Saturday, March 1, 2008

SAT-Type Algebra Challenges - How Would You or your Students do on these?

Of course this comes a bit late for all those juniors who took their SATs today but the following questions can be used to prepare for the next one OR for anyone who wants to challenge themselves or their students...

-4 ≤ P ≤ 3 and -5 ≤ Q ≤ 4

(i) What is the greatest possible value of (Q+P)(P-Q)? Explain your reasoning.

(ii) What is the least possible value of (Q+P)(P-Q)? Explain your reasoning.

(a) SAT questions don't ask for explanations but this goes beyond that.
(b) These would be known as 'grid-ins' or student-constructed response questions. On a real SAT, answers to these must be greater than or equal to zero.
(c) The intent here is to go beyond the typical 'plug-in' methods most students use. One can apply actual skills and reasoning!
(d) Even the strongest students fall into a 'trap' set in these questions. Try them in your classes!
(e) Do these kinds of questions develop algebraic reasoning and mathematical power OR are they just your typical 'tricky' SAT-type that has little value outside the test?
(f) Make up your own version of one of these or, even better, encourage your students to invent their own!


Mr. K said...

My students (low performing eighth graders) would approach this problem by trial and error.

My first assumption was that the answers would lie at the boundary points, so that I'd only have to test out 4 cases..

This is obviously not true, and is more easily seen when the expression is rewritten as P*P - Q*Q. (I will forgo further explanation, to allow for some discussion)

Hypatia said...

Thanks for an excellent challenge questions, Dave.
[1] Since the reasoning is really the same for both parts i and ii, why not ask both questions, and
then "explain your reasoning"?
[2] While the format for the SAT isn't conducive for this type of challenge, most state tests have a free response section. This would be an excellent question(s) for those tests.
[3] The math classroom is an even better place for this type of challenge and the earlier the better.
[4] Mathematics is so "useful in the real world" that we forget that is also where critical thinking skills are best developed. Dr. David Eggenschwiler, English professor emeritus from USC, expressed this far more eloquently than I ever could in an article that appeared in the LA Times. A Math Forum link for this article is:

Hypatia said...

An even better reason to read Dr. Eggenswiler's article instead of my comments, is that he no doubt edits before printing. That first line should read 'question'.

mathmom said...

Certainly, as Mr.K said, this problem is much easier if you re-write the expression. I'm not sure how many students would think to do that.

Some of my middle schoolers know how to multiply out that expression, but I think I would have to suggest it to them. Even once they had done that, probably most of them would only consider positive values to be of any use in maximizing the expression.

Without re-writing it, I think they would try to maximize both terms of the multiplication, again ignoring the possible benefit of multiplying a negative by a negative.

Dave Marain said...

Thanks everyone for the thoughtful comments about this algebra problem. The question has a life of its own doesn't it -- an elegance, almost a quiet eloquence. I wish I could say I invented it, but I did not. I saw a similar question in some practice SAT book or somewhere and I revised it. Certainly, students' reasoning and ability to think outside the box develop when we challenge them with questions such as these. A wonderful source for these kinds of questions is of course math contests such as Math Counts and the contests from the Math League. In fact my next post will probably be a question from the most recent contest from the Math League, a geometry question of which I am in awe.

Thank you Hypatia for linking us to that profound statement by Dr. Eggenschwiler. I read it, enjoyed it and would recommend it be made into a poster for the wall of every math classroom!

Mr. K, I specifically chose this question because my experience has shown me that most students use 'guess-test' or plug-in methods, usually checking boundary values! This question is initially perceived as 'nasty' or 'tricky' by some, but I don't see it that way. It requires an extra level of reasoning, not simply a 'beat the test approach!' Student insight deepens as a result of such experiences. Another neural connection...

mathmom said...

Dave, I asked my 9th grade son to solve this, but he wasn't in the mood (mostly owing to the fact that his computer currently has a trojan' though his dad has the removal of the trojan under control, we think).

But my 6th grader, who is preparing for MathCounts State competition later this month, volunteered to give it a try. I hadn't originally asked him because I wasn't sure if he knew how to expand (P+Q)(P-Q) -- it turns out he knew, but didn't think to do this. (He's currently studying some combination of pre-algebra and Algebra I)

He did realize pretty quickly that to maximize his answer he wanted both P+Q and P-Q to be negative, and he did come up with the correct answer.

For part (ii) he wanted one negative and one positive, but used Q = 4, forgetting that using Q=-5 would also do that for him, and give him a higher absolute value to work with.

After that, I asked him if he knew how to multiply it out, and once he did, of course he immediately saw that looking at it that way made the answer much more obvious.

He did appreciate what he learned from working on the puzzle. :) Let's hope he gets a MathCounts question like this. ;-)

Florian said...

At misread the post and first worked with the expression (Q-P)(P-Q), and only then tried (Q+P)(P-Q). I recommend doing it, you will see that there is a nice connection between the two and how #2 increases the complexity.

mathmom said...

That's an interesting variant, Florian, because in that case I think it is best not to expand the expression, but rather to work with it as-is.