The Quadratic The Quadratic Equations Generator Equations Generator

The authorsThe authors

Natalia BudinskiNatalia Budinski Osnovna skola i gimnazija ”Petro Osnovna skola i gimnazija ”Petro

Kuzmjak”, Ruski KrsturKuzmjak”, Ruski Krstur Novta MiroslavNovta Miroslav

MicronasNIT, Novi SadMicronasNIT, Novi Sad Djurdjica Takaci Djurdjica Takaci

Departman za matematiku i informatiku , Departman za matematiku i informatiku , Prirodno matematicki fakultet, Novi SadPrirodno matematicki fakultet, Novi Sad

The abstactThe abstact

This paper is the result of a master these about This paper is the result of a master these about types of the questionnaires in the types of the questionnaires in the mathematical education mathematical education

It contains The Generator of the It contains The Generator of the questionnaires which provides generating questionnaires which provides generating different questionnaire for each pupildifferent questionnaire for each pupil

Generating questionnaire is based on the basic Generating questionnaire is based on the basic methodical principles of teaching Quadratic methodical principles of teaching Quadratic equations in high school.equations in high school.

Among the themes in high school mathematic Among the themes in high school mathematic Quadratic equations is one with great importance. Quadratic equations is one with great importance.

Proper knowledge of solving these types of equations Proper knowledge of solving these types of equations and being familiar with their characteristics are the and being familiar with their characteristics are the base of the further mathematical education. base of the further mathematical education.

To check pupils’ knowledge we need to have proper To check pupils’ knowledge we need to have proper mean of measure their knowledge.mean of measure their knowledge.

The Questionnaires are providing quick and efficient The Questionnaires are providing quick and efficient information how the pupils overcome the theme to the information how the pupils overcome the theme to the teacher. teacher.

The Generator of Questionnaire is The Generator of Questionnaire is designed designed forfor the teacher the teacherss who can who can createcreate each questionnaire for each pupil. each questionnaire for each pupil.

For example, having 20 pupils- For example, having 20 pupils- teacher teacher can generate 20 different questionnaires. can generate 20 different questionnaires.

The questions are chosen in a way to be easy The questions are chosen in a way to be easy for generating and making numerous different for generating and making numerous different questionnaires with the same quality.questionnaires with the same quality.

It is developed in C# for .NET 2.0 Framework. It is developed in C# for .NET 2.0 Framework. The questionnaires can be easy printed and The questionnaires can be easy printed and saved by Word. saved by Word.

The Questionnaire Generator provides questionnaires The Questionnaire Generator provides questionnaires with problems which solutions are called among pupils with problems which solutions are called among pupils as “nice”. as “nice”.

The nice solution implies numbers that are not The nice solution implies numbers that are not complicated ratios, irrational numbers, or numbers with complicated ratios, irrational numbers, or numbers with decimal point, easy for “manual solving”. decimal point, easy for “manual solving”.

The manual solving of equations is important in order The manual solving of equations is important in order to accept solving techniques and developing to accept solving techniques and developing mathematical skills of pupils. mathematical skills of pupils.

The types of the questions The types of the questions are: are:

solving quadratic equations, solving quadratic equations, applying Viet`s formulae in quadratic applying Viet`s formulae in quadratic

equations, equations, solving quadratic equations using solving quadratic equations using

discriminantsdiscriminants biquadrate equations, biquadrate equations, system of equations which include system of equations which include

applying quadratic equations.applying quadratic equations.

Solving quadratic equationSolving quadratic equation

It is very important that pupils know how It is very important that pupils know how to solve the quadratic equation using the to solve the quadratic equation using the formula which is formally stated as:formula which is formally stated as:

For For axax22+bx+c=0+bx+c=0 ,a ,a≠0,≠0, the value of x is the value of x is given bygiven by

a

acbbx

2

42

2,1

The pupils have to apply formula for their The pupils have to apply formula for their equations, make sure that they did not equations, make sure that they did not drop the square root or the sign, as they drop the square root or the sign, as they sometimes do.sometimes do.

One common mistake is miscount One common mistake is miscount bb22, , forgetting that it means “square = of all b, forgetting that it means “square = of all b, including the sign”.including the sign”.

The Generator also deals with incomplete The Generator also deals with incomplete quadratic equation, which can be solved both quadratic equation, which can be solved both by formula and factoring.by formula and factoring.

Applying formula, pupils must consider which Applying formula, pupils must consider which one coefficient is missing, linear or free term to one coefficient is missing, linear or free term to apply formula properly. apply formula properly.

On the other hand, if pupil decides to solve On the other hand, if pupil decides to solve equation by factoring it will imply his previous equation by factoring it will imply his previous knowledge. knowledge.

Applying Viete`s Applying Viete`s formulae in quadratic formulae in quadratic equationsequations

The Viete`s formulae give a simple relation between The Viete`s formulae give a simple relation between the roots of a quadratic equation and its coefficients.the roots of a quadratic equation and its coefficients.

The pupils need to know that aThe pupils need to know that a sum of roots of sum of roots of reduced quadratic equationreduced quadratic equation xx22+px+q=0+px+q=0 is equal to is equal to coefficient at the first power of unknown, taken with a coefficient at the first power of unknown, taken with a back sign, i.e.                                                       back sign, i.e.                                                       

xx11+x+x22=-p=-p and a product of the roots is equal to a free term, i.e.and a product of the roots is equal to a free term, i.e.

xx11xx22=q=q

The Generator makes task where is The Generator makes task where is needed to determine the quadratic needed to determine the quadratic equation knowing the solutions.equation knowing the solutions.

