Vision - Potential
Vision Within Every Instructor - Potential Within Every Student
Newsletter of the HBCU College Algebra Reform Consortium*
Number 49, September 2003
www.ContemporaryCollegeAlgebra.org
[1] Geometry — Algebra Interplay
Welcome back to school and, in particular, to the wonderful,
challenging, and useful world of Contemporary College Algebra with its
emphasis on problem-solving. A common ingredient in everyone's
problem-solving process should be Sketch a Picture. This is where
geometry enters the picture (no pun intended). Lines, rectangles, circles,
parabolas, etc. are common geometric shapes that we have known for years.
Linking these geometric shapes with their algebraic equations (e.g., the
equation of circle with center (a,b) and radius r is (x-a)2+(y-b)2=r2) forms "bridges" that allow us to move back and
forth between geometry and algebra, each enhancing the other. For example solving (or approximating the solutions) 2x = x2 for x is an algebraic problem. However it is best
accomplished by plotting f(x) = 2x - x2 and noting where the curve touches the x-axis, a geometric problem. Another example is determining if two lines are parallel, a geometric problem. This question is best resolved by writing the equations of the lines in point slope form to see if their
slopes are equal, an algebraic problem.
Triangles, particularly right-triangles, are another basic shape
along with its algebraic counterpart, the Pythagorean theorem, that is
frequently encountered. (Pythagorean theorem relates the sides and
hypotenuse of a right triangle: The sum of the squares of the sides
equals the square of the hypotenuse.) Consider their critical role in
modeling periodic motion. Periodicity is central to our lives - the days of
the week, the seasons, hours of daylight, heartbeats, ocean tides, etc. are
a few examples. Maybe the most familiar example, is the periodic motion of
the minute hand on a clock (non-digital). The tip of the minute hand traces
out a complete circle every hour. Visualize the circle on the xy-plane with
the center at the origin and then visualize a right triangle formed by
dropping a line from the tip of the minute hand to the x-axis and a line
(hypotenuse) from the tip of the minute hand to the origin. As the tip of
the hour hand travels around the circle both the height and the base of the
right-triangle change in a periodic manner. Trigonometric functions are
defined to describe this motion (sin(q) = side opposite/hypotenuse, cos(q) = side adjacent/hypotenuse). Using these functions, the Pythagorean theorem becomes the
fundamental trigonometric identity: sin2 + cos2(q) = 1.
Draftspeople, engineers, kitchen planners, and a host of people in
other professions often encounter a need to shift a shape (circle, parabola,
etc.) from one position to another or to resize or to reorient it in their
drawings. (See Section 3.4 in Contemporary College Algebra: Data,
Functions, Modeling on "Shifting and Scaling Graphs".) The following four
articles illustrate the interplay between geometry and algebra in resolving
standard problems in drafting.
[2] Finding a Second Point
Determining the distance between two points is a straightforward
calculation using the distance formula. However, determining the coordinates
of a point on a line that is a given distance from a known point on the line
presents an interesting challenge. Consider the line defined by y = 2x+5 and
the point A: (1,7). Determine a point B on the line that is 3 units from
point A. (How many solutions should there be?) Hint: Denote the coordinates
of B by (a,b). What relationship must exist between a and b for B to lie on
the line? Set the distance between A and B equal to 3 and solve for a and b.
[3] Slopes of Perpendicular lines
Are the lines y = 3x+5 and y = -2x+6 perpendicular to each other? How can you tell? The following outline of a geometric argument provides an
answer.
Consider a line L with distinct points marked A and B on it.
Draw a horizontal line through point A and denote the angle formed by
x.
Draw a line through B perpendicular to the horizontal line forming a
right triangle ACB.
Explain why angle y = (pi / 2) - x.
Draw a line, call it H, through B perpendicular to L. Let D denote
the point of intersection of H with the horizontal line.
Explain the labelling of the angles in the right triangle BCD.
The slope of L is tan(x) = BC / AC and the slope of line H is
tan(pi - y) = -tan(y) = - (AC / BC). Therefore the slope of H = - (1 / slope of L).
Thus the slopes of perpendicular lines are negative reciprocals of
each other. (The reader should verify that the result holds if points A and
B are interchanged or if line L has a negative slope.)
