curriculum: precalculus vs. college algebra
Leonard Blackburn
What is the difference between these classes in the U.S.
with these titles (each is a class taken before calculus):
"College Algebra"
"Precalculus"
"Intermediate Algebra"
"Algebra II"
I am particularly interested in the first two. The content seems
strikingly similar with large overlap, but I am unsure.
Furthermore, I am about to give a short lecture entitled,
"An Introduction to Inverse Functions"
to an audience of pre-calculus students. Can anyone suggest
any good answers to questions that I may get like:
"What are inverse functions useful for in the real world?"
Also, can anyone suggest any particularly interesting examples
or ideas regarding teaching this subject. I already know the
mathematics involved and roughly how I want to present it.
But I want to do at least one interesting thing that will get the
students' attention.
Thanks for any input,
Leonard Blackburn
David Moran
who took College Algebra and then we had Precalculus together. He said that
the only difference he saw in the two is that precalculus presented more
details and provided more challenging problems than College Algebra did. I
cannot speak for myself because I went from Algebra II directly into
Precalculus. There were also a few topics in Precalculus that I've never
seen in a College Algebra book such as rotations of conics, tangents and
normals to conics, and vectors (which usually are not covered even if a
College Algebra book has trig).
David
"Leonard Blackburn" <blac...@math.umn.edu> wrote in message
news:aa503d8.03022...@posting.google.com...
Stan Brown
>Furthermore, I am about to give a short lecture entitled,
>"An Introduction to Inverse Functions"
>to an audience of pre-calculus students. Can anyone suggest
>any good answers to questions that I may get like:
>"What are inverse functions useful for in the real world?"
Reasoning from effects to causes.
I think inverses are particularly useful with exponential functions
because they let you answer "how long" questions. "The half-life of
Carbon-14 is {whatever it is; can't remember}. How long ago did the
owner of these bones die?" "How long will it take at x% to pay off
this debt?"
>Also, can anyone suggest any particularly interesting examples
>or ideas regarding teaching this subject.
One that my students enjoy is time of death of a body. Newton's Law
of Cooling leads to exponential decay of temperature difference
between body temp and ambient temp. If body temp is e.g. 80° and
ambient temp has been constant 68°, how long has body been dead?
--
Stan Brown, Oak Road Systems, Cortland County, New York, USA
http://OakRoadSystems.com/
"You find yourself amusing, Blackadder."
"I try not to fly in the face of public opinion."
Barb Knox
blac...@math.umn.edu (Leonard Blackburn) wrote:
That could be a big ask. By the time students get to that level either
they have some intrinsic interest in maths (in which case gee-whiz
examples won't make a lot of difference) or else they are just serving
time (in which case gee-whiz examples won't make a lot of difference).
But here's an idea that might be of some use. The students are presumably
interested in sex (and saying "sex" in a lecture theatre always produces
at least momentary attention). Maybe you could do a population genetics
example where inverse functions are used when working backwards from a
known distribution of phenotypes to the underlying distribution of
genotypes. The example that comes to mind is blood group phenotypes (O,
A, B, AB) and their underlying alleles (o, a, and b). Maybe you could
relate this to the recent transplant fiasco caused by a blood-group error.
> Thanks for any input,
> Leonard Blackburn
--
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| BBB b \ barbara minus knox at iname stop com
| B B aa rrr b |
| BBB a a r bbb |
| B B a a r b b |
| BBB aa a r bbb |
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Chergarj
necessarily be universal even throughout the U.S.A. as some institutions may
use various notational differences in identifying the courses, but generally
Algebra II is the same as Intermediate.
A particular institution used the terminology "Precalculus" and "College
Algebra & Trigonometry" approximately interchangeably; it varied from semester
to semester, but they were basically equivalent. By examining the state
university catalog, apparantly College Algebra & Trigonometry was also
equivalent to Precalculus there, too. Additionally, some universities have
separate courses for "College Algebra" and for Precalculus which includes
Trigonometry with the college algebra.
As they were presented in my education, Precalculus included very much algebra
in somewhat more detail at a slightly higher level of refinement of the
concepts than presented in Intermediate level. Precalculus and College Algebra
work heavily toward graphing and properties of polynomial functions and
transcendental functions; sequences and series; and limits. In intermediate
level, we only dealt with the inclusion of the elementary level, conic
sections, logarithmic and exponential functions, and quadratic equation
(especially formula for the general solution).
The purpose of precalculus seems to be to rigorously review the mathematics
necessary for the learning of Calculus.
G C
Dave L. Renfro
[sci.math Feb 24 2003 4:30:30:000PM]
http://mathforum.org/discuss/sci.math/m/485429/485429
wrote (in part):
> Furthermore, I am about to give a short lecture entitled, "An
> Introduction to Inverse Functions" to an audience of pre-calculus
> students. Can anyone suggest any good answers to questions that
> I may get like: "What are inverse functions useful for in the
> real world?" Also, can anyone suggest any particularly interesting
> examples or ideas regarding teaching this subject.
Solving equations is ubiquitous. Thus, anything that serves
to increase our understanding of solving equations is important.
