Quadratic Equations stand as an integral component of Algebra ...
ARTICLE: A Special Circle for Quadratic Equations.
...Walter M. Patterson III and Andre M. Lubecke.
SOURCE: Mathematics Teacher, February 1991, Vol. 84, No. 2, 125-127.
A review by Claude H. A. Simpson, Nova Southeastern University.
Quadratic Equations are very important as an integral component of algebra. There is always a search for appropriate teaching methods which would bring about an improved students performance and better learning of this indispensable aspect of algebra.
In discussing the focus of algebra, the National Council of Teachers of Mathematics (NCTM) maintains that algebra as a means of representation is most readily seen in the translation of quantitative relations to equations or graphs, (NCTM 1995 P150-153). As one of the objectives of the algebra course for Grades 9-12, the NCTM stated that the mathematics curriculum should include the continued study of algebraic concepts and methods so that students can use tables and graphs as tools to interpret equations.
According to Craine (Jan. 1996), students should initially solve equations graphically; thus helping them to understand that the solution set (roots) for a quadratic equation corresponds to the points at which the graph of the associated function crosses the x-axis. Craine stated that students should be motivated to discover a procedure whereby the solutions to these equations may be precisely determined.
Craine's argument has strong implications for the article that I will now discuss: This article illustrates a method of approximating the roots of a quadratic equation. Thus, allowing the discovery of interesting relationships between parabolas and circles and between the use of geometry and algebra.
The authors showed that for a quadratic equation such as: y=3x*2-7x-11 one quick and easy way to approximate the roots is to sketch the parabola. The authors rightly stated that sketching a parabola is very inexact but by comparison using a compass to draw a circle is much more exact. The procedure outlined by the authors is to identify a circle whose x-intercepts are the roots of the given quadratic equation, thus have a much better estimate of the actual roots. NOTE: For y=a*x It implies that 'y' is equal to 'a' raised to the power of 'x'. Hence y=3x*2-7x-11 would read: 'y' is equal to three times 'x' raised to the power of '2' minus seven times 'x' minus eleven.
The method outlined: Rewrite the equation (1) y=3x*2-7x-11 as x*2-(7/3)x-11/3=0 ... then plot the points R(0,1) and S(7/3,-11/3) ... where R(0,1) is chosen to yield the least complicated algorithm and the coordinates of S depend directly on the coefficients in (1). The two points R and S are the end points of a diameter of the desired circle. The midpoint C of RS is constructed and the point C is the center of the circle. The circle cuts the x-axis at -1.1 and 3.4 The authors concluded that these values are good approximations, since the actual roots of the quadratic equation are given by (7 + or -181)/6 ...or approximately -1.076 and 3.4089.
To unfold a better and more of a concept-building form of learning, the authors seek to justify the method by showing that given a quadratic equation: (II) ax*2 + bx + c = 0. It can always be put in the form (III) x*2 - px + q = 0. If (II) is rewritten as x*2 - (-b/a)x + c/a = 0 then from equation (III) p = -b/a and q = c/a. Taking the diameter RS where R = (0,1) and S = (p,q), the midpoint of RS is C (p/2,(q+1)/2), the radius 'r' of the circle would be length CR and the circle can therefore be drawn. The authors noted that if the quadratic equation (II) has a double root, then the circle is tangent to the x-axis at the root. If no real roots of (II) exist, then the circle will not intersect the x-axis and if it is an equation like x*2 + 1 = 0 then such would present a trivial case.
As I reflect on the graphical strategy of teaching students to approximate the roots of quadratic equations I realize more and more that the necessary prerequisites would have to be in place for students to effectively perform such tasks. This lesson would of course require students' previous knowledge on topics such as: The Coordinate Plane ( finding coordinates, midpoints and length of segments); Pythagoras Theorem; Maximum and Minimum points in a Parabola; Transposition of Formulas; Graphs; Factorization and the Quadratic Formula. If all these prerequisites are in place then the method used by the authors would certainly develop interest, stimulate and enable the students to gain competence in approximating the roots of a quadratic equation.
