Learning More by Learning More by Solving Series of Solving Series of ProblemsProblems

Martin Kollar,Martin Kollar,Comenius University, Comenius University, Bratislava Bratislava

Jozef HvoreckyJozef Hvorecky,,

University Bellevue, University Bellevue, Bratislava, SlovakiaBratislava, Slovakia

Graphical calculator Graphical calculator

CASIOCASIO ClassPad 300ClassPad 300

Flexible toolFlexible tool in in teachingteaching mathematics mathematics User friendlyUser friendly

Students are motivatedStudents are motivated

More than capable to cover high school More than capable to cover high school needsneeds

Series of problems Series of problems on the given topicon the given topic

We use graphical calculator to visualize our We use graphical calculator to visualize our ideasideasStudents can easier discover the facts Students can easier discover the facts and connections between themand connections between them

We encourage our students to make We encourage our students to make suggestions, hypotheses suggestions, hypotheses

We let the students decide whether they We let the students decide whether they need more examples and how many of the need more examples and how many of the topic giventopic given

An example: An example: L’Hospital RuleL’Hospital Rule

It is a strong rule for computing limits.It is a strong rule for computing limits.

Basically it saysBasically it says

We wanted to visualize this rule We wanted to visualize this rule

IT can do things you can’t do on the IT can do things you can’t do on the blackboardblackboard

lim limx a x a

f x f x

g x g x

Series of problems Series of problems to discover L’Hospital Ruleto discover L’Hospital Rule

Problem 1.Problem 1.

ComputeCompute

Problem 2.

FindFind

0

3lim2x

x

x

0

ln 1 3lim

sin 2x

x

x

If needed, we provide If needed, we provide studentsstudents similar couples of similar couples of problems.problems.Students discover the fact that every (differentiable) Students discover the fact that every (differentiable) function zoomed many times looks like a straight line.function zoomed many times looks like a straight line.

Captures of screen of CASIO Captures of screen of CASIO ClassPad 300ClassPad 300

We draw the graphs of the numerator and denominator functions.

Next, we zoomed the graphs in the neighbourhood of origin.

Captures of screen of CASIO Captures of screen of CASIO ClassPad 300ClassPad 300

Every picture below is 2 times zoomed previous picture.

Answers to teacher’s Answers to teacher’s questionsquestions

What line is it?What line is it?This line is the tangent in a given point.This line is the tangent in a given point.

In the first problem with lines we saw that the In the first problem with lines we saw that the result is in fact the ratio of slopes of these lines.result is in fact the ratio of slopes of these lines.

Slope of a tangent in a point is the Slope of a tangent in a point is the derivative. derivative.

““Ahaaa” effect: Ahaaa” effect: This is a L’Hospital’s rule!This is a L’Hospital’s rule!

The workshopThe workshop

Further problems on the L’Hospital’s rule.Further problems on the L’Hospital’s rule.

Sensitivity of solutions of systems of linear equations

Lines and circles in analytical geometry. Lines and circles in analytical geometry.

Animations in planar geomerty.Animations in planar geomerty.

Thank you for your attentionThank you for your attention