The most important tool for solving equations is the "Golden
Rule" -- whatever you do to one side of the equation, you must
do to the other side. A natural question is what are the allowable
things that you can do to each side of an equation? Early on,
students are taught they can add, multiply, etc. quantities to
both sides. Then they'll encounter squaring both sides, taking
the logarithm of both sides, etc. The answer to the first question
is that you can apply any function to both sides, since if f is a
function, then the implication "x_1 = x_2 ==> f(x_1) = f(x_2)" holds
(i.e. functions preserve equality). Another natural question is
under what situations can extraneous solutions arise? Students are
taught that extraneous solutions can arise when squaring both sides
of an equation or when multiplying both sides of an equation by a
variable expression. The answer to this question is that extraneous
solutions can arise if you apply a non-invertible function to
both sides of an equation. The squaring function is clearly
non-invertible. As for the situation involving multiplying both
sides by a variable expression, here you're multiplying both sides
by something that, if equal to zero, involves multiplying both
sides by zero, and f(x) = 0*x = 0 is highly non-invertible.
Invertible functions are functions that preserve *non-equality*.
If f is invertible, then x_1 not equal to x_2 implies f(x_1) not
equal to f(x_2) or, equivalently, f(x_1) = f(x_2) implies x_1 = x_2.
Dave L. Renfro
Kevin Foltinek
> [Example applications of inverse functions]
> "The half-life of
> Carbon-14 is {whatever it is; can't remember}. How long ago did the
> owner of these bones die?"
> [snip]
> One that my students enjoy is time of death of a body. Newton's Law
> of Cooling leads to exponential decay of temperature difference
> between body temp and ambient temp. If body temp is e.g. 80° and
> ambient temp has been constant 68°, how long has body been dead?
I once looked into how carbon-dating is actually done, in preparation
for the standard calculus lecture about carbon dating. Due to
variations in miscellaneous things, the percentage of C14 in a living
organism is significantly variable over the hundreds to thousands of
years of history to which C14 dating can be applied. Tables have been
constructed, based on tree rings and various other calibration data,
showing C14% as a function of age, and it is those tables that are
used to carbon-date artifacts. Nobody actually computes logarithms to
do real carbon dating.
I suspect that similar things may apply to the cooling of a body
(though I haven't looked into this topic). Various biological or
biochemical processes are probably involved, which may generate heat
and corrupt Newton's law of cooling.
I wonder whether we are doing a disservice to students when we present
these (and other) "examples" as "applications" of mathematics.
Kevin.
Leonard Blackburn
Excellent response. Thank you. Furthermore, this is something understandable
at the pre-calculus level.
Leonard
Leonard Blackburn
> Leonard Blackburn <blac...@math.umn.edu> wrote in sci.math:
>
> >Furthermore, I am about to give a short lecture entitled,
> >"An Introduction to Inverse Functions"
> >to an audience of pre-calculus students. Can anyone suggest
> >any good answers to questions that I may get like:
> >"What are inverse functions useful for in the real world?"
>
> Reasoning from effects to causes.
>
> I think inverses are particularly useful with exponential functions
> because they let you answer "how long" questions. "The half-life of
> Carbon-14 is {whatever it is; can't remember}. How long ago did the
> owner of these bones die?" "How long will it take at x% to pay off
> this debt?"
>
> >Also, can anyone suggest any particularly interesting examples
> >or ideas regarding teaching this subject.
>
> One that my students enjoy is time of death of a body. Newton's Law
> of Cooling leads to exponential decay of temperature difference
> between body temp and ambient temp. If body temp is e.g. 80° and
> ambient temp has been constant 68°, how long has body been dead?
Thank you for the good response. I shall remember "reasoning from effects
to causes" well. I will not present Newton's Law of Cooling, though,
since it is a Calculus topic (unless you present the solution to the
appropriate differential equation as the given Law).
Leonard Blackburn
You may be at least partially correct about this. But then what is the
point of teaching this material? Also, I think that nearly all students
who take pre-calculus in college probably have very little interest in
math, since they didn't already learn such material at an earlier age.
But is it possible to get them interested?
>
> But here's an idea that might be of some use. The students are presumably
> interested in sex (and saying "sex" in a lecture theatre always produces
> at least momentary attention). Maybe you could do a population genetics
> example where inverse functions are used when working backwards from a
> known distribution of phenotypes to the underlying distribution of
> genotypes. The example that comes to mind is blood group phenotypes (O,
> A, B, AB) and their underlying alleles (o, a, and b). Maybe you could
> relate this to the recent transplant fiasco caused by a blood-group error.
Thanks for this example.
Barb Knox
For me the satisfaction comes from that rare "aha!" event when someone
visibly "gets it". I found it was vital to concentrate on these few
golden moments and ignore the vast ocean of dross. YMMV.
> Also, I think that nearly all students
> who take pre-calculus in college probably have very little interest in
> math, since they didn't already learn such material at an earlier age.
> But is it possible to get them interested?
I don't think it's feasible to *make* them interested, especially those
many who have suffered some form of "maths trauma" during their
schooling. But if you can deal with students 1-1 or in small groups then
you at least have a chance of
helping them through some of their individual blocks. "Remedial
lecturing" is IMO hopeless.
> > But here's an idea that might be of some use. The students are presumably
> > interested in sex (and saying "sex" in a lecture theatre always produces
> > at least momentary attention). Maybe you could do a population genetics
> > example where inverse functions are used when working backwards from a
> > known distribution of phenotypes to the underlying distribution of
> > genotypes. The example that comes to mind is blood group phenotypes (O,
> > A, B, AB) and their underlying alleles (o, a, and b). Maybe you could
> > relate this to the recent transplant fiasco caused by a blood-group error.
>
> Thanks for this example.
You're welcome. Do let us know how it goes.
> > > Thanks for any input,
> > > Leonard Blackburn
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