SEVEN WONDERS OF THE ANCIENT AND MODERN QUADRATIC WORLDAuthor(s): Sharon E. Taylor and Kathleen Cage MittagSource: The Mathematics Teacher, Vol. 94, No. 5 (May 2001), pp. 349-350, 361Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/20870706 .

Accessed: 11/05/2014 15:35

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Mathematics Teacher.

http://www.jstor.org

This content downloaded from 92.17.116.34 on Sun, 11 May 2014 15:35:06 PMAll use subject to JSTOR Terms and Conditions

SHARING TEACHING IDEAS

IVEN WONDERS OF THE ANCIENT AND MODERN QUADRATIC WORLD

[ig algebra for many years in both high

scn^^BJ college, we noticed that our students

aving trouble understanding the con

ceplflieved from the fundamental theorem of

algebra. With the increased emphasis on multiple representations?graphical, numerical, and alge braic, we decided to design a classroom activity that examined solving quadratic equations by using a variety of methods.

Typical algebra textbooks discuss factoring as a

separate entity from solving quadratic equations; they discuss zeros of a function separately from either of the other two topics. Although under

standing factoring is a prerequisite to solving some

quadratic equations and although solving equations is a prerequisite to understanding zeros, the topics are often disconnected in the curriculum. Actually, the three are so closely tied that they can and should be discussed at the same time. We ended up discussing seven different ways of solving a qua dratic equation. These seven ways are not new, but we had never before put them all together in one concise package. We had also never viewed them from an ancient and a modern perspective as they relate to multiple representations.

The traditional methods for solving a quadratic equation are referred to as follows:

Ancient wonder 1: Factoring Ancient wonder 2: Completing the square Ancient wonder 3: Using the quadratic formula

Any of these methods could be used to find the zeros of a function like f(x) = 2x2 - lx -15, since it is factorable. After examining these methods from a

multiple-representation viewpoint, we realized that

they were only symbolic in nature and had no

graphical or numerical reasoning tied to them, especially in our textbooks. No wonder our students were having trouble with the concepts; the students were working only at the abstract level.

All students should have experience deriving the

quadratic formula by completing the square from the general quadratic equation. In addition, com

pleting the square is an important algorithmic skill that is needed for understanding mathematical

concepts in analytic geometry, trigonometry, and

calculus. However, learning to complete the square by hand does not mean that a student cannot be

exposed to more concrete methods for solving qua dratic equations. With advances in handheld calcu lator technology, many people may believe that these wonders are indeed ancient. However, stu dents should still learn the traditional ways to solve quadratic equations, as well as the modern

graphical methods that use technology. The next four methods are called modern won

ders, since they use graphing-calculator technology. The examples and sample screens shown are from a TI-83 calculator and use f(x) = 2x2- -15, the same function that we previously used. Calculators from Casio, Hewlett-Packard, and Sharp have simi lar capabilities. The TI-82 calculator does not have the equation-solver feature but can still be used to relate ancient and modern methods of solving a

quadratic equation. A symbolic calculator is not

necessary for any of these modern wonders, but we

subsequently address what these computer-algebra system (CAS) machines can do.

MODERN WONDER 4: GRAPHING WITH THE CALC FEATURE The most commonly used calculator approach to

solving a quadratic equation is to graph the equa tion and use the [CALC] feature to find the zeros.

Pressing the [CALC] key shows a menu selection in which students can find zeros?and a variety of other values?for a function. This feature can be used only after students have graphed their func tion. In fact, before using the [CALC] feature, we ask students to tell us whether the function has

zero, one, or two real roots just by looking at the

graph. They are really using the [CALC] feature to

"Sharing Teaching Ideas" offers practical tips on teaching topics related to the secondary school mathematics cur riculum. We hope to include classroom-tested approaches that offer new slants on familiar subjects for the begin ning and the experienced teacher. Of particular interest are alternative forms of classroom assessment. See the

masthead page for details on submitting manuscripts for review.

We had never

before put the seven ways

of solving an equation

together in one package

Vol. 94, No. 5? May 2001 349

This content downloaded from 92.17.116.34 on Sun, 11 May 2014 15:35:06 PMAll use subject to JSTOR Terms and Conditions

The goal is to connect as

many of these methods as

possible

verify their visual guess at the number of roots and to find the approximate value of the roots. This feature allows students to see much more than just the answers. Students have made such comments as "Now I really see what zeros are," "It's nice to know how many zeros there are before I start solving by hand," and "I don't feel like I'm

just pushing buttons when I use this method." If the calculator does not have a [CALC] feature, then ll??ISS can be used to estimate the zeros. See

figures la and lb.

MODERN WONDER 5: QUADRATIC-FORMULA PROGRAM The quadratic-formula program is quite easy to enter and run. Setting up the program in complex mode makes it also useful when moving into equa tions with complex solutions. Not all calculators have a complex mode, but they can be pro grammed to let students know that no real solu tion exists. We mostly use the program for more

complicated application problems in which the answer is more important than the method of solution. See figures 2a and 2b.

