Project Matematech – Functions

H. Mahnelová, Kovářov, 30. - 31. March 2012

Content

Information about the specification of the topic function at the czech high schools

The requirements of the student´s knowledge

An overview of the used computer programs in the Czech

Presentation of any examples

The educational content of a whole topic Addictions and relationships of function at the czech high schools

Basic knowledge about the functions – the concept of function, the properties of functions, the inverse function, the composite function, the periodic function

The educational content of a whole topic Addictions and relationships of function at the czech high schools

The algebraic functions – polynomial, linear, quadratic, power functions with the natural and integral exponent, the rational functions

The transcendent functions – exponentional, logarithmic, goniometric

The educational content of a whole topic Addictions and relationships of function at the czech high schools The basics of differential calculus –

continuity, limit of function, derivative of function, course

The basics of integral calculus – antiderivative, integral‘s methods, definite integral

The using of differential and integral calculus – in geometry and physics

The educational content of a whole topic Addictions and relationships of function at the czech high schools

Sequences – arithmetic, geometric Series – definite, indefinite, arithmetic,

geometric

The expected ascents of the student (quotation from educational document for gymnasium) The student sketches graphs of the functions

and describes basic characteristic …formulates and explains properties of the

functions and sequences …can use knowledge about the functions during

solution equations and inequality, in the process of determining quantitative relations

The expected ascents of the student

(quotation from educational document for gymnasium) …applies relations between values of

exponential, logarithmical and goniometric functions and relations between this functions

…simulates relationships of real process with help of known functions

…uses knowledge about functions and sequences by solving applications of examples

The expected ascents of the student

(quotation from educational document for gymnasium) …interprets compound interest, applies

exponential function and geometric sequences in the financial mathematic.

A computer in education of Math

It is a material didactic means, which can be use in any phase of the educational process:motivation and activatingexposure fasteningapplicationdiagnostic and evaluating

The most used programs for education of the topic Functions in the Czech high school: The standards applications programs

MS Excel The especial mathematics software

represented by CASDerive 6, Mathematica

The programs of dynamic geometryCabri Geometry

Especial programsFunctions

GeoGebra

Nowadays it is a freeware software which is localized in the czech language as well and it enables connecting CAS and DGS.

Its popularity grows at the czech schools because it is freeware and the students can work with it at home by using their computers.

My examples of the using GeoGebra From differential calculus From integral calculus Two examples of application of functions

Examples from differential calculus

It means discovering relationships between properties of function and values of first or second derivative.

der.1.ggb der.2.ggb der.3.ggb der.4.ggb

Examples of the integral calculus

I would like to present geometry application of definite integral, the meaning of lower bound and upper bound of the definite integral.

určitý integrál1.ggb určitý integrál sin(x).ggb určitý integrál2.ggb určitý integrál (f-g).ggb

The application of function – ex. 1Retail entrepreneur sells his product for 85 € per piece. The total cost of the product consists a fixed cost of 700 € and production costs 50 € per unit of the product.

Task n. 1: Forming a formula of the dependence of the amount of income on the number of sold units. (without a computer)

Task n. 2: Forming a formula of the dependence of the amount of costs on the number of sold pieces of the product. (without a computer)

Task n. 3: Creating a graphical model of revenue and costs. (with a computer) podnikatel.ggb

The application of function – ex. 1

Dynamic program provides the opportunity for a rapid solution to the troubled issues, for example

Task n. 4: Specify how many products should be sold to cover the cost of the income?

Task n. 5: How much profit or loss we get if we sold 14 pieces of the products?

Task n. 6: How many products must be sold in order to gain at least 1000 €?

The application of function – ex. 2

If the depreciation of equipment p %, the price of the equipment at the end of x-th year is given by the formula

where K is the purchased by price of the equipment at the beginning of the first year.

xp

KC

1001

The application of function – ex. 2

Task n. 1: Construct a graph of dependence on the price of the equipment with the purchased price and dependence on the number p of depreciation. (use the sliders)

odpis.ggb Task n. 2: Create a model of a situation from

which it will be possible to determine for how long will be the value of equipment on one half of the original price in the chosen value of p.

The application of function – ex. 2

And again we can quickly solve other problems, for example

Task n. 3: How long it takes to fall down the value of the original on half price at the depreciation of 4%?

Task n. 4: In what percentage of depreciation we reach the half value of the original price in 6 years?

Conclusion

GeoGebra is a program that facilitates, speeds up and

makes accurate graphical representation of the dependency variables.

gives an opportunity for a deeper examination of the context.

Student is motivated by means of dynamic experimentation to a stand-alone thinking, and the search for hypotheses.