[4] Parallel Lines
A draftsman needs to determine the equation of a line parallel to
the line L: y = 3x+4 and 3 units away from L. He knows that parallel lines
have the same slope. Therefore line y=3x+b is parallel to L for any value
of b. The problem is how to determine b so that the two lines will be 3
units apart. The draftsman also knows (having read the previous article)
that any line with slope -1/3 is perpendicular to L. Thus line y = - 1/3 x is perpendicular to L and to the arbitrary line y = 3x+b. The
question can now be rephrased to: Determine the value of b so that the
intersection points of y = - 1/3 x with line L and the arbitrary line y = 3x+b are 3 units apart. A picture helps to show the reasoning.
The coordinates of point A are found by solving the system
and the coordinates of point B are found by solving the system
. Note these coordinates are in terms of the unknown b. The value
of b can now be determined by setting the distance between points A and B
equal to 3 and solving for b.
[5] Three Points Determine a Circle
Problem: Determine the equation of the circle whose circumference
contains 3 points: A, B, and D. Thus we need to determine the radius and the
center, C.
We may begin by investigating a possible relation between the
center of the circle, C, and a chord, say AB. Lines drawn from C to A and
from C to B are radii of the circle and thus are equal. Therefore triangle
ACB is an isosceles triangle (triangle with 2 equal sides). The bisector of
the angle ACB is also the perpendicular bisector of the chord AB. (Show why
this is true.) Hence the perpendicular bisectors of chords AB and AD
intersect at C, the center of the circle.
Knowing how to determine the midpoint of a line segment and that
the slopes of perpendicular lines are negative reciprocals, allows us to
determine the two perpendicular bisectors. The coordinates for the center of
the circle are then found by solving the system consisting of these two
perpendicular bisectors. Knowing the coordinates of the center, it is easy
to find the radius and then write the equation for the circle. The reader
should pick three non-colinear points and then determine the equation of the
circle they determine.
As another approach and as a check on your computations,
substitute the coordinates of each of the three points into the general
equation of a circle centered at (a,b) with radius r, (x-a)2+(y-b)2=r2. Now solve the resulting system of three
equations for a, b, and r.
[6] Arithmetic and Geometric Mean
The arithmetic mean of two numbers, x and y, is x+y/2
and the geometric mean is
(xy)^(1/2). Discuss how the two means
compare with respect to size? That is, is the arithmetic mean always greater
than or equal to the geometric mean or is the geometric mean always greater
than or equal to the arithmetic mean or are there pairs of numbers for which
the arithmetic mean is greater than the geometric mean and other pairs for
which the reverse inequality holds? Also are there pairs of numbers for
which both means are equal? Explain your reasoning.
As a follow-on problem, define arithmetic and geometric means for
triples of numbers and then discuss how they compare.
Visit our website:
www.ContemporaryCollegeAlgebra.org
[7] Notices
-
Discussions of the Contemporary College Algebra program will be
held this Fall at the following conferences:
"Mathematics in the Sun," St. Petersburg College, Tarpon Springs, FL, Sept. 26-27, 2003, sponsored by the Florida Two Year College Assoc. Norma Agras, Paul Dirks, Don Small will make presentations. ICTCM, Chicago, IL, Oct. 30 — Nov. 2, 2003. Don Small is one of the leaders of the preconference meeting that is devoted to refocusing college algebra. AMATYC, Salt Lake City, UT, Nov. 13 — 16, 2003. Russ Lundgren, Grace Wood, Bob Johnke, Diana Hooker, Don Small will make presentations .
Fall Retreat for instructors interested in teaching
Contemporary College Algebra will be held at Cy-Fair College in northwestern
Houston, TX, October 9-11, 2003. Those interested in attending please
contact Laurette Foster (Laurette_Foster@pvamu.edu) or Don Small
(don-small@usma.edu). The program will begin with an informal supper
Thursday night and conclude with lunch on Saturday. A grant from the Brown
Foundation provides financial support for travel, room, and board for the
participants. Deadline for contributions to the October
Newsletter is Monday, October 6, 2003. Send opinion articles, suggestions
for writing assignments, small group in-class activities, small group
out-of-class projects, Queries, announcements, etc. to Don Small
(don-small@usma.edu). To subscribe to this Newsletter, write to Don Small, Dept. of
Mathematics, U.S. Military Academy, West Point, NY 10996 or contact him via
e-mail at don-small@usma.edu.
* Supported by the U.S. Military Academy, West Point, NY.
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