Activities: A Graphical Approach to the Quadratic Formula

1996 ◽  
Vol 89 (1) ◽  
pp. 34-46
Keyword(s):  
Quadratic Functions ◽  
Quadratic Formula ◽  
Graphical Approach ◽  
Quadratic Equations

Introduction: Traditionally, the solution of quadratic equations has been taught before, and in isolation from, the study of quadratic functions. The quadratic formula itself has typically been derived by completing the square. Many teachers skip the derivation, and most students who see it do not fully understand it.

Computer-Oriented Mathematics: Quadratic Equations—Computer Style

1969 ◽  
Vol 62 (4) ◽  
pp. 305-309
Author(s):  
Walter Koetke ◽  
Thomas E. Kieren
Keyword(s):  
Quadratic Formula ◽  
Quadratic Equations ◽  
Modern Mathematics

ONE of the “old” topics that has been approached in a “new” way in modern mathematics is quadratic equations. No longer do students simply memorize the quadratic formula and do hundreds of exercises using it.

Sharing Teaching Ideas: A Squeeze Play on Quadratic Equations

1982 ◽  
Vol 75 (2) ◽  
pp. 132-136
Keyword(s):  
High School ◽  
Mathematics Teacher ◽  
Quadratic Equation ◽  
Senior Year ◽  
Quadratic Formula ◽  
Mathematics Courses ◽  
Quadratic Equations ◽  
Teaching Ideas

As a mathematics teacher whose present assignment is to teach science, I was somewhat dismayed when my physics class wa unable to solve a nontrivial quadratic equation. These students are all enrolled in senior-year mathematics and had taken all lower level mathematics courses available in our small Western Kansas high school. They charged this inability to having forgotten the quadratic formula. To the e students the quadratic formula is a magic passkey to solving “unfactorable” quadratic equations. On further di scussion, l discovered that they vaguely remembered having heard of the method of completing the square, but they saw no connection between the quadratic formula and that method of solving a quadratic equation. They could solve simple quadratics by hit-and-miss factoring, but that was their only tool with which to attack this problem.

A New Look at Some Old Formulas

1985 ◽  
Vol 78 (1) ◽  
pp. 56-58
Author(s):  
Edward C. Wallace ◽  
Joseph Wiener
Keyword(s):  
Quadratic Formula ◽  
Quadratic Equations ◽  
Alternative Approaches ◽  
New Look

Interest in solving quadratic equations has occupied mathematicians for nearly four thousand years. Indeed, by 2000 b.c. the Babylonians had developed a form of the quadratic formula and a method equivalent to completing the square (Eves 1969; Smith 1951). A number of approaches to the solution of quadratic equations are possible. Let's examine some of these alternative approaches to see what new insights we might discover.

Model Solutions to Quadratic Equations

1986 ◽  
Vol 79 (5) ◽  
pp. 332-336
Author(s):  
Alastair McNaughton
Keyword(s):  
Three Dimensional ◽  
Quadratic Functions ◽  
Complex Numbers ◽  
Quadratic Equations ◽  
Model Solutions

Here is a method of representing quadratic functions by three-dimensional wire models. It enables one to form a simple geometric concept of the location of the imaginary zeros. I have been using this material with my students and have been delighted with the ease with which they respond to it. As a result, their confidence in dealing with complex numbers has increased, their concept of functions has shown much improvement, and they are attacking problems with real insight.

Approaching quadratic equations from a right angle

2017 ◽  
Vol 101 (552) ◽  
pp. 424-438
Author(s):  
King-Shun Leung
Keyword(s):  
Secondary School ◽  
Mathematics Curriculum ◽  
Quadratic Equation ◽  
Bisection Method ◽  
Quadratic Formula ◽  
Secondary School Mathematics ◽  
Quadratic Equations ◽  
Newton Raphson ◽  
Raphson Method ◽  
Comprehensive Discussion

The theory of quadratic equations (with real coefficients) is an important topic in the secondary school mathematics curriculum. Usually students are taught to solve a quadratic equation ax2 + bx + c = 0 (a ≠ 0) algebraically (by factorisation, completing the square, quadratic formula), graphically (by plotting the graph of the quadratic polynomial y = ax2 + bx + c to find the x-intercepts, if any), and numerically (by the bisection method or Newton-Raphson method). Less well-known is that we can indeed solve a quadratic equation geometrically (by geometric construction tools such as a ruler and compasses, R&C for short). In this article we describe this approach. A more comprehensive discussion on geometric approaches to quadratic equations can be found in [1]. We have also gained much insight from [2] to develop our methods. The tool we use is a set square rather than the more common R&C. But the methods to be presented here can also be carried out with R&C. We choose a set square because it is more convenient (one tool is used instead of two).

4. Quadratic equations

2015 ◽  
pp. 40-52
Author(s):  
Peter M. Higgins
Keyword(s):  
General Equation ◽  
Solution Process ◽  
Quadratic Equation ◽  
Quadratic Formula ◽  
Quadratic Equations ◽  
Quadratic Approximations

A quadratic equation is one involving a squared term and takes on the form ax2 + bx + c = 0. Quadratic expressions are central to mathematics, and quadratic approximations are extremely useful in describing processes that are changing in direction from moment to moment. ‘Quadratic equations’ outlines the three-stage solution process. Firstly, the quadratic expression is factorized into two linear factors, allowing two solutions to be written down. Next is completing the square, which allows solution of any particular quadratic. Finally, completing the square is applied to the general equation to derive the quadratic formula that allows the three coefficients to be put into the associated expression, which then provides the solutions.

