A Curriculum Unit on Polynomial Functions
Samuel Obara
EMAT7080
May 3, 2001

This paper describes and gives a rationale for the design of a curriculum
unit on polynomial functions. Its target population is high school
students. The topics covered by the unit include quadratic, cubic, and
quartic functions. Where feasible, an emphasis is placed on incorporating
historical aspects of these concepts in the unit lessons.

Table 1 gives the sequence of topics for the unit. Below each heading is
a list of specific subjects which comprise that topic. The amount of time
(in hours) allocated to each topic also is indicated.

Table 1
Sequence of Lessons in a Unit on Polynomial Functions
__________________________________________________________________
1. Quadratic Functions (5 hours)
Properties of quadratic functions (including vertex and line
of symmetry)
Graphing quadratic functions
Finding roots of quadratic functions
Quadratic functions as mathematical models
2. Graphs of Polynomial Functions; Division of Polynomials
(6 hours)
Continuous functions, roots, The Intermediate Value Theorem
Determining factors of polynomials through division
The Remainder Theorem and synthetic division
Finding factors of polynomials (The Factor Theorem)
3. Theorems about Roots (5 hours)
Finding roots of factored polynomials
Roots of polynomials with real coefficients
Integer coefficients and the Rational Roots Theorem
Rational coefficients
4. Tools for Finding Roots (3 hours)
Descartes' Rule of Signs
Bounds on roots
Approximating solutions using the method of bisection
5. Review (1 hour)
6. Test (1 hour)
__________________________________________________________________
Note. The total time is 21 hours.

NCTM's Algebra Standard for Grades 9-12 includes the following
expectations which are illustrative of the goals of our unit (NCTM, 2000,
p. 296):
1. "Understand relations and functions and select, convert flexibly
among, and use various representations for them." Geometric
representations of quadratic functions, such as those implied by certain
propositions in Euclid's Elements, provide an excellent opportunity to
investigate connections within mathematics. For instance, to find the
roots of the function y = x^2 - ax + b^2, where a and b are natural
numbers, one can employ Proposition 28 of Book VI and Proposition 5 of
Book II.
2. "Analyze functions of one variable by investigating rates of change,
intercepts, zeros, asymptotes, and local and global behavior." The zeros
of cubic equations of the form x^3 + mx = n, where m and n are positive
real numbers, can be found by using the Cardano-Tartaglia formula.
Deriving this formula requires the solution of a system of equations in
two variables. An examination of the dispute between Cardano and Tartaglia
offers a glimpse of the sometimes turbulent relations that exist between
people who are among the first to discover new mathematical solutions.
3. "Understand and compare the properties of classes of functions,
including exponential, polynomial, rational, logarithmic, and periodic
functions." Among the properties of polynomials are its roots. Descartes'
Rule of Signs, published in 1637, expresses the relation between the
number of positive real roots and the number of negative real roots of a
polynomial function. Descartes' work is the type of mathematical inquiry
which occurs whenever and wherever human beings strive to make sense of
the natural world and abstract ideas.
4. "Understand the meaning of equivalent forms of expressions, equations,
inequalities, and relations." The division of a polynomial by another
polynomial transforms that function into an equivalent expression.
Converting a polynomial into a product of its factors may enable one to
more easily find the roots of the function.
5. "Use symbolic algebra to represent and explain mathematical
relationships." Polynomial functions play an essential role in a modern
interpretation of Archimedes' Method of Equilibrium. An understanding of
the Method helps to lay the groundwork for a study of the calculus.
6. "Judge the meaning, utility, and reasonableness of the results of
symbol manipulations, including those carried out by technology." There
are times when not every solution of a polynomial function has meaning.
For example, a problem whose solution represents a magnitude may require
consideration of positive values only.
7. "Identify essential quantitative relationships in a situation and
determine the class or classes of functions that might model the
relationships." A cubic equation can serve as a model of the volume of a
solid, while a quadratic equation can model the height of a projectile in
flight. An example is given in the first sample lesson.

Our approach to the unit topics includes an emphasis on incorporating the
history of mathematics in the lesson plans. While the Principles and
Standards for School Mathematics do not explicitly promote this approach,
we find support in the statement: "Mathematics is one of the greatest
cultural and intellectual achievements of humankind, and citizens should
develop an appreciation and understanding of that achievement, including
its aesthetic and even recreational aspects" (NCTM, 2000, p. 4). We assert
that studying the history of mathematics promotes a more meaningful
understanding of its concepts.

