Tutorial:Practice Block III

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Visualizing a System of Equations

In this activity you are going to use algebraic input and commands. Make sure you know the syntax for algebraic input and commands before you begin. You might want look at the dynamic worksheet called System equations in order to find out how your students could use this construction to graphically solve a system of linear equations.

Construction Steps

1. Create sliders m_1 and b_1 using the default settings for sliders.

2. Create the linear equation l_1: y = m_1 x + b_1.

3. Create sliders m_2 and b_2 using the default settings for sliders.

4. Create the linear equation l_2: y = m_2 x + b_2.

5. Create dynamic text1: Line 1: and select l_1 from Objects.

6. Create dynamic text2: Line 2: and select l_2 from Objects.

7. Construct the intersection point A of both lines either using Tool Intersect two objects or command A = Intersect[l_1, l_2].

8. Define xcoordinate = x(A).
Note Hint: x(A) gives you the x-coordinate of point A.
9. Define ycoordinate = y(A).
Note Hint: y(A) gives you the y-coordinate of point A.

10. Create dynamic text3: Solution: x = and select xcoordinate from Objects. Type in y = and select ycoordinate from Objects.

9 equations.PNG

Challenge

Create a similar construction that allows for visualizing the graphical solution of a system of quadratic polynomials.
Note Hint: Functions need to be entered using the syntax f(x) = …
Note: Such a dynamic figure can also be used to visualize an equation in one variable by entering each side of the equation as one of the two functions.

Translating Pictures

In this activity you are going to use the following tools and commands. Make sure you know how to use each tool and command before you begin.

Tool Insert Image.gif Image
A = (1, 1)
Tool Polygon.gif Polygon
Vector[O, P]
Tool Vector between Two Points.gif Vector
Tool Translate Object by Vector.gif Translate by Vector
Tool Move.gif Move
Tool Insert Text.gif Text

Construction Steps

1. Open a new GeoGebra window. Show the Algebra View, Input Bar, coordinate axes, and grid. In the Options Menu set the point capturing to Fixed to Grid.

2. Insert picture A_3b_Bart.png into the first quadrant.

3. Create points A = (1, 1), B = (3, 1), and D = (1, 4).

4. Set point A as the first, B as the second, and D as the fourth corner point of the picture (Properties Dialog, tab Position).

5. Create triangle ABD.

6. Create point O = (0, 0) and point P = (3, -2).

7. Create vector u = Vector[O, P].
Note Hint: You could also use tool Vector.


8. Translate the picture by vector u using Translate by Vector.
Note Hint: You might want to reduce the filling of the image.

9. Translate the three corner points A, B, and D by vector u.

10. Create triangle A'B'D'.

11. Hide point O so it can’t be moved accidentally. Change the color and size of objects to enhance your construction.

Challenge

Insert dynamic text that shows

  • the coordinates of points A, B, C, A', B', and D'.
  • the coordinates of vector u.
9 bart.PNG

Constructing a Slope Triangle

In this activity you are going to use the following tools and algebraic input. Make sure you know how to use each tool and the syntax for algebraic input before you begin.

Tool Line through Two Points.gif Line
Tool Perpendicular Line.gif Perpendicular Line
Tool Intersect Two Objects.gif Intersect
Tool Polygon.gif Polygon
rise = y(B) - y(A)
run = x(B) - x(A)
slope = rise / run
Tool Insert Text.gif Text
Tool Midpoint or Center.gif Midpoint or Center
Tool Move.gif Move

Construction Steps

1. Show the Algebra View, coordinate axes and the grid. Set point capturing to Fixed to Grid and the labeling to All new objects.

2. Create line a through two points A and B.

3. Construct a perpendicular line b to the y-axis through point A.

4. Construct a perpendicular line c to the x-axis through point B.

5. Intersect perpendicular lines b and c to get intersection point C.
Note Hint: You might want to hide the perpendicular lines.

6. Create polygon ACB and hide the labels of the sides.

7. Calculate the rise: rise = y(B) - y(A)
Note Hint: y(A) gives you the y-coordinate of point A.
8. Calculate the run: run = x(B) - x(A)
Note Hint: x(B) gives you the x-coordinate of point B.

9. Enter the following equation into the input bar to calculate the slope of line a: slope = rise / run

10. Insert dynamic text: rise = and select rise from Objects, run = and select run from Objects, slope = and select slope from Objects

11. Change properties of objects in order to enhance your construction.

Challenge 1: Insert a dynamic text that contains a fraction

Using LaTeX formulas, text can be enhanced to display fractions, square roots, or other mathematical symbols.

