GeoGebra Short-Term Course Concluded

Last Saturday, June 5, 2010, we have concluded our 3-Saturday seminar on Using GeoGebra in Teaching Mathematics. Below are some of the pictures  taken from the said short-term course.

That’s me on the first pic explaining the mathematics behind the “how-to” construction. If you’re wondering why our whiteboard is high, well, I am just short. I am 5’3” feet, or maybe shorter (chuckles).

As you can see, the participants are really serious. 🙂

One of our participants constructing her GeoGebra applet.


So if you want to invite me* to conduct a lecture on GeoGebra in your school, I am just an email away.

*Philippines only (unless, of course, your school is willing to shoulder my expenses. Just kidding  🙂 ).

GeoGebra Tutorial 17 – Functions, Tangent Lines and Derivatives

This is the 17th tutorial of the GeoGebra Intermediate Tutorial Series. If this is your first time to use GeoGebra, I strongly suggest that you read the GeoGebra Essentials Series.    In this tutorial, we are going to use slider control a, b, c, d and e and graph the function f(x) = ax4 + bx3 + cx2 + dx + e.

Figure 1

We then  construct a line tangent to the function and passing through  point A and trace the graph of the point whose x-coordinate is the x-coordinate of A, and whose y-coordinate as the  slope of the tangent line. We compute for the derivative of f(x), and see if  there is a relationship between the trace and the derivative. If you want to follow the this tutorial step by step, you can open the GeoGebra window. Before following the tutorial, you may want to see the final output.

1. Open GeoGebra. We will need the Algebra view and the Axes so be sure that they are displayed. If not, use the View menu from the menu bar to show them.
2. To label points only, click the Options menu, click Labeling, and then click New Points Only.
3. To create slider a, type a = 1 in the input box and press the ENTER key. Right click the equation a = 1 in the algebra window (leftmost window pane) and click Show object from the context menu. Slider a should appear on your drawing pad.
4. Using step 3, create 4 more sliders namely b, c, d and e.
5. To graph the function f(x) = ax4 + bx3 + cx2 + dx + e, type f(x) = a*x^4 + b*x^3 + c*x^2 + d*x + e in the input box, then press the ENTER key.
6. Move the sliders and observe what happens.

Figure 2

7. To construct point A on function f, select the New Point tool and click graph of the function.
8. To construct a line tangent to f and passing through A, select the Tangents tool, click point A and click the graph of f. A tangent line should appear passing through point A.
9. Move point A on the function, and move the sliders. What do you observe?
10. To get the slope of the tangent line, select the Slope tool and then click the tangent line. This will produce m (see Algebra window). Given similar values of the numerical coefficients and the right place for point A, your graph should look like Figure 3.

Figure 3

11. We now create point D, which will trace the ordered pair (x(A),m) where m is the slope of the tangent line. Note that x(A) means the x-coordinate of A and m was automatically assigned by GeoGebra to the value of the slope.  To create the point type D = (x(A), m).
12. We now change the color of point D. To do this, right click point D, and click Object Properties from the context menu. In the dialog box, select the Color tab and select a color you want from the color palette. Next, select the Basic tab, be sure to check the Show trace check box, then click the Close button.

Figure 3

13. Move point A along the function. What do you observe about the traces of point D?
14. To graph the derivative of f(x), type f’(x) = derivative[f], then press the ENTER key. What do you observe about the derivative of the function f?
15. Right click the derivative function and click Properties. In the dialog box, go to the Color tab and select the color you want, preferably the same color as point D. Drag point A. After this step, your drawing should look like the one shown in Figure 1.
16. What can you say about the relationship of the derivative and the path traced by point D and the derivative function?
17. How can you relate the tangent function, its slope, the derivative function and the line traced by point D?
18. Based on the activity above, how will you describe the derivative of a function at a particular point and derivative in general?

GeoGebra Tutorial 9 – Vector and Translation

This is the ninth tutorial in the GeoGebra Intermediate Tutorial Series. If this is your first time to use GeoGebra, you might want to read the GeoGebra Essentials Series.

In this tutorial, we use the Vector between Two Points tool to translate a triangle and investigate the relationship between its preimage and image. We will also use the grid in this tutorial.

If you want to follow this tutorial step-by-step, you may open the GeoGebra window in your browser by clicking here.

1.) Open GeoGebra and select the Algebra & Graphics view from the Perspectives menu.
2.) Display the grid by clicking the View menu and choosing Grid.


3.) Click the New Point tool and place the points on the coordinates given: A on (2,3), B on (4,1) and C on (5,2).
4.) Next, we draw triangle ABC using the Polygon tool. To do this, click the Polygon tool and click the points in the following order: point A, point B, point C and point A again to close the polygon.
5.) To display the label and the coordinates of the points, right click the points then click Object Properties to display the Preferences dialog box.
6.) In the Basic tab of the Preferences dialog box, check the Show label check box, and choose the Name & Value option in the drop-down list box, and then close the window. Your drawing should look like the figure below.

Figure 1

7.) The only remaining part of the construction is the vector tool. To construct vector DE, select the Vector between Two Points tool, click the origin and click the coordinate (1,2). After this step, your drawing should look like the one shown in Figure 2.

Figure 2

8.) To translate the object using the vector, select the Translate Object by Vector tool, click the triangle and then click the vector. Notice that a translated triangle appears after clicking the vector tool.
9.) What can you say about the preimage of the triangle object and the translated object?
10.) If the coordinates of the vertices of the translated triangle is not displayed, display it using the steps we have done in step 5 and 6.
11.) What do you observe about the relationship of the coordinates of the points of the original triangle and the translated triangle?
12. Move the terminal point (point E) of the vector. Does your observation in (11) still hold?
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