CALCULATORS (WAS RE:
Sheila King
I really enjoyed your response about "real world" problems along with
"messy" numbers. However, I DO realize that I can have the students
measure anything (such as their desks) and come out with numbers that
are not integers. This isn't what I had in mind. What I meant, when I
asked for someone to provide me with messy number-problems, was more
like systems of equations that it is very difficult to see whether they
intersect or not (the values are very CLOSE to some system that does
intersect, but perhaps these are just enough off that they don't. Doing
these on a graphing calculator can be interesting). However, after I
posted that I figured I could just model problems off of ones in the
text and change the numbers a bit until I get something like that.
Of course, your examples (about painting the school desks, baseball
trajectories and the moon) are excellent problems. I tell you, even one
of those would blow my students minds away. Well, my honors geometry
class might enjoy the desk painting one. I could see them getting into
that. But the other two that you suggest are more Algbera 2 material. I
can see my students just staring at the paper with no idea where to
start.
-> You want a quadratic equation? Ask them to figure out how high a
-> baseball would rise if it were thrown so as to travel in a parabolic
-> arc from third base to home plate with a certain initial velocity.
-> Pick a "messy" initial velocity, like 17.35 meters per second, and
-> don't let them round off the gravitational acceleration to less than
-> three digits.
Ok, am I supposed to teach them how to use the physics formulas for this
and provide them with the formulas? I always feel so torn, confused and
unsure in this area. We have VERY short instructional time at our
school. I will be *lucky* if I get to finish teaching logarithms in
June. Doing a problem like the one you suggest seems to me a very
valuable experience too. But in order to get my kids to be able to do it
I'd have to spend about 3 days preparing them for it. And would it be
the only such type of problem I would do with my class all year? I
should think if I was going to do one, I ought to do three or four in
order to really reap benefits from it. Assuming each of the three or
four "real" problems was of a different nature, it might take out one to
two weeks of my instructional time. So, I would cover one less Chapter
than I do now. (There go logarithms.) Adding pressure to the whole
situation is the fact that we give departmental exams at our school, and
so all the teachers of a particular subject try to stay at about the
same place in the text. (Of course, this isn't absolutely necessary. I
suppose I could be a Chapter behind everyone else at the end of the
year.)
Last year in my honors geometry course I did do four of these types of
problems and decided to abandon the coordinate geometry chapter for the
year. I thought the benefit to be gained from these problems was great
and needed to be done. Actually, I think you'd be proud of me. We used
real "messy" numbers. I provided that students with a xerox of a circle
on a piece of paper with a radius somewhere around 6 and 1/2 inches (it
wasn't exact). They cut it out and used it to model cones. They had to
play around with it to find out what was the cone of maximum volume that
they could construct from that paper. It involved all sorts of messy
decimal computations. They graphed the points on graph paper (I wanted
them to see the parabola and where the point of maximum volume lay on
the curve.) I only allowed one class period for this exercise, and it
was *barely* enough. Could easily have spent two days on it.
I also had another problem they worked on where I asked them (using
right triangle trigonometry) to calculate the perimeter of a regular
polygon inscribed in a circle with a diameter of one, as the number of
sides of the polygon kept increasing. Again, messy calculations, but I
wanted them to see that the value approached the limit pi. Well, I'm
almost afraid to share these, because there is some reason that you'll
think this is stupid too. But I was quite proud of the problems. It was
*something* different that I thought of doing on my own. That moon
problem is something I would NEVER think of, unless someone (like
yourself) shared it with me.
Well, I am not really in search of "messy" number problems in general to
just throw into the curriculumn somewhere. I just was thinking of messy
numbers in line with graphing and conic sections and other traditional
Algebra 2 topics.
-> "Find an exact answer and then round it off to whatever accuracy the
-> machine happened to be built for.
-> Just do it because I said to." This is not useful.
OK, you're right that it's not the best motivation for getting the
decimal aprox. ("because I said to" is never a very good motivational
tool, and in this particular case it does not tie in pedagogically with
the course). However, won't there at some time in the future very likely
be a need for the student of today to be able to use his calculator to
come up with the decimal aproximation? Mightn't he need to be able to do
that someday? Because most of the Algebra 2 students in my class need
some instruction in how to use the calculator properly. Many of the can
get the correct exact answer, but have difficulty using the calculator
to get an aproximation. Shouldn't they come out of my class being able
to do both, and don't I have a responsibility to make sure that they
can, and to make sure that they understand which one is the exact answer
and which one is the aproximation? I'm sorry my motivation for learning
how to get the aproximation is so shallow.
BTW, you must eralize that your experience in the field is certainly
much greater than many of ours. (Certainly mine, anyway. I've had 4
semesters of university math, that's all. I'm looking to correct the
situation, and I have great enthusiasm for my subject. I look upon my
participation in this echo as one way of broadening my horizons and
learning a bit about my field.) I realize that I have a lot to learn.
Please don't be too harsh on me. I may be ignorant and niave in some
areas of math, but at least I'm trying to learn.
Sheila
coming to you from Diamond Bar, CA (in Los Angeles County)
--
uucp: uunet!m2xenix!puddle!103!315!Sheila.King
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Tom Scavo
>What I meant, when I
>asked for someone to provide me with messy number-problems, was more
>like systems of equations that it is very difficult to see whether they
>intersect or not (the values are very CLOSE to some system that does
>intersect, but perhaps these are just enough off that they don't. Doing
>these on a graphing calculator can be interesting).
Here's one that seems to be receiving a lot of attention these days:
Graph the functions y = x^3 - 2x and y = 2 cos x . Find the
point(s) of intersection.
--
Tom Scavo
sc...@cie.uoregon.edu