Tutorial:Algebraic Input, Functions & Export of Pictures to the Clipboard

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Parameters of a Linear Equation

In this activity you are going to use the following tools, algebraic input, and commands. Make sure you know how to use them before you begin with the actual construction.

Tool Slider.gif Slider
line: y = m x + b
Tool Segment between Two Points.gif Segment
Intersect[line, yAxis]
Tool Intersect Two Objects.gif Intersect
Tool Slope.gif Slope
Tool Show Hide Object.gif Show / Hide Object
Tool Move.gif Move

Preparations

  • Open a new GeoGebra window.
  • Switch to Perspectives - Algebra & Graphics.
6 linear equation.PNG

Construction Steps

1. Enter: line: y = 0.8 x + 3.2

Tasks

  • Move the line in the Algebra View using the arrow keys. Which parameter are you able to change in this way?
  • Move line in the Graphics View with the mouse. Which transformation can you apply to the line in this way?

Construction Steps 2

2. Delete the line.

3. Create sliders m and b using the default settings of sliders.

4. Enter line: y = m x + b.
Note Hint: Don’t forget to use an asterisk or space to indicate multiplication!
5. Create the intersection point between the line and the y-axis.
Note Hint: Use tool Intersect or command Intersect[line, yAxis].

6. Create a point at the origin and draw a segment between these two points.

7. Use tool Slope and create the slope (triangle) of the line.

8. Hide unnecessary objects and modify the appearance of the other ones.

Tasks

Write down directions for your students that guide them through examining the influence of the equation’s parameters on the line by using the sliders. These directions could be provided on paper along with the GeoGebra file.

Library of Functions – Visualizing Absolute Values

Apart from polynomials there are different types of functions available in GeoGebra (e.g. trigonometric functions, absolute value function, exponential function). Functions are treated as objects and can be used in combination with geometric constructions.

Note: Some of the functions available can be selected from the menu next to the Input bar. Please find a complete list of functions supported by GeoGebra in the GeoGebra Wiki.

Preparations

  • Open a new GeoGebra window.
  • Switch to Perspectives – Algebra & Graphics.
4 absolute.PNG

Construction Steps

1 f(x) = abs(x) Enter the absolute value function f
2 g(x) = 3 Enter the constant function g
3 Tool Intersect Two Objects.gif Intersect both functions
Note Hint: You need to intersect the functions twice in order to get both intersection points.
Note Hint: You might want to close the Algebra View and show the names and values as labels of the objects.


Back to school...

(a) Move the constant function with the mouse or using the arrow keys. The y-coordinate of each intersection point represents the absolute value of the x-coordinate.

(b) Move the absolute value function up and down either using the mouse or the arrow keys. In which way does the function’s equation change?

(c) How could this construction be used in order to familiarize students with the concept of absolute value?
Note Hint: The symmetry of the function graph indicates that there are usually two solutions for an absolute value problem.


Library of Functions – Superposition of Sine Waves

Excursion into physics

Sound waves can be mathematically represented as a combination of sine waves. Every musical tone is composed of several sine waves of the form y(t) = a sin(ω t + φ) .

The amplitude a influences the volume of the tone while the angular frequency ω determines the pitch of the tone. The parameter φ is called phase and indicates if the sound wave is shifted in time.

If two sine waves interfere, superposition occurs. This means that the sine waves amplify or diminish each other. We can simulate this phenomenon with GeoGebra in order to examine special cases that also occur in nature.

Preparations

  • Open a new GeoGebra window.
  • Switch to Perspectives – Algebra & Graphics.

Construction Steps

1 f(x) = abs(x) Create three sliders a_1, ω_1, and φ_1
Note Hint: a_1 produces an index. You can select the Greek letters from the menu next to the text field name in the Slider dialog window.
2 g(x)= a_1 sin(ω_1 x + φ_1) Again, you can select the Greek letters from a menu next to the Input Bar.
3 Tool Slider.gif Create three sliders a_2, ω_2, and φ_2
Note Hint: Sliders can only be moved when the Slider Tool is activated.
4 h(x)= a_2 sin(ω_2 x + φ_2) Enter another sine function h
5 sum(x) = g(x) + h(x) Create the sum of both functions
6 Change the color of the three functions so they are easier to identify.

Back to school...

(a) Examine the impact of the parameters on the graph of the sine functions by changing the values of the sliders.

(b) Set a_1 = 1, ω_1 = 1, and φ_1 = 0. For which values of a_2, ω_2, and φ_2 does the sum have maximal amplitude?
Note Hint: In this case the resulting tone has the maximal volume.
(c) For which values of a_2, ω_2, and φ_2 do the two functions cancel each other?
Note Hint: In this case no tone can be heard any more.


