(11) Graphs and Ellipses
Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern
This lesson plan supplements: "Graphs and Ellipses," section #11a: on disk Sellipse.htm, on the web |
Goals: The student will|
Stories and extras: The focusing property of an ellipsoid, in particular the focusing of whispers in the old chamber of the US House of Representatives. Also the painting of that chamber by Samuel Morse, inventor of the telegraph.
Guiding questions and additional tidbits:
Start this lesson by explaining that the most useful and most common use of cartesian coordinates is to create graphs. On the board:
"A graph is a graphical representation of a mathematical relationship."
It is the bridge connecting shapes of lines, as seen by our eyes, with mathematical relationships and formulas.
Before getting into graphs (with which many of you here are already familiar, maybe all of you), let us first review what we know about Cartesian coordinates:
Start the discussion of graphs by a review of coordinates
Guiding questions and additional tidbits about cartesian coordinates, with answers and extensions. (All this material is discussed in section #5a.).
-- What are "systems of coordinates"?
-- What are the "cartesian coordinates" of a point on a flat plane?
-- Define the x and y coordinates of a point on a flat plane.
x is the distance measured parallel to the x axis. It is measured from the y axis--to the right it is positive, to the left, negative.
y is the distance measured parallel to the y axis. It is measured from the x axis--up is positive, down is negative.
-- What are the coordinates of the origin O?
--Can systems other than the cartesian be used to label points on a plane?
-- Describe one such system, polar coordinates in the plane.
Graphs--the material of section (11a).
[Note to the teacher: It is easier for the student to start with concrete examples than with abstract formulas, which need mental translation]
A graph is a way of using coordinates to present visually the relationship between quantities. The relationship can be something observed--for instance, stock market prices (for example, as given by the "Dow Jones Index") against time, or the temperature of a patient in a hospital against time, etc. When either of these graphed quantities goes up or down, the graph will instantly show it, also telling how steep and how big the change is.
You should be familiar with graphs, they are widely used (if the students use graphing calculators, bring that up). Graphs are even more useful for mathematically defined variations, and can be used to represent many kinds of shapes--including ellipses.
Then present section (11a), using the questions below in the presentation and/or for review.
-- What is a graph?
-- In a system of (x,y) coordinates, we connect all points where x is the time in years and y is the population of the United States. Is that a graph?
-- What is the graph of all points with y = –(2/3)x + 2?
[That is the example in the lesson. Draw the line on the blackboard, but don't label the axes, only the origin. Then as answers come in (below) label also the intersections with the axes with their values of y and x].
-- How does one use such a formula to get its graph?
The collection of all such pairs describes a line, which turns out to be straight.
--Where does that graph cross the y-axis?
-- Where does the graph cross the x-axis?
Add (2/3)x to both sides: 2 = (2/3)x
Divide by 2: 1 = (1/3)x
Multiply by 3: x=3.
-- Are all lines defined in this manner straight?
They are not with other relations, e.g. y = 3x2 which is a parabola, or y = 3/x which is a hyperbola [also, if you replace "3" by any other number, positive or negative]
--[Riddle] Say in y = ax + b you choose a=0, b=2, giving y = 2. Is this a straight line?
[One may add a comment on the word "linear" in mathematics. The equation of a graph giving straight line may also be written
"ax + by = c", and mathematicians call this a "linear" expression.
-- What is the graph y = 4x2 ?
[The example below is given for illustration. It should not be on any test, and is optional material].
-- What is the graph y = 12/x ?
Point out that at x = 0, y is not defined--it is + infinity if we approach from the right, - infinity if we come from the left, in either case the point cannot be drawn.
[The next question is best left for the teacher to answer]
--Does the equation of a line always have the form y = f(x) , where f(x) is "some expression involving x"?
The "expression involving x" is called "a function of x" which is why the shorthand for it is f(x).
However, any equation connecting x and y can be used. In such cases, if we choose x, we may need some extra work to get y.
[The next example shows one of them.]
Draw a circle on the board, mark its center with O, put a system of Cartesian axes through the center, select a point P on the circle, draw its radius R and its projection A on the x axis. Mark AP as y, OA as x.
--What is the relation between x and y on this circle, for radius R=5?
--Why do the values of x and y for any point P on the circle obey this relation?
--At what x does the circle cut the x-axis?
--At what y does the circle cut the y axis?
--What is the graph whose equation is x2/25 + y2/25 = 1?
--What about the graph is x2/64 + y2/16 = 1?
[Optional: Where does it cut the axes?
--What do we mean by the "major and minor axes" of an ellipse?
--How long are these axes here?
--How did the Greeks define the ellipse, 1800 years or so before Descartes introduced his axes?
--How are these special points called?
--Why are the foci of an ellipse of interest here?
[Tell story of old chamber of the House of Representatives in the US Capitol. First however ask the class if anyone had been to the capitol, and ask those who have, if they remember something special about the big room where statues were collected. Let a student tell it, later, if necessary, fill in more details.]
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Author and Curator: Dr. David P. Stern
Mail to Dr.Stern: audavstern("at" symbol)erols.com .
Last updated: 12.17.2001