Geogebra for Interactive Teaching and Learning

5:00 PM

Outline

1.Importance of Visualization

2. Geogebra and its capability

4. Applets for Visualizing 1D and 2D functions

5. Applets for Activity based Learning

6. Conclusion

3. Funny and Motivating Example

Imagination is more important than knowledge. Knowledge is limited. Imagination encircles the world.

-Einstein

The game I play is a very interesting one. It’s imagination, in a tight strait-jacket.

Richard Feynman

I like to play about with equations, just looking for beautiful mathematical relations which maybe don’t have any physical meaning at all. Sometimes they do.

Paul Dirac

The universe is

full of

objects

Physical

Mathematical

and

their Interactions

So is Geogebra !

Funny and Motivating Example

Car is an Object in the universe

(Point in the coordinate system)

Moving is an (inter)action

(Change in Coordinate of the point)

Collision is an (Inter)action

(Overlapping Coordinate Points)

Click on the Buttons to Start/Stop driving

With this Object Oriented view, I Hope, we can (Ideally) visualize anything in Geogebra!

You can Interact with the applets in the upcoming slides and get a feel for it.

To illustrate the capability of geogebra, I have cherry-picked examples that we may already be familiar with.

Know the objects involved and their Interactions (Relations)

Taylor Series

Please do not run away !

What is Taylor Series?

Taylor series is a way of approximating any continuously differentiable function \(\mathscr{f(x)}\) using polynomials of degree \(n\). The higher the degree the better the approximation!.

\(\mathscr{f}(x)=\mathscr{f}(p)+\frac{\mathscr{f'}(p)}{1!}(x-p)+\frac{\mathscr{f''}(p)}{2!}(x-p)^2+\frac{\mathscr{f'''}(p)}{3!}(x-p)^3+ \cdots\)

\(\eta \in (0,1) \) is a parameter to adjust the gradient value to obtain a better approximation around point \(p\)

\mathscr{f}(x)

Linear Approximation (\( n=1 \))

\( \mathscr{f}(x)=\mathscr{f}(p)+\eta \frac{\mathscr{f'}(p)}{1!}(x-p) \)

Quadratic Approximation (\(n=2\))

\(\mathscr{f}(x)=\mathscr{f}(p)+\eta \frac{\mathscr{f'}(p)}{1!}(x-p)+\eta^2 \frac{\mathscr{f''}(p)}{2!}(x-p)^2 \)

Try with different \( f(x) \) such as \(x^2,exp(x),sin(x),x^3 \cdots \) and see how closely taylor series approximate the function at the point \(x = X\) and around its neighbour \(\epsilon \)

You can drag the point on the sliders

You can increase/decrease Polynomial

Degree by clicking +/- symbol

Let us extend the Taylor series to 2D functions and Observe how it approximates the surface in and around the point \(\mathbf{x}\)

Approximation in 2D

Adjust the slider \(p\) to see the Linear approximation of \(f(x)\)at the point \(p\)

Notice the gradient (\(dp\)) value.

Change the value of \(p\) according to the gradient value. That is, take a step \(p\pm dp\) (Enter the new value for \(p\) in the input box only)

After few adjustments, did you reach the local minimum?

If no, Repeat the game and make necessary

changes to land in local minima.