The Quadratic Equations Generator

The authors  Natalia Budinski Osnovna skola i gimnazija ”Petro Kuzmjak”, Ruski Krstur Osnovna skola i gimnazija ”Petro Kuzmjak”, Ruski Krstur  Novta Miroslav MicronasNIT, Novi Sad  Djurdjica Takaci Departman za matematiku i informatiku, Prirodno matematicki fakultet, Novi Sad Departman za matematiku i informatiku, Prirodno matematicki fakultet, Novi Sad

The abstact  This paper is the result of a master these about types of the questionnaires in the mathematical education  It contains The Generator of the questionnaires which provides generating different questionnaire for each pupil  Generating questionnaire is based on the basic methodical principles of teaching Quadratic equations in high school.

 Among the themes in high school mathematic Quadratic equations is one with great importance.  Proper knowledge of solving these types of equations and being familiar with their characteristics are the base of the further mathematical education.  To check pupils’ knowledge we need to have proper mean of measure their knowledge.  The Questionnaires are providing quick and efficient information how the pupils overcome the theme to the teacher.

 The Generator of Questionnaire is designed for the teachers who can create each questionnaire for each pupil.  For example, having 20 pupils- teacher can generate 20 different questionnaires.

 The questions are chosen in a way to be easy for generating and making numerous different questionnaires with the same quality.  It is developed in C# for.NET 2.0 Framework. The questionnaires can be easy printed and saved by Word.

 The Questionnaire Generator provides questionnaires with problems which solutions are called among pupils as “nice”.  The nice solution implies numbers that are not complicated ratios, irrational numbers, or numbers with decimal point, easy for “manual solving”.  The manual solving of equations is important in order to accept solving techniques and developing mathematical skills of pupils.

The types of the questions are:  solving quadratic equations,  applying Viet`s formulae in quadratic equations,  solving quadratic equations using discriminants  biquadrate equations,  system of equations which include applying quadratic equations.

Solving quadratic equation Solving quadratic equation  It is very important that pupils know how to solve the quadratic equation using the formula which is formally stated as: For ax 2 +bx+c=0,a≠0, the value of x is given by For ax 2 +bx+c=0,a≠0, the value of x is given by

 The pupils have to apply formula for their equations, make sure that they did not drop the square root or the sign, as they sometimes do.  One common mistake is miscount b 2, forgetting that it means “square = of all b, including the sign”.

 The Generator also deals with incomplete quadratic equation, which can be solved both by formula and factoring.  Applying formula, pupils must consider which one coefficient is missing, linear or free term to apply formula properly.  On the other hand, if pupil decides to solve equation by factoring it will imply his previous knowledge.

Applying Viete`s formulae in quadratic equations  The Viete`s formulae give a simple relation between the roots of a quadratic equation and its coefficients.  The pupils need to know that a sum of roots of reduced quadratic equation x 2 +px+q=0 is equal to coefficient at the first power of unknown, taken with a back sign, i.e.  The pupils need to know that a sum of roots of reduced quadratic equation x 2 +px+q=0 is equal to coefficient at the first power of unknown, taken with a back sign, i.e. x 1 +x 2 =-p x 1 +x 2 =-p  and a product of the roots is equal to a free term, i.e. x 1 x 2 =q x 1 x 2 =q

 The Generator makes task where is needed to determine the quadratic equation knowing the solutions.  The generated solutions are the ratio numbers. This task can be solved by using the Viete`s formulae.

Solving quadratic equations using discriminants  When a quadratic equation is in standard form, ax 2 +bx+c=0, a≠0, the expression, b 2 -4ac=0, that is found under the square root part of the quadratic formula is called the discriminant.  When a quadratic equation is in standard form, ax 2 +bx+c=0, a≠0, the expression, b 2 -4ac=0, that is found under the square root part of the quadratic formula is called the discriminant.  Knowing the characteristics of the discriminant pupils can tell how many solutions there are going to be and if the solutions are real numbers or complex imaginary numbers.

 The generator provides tasks in which is needed to determine the parameter m, to provide real solutions of quadratic equation.  The pupils would plug in values of the coefficient a, b, c in the discriminant formula, and solve the received equation which is linear equation.

The biquadrate equations  To solve the biquadrate equations, pupils have to know how to transform it to the quadratic equation.  Biquadrate equation may be quadratic in form, such as: ax 4 +bx 2 +c=0,a≠0 ax 4 +bx 2 +c=0,a≠0 which can be written as au 2 +bu+c=0 which can be written as au 2 +bu+c=0 where u=x 2 where u=x 2

 It is needed for pupils to note that the highest exponent is twice the value of the exponent of the middle term.  If they are familiar with that they will be able to resolve the equation directly or with a simple substitution, using the methods that are available for the quadratic.

The system of equations which include applying quadratic equations  The generator is running with a system of two equations which solving includes applying quadratic equation.  The system is in the form of: xy=ax+y=b

 The easiest way to find the solution is to solve one of the variables in the linear equation, than to substitute that variable in the “xy “ equation, and solve the resulting equation which is quadratic.  After calculating the values for e.g. the x, pupil need to find the corresponding values for y. It can be calculated by substituting each value of x in to the linear equation.  Finally, pupils have to specify the solution set for the system.  The system can be also solved by applying the Viete`s formulae.

 The generator modulates the equations which are not complicated for simplifying.  The importance of this type of system is that they can be applied in real situation tasks, like e. g. : Of which numbers is sum -12, and product is 35? Of which numbers is sum -12, and product is 35?

The questionnaire contains the basics characteristics of a good questionnaire:  Questions are worded simply and clearly, not ambiguous or vague  Attractive in appearance (questions spaced out, and neatly arranged)  An introduction is written to the questionnaire  Questions are in order of easiness and logical sequence  Questionnaire is easy to complete  Phrase questions are the same for all pupils

The questionnaire also satisfies the features of questionnaires like:  validity It is based on relevant mathematical concepts which ought to be adopted by pupils. It is based on relevant mathematical concepts which ought to be adopted by pupils.  The quadratic equation is mathematical concept which is observed in high school education.  The quadratic equation is applied in various mathematical concepts.  The questionnaire contains questions about the main characteristics of quadratics equations.

 Objectivity  What is more, the questionnaire provides solutions of the questions.  Therefore, every possibility of unobjectivity is excluded.

 Sensitivity  This questionnaire will detach pupils who overcome the theme from the pupils who did not overcome the quadratic equations  Economically  Providing the solution, the teacher`s time is saved by the easiness of the test skimming

 According to personal experiences, different examples for each pupil increase their motivation.  They are more interested in solving tasks that are especially created for each of them.  That was the main idea in writing this paper and constructing Questionnaire Generator.

 These types of questionnaire can actually be helpful to the teachers, because they could in a quick and efficient way check the knowledge level of their pupils.