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Visualizzare un sistema di equazioni

In questa attività verranno utilizzati l'input algebrico e i comandi. Prima di iniziare, apprendere la sintassi relativa all'input algebrico e ai comandi . Fare riferimento al foglio di lavoro dinamico System Equations in modo da avere indicazioni su come gli studenti possano utilizzare questa costruzione per risolvere graficamente un sistema di equazioni lineari.

Processo di costruzione

1. Creare gli slider m_1 e q_1 utilizzando le impostazioni predefinite.

2. Creare l'equazione lineare l_1: y = m_1 x + q_1.

3. Creare gli slider m_2 e q_2 utilizzando le impostazioni predefinite.

4. Creare l'equazione lineare l_2: y = m_2 x + q_2.

5. Creare il testo dinamico testo1: "Retta 1 e selezionare l_1 da Oggetti.

6. Creare il testo dinamico testo2: "Retta 2 e selezionare l_2 da Oggetti.

7. Costruire il punto di intersezione A delle due rette utilizzando lo strumento Intersezione di due oggetti o il comando A = Intersezione[l_1, l_2].

8. Definire ascissa = x(A). Suggerimento: x(A) restituisce l'ascissa del punto A.

9. Definire ordinata= y(A). Suggerimento: y(A) restituisce l'ordinata del punto A.

10. Creare il testo dinamico testo3: Soluzione: x = e selezionare ascissa da Oggetti, quindi digitare y = e selezionare ordinata da Oggetti.

Challenge

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

Construction Steps

1. Open a new GeoGebra window. Show the Vista Algebra, barra di inserimento, 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 punti 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 (finestra di dialogo Proprietà, tab Position).

5. Create triangle ABD.

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

7. Create vector u = Vector[O, P]. {{hint|You could also use [[tool Vector Between Two punti.}}

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

9. Translate the three corner punti 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 punti A, B, C, A', B', and D'.
• the coordinates of vector u.

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.

Construction Steps

1. Show the Vista Algebra, 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 punti 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. Suggerimento: 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) Suggerimento: y(A) restituisce you the y-coordinate of point A.

8. Calculate the run: run = x(B) - x(A) Suggerimento: x(B) restituisce you the x-coordinate of point B.

9. Enter the following equazione into the barra di inserimento 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 Insert text and click on the Vista Grafica.
2. Type slope = into the Insert text window’s barra di inserimento.
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 finestra di dialogo Proprietà 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 finestra di dialogo Proprietà and set point E as starting point.
5. Hide the midpunti D and E.

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.

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 (finestra di dialogo Proprietà).

4. Create a line through two punti with the first point at the base and the second point at the vertex of the pyramid. Suggerimento: Change the properties of line to improve its visibility.

5. Use the Slope Tool to get slope triangle of line. Suggerimento: 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.

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 punti C and B at the pyramid’s vertex. Suggerimento: 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. Suggerimento: Hide the auxiliary lines.

3. Use segments to connect point D with punti B and C. Suggerimento: Change the properties of the segments to increase their visibility. Suggerimento: You might want to rename the vertical segment to height and the horizontal segment to halfBase.3

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. {{hint|Use the slope triangle and your new triangle. Remember that the base side length is 35 m.

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

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