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Mathematics & Pedagogy - Solving Quadratics

teaching quadratics

Date : 08/09/2014

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Nadia

Uploaded by : Nadia
Uploaded on : 08/09/2014
Subject : Maths

A quadratic expression is the form ax2 + bx + c where a, b, c, are constant. The simplest quadratic form is when a=1 so we have x2 + bx + c. There are a number of different teaching methods used in explaining how to solve quadratic equations. Students tend to find "quadratic relations are one of the most conceptually challenging aspects of the high school curriculum" (Kotsopoulos, 2007, pg. 6) One of the methods used in solving quadratic equations is factorising; we need find two numbers which when multiplied gives ac and when added together gives b, these two numbers are called factors of the quadratic expression. In order to factorise quadratics, one must have previous knowledge of multiplication which links in with this method, however some students have difficulties with basic multiplication, and this consequently "directly influences students` ability to engage effectively in factorisation of quadratics, since factorisation is a process of finding products within the multiplication table." (Kotsopoulos, Donna, 2007, pg. 6) Students will turn to other methods in solving quadratic equations after being discouraged by factorizing, the concept of thinking of two numbers simultaneously to give different answers when multiplying and adding seems quite complex for some, as Bossé and Nandakumar (2005) mention, "students attempt to utilize either completing the square or the quadratic formula as techniques to solve a quadratic equation only after frustration with factoring has arisen." These other techniques will also be analysed in more detail below.

A second approach of solving quadratic equations is via graphical means. A graph can be used to deduce the factors of a quadratic equation; the two factors are the two points that intercept the x-axis, it is vital to know that there are no solutions or factors of a quadratic equation that does not intercept the -axis. When using this method, student develop and grasp onto a wider range mathematical topics such as interpreting graphs, as Tsamir and Reshef (2006) state "The graphic method involves the interpretation of graphic representations, e.g., using parabolas to solve quadratic inequalities" With this approach there are also common difficulties amongst students, one of which being distinguishing how to draw the graph, if drawn incorrectly this can cause errors, i.e. whether the graph is of x² or -x².

To find solutions of any quadratic equation we can use the formula method, knowing the solutions we can work out the factors of the quadratic equation. This links into other mathematical sectors such as using formulae correctly along with understanding the interpretation of symbols and letters. Students sometimes have difficulty in distinguishing what symbols mean when using the formula, causing them to make errors and become discouraged in learning the topic, Collis (1969, 1971, 1973c, 1975a, b), "argued that one of the difficulties which children have in algebra relates to the abstract nature of the elements used." (Algebra: Children's Strategies and Errors, Booth L.R. et al, 1984, pg. 4)

Along with these processes, another technique of solving quadratic equations is by completing the square; this can be used to give answers to a certain number of decimal places or in surd from. This approach can be quite confusing for many students because the steps involved can be a bit tricky to understand, i.e. which number to halve or multiply. Other areas of mathematics can be easily understood when one understands the method of completing the square, the concept of attaining algebraic solutions of cubic equations can be solved by students who have this knowledge, as mentioned by (Ward, 2003) "this method is readily accepted by students already familiar with completion of the square as a method for quadratic equations."

All the methods mentioned above are used day-to-day in solving quadratic equations, the key issue that arises is hat which of these methods is the best way, if any, a study showed that when presented with a number of methods students excelled in solving quadratic equations by using the method they were most comfortable with, the study concluded that "almost all students correctly solved the different inequalities, and most liked being presented with several methods." (Tsamir and Reshef, 2006) This indicates that students succeed far greater when the choice is given to them of solving quadratic equations via a number of ways as they are able to relate ideas from other sects of mathematics in solving the equation. Furthermore, each individual has a different way of thinking, so the way in which students intake and interpret information may cause errors, "problems with quadratic relations might potentially be linked to the ways in which the brain constructs cognitive representations." (Kotsopoulos, 2007, pg. 6) It is vital to look into why miscalculations occur amongst students in solving quadratics in order to find resolutions, "A study of such errors is important because of the information is provides concerning the ways in which the child views the problem, and the procedures that are used in attempting to solve the problem." (Algebra: Children's Strategies and Errors, Booth L.R. et al, 1984, pgs. 2-3)

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