Introducing Derivatives – The Slope Function

In this activity you are going to use the following tools, algebraic input, and commands. Make sure you know how to use them before you begin with the actual construction.

f(x) = x^2/2 + 1
Tool New Point.gif New Point
Tool Tangents.gif Tangent
slope = Slope[t]
S = (x(A), slope)
Tool Segment between Two Points.gif Segment
Tool Move.gif Move

Preparations

  • Open a new GeoGebra window.
  • Switch to Perspectives - Algebra & Graphics.

Construction Steps

1. Enter the polynomial: f(x) = x^2/2 + 1

2. Create new point A on function f.
Note Hint: Move point A to check if it is really restricted to the function graph.

3. Create tangent t to function f through point A.

4. Create the slope of tangent t using: slope = Slope[t]

5. Define point S: S = (x(A), slope)
Note Hint: x(A) gives you the x-coordinate of point A.

6. Connect points A and S using a segment.

Back to school...

(a) Move point A along the function graph and make a conjecture about the shape of its path, which corresponds to the slope function.

(b) Turn on the trace of point S. Move point A to check your conjecture.
Note Hint: Right click point S (MacOS: Ctrl + click) and select Trace on.

(c) Find the equation of the resulting slope function. Enter the function and move point A. If it is correct the trace of point S will match the graph.

(d) Change the equation of the initial polynomial f to produce a new problem.

Exploring Polynomials

Preparations

  • Open a new GeoGebra window.
  • Switch to Perspectives – Algebra & Graphics.

Construction Steps

1 f(x) = 0.5x³ + 2x² + 0.2x - 1 Enter the cubic polynomial f
2 R = Root[ f ] Create the roots of polynomial f
Note Hint: If there are more than one root GeoGebra will produce indices for their names if you type in R = (e.g. R1, R2, R3).
3 E = Extremum[ f ] Create the extrema of polynomial (English UK: Create the turning points of polynomial f: E = TurningPoint[f])
4 Tool Tangents.gif Create tangents to f in E1 and E2
5 I = InflectionPoint[ f ] Create the inflection point of polynomial f
Note Hint: You might want to change properties of objects (e.g. color of points, style of the tangents, show name and value of the function).


Exporting a Picture to the Clipboard

GeoGebra’s graphics view can be exported as a picture to your computer’s clipboard. Thus, they can be easily inserted into text processing or presentation documents allowing you to create appealing sketches for tests, quizzes, notes or mathematical games.

GeoGebra will export the whole Graphics View into the clipboard. Thus, you need to make the GeoGebra window smaller in order to reduce unnecessary space on the Graphics View:

  • Move your figure (or the relevant section) to the upper left corner of the Graphics View using the Move Graphics View tool (see left figure below).
    Note Hint: You might want to use tools Tool Zoom In.gif Zoom In and Tool Zoom Out.gifZoom Out in order to prepare your figure for the export process.
  • Reduce the size of the GeoGebra window by dragging its lower right corner with the mouse (see right figure below).
    Note Hint: The pointer will change its shape when hovering above an edges or corner of the GeoGebra window.


5 size.PNG

Use the File Menu to export the Graphics View to the clipboard:

  • Export – Graphics view to Clipboard
    Note Hint: You could also use the key combination Ctrl + Shift + C

.

  • Your figure is now stored in your computer’s clipboard and can be inserted into any word processing or presentation document.
5 toclipboard.PNG

Inserting Pictures into a Text Processing Document

After exporting a figure from GeoGebra into your computer’s clipboard you can now paste it into a word processing document.

Inserting pictures from the clipboard to MS Word

  • Open a new text processing document
  • From the Home tab select Paste. The picture is inserted at the position of the cursor.
    Note Hint: You can use the key combination Ctrl + V instead.


5 word.PNG

Reducing the size of pictures in MS Word

If necessary you can reduce the size of the picture in MS Word:

  • Double click the inserted picture.
  • Change the height/width of the picture using the Size group on the right.
5 word size.PNG
Note: If you change the size of a picture, the scale is modified. If you want to maintain the scale (e.g. for your students to measure lengths) make sure the size of the picture is 100%.
Note: If a picture is too big to fit on one page MS Word will reduce its size automatically and thus, change its scale.

Inserting pictures from the clipboard to OO Writer

  • Open a new text processing document
  • From the Edit menu select Paste or use the key combination Ctrl + V.
5 openoffice.PNG

Reducing the size of pictures in OO Writer

  • Double click the inserted picture.
  • Select the Type tab in the appearing Picture window.
  • Change width/height of the picture.
  • Click OK.

Challenge of the Day: Creating Instructional Materials

Pick a mathematical topic of your interest and create a worksheet / notes / quiz for your students.

  • Create a figure in GeoGebra and export it to the clipboard.
  • Insert the picture into a word processing document.
  • Add explanations / tasks / problems for your students.
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