Saturday, December 20, 2008

A Traditional Algebra 2 Rational Equation for the Holidays or ...

Solve for x:

\frac{4x-1}{2x}+\frac{2x}{4x-1}=2

Comments:
(1) The traditional procedure is time-consuming but part of the rites of passage for Algebra 2 students. This is the kind of exercise that "trains the brain!"
(2) Note that the form of the equation is a + (1/a) = 2. Does this suggest anything to you?
(3) Would this question be appropriate for SATs? Algebra 2 End of Course Exams?

9 comments:

roadster5555 said...
This comment has been removed by the author.
roadster5555 said...

it may be more useful to see the pattern as a/b + b/a = 2 and solve that as an initial step IMHO
cheers
Rob

Dave Marain said...

Rob,
Happy Holidays!
Yes, I agree that recognition of that pattern would make this problem a one-step exercise.

If we took a random sample of 100 Algebra 2 students (including honors level), how many do you think would solve it by this short-cut? Some would surely notice the a/b + b/a pattern (or in the form x + (1/x)), but would they know that the only solution for this equation would be a/b = 1, i.e., a = b?? If students had been exposed to this idea earlier they might recall it and apply it, perhaps...

My message here is that we need to provide these kinds of challenges if we want our students to develop their reasoning, not to mention learning useful techniques and some beautiful elegant mathematics. again, Rob, thank you for the comment.

roadster5555 said...

Good point if we as teachers can use patterns then there is a chance we can teach them - agree few would recognise the pattern. Another simple ex. mult by 11 through adding multiplicand digits - few teacher seem to know the trick - Dave - great blog!

jd2718 said...

I often agree in advance, on harder problems, to do some piece of the work for the students.

In this case I would readily agree to do the check, or at least the last line of the check...

Happy holidays!

Jonathan

Dave Marain said...

Happy Holidays to you too, Jonathan!
B ythe 'check', you're referring to the standard 'plug it back in to make sure the denominators are not zero' step? OR something else?

The issue of the sum of a number and its reciprocal equaling two is of course a lesson in itself. BUT I'm planting seeds here...

jd2718 said...

Indeed, I like to claim pieces of advanced arithmetic for myself... in this case, pronouncing "one plus one equals two" and then claiming my fair share of the credit for solving the problem (I like to tease).

Not as slick as the sum of a number and its reciprocal (and following roadster): (a^2 + b^2)/ab = 2,
multiply through by ab, move to one side, factor: (a - b)^2 = 0

Really, not as slick, but much easier than "brute force," and provides some standard work in a slightly non-standard situation.

Jonathan

roadster5555 said...

I am struck by the discussion because it illustrates the way algebra is actually the/a superstructure of/on numbers. It raises so many issues.

Cheers for holidays!
Rob

Dave Marain said...

Thanks Rob and Jonathan...
Yes, Jonathan -- it's as "easy" as 1 + 1 = 2!
And, Rob, I agree with your insight that algebra is the language of generalized number patterns. I wonder if most students appreciate that.
Cheers!
Dave