Enhancing Pedagogical Design in Mathematics Using Geogebra

Enhancing Pedagogical Design in Mathematics Using Geogebra

Enhancing Pedagogical Design in Mathematics using GeoGebra

1.Introduction and Basic Operations

Anthony C.M. Or

Assessment and HKEAA Section,

Education Infrastructure Division, Education Bureau.

Jan 2011

1.. Introduction to GeoGebra

GeoGebra is a free dynamic mathematics software developed by Markus Hohenwarter and his team. It joins together geometry, algebra, and calculus. On the one hand, you can construct geometric objects in the geometry window using the given geometry tools with the mouse. On the other hand, you can also create objects by using the keyboard to enter directly coordinates, equations, functions or commands into the input field. While the graphical representation of all objects is displayed in the graphics window, their algebraic numeric representation is shown in the algebra window. Objects can be changed dynamically with the relations defining them preserved.

The toolboxes of GeoGebra

Each toolbox contains a number of tools. To use a tool, click on the toolbox, or click the small triangle at the lower right corner, and click the required tool.

2.. Installing GeoGebra

GeoGebra official website:

WebStart (with Internet access):

Install and start GeoGebra on your computer.

Applet Start (with Internet access):

Open a fully functional GeoGebra applet in your web browser. Nothing will be installed on your computer.

Offline Installers:

GeoGebra Portable:

GeoGebra Portable runs on every computer without installation. Just download a portable package and extract it to your USB drive to have your GeoGebra on the go.

3.. Drawing a Square

  1. Right-click at any position in the Geometry Window. Uncheck “Axes” to hide the axes.
  2. Use the “Segment” tool to draw the line segment AB in the Geometry Window.
  3. Use the “Perpendicular Line” tool, click on A and then the segment to draw a line through A perpendicular to the segment.
  4. Double-click the perpendicular line to see its command: PerpendicularLine[A, a]
  5. In the Input Field, type
    PerpendicularLine[B, a]
    to construct a perpendicular line passing through B.
  6. Use the “Circle with Centre through Point” tool to draw a circle of centre A and passes through B.
  7. Use the “Intersect” tool to find the intersection between the circle and the perpendicular at A.
  8. Use the “Parallel Line” tool to draw a line through the intersection and parallel to AB.
  9. Use the “Intersect” tool the remaining vertex of the square.
  10. Use the “Selection” tool. Press and hold the Ctrl key, click the circle, the lines and point C, right-click either one of the selected item, uncheck the “Show Object” check box to hide them.
  11. Use the “Segment” tool to join the other three sides of the square. Right click the vertex to rename the square as ABCD. Drag A or B to see if the square is robust, i.e. it is always a square under dragging.

4.. Drawing a Square and change it to an Equilateral Triangle

  1. Open a new window (Ctrl+N). Use the “Regular Polygon” tool. Click at any two positions in the Geometry Window, enter 4 to construct a square.
  2. Double-click the shaded region to see its command: Polygon[A, B, 4]
  3. Change 4 to 3 in the “Redefine” window to change the square to an equilateral triangle.

5.. The Polygon by Joining the Midpoints of the Sides of a Quadrilateral

  1. Open a new window (Ctrl+N). Use the “Segment” tool to draw a quadrilateral ABCD.
  2. Use the “Midpoint” tool. Click each side to create its midpoint.
  3. Use the “Polygon” tool, click the midpoints respectively in anticlockwise direction to create a quadrilateral.
  4. Use the “Angle” tool. Click G, F, E respectively to measure ∠EFG. Similarly, measure the other three angels of the new quadrilateral.
  5. Drag the vertices. What is this polygon?

6.. Graph of Quadratic Function with Coefficient Sliders

  1. Open a new window (Ctrl+N). Enter the following commands in the input field:

a = 1

b = 0

c = 0

y = a x² + b x + c

  1. Right-click the parabola, choose “Properties”. In the “Show Label” pull-down menu, choose “Value”.
  2. In the Algebra Window, check the buttons of a, b and c to show the sliders controlling their values.
    (Fig. 1 below.)
  3. Right click on the empty space of the Geometry Window. Choose “Grid”. Move the sliders to appropriate positions, drag a, b or c to see how the graph changes accordingly. (Fig. 2 below.)

Fig. 1 Fig. 2

7.. Graph of Trigonometric Function

  1. Open a new window (Ctrl+N). Enter the following commands in the input field: sin(x°)
    Choose “°” from the input options as shown in the figure.
  2. Right click on the empty space of the Geometry Window. Choose “Properties”.
    Input according to the following figures for the tabs “xAxis”, “yAxis” and “Grid”.
  1. Drag the curve to see that it is movable (and its equation changes accordingly). To fix it, press Ctrl+Z to undo the action, then right-click the curve and choose “Properties”. In the “Basic” tab check the “Fix Object” box. (See the figure.)
  2. In the same tab, in the “Caption” field, enter
    “y = sin x”, and choose “Caption” in the “Show Label”. (See the figure.)
  1. If we wish to confine the curve in the range 0° ≤ x ≤ 360°, uncheck the “Fix Object” box. Double-click the curve, in the “Redefine” window type
    Function[sin(x °), 0, 360]

