Tutorial:Basic Algebraic Input, Commands and Functions

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Tips and Tricks

  • Name a new object by typing name = into the input bar in front of its algebraic representation.
    Example: P = (3, 2) creates point P.
  • Multiplication needs to be entered using an asterisk or space between the factors.
    Example: a*x or a x
  • GeoGebra is case sensitive! Thus, upper and lower case letters must not be mixed up.
    • Points are always named with upper case letters
      Example: A = (1, 2)
    • Vectors are named with lower case letters
      Example: v = (1, 3)
    • Segments, lines, circles, functions… are always named with lower case letters.
      Example: circle c: (x – 2)^2 + (y – 1)^2 = 16
    • The variable x within a function and the variables x and y in the equation of a conic section always need to be lower case.
      Example: f(x) = 3*x + 2
  • If you want to use an object within an algebraic expression or command you need to create the object prior to using its name in the input bar.
    • y = m x + b creates a line whose parameters are already existing values m and b (e.g. numbers / sliders).
    • Line[A, B] creates a line through existing points A and B.
  • Confirm an expression you entered into the input bar by pressing the Enter key.
  • Open the help window for using the input bar and commands by selecting Help from the Help Menu (or shortcut F1).
  • Error messages: Always read the messages – they could possibly help to fix the problem!
  • Commands can be typed in or selected from the list next to the Input Bar.
    Note Hint: If you don’t know which parameters are required within the brackets of a certain command, type in the full command name and press key F1 to open the GeoGebra Wiki.
  • Automatic completion of commands: After typing in the first two letters of a command into the Input Bar, GeoGebra tries to complete the command.
    • If GeoGebra suggests the desired command, hit the Enter key in order to place the cursor within the brackets.
    • If the suggested command is not the one you wanted to enter, just keep typing until the suggestion matches.

Constructing Tangents to a Circle (Part 1)

Open the dynamic worksheet Tangents to a Circle. Follow the directions on the worksheet in order to find out how to construct tangents to a circle.

Discussion

  • Which tools did you use in order to recreate the construction?
  • Were there any new tools involved in the suggested construction steps? If yes, how did you find out how to operate the new tool?
  • Did you notice anything about the toolbar displayed in the right applet?
  • Do you think your students could work with such a dynamic worksheet and find out about construction steps on their own?

Constructing Tangents to a Circle (Part 2)

What if my Mouse and Touchpad wouldn’t work?

Imagine your mouse and / or touchpad stop working while you are preparing GeoGebra files for tomorrow’s lesson. How can you finish the construction file?

GeoGebra offers algebraic input and commands in addition to the geometry tools. Every tool has a matching command and therefore could be applied without even using the mouse.

Note: GeoGebra offers more commands than geometry tools. Therefore, not every command has a corresponding geometry tool!

Preparations

Construction Steps

1 A = (0, 0) Point A
2 (3, 0) Point B
Note Hint: If you don’t specify a name objects are named in alphabetical order.
3 c = Circle[A, B] Circle with center A through point B
Note Hint: Circle is a dependent object
Note: GeoGebra distinguishes between free and dependent objects. While free objects can be directly modified either using the mouse or the keyboard, dependent objects adapt to changes of their parent objects. Thereby, it is irrelevant in which way (mouse or keyboard) an object was initially created!
Note Hint: Activate Move mode and double click an object in the Algebra View in order to change its algebraic representation using the keyboard. Hit the Enter key once you are done.
Note Hint: You can use the arrow keys in order to move free objects in a more controlled way. Activate Move mode and select the object (e.g. a free point) in either window. Press the up / down or left / right arrow keys in order to move the object into the desired direction.


4 C = (5, 4) Point C
5 s = Segment[A, C] Segment AC
6 D = Midpoint[s] Midpoint D of segment AC
7 d = Circle[D, C] Circle with center D through point C
8 Intersect[c, d] Intersection points E and F of the two circles
9 Line[C, E] Tangent through points C and E
10 Line[C, F] Tangent through points C and F

Checking and Enhancing the Construction

  • Perform the drag-test in order to check if the construction is correct.
  • Change properties of objects in order to improve the construction’s appearance (e.g. colors, line thickness, auxiliary objects dashed,…)
  • Save the construction.

Discussion

  • Did any problems or difficulties occur during the construction steps?
  • Which version of the construction (mouse or keyboard) do you prefer and why?
  • Why should we use keyboard input if we could also do it using tools?
    Note Hint: There are commands available that have no equivalent geometric tool.
  • Does it matter in which way an object was created? Can it be changed in the Algebra View (using the keyboard) as well as in the Graphics View (using the mouse)?

Exploring Parameters of a Quadratic Polynomial

In this activity you will explore the impact of parameters on a quadratic polynomial. You will experience how GeoGebra could be integrated into a "traditional" teaching environment and used for active, student-centered learning.

  1. Open a new GeoGebra window
  2. Type in f(x) = x^2 and hit the Enter key. Which shape does the function graph have? Write down your answer on paper.
  3. In Tool Move.gif Move mode, highlight the polynomial in the algebra view and use the ↑ up and ↓ down arrow keys.
    • How does this impact the graph of the polynomial? Write down your observations.
    • How does this impact the equation of the polynomial? Write down your observations.
  4. Again, in Move mode, highlight the function in the Algebra View and use the ← left and → right arrow keys.
    • How does this impact the graph of the polynomial? Write down your observations.
    • How does this impact the equation of the polynomial? Write down your observations.
  5. In Move mode, double click the equation of the polynomial. Use the keyboard to change the equation to f(x) = 3 x^2. Use an asterisk * or space in order to enter a multiplication.
    • Describe how the function graph changes.
    • Repeat changing the equation by typing in different values for the parameter (e.g. 0.5, -2, -0.8, 3). Write down your observations.

Discussion

  • Did any problems or difficulties concerning the use of GeoGebra occur?
  • How can a setting like this (GeoGebra in combination with instructions on paper) be integrated into a ‘traditional’ teaching environment?
  • Do you think it is possible to give such an activity as a homework problem to your students?
  • In which way could the dynamic exploration of parameters of a polynomial possibly affect your students’ learning?
  • Do you have ideas for other mathematical topics that could be taught in similar learning environment (paper worksheets in combination with computers)?
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