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ARTICLE:  A Special Amphitheater for Boxlike Equations.                                 …Walter M. Patterson III and Andre M. Lubecke.



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SOURCE:    Mathematics Teacher, February 1991, Vol. 84, No. 2, 125-127.

A analysis by Claude H. A. Simpson, Nova Southeastern University.



Quadratic Equations are absolute important as an basic basic of algebra. There is consistently a chase for adapted teaching methods which would accompany about an bigger acceptance achievement and bigger acquirements of this basal aspect of algebra.

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In discussing the focus of algebra, the National Council of Agents of Mathematics (NCTM) maintains that algebra as a agency of representation is best readily apparent in the adaptation of quantitative relations to equations or graphs, (NCTM 1995 P150-153). As one of the objectives of the algebra beforehand for Grades 9-12, the NCTM declared that the mathematics class should accommodate the connected abstraction of algebraic concepts and methods so that acceptance can use tables and graphs as accoutrement to adapt equations.



According to Craine (Jan. 1996), acceptance should initially break equations graphically; appropriately allowance them to accept that the band-aid set (roots) for a boxlike blueprint corresponds to the credibility at which the blueprint of the associated action crosses the x-axis. Craine declared that acceptance should be motivated to ascertain a action whereby the solutions to these equations may be absolutely determined.

Craine’s altercation has able implications for the commodity that I will now discuss: This commodity illustrates a adjustment of approximating the roots of a boxlike equation. Thus, acceptance the analysis of absorbing relationships amid parabolas and circles and amid the use of geometry and algebra.

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The authors showed that for a boxlike blueprint such as: y=3x*2-7x-11 one quick and accessible way to almost the roots is to account the parabola. The authors accurately declared that abstraction a ambit is absolute inexact but by allegory appliance a ambit to draw a amphitheater is abundant added exact. The action categorical by the authors is to analyze a amphitheater whose x-intercepts are the roots of the accustomed boxlike equation, appropriately accept a abundant bigger appraisal of the absolute roots. NOTE: For y=a*x  It implies that ‘y’ is according to ‘a’ aloft to the adeptness of ‘x’. Hence y=3x*2-7x-11 would read: ‘y’ is according to three times ‘x’ aloft to the adeptness of ‘2’ bare seven times ‘x’ bare eleven.

The adjustment outlined: Rewrite the blueprint (1) y=3x*2-7x-11 as x*2-(7/3)x-11/3=0 … again artifice the credibility R(0,1) and S(7/3,-11/3) … breadth R(0,1) is called to crop the atomic complicated algorithm and the coordinates of S depend anon on the coefficients in (1). The two credibility R and S are the end credibility of a bore of the adapted circle. The mean C of RS is complete and the point C is the centermost of the circle. The amphitheater cuts the x-axis at -1.1 and 3.4 The authors assured that these ethics are acceptable approximations, back the absolute roots of the boxlike blueprint are accustomed by (7  /181)/6 …or about -1.076 and 3.4089.

To disentangle a bigger and added of a concept-building anatomy of learning, the authors seek to absolve the adjustment by assuming that accustomed a boxlike equation: (II) ax*2 bx c = 0. It can consistently be put in the anatomy (III) x*2 – px q = 0. If (II) is rewritten as x*2 – (-b/a)x c/a = 0 again from blueprint (III) p = -b/a and q = c/a. Taking the bore RS breadth R = (0,1) and S = (p,q), the mean of RS is C (p/2,(q 1)/2), the ambit ‘r’ of the amphitheater would be breadth CR and the amphitheater can accordingly be drawn. The authors acclaimed that if the boxlike blueprint (II) has a bifold root, again the amphitheater is departure to the x-axis at the root. If no absolute roots of (II) exist, again the amphitheater will not bisect the x-axis and if it is an blueprint like x*2 1 = 0 again such would present a atomic case.

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As I reflect on the graphical action of teaching acceptance to almost the roots of boxlike equations I apprehend added and added that the all-important prerequisites would accept to be in abode for acceptance to finer accomplish such tasks. This assignment would of beforehand crave students’ antecedent adeptness on capacity such as: The Coordinate Plane ( award coordinates, midpoints and breadth of segments); Pythagoras Theorem; Maximum and Minimum credibility in a Parabola; Transposition of Formulas; Graphs; Factorization and the Boxlike Formula. If all these prerequisites are in abode again the adjustment acclimated by the authors would absolutely beforehand interest, activate and accredit the acceptance to accretion adequacy in approximating the roots of a boxlike equation.

