Rational expressions, equations, and functions

[tomergal]

I find the standards unclear regarding the scope for working with rational expressions. In general, I identified only four standards that concern rational expressions:

– A-APR.7, regarding the analogy between rational expressions and rational numbers, implies that students should know how to add/subtract/multiply/divide rational expressions.
– A-REI.2 only calls for solving “simple” rational equations, although it’s unclear what “simple” implies. Also, the standard is mainly concerned with extraneous solutions, where rational equations are just “means to an end.”
– A-CED.1 again calls for using “simple” rational functions, this time for the purpose of modeling.
– F-IF.7d calls for graphing rational functions. The sentence “identifying zeros and asymptotes when suitable factorizations are available” seems to imply the functions are not necessarily very “simple.”

It’s very easy to get really messy and complicated when dealing with rational expressions, equations, and functions. I wonder what’s the standards’ intention for those. Should we limit ourselves to linear denominators? Quadratic denominators? Constant numerators? Any guidance is welcome here.

[lhwalker]

One thing have found helpful is to occasionally take placement tests at colleges (I took two in April). I consistently see traditional cases of arithmetic with rational expressions (including the need to factor), so I continue to teach that in Algebra 2 and above until I hear otherwise. The ACT practice problems include an addition problem where the student must get a common denominator for x and (x + 5). As you pointed out, rational expressions present a challenge for making them seem relevant instead of a “means to an end.” I just finished writing a lesson sequence that bounces off gas laws and capacitors in series. While my disaffected seniors may not have dreams of becoming engineers, it is certainly not a huge stretch for them to consider working in a technical environment, and having a clue how those formulas work has its advantages. I include brief videos of things blowing up when maximums are exceeded and am confident I will at least get their attention. In another lesson, they modify a 8-line calculator program that calculates D=rt to t = D/r to see one is a linear function and the other is rational. Please email me if you want the series: lane.walker@fhsdschools.org But, yes, I am anxious to see if Dr. McCallum has any words of wisdom on this.

[lhwalker]

I just found the explanation for “simple” rational and radical functions. It’s a footnote in Appendix A, page 36: http://www.corestandards.org/assets/CCSSI_Mathematics_Appendix_A.pdf

[Cathy Kessel]

Note that A-APR.7 is a plus standard and the others mentioned are not.

[Cathy Kessel]

Also, note discussion of “complicated” here: http://mathematicalmusings.org/forums/topic/definitions-complicated-expression/

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