Notes
on MTHT 466
Monday, 8/24/2009
The course is an introduction to the ideas and methods of
calculus for
preservice elementary and middle school teachers.
Our subject has roots in two geometric problems that were posed in
antiquity:
(1) construct a line tangent to a given figure in the plane,
(2) calculate the area of a given figure.
An example of the first problem is to construct the line tangent to a
circle at a given point on the circle.
An example of the second problem is to calculate the area of a
rectangle with sides of lengths l
and m.
Two other concepts play a major role: limit and function
including the graph of a function.
In our first class we looked at the construction of a tangent to a
circle. Using a compass to draw a circle,
the set of points on the board all the same
distance from a given point called the center, O.
Given a point P
on the circle, draw the radial line
through O
and P.
Then construct the line t
perpendicular to the radial line
at P.
What do we mean by saying the line t is tangent to the
circle? One property which seems to be important
in the case is that any
point Q on the
circle other than P
is further
from O than
P is, and hence lies
outside the
circle. This is because the triangle OPQ has of a a right angle
at P. Then by
Pythagoras's
theorem, the segment OQ is longer than OP. We will l;ater
need a more general notion of tangent.
Wednesday, 8/26
We looked at a paper-folding construction of a family of line tangent
to a parabola. Choose one straight edge
of a piece of paper, label this edge e, and mark a point
F near the middle of
this edge and an inch or so above it.
Fold the paper so that some point X on e lies
on top of F and crease
the fold line. Label this line
t.
Repeat this step to make a dozen more fold lines, hold the paper up to
better see the folds, and look for a parabola.
Fold the paper so that the edge
e folds over on itself
and X lies on the
new fold line to get a line through
X
perpendicular to the edge. Mark the point P where this vertical fold
line meets the
line t. Using
congruent
triangles, we argued that the distance FP
equals the distance PX
and that PX is
the
shortest line segment from
the point P
to the line e.
A parabola is defined geometrically as the set of points P
for which the segments FP
and PX have the
same
length, that is, P
is the same distance from F as from e. Given e and F and any point X
on e, we can
construct the point P
on the parabola
defined by e
and F.
Friday, 8/28
On a clean sheet of paper we repeated some steps from Wednesday's work
to get a single point P.
We started
with an edge e and
a fold perpendicular to e.
Then we placed a point F
on this fold line. We placed a point X
on e to the right
of F and drew the
straight line from F
to X. We folded
so that X lay on
top of F and
labelled
the new fold line t.
The point where FX
intersects t is
the midpoint, M,
of FX. Next
fold e on intself
with
the fold line going through X
to get a line perpendicular to e.
This perpendicular line through X meets t
in a point P that
lies on the parabola defined by e
and F. I argued
that the fold line t
meets the parabola at P
but at all other points lies below the parabola. This fits our
intuitive idea of a tangent line to the parabola at P.
Next time
we will introduce coordinates and use geometry to find an
equations for the parabola.
This is a step taken by Rene Descartes in his analytic geometry about
1637 or about 2000 years after Achimedes.
There was progress during this long period, but the basic ideas are not
obvious, they took a long time to discover and develop.
For students, understanding these ideas so that they themselves can
work with them is not easy and also takes time.
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