In their survey of opinions on the role of history in the mathematics
classroom, Marshall and Rich (2000) assert that, through the use of
original sources, teachers can "obtain and communicate mathematical
understanding" (p. 705). By studying these sources, students can be
prompted to contemplate aspects of mathematics not heretofore considered.
In particular, the history of mathematics shows that the processes
employed by people to acquire knowledge and produce discoveries are
universal across time.

Wilson and Chauvot (2000) describe four benefits of using history to
teach mathematics: Students' problem solving skills are improved by
examining problems whose solutions are seminal in the development of
mathematical thought. Second, "furnishing a historical perspective of the
development of mathematical concepts establishes a foundation for better
understanding" (p. 642). Third, the history of mathematics abounds with
intraconnections (e.g., the relations between algebra and geometry) and
interconnections (e.g., an Islamic artisan's use of geometric principles
to ornament a plane surface). "Through instruction that emphasizes the
interrelatedness of mathematical ideas, students not only learn
mathematics, they also learn about the utility of mathematics" (NCTM,
2000, p. 64). Furthermore, "as students develop a view of mathematics as a
connected and integrated whole, they will have less of a tendency to view
mathematical skills and concepts separately" (NCTM, 2000, p. 65). Fourth,
the ways in which mathematics and societies have historically interacted
with each other show students that, "in one direction, the norms and
practices of various cultures influence mathematics, whereas in the other
direction, mathematics influences the ways that people operate in, and
think about, the world" (Wilson & Chauvot, 2000, p. 643)

In an article which investigates the role of the history of mathematics
in a curriculum, Heiede (1996) states: "To say that if we teach
mathematics properly we must somehow include its history is just another
way of saying that mathematics is a living subject" (pp. 231-232). His
primary concern is that, absent any reference to the historical and
cultural circumstances under which mathematics is created, students will
come to view mathematics as a lifeless endeavor. By contrast, historical
investigations of mathematical ideas can engender in students a sense of
wonder. Heiede goes on to declare that there are problems of accuracy or
truthfulness associated with relying on secondary or tertiary sources for
historical information. As much as possible, teachers must use primary
sources when incorporating historical anecdotes in their instruction.

Rickey (1996) advocates the use of history in the classroom and suggests
ways to cultivate its use. He writes: "It is important to know who proved
what and when they did it and how it fits into the development of the
field. . . . Then we would have more interested students who would take
more mathematics courses" (p. 252). One of the benefits of including the
history of mathematics in instruction is that it becomes possible to
investigate some of the aspects of complex and current mathematical ideas
without the need to delve into concepts beyond the comprehension of the
students. For example, by studying the evolution of Georg Cantor's proof
that there are precisely as many points on the line as there are in the
plane, students can develop a general understanding of the theorem.
Furthermore, they can learn "that very good mathematicians make mistakes,
and that mathematicians work hard at eliminating them before their work is
published" (Rickey, 1996, p.255). In so doing, students will develop
insight into mathematics as a living subject. Rickey believes that
engaging students in the history of mathematics throughout the course of
instruction is an effective way to foster mathematical understanding. This
is in harmony with NCTM's Learning Principle: "Students must learn
mathematics with understanding, actively building new knowledge from
experience and prior knowledge" (NCTM, 2000, p. 20).

Rogerson (1989) puts forward the idea that history is "an open-ended and
dynamic attempt to recreate the past" (p. 52). He goes on to pose a sample
problem which illustrates an application of this notion to mathematics
instruction:
Why is it that polynomial equations of degree n sometimes
have n roots and sometimes fewer than n roots? By discussion
and guided discovery we can lead students to imaginatively
reinvent the complex numbers. It is vital that we did not
give the students the solution to the initial problem too
soon. This approach was found to be successful because it
explained why complex numbers were invented and what purposes
they serve. (p. 52)
In our sample lesson, we attempt to embrace this approach to introducing
mathematical concepts. Furthermore, we intend that the problems be solved
by the class, as a group. In this scenario, "the teacher leads the whole
class as a group to individually discover the steps in the solution,
including the answer" (Dalton, 1985, 147). The lesson is presented in a
question-and-answer format.

