The Quadratic Equations Generator

1. The Quadratic Equations Generator

2. The authors • Natalia Budinski 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

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. 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 ax2+bx+c=0 ,a≠0, the value of x is given by

10. 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 b2, forgetting that it means “square = of all b, including the sign”.

11. 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.

12. 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 equationx2+px+q=0 is equal to coefficient at the first power of unknown, taken with a back sign, i.e.                                                        x1+x2=-p • and a product of the roots is equal to a free term, i.e. x1x2=q

13. 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.

14. Solving quadratic equations using discriminants • When a quadratic equation is in standard form,ax2+bx+c=0 , a≠0, the expression, b2-4ac=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.

15. 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.

16. 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: ax4+bx2+c=0 ,a≠0 which can be written as au2+bu+c=0 where u=x2

17. 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.

18. 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=a x+y=b

19. 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.

20. 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?

21. 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

22. The questionnaire also satisfies the features of questionnaires like: • validity 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.

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

24. 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

25. 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.

26. 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.