Finding Meaning in the Quadratic Formula

2019 ◽  
Vol 112 (4) ◽  
pp. 258-261
Author(s):  
Thomas G. Edwards ◽  
Kenneth R. Chelst
Keyword(s):  
Graphic Representation ◽  
Quadratic Functions ◽  
Quadratic Formula ◽  
Finding Meaning

Connecting the formula to the graphic representation of quadratic functions makes the mathematics meaningful to students.

The Quadratic Formula—an Enrichment Approach

1972 ◽  
Vol 65 (5) ◽  
pp. 472-473
Author(s):  
Kenneth Stilwell
Keyword(s):  
Analytic Geometry ◽  
Quadratic Formula ◽  
Quadratic Equations ◽  
Simple Transformation

The procedure of completing the square, used in deriving the quadratic formula, is pervasive in mathematics. The following is presented as an alternate method of derivation of the quadratic formula for students studying analytic geometry. It is not intended to replace the traditional derivation, but rather is presented as an enrichment topic based on the student's ability to solve pure quadratic equations (that, is, equations of the form ax2 + b = 0, a ≠ 0) and perform a simple transformation.

scholarly journals ANALISIS TINGKAT KESULITAN SISWA DALAM KEMAMPUAN MENYELESAIKAN MASALAH MATEMATIKA MATERI PERSAMAAN KUADRAT

2020 ◽  
Vol 4 (1) ◽  
pp. 139
Author(s):  
Lala Intan Komalasari
Keyword(s):  
High School ◽  
Quadratic Functions ◽  
Quadratic Equations ◽  
Teachers And Students ◽  
Mathematical Problems ◽  
Factor Theorem

Penelitian ini bertujuan untuk menganalisis kesulitan – kesulitan siswa dalam menyelesaikan masalah matematika pada materi Persamaan Kuadrat Dan Fungsi Kuadrat, Teorema Faktor Dan Teorema Sisa Metode yang digunakan adalah menggunakan tes dan wawancara. Tes dilakukan kepada siswa sedangkan wawacara dilakukan kepada guru dan siswa. Sekolah yang merupakan tempat penelitian adalah SMA Advent Purwodadi sekolah ini adalah merupakan sekolah satu atap (SATAP) yang terdiri dari 300 siswa dari berbagai daerah Hasil wawancara guru menyatakan bahwa guru tidak terlalu mengalami kesulitan dalam mengajar materi Persamaan Kuadrat Dan Fungsi Kuadrat, Teorema Faktor Dan Teorema Sisa guru akan mengalami kesulitan apabila sudah masuk pada bentuk akar, sedangkan kesalahan yang dilakukan siswa bervariasi yaitu kesalahan fakta, kesalahan konsep, kesalahan prinsip dan kesalahan operasi. Solusi yang di tawarkan adalah pembelajaran dengan memberikan soal open- ended pada materi persamaan kuadrat dan untuk menentukan grafik fungsi kuadrat yaitu dengan mengkontruksi prinsip.ABSTRACTThis study aims to analyze the difficulties of students in solving mathematical problems in the material Quadratic Equations and Quadratic Functions, Factor Theorem and Time Theorem The method used is to use tests and interviews. Tests are conducted on students while interviews are conducted on teachers and students. The school which is a place of research is Adventist Purwodadi High School. This school is a one-roof school (SATAP) consisting of 300 students from various regions. The teacher's interview results state that the teacher has no difficulty in teaching the material Quadratic Equations and Quadratic Functions, Theorem Factors and Theorems The rest of the teachers will experience difficulties if they have entered the root form, while the mistakes made by students vary, namely fact errors, concept errors, principle errors and operating errors. The solution offered is learning by giving open-ended questions to the material in quadratic equations and to determine the graph of quadratic functions, namely by constructing the principle.

Solving and Graphing Quadratics with Symmetry and Transformations

2016 ◽  
Vol 110 (5) ◽  
pp. 394-397
Author(s):  
Michael Weiss
Keyword(s):  
Common Core ◽  
Common Core State Standards ◽  
Quadratic Function ◽  
State Standards ◽  
Quadratic Functions ◽  
List Type ◽  
Core State ◽  
Quadratic Equations ◽  
The Common ◽  
Axis Of Symmetry

One of the central components of high school algebra is the study of quadratic functions and equations. The Common Core State Standards (CCSSI 2010) for Mathematics states that students should learn to solve quadratic equations through a variety of methods (CCSSM A-REI.4b) and use the information learned from those methods to sketch the graphs of quadratic (and other polynomial) functions (CCSSM A-APR.3). More specifically, students learn to graph a quadratic function by doing some combination of the following: Locating its zeros (x-intercepts)Locating its y-interceptLocating its vertex and axis of symmetryPlotting additional points, as needed

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