  1. Activate tool Text and click on the Graphics View.
  2. Type slope = into the Insert text window’s input bar.
  3. Check LaTeX formula and select Roots and Fractions a/b from the dropdown list.
  4. Place the cursor within the first set of curly braces and replace a by number rise from the Objects drop-down list.
  5. Place the cursor within the second set of curly braces and replace b by number run from the Objects drop-down list.
  6. Click OK.

Challenge 2: Attach text to an object

Whenever an object changes its position, attached text adapts to the movement and follows along.

  1. Create midpoint D of the vertical segment using tool Midpoint or center.
  2. Create midpoint E of the horizontal segment.
  3. 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.
  4. Select text2 (run = …) in the Properties Dialog and set point E as starting point.
  5. Hide the midpoints D and E.
9 slope.PNG

Exploring the Louvre Pyramid

In this activity you are going to use the following tools and some algebraic input. Make sure you know how to use every single tool and the syntax for algebraic input before you begin. Also, check if you have the picture A_3d_Louvre.jpg saved on your computer.

Tool Insert Image.gif Image
Tool Line through Two Points.gif Line Through Two Points
Tool Slope.gif Slope
Tool Angle.gif Angle
Tool New Point.gif Point
Tool Perpendicular Line.gif Perpendicular Line
Tool Intersect Two Objects.gif Intersect
Tool Show Hide Object.gif Show / Hide Object
Tool Segment between Two Points.gif Segment
Tool Move.gif Move

The Louvre in Paris is one of the most visited and famous art museums in the world. The building holds some of the world's most famous works of art, such as Leonardo da Vinci's Mona Lisa. In 1989 the main entrance of the museum was renovated and a glass pyramid was built (from http://en.wikipedia.org/wiki/Louvre, February 20, 2008).

Determine the slope of the pyramid’s faces

1. Set point capturing off. Set the decimal places to 1. Change the labeling setting to All new objects (menu Options).

2. Insert the picture A_3d_Louvre.jpg into the first quadrant of the coordinate system. The left lower corner should match the origin.

3. Reduce the filling of the picture (about 50%) and set it as background image (Properties Dialog).

4. Create a line through two points with the first point at the base and the second point at the vertex of the pyramid.
Note Hint: Change the properties of line to improve its visibility.
5. Use the Slope Tool to get slope triangle of line.
Note Hint: Change the properties of slope triangle to improve its visibility. The slope triangle is attached to the point created first.

6. Task: Determine the slope of the pyramid’s faces in percent.

7. Create the angle between the x-axis and the line. Task: Determine the inclination angle for the pyramid’s face.

9 pyramid1.PNG

Challenge

The pyramid’s base is a square with a side length of 35 meters. Determine the height of the pyramid using similar triangles.

1. Create a new point C on the line.

2. Construct the slope triangle of the line using points C and B at the pyramid’s vertex.
Note Hint: Create a line perpendicular to the y-axis through point C and a line perpendicular to the x-axis through point B at vertex of pyramid. Create the intersection point D of the two lines.
Note Hint: Hide the auxiliary lines.
3. Use segments to connect point D with points B and C.
Note Hint: Change the properties of the segments to increase their visibility.
Note Hint: You might want to rename the vertical segment to height and the horizontal segment to halfBase

4. Move point C along the line until the horizontal segment of the triangle matches the level of the road in front of the pyramid.

5. Task: Calculate the height of the pyramid using similar triangles.
Note Hint: Use the slope triangle and your new triangle. Remember that the base side length is 35 m.


Check your answer with GeoGebra

6. Show name and values of segments height and halfBase.

7. Drag point C until the horizontal segment has length 35/2 = 17.5.
Note Hint: You might need to zoom out of the construction and / or move the graphics view in order to be able to do this.

8. Check if the height of the pyramid matches your answer.

9 pyramid2.PNG

Comment

By implementing the instructions above you were able to graphically determine the approximate value for the pyramid’s height. In reality, the Louvre pyramid has a base length of 35 m and a height of 21.65 m. Its faces have a slope of 118% and an inclination angle of bout 52° (from http://de.wikipedia.org/wiki/Glaspyramide_im_Innenhof_des_Louvre#Daten, February 22, 2008).

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