though the final result is usually a nice rational. Technically, ALL variables on the SATs represent real numbers only - it says so!
I devised this problem to demonstrate this issue and to lead the student to a deeper analysis than just a quick method to 'beat' the queston. We reap what we sow. If we stress in class only the clever efficient method, then our students will look only for the most expedient approach, rather than a more thorough analysis.
I know that you understand that I'm writing these questions to serve as models so that math educators can go beyond the SAT problems and beyond the superficialities one finds in most texts. I'm not sure any student will appreciate these open-ended questions, but it's all about planting seeds, isn't it Eric?
Certianly. That is too mean a trick for the SATs though; people would be angered by the form of parts b and c.
On the other hand, there's another lesson about the world in all this. People make mistakes. Teachers, professors, supervisors, and engineers are not immune to errors. I made a silly mistake on this blog earlier in the week. One of the skills your students should learn is how to discern when a problem is erroneous. A student in a chemistry lab who can't tell that the numbers he is given are obviously wrong will not last long. A student who 'knows' the correct answer can manipulate bad data to get the desired answer, even if his assumptions be wrong.
Here's an example from organic chemist Derek Lowe's blog, In the Pipeline:
"We were figuring out the molecular weight of benzoic acid by adding increasing amounts of it to a solution (toluene, I seem to recall) and seeing how much the boiling point increased. We then plotted this out, running it through Raoult's Law to get the answer."
The correct answer is 122, but he and his fellow students were getting 244. Then, one of the students realized that benzoic acid forms a dimer, a compound of two acid molecules, thus making the weight 2·122=244. A good lesson, no?
Some other students chose to use only observations that were close to 122 to get the answer they 'knew' was correct. One hopes they learned a lesson from their mistakes, but one doubts it.
One can learn how to discern when the data one is given is invalid, either by being totally mistaken, or by being subtly manipulated. But, one needs to have the possibility in mind before one can recognize it. Numbers can lie if one choose them incorrectly. Numbers can be wrong if they are calculated incorrectly. The wise engineer and scientist can sometimes discover this before it's too late.
When engineers and scientists accept bad data, or are ineffective in announcing that the data are bad, people die. Consider Challenger.
When scientists and politicians allow themselves to be fooled by maliciously-created or chosen data, much suffering can occur. Consider Cyril Burt.
But, I wouldn't post such a problem on the SAT. Incidentally, a perfect example of the way out-of-domain values can appear in the middle of a calculation comes from the equation for finding roots of a cubic polynomial. It turns out that if c(x) is a cubic in x that has three distinct real roots, Cardano's formula for the solutions will always involve complex numbers. You cannot calculate the answer without leaving the real domain temporarily. And, people back then had problems accepting negative numbers!
Eric--
Pretty harsh words there! I chose this problem not as an exact replica of an SAT problem (which I always avoid for copyright reasons) but because I wanted to present an example that was multifaceted. On the surface, a problem that could easily be handled by good algebraic techniques (students would call this a 'trick' but of course it isn't!). Then I left the realm of SAT problems and asked the student to delve deeper and actually solve the system, which leads to imaginary solutions, and to do the messy calculations to verify it. I think my post might have been more effective if I omitted (b) and (c) since the graphical piece below is fairly powerful, particularly the asymptote questions. However, I left it in precisely because it is messy and 'complex' (double entendre is intentional). I'm not proposing that this question would ever 'make the cut' and actually appear on the test. I was using the question merely as a vehicle.
Now onto your other profound points. Errors are made on standardized tests, although rarely on the actual SAT because of the number of individuals editing and checking for validity as well as the field testing of questions. The errors on the problems I write or the solutions I attempt are far more likely given the lack of quality control, i.e., no one else gets to edit them before I publish Fortunately I have you and a few others to check for accuracy and correct my errors!
The problem I posted is a common type. In some instances I've seen, it was possible to guess the values of x and y as they were small positive integers. However, in several Algebra 2 text books (containing sample standardized test questions) and in SAT prep books, I have seen several instances of 'impossible' equations leading to straightforward results. This happens most likely because the author did not take the time to check the actual values of the variables, an easy oversight to make when rushed (and in our society most of us are rushed to complete tasks and meet deadlines!). It's all about quality control, isn't it!
