Preparations Open a new GeoGebra window. Switch to Perspectives Algebra & Graphics. Construction Steps 1 A = (0,0) Create point A. Hint: The parentheses are closed automatically. 2 (3,0) Create point B. Hint: If you don’t specify a name objects are named in alphabetical order. 3 Circle[A,B] Construct a circle c with center A through point B Hint: This 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! Task 1 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. Task 2 Use the arrow keys in order to move free objects in a more controlled way. Activate Move mode and select an object (e.g. a free point) in either window. Press the up / or left / right arrow keys in order to move the object into the desired direction. 4 C = (5, 4) Create point C. 5 s = Segment[A, C] Create segment AC. 6 D = Midpoint[s] Create midpoint D of segment AC.
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 a1 = 1, ω1 = 1 and φ1 = 0. For which values of a2, ω2 and φ2 does the sum have maximal amplitude? Note: In this case the resulting tone has the maximal volume. (c) For which values of a2, ω2, and φ2 do the two functions cancel each other? Note: 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 S = (x(A), m) Point Segment Tangents Move m = Slope[t] Hint: You might want to have a look at the link to the dynamic worksheet “Introducing Derivatives - The Slope Function” first. Preparations Open a new GeoGebra window. Switch to Perspectives - Algebra & Graphics. Introduction of new tool Tangents New! Hint: Click on a point on a function and then on the function itself. Hints: Don’t forget to read the Toolbar help if you don’t know how to use the tool. Try out the new tool before you start the construction.
Creating a ‘Function Domino’ Game In this activity you are going to practice exporting function graphs to the clipboard and inserting them into a word processing document in order to create cards for a ‘Function Domino’ game. Make sure you know how to enter different types of functions before you begin with this activity. Preparations Open a new GeoGebra window. Switch to Perspectives – Algebra & Graphics. Construction Steps for GeoGebra 1 Enter an arbitrary function. Examples: e(x) = exp(x) or f(x) = sin(x) 2 Move the function graph into the upper left corner of the Graphics View. 3 Reduce the size of the GeoGebra window so it only shows the desired part of the Graphics View. 4 Export the Graphics View to the clipboard. Hint: Menu File – Export – Graphics View to Clipboard. Construction Steps for MS Word 1 Open a new word processing document (e.g. MS Word). 2 Create a table with 2 columns and several rows. Hint: Menu Insert – Table… 3 Highlight the entire table (all cells) and open the Table Properties dialog. Hint: right-click – Table Properties… 4 Click on tab Row and specify the row height as 2 inches. 5 Click on tab Column and set the preferred width of the columns to 2 inches. 6 Click on tab Cell and set the vertical alignment to Center. 7 Click the OK button. 8 Place the cursor in one of the table cells. Insert the function graph picture from the clipboard.
Polygon Rotate around Point New! Point Segment Rename Angle Slider Move Preparations Open a new GeoGebra window. Switch to Perspectives – Geometry and show the coordinate axes. Click on Preferences in the Toolbar and choose Graphics to open the Properties dialog for the Graphics View. On tab xAxis change the Distance to 1. On tab yAxis change the Distance to 1. Introduction of new tool Rotate around Point New! Hint: Click the object to be rotated, the center of rotation and enter the angle in the appearing dialog window. Hints: Don’t forget to read the Toolbar help if you don’t know how to use the tool. Try out the new tool before you start the construction. Construction Steps 1 Create an arbitrary triangle ABC in the second quadrant placing the vertices on grid points. 2 Create a new point D at the origin of the coordinate system. 3 Rename the point D to O. Hint: Just type “O” to open the Rename dialog. 4 Create a slider for angle α. Hint: In the Slider dialog window check Angle and set the Increment to 90˚. Make sure you don’t delete the ˚ symbol. 5 Rotate triangle ABC around point O by angle α. Hint: Check counter clockwise rotation. 6 Create segments AO and A’O. 7 Create angle AOA’.