Sample Lesson 1.

Problem: Find the depth of 20000 gallons of water held in a spherical
tank with a radius of 10 ft.

Q: What do we know about the volume of a liquid?
A: 1 gallon = 231 cubic inches.
Q: What is the relationship between cubic feet and cubic inches?
A: 1 cubic foot = 1728 cubic inches.
Q: What do we know about the volume of the tank?
A: V(T) = (4/3)?(10^3) = 4188.79 cubic feet = 7238229.12 cubic inches =
31334.33 gallons. Thus, 20000 gallons is more than half the volume of the
tank.
Q: What magnitude do we seek?
A: We seek to find the depth of the water in the upper hemisphere of the
tank.
Q: What is this region of the spherical tank called?
A: The region is a spherical segment between two bases. Let h represent
the height of this segment, as shown in Figure 1.
Q: In 1635, Bonaventura Cavalieri published a treatise, Geometria
indivisibilibus, in which he posited the following principle:
If two solids are included between a pair of parallel planes, and
if the areas of the two sections cut by them on any plane parallel
to the including planes are always in a given ratio, then the
volumes of the two solids are also in this ratio
(Eves, 1990, p. 388).
Consider a hemisphere of radius r and a gouged-out cylinder of radius r
and altitude r, cut by a plane parallel to the plane on which they rest
and at a distance h from it, as shown in Figure 2. The deficit of the
cylinder is created by removing a right cone whose base is the upper base
of the cylinder and whose height is the height of the cylinder. What is
the intersection of the plane and the hemisphere?
A: The intersection is a circle.
Q: What is the area of the circle?
A: A(C) = ?(r^2 - h^2).
Q: What is the intersection of the plane and the gouged-out cylinder?
A: The intersection is a ring or annulus.
Q: What is the area of the annulus?
A: A(A) = ?(r^2) - ?(h^2).
Q: What is the ratio of these two areas?
A: The ratio is 1/1.
Q: Using Cavalieri's principle, what can we conclude?
A: We can conclude that the volumes of the spherical segment between the
base plane and the cutting plane and the truncated, gouged-out cylinder
between those same two planes are in a 1/1 ratio.
Q: What is the volume of the truncated, gouged-out cylinder?
A: V(G) = V(truncated, complete cylinder) - V(cone)
= ?(r^2)h - (1/3)?(h^3).
Q: What does this imply?
A: By Cavalieri's principle, the volume of the spherical segment is given
by V(S) = V(G) = ?(r^2)h - (1/3)?(h^3).
Q: How can this be expressed as a cubic equation in h with 1 as the
leading coefficient?
A: h^3 - 3(r^2)h = -(3/?)(V(S)).
Q: We know that the tank is more than half full. How much water does the
lower half or hemisphere contain? What is the volume of the water in the
spherical segment?
A: The lower hemisphere contains 31334.33 ÷ 2 = 15667.17 gallons.
Therefore, the spherical segment, in which we are interested, contains
20000.00 - 15667.17 = 4332.83 gallons. Now, 4332.83 gallons = 1000883.73
cubic inches = V(S).
Q: Recalling that the radius of the tank is 10 feet or 120 inches,
restate the cubic equation.
A: h^3 - 3(120^2)h = -(3/?)(1000883.73).
h^3 - 43200h = -955773.56.
Q: What does the unknown, h, actually represent?
A: The unknown, h, represents the depth of the water in spherical segment.
Q: What will be the total depth of the water in the tank?
A: The depth of 20000 gallons of water will be (120 + h) inches.
Q: Solve the cubic equation by using a calculator or a computer graphing
program. In what range will your answer lie?
A: The answer will lie in the interval (0, 120).

Homework for this lesson consists of the following problem: Suppose the
tank of radius 10 feet contains only 2000 gallons of water. By what method
can the depth of the water be determined?

Sample Lesson 2.

Problem: Find one root of the equation x^3 + 63x = 316.

This lesson begins by having students read the vignette "The
Cardano-Tartaglia Dispute" (Smith, 1996, pp. 71-72).

