difficulties and misconceptions in learning of quadratic equation
Introduction
The main aim of this report was to define the teaching methods used, difficulties and misconceptions in learning of quadratic equation. Mathematics, which includes quadratic equations, as a subject is aimed at finding basic to complex solutions for problems mainly in numbers. To come up with these solutions students need to acquire necessary skills. However, errors occur in the process of coming up with solutions. The errors vary in degree from student to student. A study by Nakamura and Prakitipong (2006), outlines two main obstacles which hinder students from arriving at the correct answers. The two obstacles were linguistic frequency and problems in mathematical processing.
Concepts and skills
Concepts
Quadratic equation makes use of a wide range of mathematical concepts. Quadratic equations are classified as polynomials of the second order for a single variable. This classification implies that according to algebra rules, at least one solution can be found to the equation. These concepts include basic operations such as addition and subtraction. A quadratic equation takes up the form shown by the equations below.
Where a, b and c are the coefficients of the equation. The aim of solving quadratic equations is to find their roots. The roots are given by the equation below.
These roots can either be complex or real. If then the roots are real and if the value is less than 0 then the roots are complex. If the solutions of the above equation are a and b, the equation can be written in the form.
Skills
In order to acquire competence in solving of quadratic equations, students need to acquire a particular set of skills. The solving of quadratic formulas is a step by step procedure. The students should have skills that enable them to expand quadratic equations. These include skills in basic algebra. There are different techniques that can be used for factorization which includes using of common factors. In factorization of the equation, completing squared method can be applied. The students need skills to help understand the relationship between algebraic representations of quadratic solutions and the originating graphical representations. The student should demonstrate the ability to plot of various equations’ graphs and applying transformations. There are three techniques which can be used to acquire the solutions to equations. These methods include use of the graphical procedure, using the factoring method and using the quadratic formula. Students should acquire skills that will facilitate the use of the techniques to find a solution
Understanding for the content descriptions and elaborations above content descriptions
For the student to acquire competent skills in coming up with solutions for quadratics, the students should be able to expand binomial products and factorization of monotonic expressions. To acquire these knowledge there are resources that the students can acquire to help them learning. These resources include revision questions. One of the resources for the learning is the book Key to Algebra 10 Practice Test pg. 36-37. Examples for the coverage of this content include: Expand the equation (x+3) (x+4) =0, Find the factors of the equation.
Students should be able to grasp concepts that enable them establish a relationship between algebraic representations of an equation and its subsequent graphical representation. The student should have the concept of drawing parabolas as most of the graphs for quadratic equations take up the form of parabolas. Application of transformations on the equation to help generate an equation that will help generate x and y values for plotting. To help in grasping these concepts, there are resources available such Lecture 6 pg. 1q 1-6. Examples of problems that should may describe or receive help in describe include: Graph the quadratic equation and Graph.
The learning of the concepts above by students is to enable them generate solutions for quadratic equations. Solutions can be either complex or real and are attained using different techniques. The student should be able to learn this techniques and have the ability to determine situations where each technique is suitable. To assist in learning of these techniques, the student can acquire resources such as MyMath 9 Exercise 4A pg. 155Q. 1-4 and MyMaths10 Exercise 5A pg. 167 Q.1-10. Examples that students can solve include: Finding a solution for using the graphical method. Solving for x in the equation using the factors method.
Misconceptions and Difficulties
Finding solutions for quadratic equations is a process that requires skills and possess a problem to people. There are many common misconceptions when learning quadratic equations that may affect the understanding of the students.
Misconceptions on negative roots
As mentioned earlier equations take various forms. In some instances the students have a very high tendency of ignoring one of the factors especially when it is zero. Quadratic equations have two solutions. Taking the equation Most students are likely to follow the procedure outlined below.
Taking a look at these procedure, there seems to be only one solution to the quadratic equation which is x=3. The solution found is the positive square root. One of the explanations for why the students might only consider the positive roots is the fact that most students are introduced to positive roots only at the beginning of their education while negative roots due to their complexity come at later stages in the educational cycle.