The generated solutions are the ratio The generated solutions are the ratio numbers. This task can be solved by numbers. This task can be solved by using the using the Viete`s formulaeViete`s formulae..

SSolving quadratic equations olving quadratic equations using discriminantsusing discriminants

When a quadratic equation is in standard form,When a quadratic equation is in standard form, axax22+bx+c=0+bx+c=0 , a , a≠0≠0, the expression,, the expression, b b22-4ac=0-4ac=0 ,  , that is found under the square root part of the that is found under the square root part of the quadratic formula is called the discriminant. quadratic formula is called the discriminant. 

Knowing the characteristics of the discriminant Knowing the characteristics of the discriminant pupils can tell pupils can tell how many solutions there are how many solutions there are going to be and if the solutions are real going to be and if the solutions are real numbers or complex imaginary numbers.numbers or complex imaginary numbers.

The generator provides tasks in which is The generator provides tasks in which is needed to determine the parameter m, to needed to determine the parameter m, to provide real solutions of quadratic equation.provide real solutions of quadratic equation.

The pupils would plug in values of the The pupils would plug in values of the coefficient a, b, c in the discriminant formula, coefficient a, b, c in the discriminant formula, and solve the received equation which is linear and solve the received equation which is linear equation. equation.

The biThe biquadratquadratee equations equations

To solve the biquadratTo solve the biquadratee equations, pupils equations, pupils have to know how to transform it to the have to know how to transform it to the quadratic equation. quadratic equation.

Biquadrate equation may be quadratic in Biquadrate equation may be quadratic in form, such as:form, such as:

axax44+bx+bx22+c=0+c=0 ,a ,a≠0≠0 which can be written as which can be written as auau22+bu+c=0+bu+c=0 where u=xwhere u=x22

It is needed for pupils to note that the highest It is needed for pupils to note that the highest exponent is twice the value of the exponent of exponent is twice the value of the exponent of the middle term. the middle term.

If they are familiar with that they will be able to If they are familiar with that they will be able to resolve the equation directly or with a simple resolve the equation directly or with a simple substitution, using the methods that are substitution, using the methods that are available for the quadratic.available for the quadratic.

The system of equations The system of equations which include applying which include applying quadratic equationsquadratic equations

The generator is running with a system of The generator is running with a system of two equations which solving includes two equations which solving includes applying quadratic equation. applying quadratic equation.

The system is in the form of:The system is in the form of:

xy=axy=a

x+y=bx+y=b

The easiest way to find the solution is to solve one of the variables 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 “ in the linear equation, than to substitute that variable in the “xy “ equation, and solve the resulting equation which is quadratic.equation, and solve the resulting equation which is quadratic.

After calculating the values for e.g. the x, pupil need to find the After calculating the values for e.g. the x, pupil need to find the corresponding values for y. It can be calculated by substituting corresponding values for y. It can be calculated by substituting each value of x in to the linear equation.each value of x in to the linear equation.

Finally, pupils have to specify the solution set for the system.Finally, pupils have to specify the solution set for the system.

The system can be also solved by applying the Viete`s formulae. The system can be also solved by applying the Viete`s formulae.

The generator modulates the equations The generator modulates the equations which are not complicated for simplifying.which are not complicated for simplifying.

The importance of this type of system is The importance of this type of system is that they can be applied in real situation that they can be applied in real situation tasks, like e. g. :tasks, like e. g. :

Of which numbers is sum -12, and Of which numbers is sum -12, and product is 35?product is 35?

The questionnaire contains the The questionnaire contains the basics characteristics of a good basics characteristics of a good questionnaire:questionnaire:

Questions are worded simply and clearly, not Questions are worded simply and clearly, not ambiguous or vagueambiguous or vague

Attractive in appearance (questions spaced Attractive in appearance (questions spaced out, and neatly arranged)out, and neatly arranged)

An introduction is written to the questionnaireAn introduction is written to the questionnaire Questions are in order of easiness and logical Questions are in order of easiness and logical

sequencesequence Questionnaire is easy to completeQuestionnaire is easy to complete Phrase questions are the same for all pupilsPhrase questions are the same for all pupils

The questionnaire also satisfies the The questionnaire also satisfies the features of questionnaires like: features of questionnaires like:

validity validity It is based on relevant mathematical concepts which ought to be It is based on relevant mathematical concepts which ought to be

adopted by pupils.adopted by pupils.

The quadratic equation is mathematical concept which is The quadratic equation is mathematical concept which is observed in high school education. observed in high school education.

The quadratic equation is applied in various mathematical The quadratic equation is applied in various mathematical concepts. concepts.

The questionnaire contains questions about the main The questionnaire contains questions about the main characteristics of quadratics equations.characteristics of quadratics equations.

ObjectivityObjectivity What is more, the questionnaire provides What is more, the questionnaire provides

solutions of the questions. solutions of the questions. Therefore, every possibility of Therefore, every possibility of

unobjectivity is excluded.unobjectivity is excluded.

SensitivitySensitivity This questionnaire will detach pupils who This questionnaire will detach pupils who

overcome the theme from the pupils who did overcome the theme from the pupils who did not overcome the quadratic equationsnot overcome the quadratic equations

EconomicallyEconomically Providing the solution, the teacher`s time is Providing the solution, the teacher`s time is

saved by the easiness of the test skimmingsaved by the easiness of the test skimming

According to personal experiences, different According to personal experiences, different examples for each pupil increase their examples for each pupil increase their motivation.motivation.

They are more interested in solving tasks that They are more interested in solving tasks that are especially created for each of them.are especially created for each of them.

That was the main idea in writing this paper That was the main idea in writing this paper and constructing Questionnaire Generator.and constructing Questionnaire Generator.

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