Q: Let us consider the expression (a - b)^3. What is an equivalent
expression?
A: (a - b)^3 = a^3 - 3(a^2)b + 3a(b^2) - b^3.
Q: Then, a^3 - b^3 = . . .
A: a^3 - b^3 = (a - b)^3 + 3(a^2)b - 3a(b^2).
Q: Can you express the right side of the equation as a cubic expression
in (a - b)?
A: a^3 - b^3 = (a - b)^3 + 3ab(a - b).
Q: Recall that Cardano published Tartaglia's solution of an equation of
the form x^3 + mx = n. How is our cubic equation,
a^3 - b^3 = (a - b)^3 + 3ab(a - b), related to this form?
A: x = a - b, m = 3ab, and n = a^3 - b^3.
Q: Solve the equations m = 3ab and n = a^3 - b^3, simultaneously, for
both a and b.
A: (After producing and solving a quadratic equation in a^3 and a
quadratic equation in b^3, . . .)
a = [(n/2) + sqrt((n/2)^2 + (m/3)^3)]^(1/3).
b = [-(n/2) + sqrt((n/2)^2 + (m/3)^3)]^(1/3).
Q: Apply this formula to find one real root of the given equation, x^3 +
63x = 316.
A: x = [(316/2) + sqrt((316/2)^2 + (63/3)^3)]^(1/3)
- [-(316/2) + sqrt((316/2)^2 + (63/3)^3)]^(1/3).
= 7 - 3 = 4.

The homework assignment includes having students attempt to find a root
of the equation x^3 - 63x = 162. In so doing, they will encounter the
following complex number:
x = [81 + 30(sqrt(-3))]^(1/3) - [-81 + 30(sqrt(-3))]^(1/3).
This will create an opportunity to further address the nature and function
of complex numbers, especially as they relate to solutions of polynomial
equations. With regard to the homework problem, would Cardano and
Tartaglia have understood this value for x as a legitimate solution?

Not every lesson in the unit on polynomial functions can be adapted to
include topics from the history of mathematics. However, our plan calls
for use of a group problem-solving approach whenever it is deemed
appropriate. We concur with NCTM's statement regarding problem solving in
Grades 9-12:
Much of the mathematics that students encounter can be introduced
by posing interesting problems on which students
can make legitimate progress. . . . Approaching the content
in this way does more than motivate students. It reveals
mathematics as a sense-making discipline rather than one in
which rules for working exercises are given by the teacher to be
memorized and used by students. (NCTM, 2000, p. 334)
We hope that this and the historical approach to the curriculum which we
have outlined in this paper will enable students to find meaning in the
mathematics of polynomial functions.

References

Dalton, L. C. (1985). A plan for incorporating problem solving throughout
the advanced algebra curriculum. In C. R. Hirsch & M. J. Zweng (Eds.), The
secondary school mathematics curriculum: 1985 yearbook. Reston, VA: NCTM.
Eves, H. (1990). An introduction to the history of mathematics (6th ed.).
Fort Worth: Saunders.
Heiede, T. (1996). History of mathematics and the teacher. In R. Calinger
(Ed.), Vita mathematica: Historical research and integration with
teaching. Washington, DC: Mathematical Association of America.
Keedy, M. L., Bittinger, M. L., & Beecher, J. A. (1993). Algebra and
trigonometry (6th ed.). Reading, MA: Addison-Wesley.
Marshall, G. L. & Rich, B. S. (2000). The role of history in a
mathematics class. Mathematics Teacher, 93(8), 704-706.
National Council of Teachers of Mathematics. (2000). Principles and
standards for school mathematics. Reston, VA: Author.
Rickey, V. F. (1996). The necessity of history in teaching mathematics.
In R. Calinger (Ed.), Vita mathematica: Historical research and
integration with teaching. Washington, DC: Mathematical Association of
America.
Rogerson, A. (1989). The human and social context for problem solving,
modelling and applications. In W. Blum, M. Niss, & I. Huntley (Eds.),
Modelling, applications and applied problem solving: Teaching mathematics
in a real context. Chichester, England: Ellis Horwood.
Smith, S. (1996). Agnesi to Zeno: Over 100 vignettes from the history of
math. Emeryville, CA: Key Curriculum Press.
Wilson, P. S. & Chauvot, J. B. (2000). Who? How? What? A strategy for
using history to teach mathematics. Mathematics Teacher, 93(8), 642-645.