Misconception of the coefficients when using the formula
There is a misconception that using the formula is the easiest method since it involves only one step. However, students have an issue when identifying the values of a, b and c from the quadratic equation. The formula for solving a quadratic equations is . Fitting the wrong values of a, b and c in the formula results to the wrong roots. Taking the example of the quadratic equation, students may assume that the value of a is 0 since there is no coefficient before. This implies that the formula will result to both roots being equal to zero. The equation above is of the general form
Misconception on the purpose of learning algebraic equations
Some techniques used in the solving quadratic equations do not allow the students to understand what is behind the quadratic equation. A technique such as factorization which is procedural and other techniques which use formulas encourages the students to cram the procedures and formulas. This implies that the students are likely to categorize the learning of quadratic equations as instrumental where the students learn the rules but do not have the understanding of why the rules were applied. The categorization of the learning the quadratic equations as instrumental implies that there is failure to establish a relational understanding on the topic. Instrumental is rarely impactful in finding solutions for real world problems which require quadratic equations.
Common difficulties
Solving quadratic equations is a conceptually challenging topic for many students in high schools (Kotsopoulos, 2007).
Solving non-standard form equations.
Most students encounter problems when trying to come up with solutions for quadratic equations which have not been presented in the standard form. The standard form of a quadratic equation is. Taking the example of a quadratic equation written as, at first glance the equation looks complex but it can be rearranged if the students ha gained competent skills in finding solutions for algebraic problems. From the form of the quadratic equation above, it can be concluded that the values of a, b and c for the quadratic formula are not clearly described and various transformations need to be performed to make the equation standard. Students may have issues performing these transformations in order to rearrange the equation to standard form.
Derivation and solving of algebraic quadratic equations from statements.
In the process of learning algebraic solutions, students are mostly exposed to already derive algebraic equations. However, in the real world and in some test questions for the students the quadratic equations are represented in statements that represent real life situations. Quadratic equations cannot be solved in statement form. Therefore, the students face problems in interpreting the statements to from quadratic equations. Without guiding statements in the problems which indicate that the required solution follows a quadratic form, students may face a problem if the identification and formulation of any equation.
Factoring technique problem.
Most students face issues in the use of the factoring techniques (Bosse and Nandakumar, 2005). Factoring of the quadratics looks like a complex process due to the many steps that are involved in finding the solution. Complexity increases depending on the value of the leading coefficient. If it has many pairs of possible factors the more complex it is. Taking the example of the equation, the leading coefficient is 64 which has many factors. Using the factoring techniques is complex for this problem. There are also some situations when the factoring techniques is impossible to use. There are some equations which cannot be factored. For these equations, the students has to apply other methods such as the quadratic formula to find roots.
4.0 Teaching
As mentioned earlier different techniques such as the method of completing squares and the quadratic formula exist to help in finding the solution to quadratic equations.
For the graphical method a plot is expected from the student. The plot follows values of the quadratic equations. The student is first asked to basically translate the quadratic equation in a graph. Taking the basic form of the quadratic equation, the 0 on the left hand of the equation is replaced with y in order to help generate points to be plotted on the x, y axis. The equation becomes, .To generate points replace x with random values with a certain range of integers such as -1,1,0,1,2 to acquire values of y. Plot the values of x against y. If drawn accurately, the intersections of the lines that are generated by plotting the points and the x axis gives the roots of the equation.
Solve for x in the equation using the graphical method.
In order to find a solution for this equation, we find points that satisfy the quadratic equation above written in the form. We fill the table as follows.
| X | -2 | -1 | 0 | 1 |
| Y | 15.4 | 13.2 | 10 | 5.8 |
The points can be plotted on the x, y axis as shown in the figure below.
From the figure above the roots of the quadratic equation are. These values are the solutions for the quadratic equation.
Task: Solve for x in the quadratic equation using the graphical method.