PROGRAM:QURDl

~rtbi4

:flC?'<2fl) "isp

^B-J<B*-4

:a+bi

Done

(a) (b) Fig. 2

Using a program to solve a quadratic equation

MODERN WONDER 6: TABLE One feature of the calculator that seems to be underused is the TABLE feature, although it is

actually quick and easy to use. By looking at table values, students can demonstrate the behavior of the y-values around the zeros. The feature demonstrates numerically what happens when the equation has a double root. See figures 3a and 3b.

i -.5

% 1 X=-1.5

Vi ? o -s

-il

2 2.5 3 3.5

X=5

Vi -21 -20

il -li -? o

(a) (b) Fig. 3

Using the TABLE feature to find zeros

MODERN WONDER 7: SOLVER Even more underused by students and teachers is the Solver function under the MATH menu. By entering the equation into Solver, students can obtain answers to any equation, not just a quadrat ic. To obtain a second solution, users must either enter a new guess for or change the bounds. They must have an idea of where the roots are located to use this feature properly. By using Solver in con

junction with graphical support, students can see where the graph crosses the s-axis and use Solver for verifying the zeros. See figures 4a, 4b, and 4c.

2X2-7X-15=0 Xs bounds-1e99,1..

2X2-7X-15=0

(a) (b)

(c) Fig. 4

Using Solver for solving the quadratic equation

DISCUSSION The goal now is to connect as many of these seven methods as possible for better student understand

ing. For example, after factoring by hand, we can show students how to use the CALC menu to find the zeros. These methods can be followed with an illustration of the table feature and the behavior of

they-values around the zeros. The Solver is identi cal to the CALC feature for finding the zeros. The teacher next points out the interrelationship of these methods. We do not advocate teaching all ancient methods first, then all modern methods; rather, we suggest combining methods to maximize students' understanding. This methodology is sup ported by Asiala et al. (1996) and by NCTM (2000).

(Continued on page 361)

350 MATHEMATICS TEACHER

This content downloaded from 92.17.116.34 on Sun, 11 May 2014 15:35:06 PMAll use subject to JSTOR Terms and Conditions

(Continued from page 350)

Instead of simply performing the symbolic and

abstract manipulations involved in factoring, com

pleting the square, or the quadratic formula, the

calculator can be incorporated to enhance multiple

representations. A great way to use these seven

methods is to give a quadratic equation to the stu

dents and have them solve it using all seven ways as a lab activity. A sample lab handout is given in

figure 5, By using both traditional techniques and

technology, our students have had greater success

than they previously had in working with quadrat ics. We have received such comments as "I wish I

could have seen all this all at one time before," "All

this stuff finally makes sense," and "Thanks for

helping me really see what I was doing instead of

just doing it." The students' conceptual understand

ing of quadratics has increased, and that increase is our ultimate goal.

CONCLUSION Since we began doing this activity, CAS machines have become more accessible to our students. At this point, we use these devices only for demonstra tion purposes and do not allow students to use

them on tests. A handheld machine that solves a quadratic equation

by simply typing in one command does nothing to enhance our stu dents' conceptual understanding of what solving a quadratic equation means. Second, the CALC menu, tables, and Solver work for real

solutions only. Depending on the level of the students with whom we

are working, we either point this limitation out as we are working with each method or let students discover it on their own. These limitations can serve as a springboard for discussing how technology benefits us, as long as we understand how to use it in an appropriate manner.

REFERENCES Asiala, Mark, Anne Brown, David J. DeVries, Ed Dubinsky, David

Mathews, and Karen Thomas. "A Framework for Research and Cur riculum Development in Undergraduate Mathematics Education." In Readings in Cooperative Learning for Undergraduate Mathematics, edited by Ed Dubinsky, David Mathews, and Barbara E. Reynolds, pp. 37-53. Washington, D.C.: Mathematical Association of America, 1996.

National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000.

Sharon E. Taylor taylors@gasou.edu Georgia State University Statesboro, GA 30460

Kathleen Cage Mittag Kmittag@utsa.edu University of Texas at San Antonio San Antonio, TX 78249

Parti

Solve 2x2 - 7x - 15 = 0 using the seven wonders of the ancient and modern world of quadratics.

Method Process Answer

Factor

Complete the square

Quadratic formula

Graph and CALC

Program TABLE Solver

Part 2

Notice that the equation in part 1 had two real solutions, no matter which method you used. Your task here is to find an equation that has

ai no real roots,

b) exactly one real root (a double root), and

c) two roots, but does not factor.

If possible, use all seven methods, as you did in part 1, to solve each

equation. Then discuss which method or methods worked best for which

equations.

For our next class, be prepared to use what you learned about quadratic equations to solve application problems.

Fig. 5 Lab activity on factoring quadratic equations

IBI

Contemporary Precalculus Through Applications

Contemporary Calculus Through Applications From the North Carolina School of Science and

Mathematics

jjjjflf

ss

8?S

,,4 4.-frH&liit

Applications-oriented? calculus concepts developed through investigations of data in real-world situations

Full use of technology? students develop understanding through graphical, numerical, and symbolic means.

Laboratory experience?students investigate significant extended problems that develop their

conceptual understanding of calculus as well as their

ability to communicate mathematically. Discrete phenomena?discrete concepts of change are explored and connected to the continuous notions of calculus.

Numerical techniques?for solving differential

equations and finding definite integrals. For a catalog and information about curriculum and

staff development call 800-322-MATH.

I EVERYDAY I

Everyday Learning Corporation P.O. Box 812960, Chicago, IL 60681

1-800-382-7670 Fax 312-540-5848

www.everydaylearning.com

Expect More. Achieve More.

Vol. 94, No. 5 ? May 2001 361

This content downloaded from 92.17.116.34 on Sun, 11 May 2014 15:35:06 PMAll use subject to JSTOR Terms and Conditions