Switch to Perspectives – Geometry. Show the Input Bar (View menu). Set the number of decimal places to 0 (menu Options – Rounding). Introduction of new tool Midpoint or Center New! Hint: Select two points, one segment, circle or conic to get the midpoint or center. Hints: Don’t forget to read the Toolbar help if you don’t know how to use a tool. Try out the new tool before you start the construction. Construction Steps 1 Create a triangle ABC with counter clockwise orientation. 2 Create the angles α, β and γ of triangle ABC. 3 Create a slider for angle δ with Interval 0˚ to 180˚ and Increment 10˚. 4 Create a slider for angle ε with Interval 0˚ to 180˚ and Increment 10˚. 5 Create midpoint D of segment AC and midpoint E of segment AB. 6 Rotate the triangle around point D by angle δ (setting clockwise). 7 Rotate the triangle around point E by angle ε (setting counterclockwise). 8 Move both sliders δ and ε to show 180˚. 9 Create angle ζ using the points A’C’B’. 10 Create angle η using the points C'1B'1A'1 . 11 Enhance your construction using the Stylebar. Hint: Congruent angles should have the same color. 12 Create dynamic text displaying the interior angles and their values (e.g. α = and select α from Objects). 13 Calculate the angle sum using sum = α + β + γ 14 Insert the angle sum as a dynamic text: α + β + γ = and select sum from Objects. 15 Match colors of corresponding angles and text. Fix the text that is not supposed to be moved.
8 Calculate the run: run = x(B) - x(A). Hint: x(B) gives you the x-coordinate of point B. 9 Insert dynamic text1: rise = and select rise from Objects. 10 Insert dynamic text2: run = and select run from Objects. 11 Calculate the slope of line a: slope = rise / run. 12 Insert dynamic text3: slope = and select slope from Objects. 13 Change properties of objects in order to enhance your construction and fix text that is not supposed to be moved. Dynamic Fractions and Attaching Text to Objects Inserting dynamic fractions Using LaTeX formulas, text can be enhanced to display fractions, square roots, or other mathematical symbols. Enhance your construction of the slope triangle by entering a fraction showing how to calculate the slope of a line. Activate tool Text and click on the Graphics View. Type slope = into the text window’s Input Bar. Check LaTeX formula and select Roots and Fractions a/b from the list. Place the cursor within the first set of curly braces and replace a by number rise from the Objects drop-down list. Place the cursor within the second set of curly braces and replace b by number run from the Objects drop-down list. Click OK. Attaching text to objects Whenever an object changes its position, attached text adapts to the movement and follows along. Enhance your construction of a slope triangle by attaching text to the sides of the slope triangle. Create midpoint D of the vertical segment using tool Midpoint or Center. Create midpoint E of the horizontal segment. Open the Properties dialog and select text1 (rise = …). Click on tab Position and select point D from the drop-down list next to Starting Point. Select text2 (run = …) in the Properties dialog and set point E as Starting Point. Hide the midpoints D and E.
Visualizing Integer Addition on the Number Line Preparations Open a new GeoGebra window. Switch to Perspectives – Geometry. Show the Input Bar (View menu). In the Options menu set Labeling to All New Objects. Construction Steps 1 Open the Properties dialog for the Graphics View. Hint: Choose Preferences and then Graphics. 2 On tab xAxis set the distance of tick marks to 1 by checking the box Distance and entering 1 into the text field. 3 On tab Basic set the minimum of the x-Axis to -21 and the maximum to 21. 4 On tab yAxis uncheck Show yAxis. 5 Close the Properties dialog for the Graphics View. 6 Create a slider for number a with Interval -10 to 10 and Increment 1. 7 Create a slider for number b with Interval -10 to 10 and Increment 1. 8 Show the value of the sliders instead of their names. Hint: Stylebar - Set label style - Value 9 Create point A = (0 , 1). 10 Create point B = A + (a , 0). Hint: The distance of point B to point A is determined by slider a.
14 Hide all lines, circle d and segment a. 15 Hide labels of segments (Stylebar). 16 Set both sliders Columns and Rows to value 10. 17 Create a list of vertical segments. Sequence[Segment[A+i*(1, 0), C+i*(1, 0)],i ,1 ,Columns] Note: A + i*(1, 0) specifies a series of points starting at point A with distance 1 from each other. C + i*(1, 0) specifies a series of points starting at point C with distance 1 from each other. Segment[A + i*(1, 0), C + i*(1, 0)] creates a list of segments between pairs of these points. Note, that the endpoints of the segments are not shown in the Graphics View. Slider Column determines the number of segments created. 18 Create a list of horizontal segments. Sequence[Segment[A+i*(0, 1), B+i*(0, 1)], i, 1, Rows] 19 Move sliders Columns and Rows to check the construction. 20 Insert static and dynamic text that state the multiplication problem using the values of sliders Columns and Rows as the factors: text1: Columns text2: * text3: Rows text4: = 21 Calculate the result of the multiplication: result = Columns * Rows 22 Insert dynamic text5: result 23 Hide points A, B, C and D. 24 Enhance your construction using the Stylebar. Challenge of the Day: String Art Based on Bézier Curves Bézier curves are parametric curves used in computer graphics. For example, they are used in order to create smooth lines of vector fonts. Let’s create some ‘string art’ based on Bézier curves. Preparations Open a new GeoGebra window.