Thursday, June 28, 2007

Some Algebraic Reasoning (Math Contest/SAT-Type) to Whet Your Appetite

While we're waiting for Carnival of Mathematics #11 and Part II of Recursive Sequences, here are a few algebra problems to store away for your students...
Using math contest problems (AMC, MathCounts, etc.) in the classroom can be very helpful in raising the level of expectation for many students. One thing experienced teachers know is that students will only learn how think at higher levels when we provide these kinds of challenges. Math educators are always seeking more examples of this nature. Look no further than released math contests, samples of which can be found online. Use these problems as models which can be revised to match the skill and conceptual level of your group. This is not easy but it's worth the effort. Assuming that more basic level math students would have little or no chance at these (or will become too frustrated) can be a self-fulfilling prophecy. Those of you who have been following this blog since its inception 6 months ago, know that I challenged just such a group of 9th graders this year. Deciding what prerequisite skills were needed and how to develop some of the challenges incrementally required considerable thought and planning but their engagement in the activities and a feeling of accomplishment on their part made it worthwhile.

Questions 1 and 2 below are appropriate for 1st or 2nd year algebra students. The objective of these conceptually-based problems is to develop algebra sense rather than provide mechanical practice with algorithms.

1. For how many values of x is
(x-3)(x-4)(x-5)(x-6) = (3-x)(4-x)(5-x)(6-x)?
(A) 0 (B) 2 (C) 3 (D) 4 (E) more than 4

2. For how many values of x is
(x-3)(x-4)(x-5)(x-6)(x-7) = (3-x)(4-x)(5-x)(6-x)(7-x)?
(A) 0 (B) 3 (C) 4 (D) 5 (E) more than 5

3. What is the greatest integer value of N, less than one million, for which
√(1+√N) is a positive integer?

Note: Is estimation worthwhile here as a starting point? How would some students use a calculator to 'solve' this? Watch them! Do algebra textbooks provide methods and practice for solving positive integer problems? Are there special methods one needs for these?


novemberfive said...

Here goes. I hope I'm as good as a middle-schooler.

1. (x-3)(x-4)(x-5)(x-6) = (1)(1)(x-3)(x-4)(x-5)(x-6) = (-1)(-1)(-1)(-1)(x-3)(x-4)(x-5)(x-6) = (-1)(x-3)(-1)(x-4)(-1)(x-5)(-1)(x-6)
= (3-x)(4-x)(5-x)(6-x)

So the equation is true for all x, and the answer is therefore E (more than 4).

2. Using logic similar to the above, we can show that the equation is equivalent to (x-3)(x-4)(x-5)(x-6)(x-7) = -(x-3)(x-4)(x-5)(x-6)(x-7).

The only number that equals its own negative is zero, so the equation is true just when (x-3)(x-4)(x-5)(x-6)(x-7) = 0. This happens just when x is 3,4,5,6 or 7, and the answer is therefore D (5).

3. Sqrt[1+Sqrt[N]] is a positive integer just when there exists a positive integer x such that x^2 = 1 + Sqrt[N]. Solving for N, we find (x^2 - 1)^2 = N <= 10^6. Thus x^2 - 1 <= 10^3. Now 32^2 = 1024 > 10^3, so we try 31 and find that it works. The answer is therefore (31^2 - 1)^2 = 921600.

Dave Marain said...

We agree! See, I don't need to post solutions after all! How do you feel about these kinds of challenges for our students? Is there a place for them in our curriculum? Do questions such as 1 and 2 deepen student understanding? Someone asked me to define algebraic reasoning. I think your explanations did exactly that!! Thanks...

novemberfive said...

How do you feel about these kinds of challenges for our students?

These kinds of problems are essential. They should be central to the curriculum. I think the curriculum goes too fast, and should become proficient at non-routine applications of basic material (there can be a surprising amount of depth to "middle-school math" problems) before moving on to calculus.

I agree with the authors of The Calculus Trap who write: Rather than learning more and more tools, avid students are better off learning how to take tools they have and apply them to complex problems. They refer to "avid" students but I think the statement is true for all students.

Do questions such as 1 and 2 deepen student understanding?