8.. Drawing a Graph Paper

  1. Open a new window. According to the figures, set the “Distance” of x-Axis and y-Axis be “1” “Line Style” of the axes be “Bold”, and choose the “Grid” Line Style as shown in the figure.
  1. Type the following commands in the Input Field.
    S = (floor(x(Corner[1])), ceil(y(Corner[4])))
    T = (floor(x(Corner[1])), floor(y(Corner[1])))
    U = (ceil(x(Corner[2])), floor(y(Corner[1])))
    V = (ceil(x(Corner[2])), ceil(y(Corner[4])))
    int = 0.5
    list1 = Sequence[Segment[(x(S) + int i, y(S)), (x(T) + int i, y(T))], i, 0, (x(U) - x(T))/int]
    list2 = Sequence[Segment[(x(T), y(T) + int i), (x(U), y(U) + int i)], i, 0, (y(S) - y(T))/int]
  2. Change the “Line Thickness” of list1 and list2 to 2, colour to (102, 102, 102).
    The following graph would appear. See also the explanatory notes of the above commands below.

floor(x): largest integer ≤ x;ceil(x): smallest integer ≥ x.

list1: The list of vertical lines. When i = 0, we have TS; when i = , we have UV.

list2: The list of horizontal lines. When i = 0, we have TU; when i = , we have SV.

  1. Save your file as “graph-paper.ggb”.

9.. Graphical Solutions of Simultaneous Equations

  1. Use your file “graph-paper.ggb” created in the previous task. Input the following equations:
    3x – 2y = 8
    4x – 7y + 1 = 0
    Label them by their values (i.e. equations). Choose “3” as their line thickness.
  2. Find graphically the solution of the equations. If necessary, use the “Move” () tool (or Ctrl+drag), “Zoom In” () tool (mouse wheel down) and “Zoom Out” () tool (mouse wheel up).

10..Output to HTML with Customized Toolbar

  1. Open your file “graph-paper.ggb”.
  2. Close the Algebra Window by choosing in the Menu Bar “View| Algebra View” (Ctrl+Shift+A).
  3. Choose “Tools| Customize Toolbar”. Remove all the tools except . Add back the tools , and and click “Apply”, as shown in the figure.
  4. Choose “File| Export| Dynamic Worksheet as Webpage (html) … Ctrl+Shift+W”.
  5. Check the boxes in the “Advanced” tab as shown in the figure.
  6. Save the html file as “graph.html”. It would be opened in your browser. Note that besides the html file, the file “graph.ggb” and other jar files are generated in your folder. These files are necessary for opening the html file.
  7. It can be used as a web-based tool for students to solve simultaneous equations graphically, by entering the equations in the Input Field.

11..Derivative and Tangent of a Function

  1. Open a new window. Input the polynomial
    x^3 – 6x^2 + 9x + 1
    Label the curve as “y = x3 – 6x2 + 9x + 1”.
  2. Use the “New Point” () tool, click on the curve to create a movable point A on the curve.
  3. Input
    Tangent[A, f(x)]
    to construct the tangent to the curve
    y = x3 – 6x2 + 9x + 1 at A. Unlabel it.
  4. Input
    Slope[a]
    to show the slope of the tangent (named a). Label it with its value.
  5. Drag A to see how the slope varies, especially at the turning points and the point of inflection.

12..Riemann Sum, Trapezoidal Rule and Definite Integral

  1. Open a new window. Input the following:
    4x / (x^2 + 1)
    n = 8
    Show the slider for n. Change its “Properties” as shown in the figure.
  2. Input the Lower and Upper Riemann Sums, the Trapezoidal Sum and the definite integral in the interval [0, 4] by the following commands.
    LowerSum[f(x), 0, 4, n]
    UpperSum[f(x), 0, 4, n]
    TrapezoidalSum[f(x), 0, 4, n]
    Integral[f(x), 0, 4]
  3. Label the four quantities by their values. Change the colour of the LowerSum (a) to blue (0, 0, 255); the UpperSum (b) to green (0, 255, 0), the Trapezoidal Sum to magenta (255, 0, 255) and the definite integral to red (255, 0, 0).
  4. Put the Lower Sum to layer 3, the definite integral to layer 2, the Trapezoidal Sum to layer 1 in the “Advanced” tab.
  1. Use the “Text” () tool. Check the “LaTex formula” box and enter the following.
    "\int_{0}^{4} \frac{4x}{x^{2}+1} dx = " + h

Click OK afterwards. Unlabel h (the definite integral).

  1. Use the “Check Box” () tool. Click a position in the Geometry Window. In the “Caption” field enter “Lower Riemann Sum” and select “Number a: LowerSum[f(x), 0, 4, n]” from the list. Click “Apply” afterwards.
    Similarly, create the checkboxes for “Upper Riemann Sum”, the “Trapezoidal Rule” and “Definite Integral”.
  2. Click the check boxes to show or hide the different sums and the definite integral. Drag n to see how the sums approach the definite integral.

Exercise

  1. Draw rectangles, parallelograms and rhombi with variable sides and shapes.
  2. Draw the graphs of y = cos x° and y = tan x°.
  3. Draw the graph of the general sine curve y = a sin b(x° – c) + d, with sliders for –2 ≤ a, b, d ≤ 2;
    –120° ≤ c ≤ 120°.
  4. Draw an arbitrary triangle. Draw its circumcircle an incircle.

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