Your references to the Challenger and Cyril Burt were fascinating, again highlighting the importance of quality control. I read the background on Burt. It appears that the verdict is still out on the degree to which his conclusions were based on 'cooked' data although a majority have dimissed him as a fraud. My instinct is that he and all the other Bell Curve-type theorists of human intelligence have a preconceived idea of the genetic factors in intelligence and then proceed to 'find' the data to support their beliefs. I took on John Derbyshire early on in my blog to dispute his claims about the preeminence of heritabilty factors, although my arguments surely had no effect on him. I do believe however that there are definitely factors of intelligence that are inherited. It's when we discuss 'general intelligence' that my blood boils. Each human being, in my opinion, has unique intellectual strengths, not always the kind that is tested on some IQ test or standardized test. Gardner's theories on Multiple Intelligences have, IMHO, dealt a severe blow to the IQ gurus.
This discussion merits far more than a footnote to a system of algebraic equations and an SAT exercise, doesn't it? I'll stop here for now...
I wasn't trying to be harsh to you. That was never my intention. I was pointing out that, rarely on standardized tests, but more often in real life, one is faced with invalid data. Sometimes, the data comes from poor procedures. Sometimes, the data comes from incorrect selections from valid data. And, sometimes the data are contrived for effect. It is part of one's job to discern when data make no sense and are erroneous.
It isn't easy, however. Do you remember the old task from a probability and statistic course? A teacher tells the students to take a coin and either
1) Flip the coin 100 times, recording the results, or
2) Fake the process, writing 100 Ts or Hs, trying to simulate coin flips.
The student must remember which procedure he took, but should keep it secret. The teacher can almost always tell which procedure the student did. Fakers tend not to have long runs of Ts or Hs; yet you'd expect at least one run of 6.
Faking scientists are more subtle. Even Mendel probably massaged the figures in his original paper; the ratios were too close to 1:2:1.
Remember, tests are important, but they are only proxies for real life.
Again, I was not trying to criticize your work; I was trying to extend upon it. And I know all too well that there are ideas and subjects that cannot be put on an exam. Back in my teaching days, I remember hearing "Will this be on the exam, Professor Jablow?" all too many times.
Eric--
No offense taken! I certainly agree that there are many errors of omission and commission when interpreting data. The title of the classic book, "How To lie With Statistics" says it all.
As educators we have an obligation to help our students discern valid statistical inferences from invalid. I've always believed that the success of a democratic society requires an informed citizenry, a statement that sounds cliche and overblown, but is it?
Finally, here are some solutions...
(a) From the 2nd equation, x+y = 3.
From the first eqn, (x+y)/xy = 1/4.
Substituting, we obtain 3/xy = 1/4 or
xy = 12. This happens to be REAL MATH, not just some 'trick' to beat an SAT problem!
(b)
Solve for y in x+y=3 and substitute into xy=12:
x(3-x) = 12.
Solving the resulting quadratic equation leads to
x = (3+- i √ 39)/2
y = (3-+ i √ 39)/2.
I'll skip (c).
(d) Since all variables on SATs are assumed to represent only real values (this is explicitly stated in the instructions), this problem would be invalid even though the final result is real.
(e) Skipped
(f) Here's a way of obtaining the asymptotes using conceptual understanding rather than solving for y first as is traditionally taught:
For horizontal asymptotes, we are considering 'end behavior' of y as x → ∞
From the equation 1/x + 1/y = 1/4,
as x becomes large, 1/x → 0, therefore, 1/y → 1/4 or y approaches 4. Thus y=4 is a horizontal asymptotes (similar argument in other direction).
The vertical asymptote argument goes as follows:
For x=c to be a vertical asymptote,
it is necessary that y → ∞ as x approaches 'c' from one or both sides.
Note that if x → 4, 1/y → 0, therefore y increases (or decreases) without bound. I know this will be viewed as confusing compared to the traditional approach of expressing y as a rational function, but I offer it as an alternative.
That's all for now...
HI Dave ....
I am a simple retired Math teacher and a present Math tutor ... I am not going to go into big mathematical discussions about all of this ...
However ....
What a lovely problem!! I understand exactly what you are trying to show and say. I really enjoyed working my way through your questions. Well done.
A friend of mine sent me this link .... I have only read this one post .... I'll try to read more when I have time ...
Thanks for this blog!
cincin 21--
Sorry, I didn't reply sooner! Thank you for the words of support. Nothing is more gratifying than validation and encouragement from someone who has been on the front lines. Keep visiting!By the way, I'm joining the ranks of the retired in 23 days 12 hours, 57 minutes and 3,2,1,... seconds, but who's counting!
Dave
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