How to teach function by using of

DERIVE

Ingrida KraslanováMária Slavíčková

Faculty of mathematics, physics and informaticsComenius University, Bratislavahttp://www.ddm.fmph.uniba.sk

Introduction Using Derive in Austria

1993-2005: CAS I - V projects, CAS VI in preparation

CAS educated students better results than non-CAS students (always)

Textbooks – CAS in appendix Using Derive in Slovakia

Every secondary school – at least 1 PC room with DERIVE

Lack of teacher interested in CAS

Secondary school

1st phase (pilot study): 64 participants PC1- 33 NPC1 – 31

2nd phase: 63 participants PC2 – 33NPC2 - 30

Aims of the experiment to teach in both groups the subject matter

Graphs of goniometric functions and

to prepare a test for all respondents of experiment,

to evaluate the test and to compare the results of two types of groups (experimental and control group, 1st and 2nd phase),

to find out whether the computer aided teaching is more effective than the traditional.

dcxbay ).sin(. dcxbay ).cos(.

0b,0a,Rd,c,b,a

Investigating the family of functions

ExampleIn the figure below is sketched graph of function

Find the coordinates of signed point A

3x2cos4y

Example of the test questionIn the figure below is sketched graph of function .Choose one of 7 lines (k, l, m, n, o, p, q), which represents the x-axis and one line which represents the y-axis?

2cos1 xy

Evaluation

2nd phase Pre-test Test

PC 2 72,84% 47,40%

NPC 2 73,85% 50,00%

1st phase Test Post-test

PC 1 56,49% 25,32%

NPC 1 38,25% 17,51%

Worksheets1/4

Worksheets2/4

Worksheets 3/4

f´(x) > 0

f´(x) < 0

Exercise 3Investigate the function f: y = x4-8x2-12

Worksheets 4/4

University levelFind the bounded extreme of the function

on border 5, yxxyyxf

1: yxg

Finding a global extremesFind the lowest and highest value of the function on the close area

322, 22 yyxxyxf

1,0,0: yxyxM

Developing conceptual and procedural knowledge1st step: graph of real function of the two

real variables + modeling using some plastic mass

2nd step: teaching formula and procedures + using CAS for calculating derivates

3rd step: counting on the paper, CAS = control function

Structure

1st level: CAS as a tool for teaching/learning

2nd level: CAS as a tool for helping to making

calculation

3rd level: CAS as tool for controlling our own

calculation

Developing a conceptual knowledge – students must understand, what is going on

Using conceptual and procedural knowledge for solving similar tasks, or tasks where the students canuse these knowledge for solving tasks from specific topics

Developing a procedural knowledge – students must learn a procedure for solving some type of the tasks

Some ideas of solving the problems with mentioned topic Printed graphs

Using notebook and projector + CAS

CAS + some plastic mass

Worksheets and homework

Worksheets

Lets have a function f(x,y)=px2+qxy+ry2 and point A[0, 0]. Find the values for p, q, r to fulfill:

a) The function has in A saddle point

b)The function has in A maximum

c) The function has in A minimum

Results Almost excellent procedural knowledge of

the students Problems with basic operation Problems with understanding basic terms or

sketching basic graphs Better imagination of 3D Generalization of the knowledge about the

function from 2D to 3D and 4D and so on Better understanding of the procedure

Thank you for your attention!

Web: http://www.ddm.fmph.uniba.sk

E-mail addresses:ingrida.kraslanova@gmail.commaria.slavickova@centrum.sk