Without a doubt. I think the essence of understanding math is being able to use it to solve problems. Math problems are like chess problems in that both have a start state and an end state and that a solution consists of a sequence of legal moves. Non-routine problems require you to think a few moves ahead, but if you can't do that, you don't really understand the moves/material.

I wish someone had explained this to me when I was in school.

Someone asked me to define algebraic reasoning. I think your explanations did exactly that!!

Well, they might be an example of it... :) (I hope pedantry is appropriate here.)

Dave Marain said...

How is it that, in 37 years, I could not find more than a small handful of educators who shared my beliefs about math education, yet through this blog I read comments that echo my innermost feelings almost every day?
For years, at conferences, at department meetings, at workshops I stressed the importance of 'less is more', but I always felt that few understood or even listened. Content coverage was always the driving force for the secondary curriculum. Again, this is why Professor Schmidt coined his now famous description of our math curriculum: "One inch deep, one mile wide."
Your analogy to chess problems is compelling. Interestingly, I went through a 'Mate in 3' phase earlier on in life. The inventors of these puzzles left me awestruck. Solving the puzzles required virtually the same processes and qualities required to solve difficult math problems. Looking ahead is important, however, we can't all be Bobby Fischer's! I believe that I was able to solve most of these puzzles by a combination of persistence, stubbornness, a belief that if I kept working at it I would find a way, experience from doing many others, etc. Imagine if we allowed our students the opportunity to challenge themselves and become so engaged.

There are no shortcuts to learning or improving one's problem-solving abilities. Thank you for your profound reply to my 'innocent' couple of algebra problems. It is gratifying to find someone who sees why I write these. As far as pedantry and other aspects of algebraic reasoning, of course I was oversimplifying. But your explanations painted a very very good picture and one picture is worth a ...

jonathan said...

For #2, consider the graphs of y=the left side and y=the right side. Since they are reflections of one another...

novemberfive said...

How is it that, in 37 years, I could not find more than a small handful of educators who shared my beliefs about math education ....

This takes us away from mathematics and into economics and politics. It's hard enough to make a true statement in the abstract, formal world of mathematics, let alone say something true about the messy, sweaty real world of interacting human beings. Yet I will try.

The stultifying inertia you observed is a natural result of our overly-federalized public school system. Soviet-style central planning works as poorly in education as it does in food production.

I went through a 'Mate in 3' phase earlier on in life. [...] Looking ahead is important, however, we can't all be Bobby Fischers!

If solving math contest problem is akin to solving mate-in-3 chess problems, then math research is more like to writing a book on how to find the best strategic/positional move in complex positions. So while we can't all be math researchers, I think it's reasonable to expect students to solve some of the easier math contest problems.

jonathan said...

But november 5, the soviet system produced good math. Beat us into space, too.

There were lots of things wrong, but in this case, it doesn't make for a strong analogy.

novemberfive said...

The Soviets are an amazing people and they managed to achieve awe-inspiring results under communist rule. But that doesn't change the fact that central economic planning simply doesn't work.

Dave Marain said...

novemberfive, jonathan--
Although every educational decision can be linked to politics, we are now entering the OUTER LIMITS of this blog! BTW, that was going to be the original name of my blog but I settled on MathNotations because I decided it would make more sense to have 'Math' embedded somewhere in the title! Hey, anyone who wants to use Outer Limits has to ask me for permission! (or has someone already thought of it!).

Seriously, guys, we're way off track here and I'm going to ask you to bring it back to our theme of infusing more challenging problem-solving into K-12 curriculum , particularly at the upper elementary and middle school level.
Without delving too far beneath the surface and looking for dark political explanations, suffice it to say that we need strong educational leaders to raise the bar. The best research mathematicians, math education specialists, math curriculum specialists, math professors and classroom practitioners from all levels of math should have been brought together to form the National Math Panel. This group should represent many points of view but should all have one goal: To improve the quality of math education in our country. Professor Schmidt has called for a world-class national math curriculum and you know I have been advocating this for some time. Regardless of our fears of government, we need to accept the reality here. Our K-12 math curriculum is simply not competitive enough. It is superficial -
"one inch deep and one mile wide."
Those contributing wonderful ideas to this blog know this is true. The question is: HOW DO WE BEGIN TO CHANGE IT? Perhaps we are...