The equation is introduced to the student in the general form to enhance learning. The main aim of solving quadratic solutions is finding roots. Roots provide the viable solutions for the quadratic equations. In order to plot the line representing the quadratic equation, various points which satisfy the quadratic equation are generated. These points form the shape of a parabola. If drawn correctly the parabola cuts through the x axis twice providing the solutions of the quadratic equation. In the sample question the student is expected to plot a graph for an algebraically represented quadratic equations. This allows the student to gain experience in plotting quadratics and facilitates the solution of subsequent problems.
Another method for solving quadratic equations is the use of the quadratic formula. For the quadratic equation to be solved successfully using the quadratic formula, the equation must be written in the standard form i.e. . These method is important especially where a quadratic equation has complex roots. It is the only formula that gives the value of the complex roots. The quadratic formula is represented by the equation below.
Where the values of a, b and c are given by the equation written in standard form. Let the value of. When the equation has real roots. However if then the equation has complex roots.
Taking an instance where one is supposed to solve the equation using the quadratic formula.
First, the equation has to be written in standard form. We begin be opening up the brackets on the left hand side and then taking the values on the right hand side to the left.
The equation is now in standard form. The values of a, b and c from the equation are 1, 11 and 20. Replacing these values in the quadratic equation in order to find roots.
The solutions for the equation are
Task: Use the quadratic formula to solve for values of x in
Solving quadratic equations using the quadratic formula is a straight forward process as it involves identification of coefficients and placing them into the equation. However, the process becomes complicated if the problem is not presented to students in the standard form. The non-standard form is demonstrated in the example above. From the example above the student is given a description of how to arrange the equation in standard form. Once the equation is arranged in standard form, the identification of coefficients and substitution in the formula is performed. The formula is arranged in such a way that it allows the student to acquire both positive and negative roots. The equation has the option of using +/-. The application of one of these identities yields different results from the other.
The third method of solving quadratic equation is the use of factors. Factors allow algebraic quadratic equations to be represented as a multiplication of two linear representations. For this techniques to work the equation has to be represented in the factors form i.e., .The factoring method makes use of the zero product property. The property implies that at any instance where the product of two or more items is zero, then one or both of those items are zero. This implies tan when the quadratic equation is factored to the form, then either or
Find solutions for x using the factoring method in the equation.
For this instance the equation is not written in a factored form. The equation is in the quadratic form. We find the factors of , which can be added to produce a value equal to b.
The factors of which add up to are -3 and 2. The solution for x and y for the equation are -3 and 2. For this instance the quadratic equation is forward and the product of and c is relatively small hence the less complexity in finding a solution for the problem. However, solving quadratic equation where the product of a and c is large and has many factor is a long process.
Task: Solve for x in using the method of factors.
The method of factors has a relatively high number of steps which depends on the product of the coefficients. The example given to the students shows a problem where the student is required to find factors. The equation is arranged in the standard form. Factors are found by using the method of the greatest common divisor (GCD). The student has to learn how to select the factors and which factors should be combined in order to get the factors that add up to the value of the coefficient of x in the quadratic equation.
Conclusion
Learning of basic concepts of quadratics is very important. Rewriting all quadratic problems to the standard form is applied in almost all the formulas used in learning. A grasp of the standard form allows the students to gain relational knowledge on quadratic formulas. Relational knowledge allows the students to link knowledge on quadratic rules and formulas and why how they are applied from one situation to another (Skemp, 2002).
References
Bossé, M. J., & Nandakumar, N. R. (2005). The factorability of quadratics: motivation for more techniques. Teaching Mathematics and its Applications, 24(4), 143-153.
Kotsopoulos, D. (2007). It’s like hearing a foreign language. Mathematics teacher, 101(4), 301-305.
Prakitipong, N., & Nakamura, S. (2006). Analysis of mathematics performance of grade five students in Thailand using Newman procedure. Journal of International Cooperation in Education, 9(1), 111-122.
King, J. A., & Rasmussen, P. (1990). Graphs. Key Curriculum Press.