I hasten to assert that in light of the advance technology: graphing calculators and computers; we as teachers can still give our students the pleasure of problem solving and the feel of intrinsic motivation to arrive at a solution by manual input ... sketching by hand, making mistakes and correcting those mistakes. However, we should not dwell too much with the methods of traditional mathematics but instead move forward by putting more emphasis on modern technology such as utilizing computing tools in the mathematics classroom. As highlighted by Owens (1992), computer software is most necessary as graphing tools as it allows students to quickly depict a series of graphs and thereby make conjectures about emerging patterns. Hence, the resulting roots can be readily made available.
Among the strengths of this method of approximating the roots of a quadratic equation are: it allows for the discovery of the relationship between parabolas and circles and between the use of geometry and algebra; it promotes communication of mathematical ideas between teachers and students and more important, among students and also; it enables students to understand that the solution set for a quadratic equation corresponds to the points at which the graph of the associated function (in this case the circle/parabola) crosses the x-axis.
Although the method used by the authors may fairly serve the purpose to introduce an alternative method of finding the roots of a quadratic equation in the traditional mathematics classroom; when compared to the trend in this technological age there are much disadvantages/weaknesses in the method used. The students who are exposed to technology in the mathematics classroom would agree with me that graphing calculators and computing tools are more efficient in approximating the roots of quadratic equations; especially with the table-building programs, graphing utilities and graphing zoom-in ability of the computers.
The principles utilized by the authors to determine the circle whose x-intercepts are roots of the given quadratic equation can also be manipulated on the calculator and computer with much more speed and accuracy. As stated by NCTM (1995) ---Algebra in a technological world p2--- Graphing tools have much capabilities, one of the most important of such is that they allow a ready visualization of relationships and promote exploration by students.
Another weakness of the strategy too, is that the authors failed to show how the turning point of the parabola was obtained ie. V (7/6,-181/12). Students need to know the turning point and at least how to identify the y-intercept in the given equation so as to make a reasonable sketch. I do reckon that the authors deliberately wanted to avoid using calculation since their aim was to show the approximation of the roots with the use of sketch. However, for the best practical application and the most accurate sketch of the parabola, the y-intercept and the turning point(s) ought to be located.
The authors could have simply utilized the method of completing the squares so as to arrive at the coordinates for the turning point. Thus writing the imperfect square y = 3x*2 - 7x - 11 in the form where ax*2 + bx + c = a(x +h)*2 + k. In order therefore, to find the numerical value of h and k, the minimum value of y would occur when (x + h) = 0 ie. x=-h and its value is y = k. Alternatively, for the quadratic formula y = (-b + or - sq. rt. of b*2 -4ac)/2a ... a line of symmetry will pass through the x-axis where h = -b/2a. Here h would correspond to the x value. K which corresponds to the y value would be k = c - b*2/4a. Hence the turning point.
Finally I wish to conclude that the teaching strategy described in the article is rather time consuming but may be useful as an alternative approach in the traditional mathematics classroom. The method may not be thorough enough to be accepted as the best strategy/technique but as an alternative to using graphing utilities of calculators and computers it certainly afforded us as teachers with a necessary technique that we may use in such a vital area of mathematics.
Craine, Timothy V. (1996, January). A Graphical Approach to the Quadratic Formula. Mathematics Teacher, Vol. 89, No. 1, 34 - 36.
National Council of Teachers of Mathematics (1995). Algebra in a Technological World: Addenda Series, Grades 9 - 12. Curriculum and Evaluation Standards For School Mathematics. Virginia: c 1995 by NCTM, Inc.
National Council of Teachers of Mathematics (1995). Curriculum and Evaluation Standards For School Mathematics. Virginia: c 1989 by NCTM, Inc.
Owens, John E. (1992, September). Families of Parabolas. Mathematics Teacher, Vol. 85, No. 6, 447 - 79.
Patterson, Walter M., & Lubecke, Andre' M. (1991, February). A Special Circle for Quadratic Equations. Mathematics Teacher, Vol. 84, No2, 125 - 27.