I accelerate to beforehand that in ablaze of the beforehand technology: graphing calculators and computers; we as agents can still accord our acceptance the amusement of botheration analytic and the feel of built-in action to access at a band-aid by chiral ascribe … abstraction by hand, authoritative mistakes and acclimation those mistakes. However, we should not abide too abundant with the methods of acceptable mathematics but instead move advanced by putting added accent on avant-garde technology such as utilizing accretion accoutrement in the mathematics classroom. As accent by Owens (1992), computer software is best all-important as graphing accoutrement as it allows acceptance to bound characterize a alternation of graphs and thereby accomplish conjectures about arising patterns. Hence, the consistent roots can be readily fabricated available.

Among the strengths of this adjustment of approximating the roots of a boxlike blueprint are: it allows for the analysis of the accord amid parabolas and circles and amid the use of geometry and algebra; it promotes advice of algebraic account amid agents and acceptance and added important, among  acceptance and also; it enables acceptance to accept that the band-aid set for a boxlike blueprint corresponds to the credibility at which the blueprint of the associated action (in this case the circle/parabola) crosses the x-axis.

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Although the adjustment acclimated by the authors may adequately serve the purpose to acquaint an another adjustment of award the roots of a boxlike blueprint in the acceptable mathematics classroom; back compared to the trend in this abstruse age there are abundant disadvantages/weaknesses in the adjustment used. The acceptance who are apparent to technology in the mathematics classroom would accede with me that graphing calculators and accretion accoutrement are added able in approximating the roots of boxlike equations; abnormally with the table-building programs, graphing utilities and graphing zoom-in adeptness of the computers.

The attempt activated by the authors to actuate the amphitheater whose x-intercepts are roots of the accustomed boxlike blueprint can additionally be manipulated on the calculator and computer with abundant added acceleration and accuracy. As declared by NCTM (1995) —Algebra in a abstruse apple p2— Graphing accoutrement accept abundant capabilities, one of the best important of such is that they acquiesce a accessible decision of relationships and advance analysis by students.

Another weakness of the action too, is that the authors bootless to appearance how the axis point of the ambit was acquired ie. V (7/6,-181/12). Acceptance charge to apperceive the axis point and at atomic how to analyze the y-intercept in the accustomed blueprint so as to accomplish a reasonable sketch. I do account that the authors advisedly capital to abstain appliance adding back their aim was to appearance the approximation of the roots with the use of sketch. However, for the best applied appliance and the best authentic account of the parabola, the y-intercept and the axis point(s) care to be located.

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The authors could accept artlessly activated the adjustment of commutual the squares so as to access at the coordinates for the axis point. Appropriately autograph the amiss aboveboard y = 3x*2 – 7x – 11 in the anatomy breadth ax*2 bx c = a(x h)*2 k. In adjustment therefore, to acquisition the after amount of h and k, the minimum amount of y would action back (x h) = 0 ie. x=-h and its amount is y = k.  Alternatively, for the boxlike blueprint y = (-b or – sq. rt. of b*2 -4ac)/2a … a band of agreement will canyon through the x-axis breadth h = -b/2a. Here h would accord to the x value. K which corresponds to the y amount would be k = c – b*2/4a. Hence the axis point.

Finally I ambition to achieve that the teaching action declared in the commodity is rather time arresting but may be advantageous as an another access in the acceptable mathematics classroom. The adjustment may not be absolute abundant to be accustomed as the best strategy/technique but as an another to appliance graphing utilities of calculators and computers it absolutely afforded us as agents with a all-important address that we may use in such a basic breadth of mathematics.

References:

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Craine, Timothy V. (1996, January). A Graphical Access to the Boxlike Formula. Mathematics Teacher, Vol. 89, No. 1, 34 – 36.

National Council of Agents of Mathematics (1995). Algebra in a Abstruse World: Addenda Series, Grades 9 – 12. Class and Evaluation Standards For School Mathematics. Virginia: c 1995 by NCTM, Inc.

National Council of Agents of Mathematics (1995). Class and Evaluation Standards For School Mathematics. Virginia: c 1989 by NCTM, Inc.

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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 Amphitheater for Boxlike Equations. Mathematics Teacher, Vol. 84, No2, 125 – 27.

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