Tutorial:Practice Block IV

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Area Relations of Similar Geometric Figures

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

a = 2
Tool Segment with Given Length from Point.gif Segment With Given Length
Tool Regular Polygon.gif Regular Polygon
Tool Move.gif Move


In this activity you will recreate the following worksheet for your students. It allows them to discover the special relationship between the area of squares whose side lengths are a, a/2, and 2a.

Area Relations

  1. Measure the side length of the three squares below. Compare the side length of the blue square to the side lengths of the red and green squares. Which relation can you find?
  2. Calculate the areas of the three squares. Compare the area of the blue square to the areas of the red and green squares. Which relation can you find?
  3. Formulate a conjecture that compares the side length and area of the blue square to these of the red and green squares.
  4. Try to proof your conjecture.
    Note Hint: Assume that the side length of the blue square is a and calculate the areas of the corresponding squares.

13 squares.PNG

Construction Steps

1. Start your construction in GeoGebra by creating the number a = 2.

2. Construct the blue square starting with Segment With Given Length a. Then, use the two endpoints of this segment to create a regular polygon with 4 vertices.

3. In the same way construct the red square with side length a/2 and the green square with side length 2a.

4. Rename the vertices and change the properties of the squares (e.g. color, line thickness).

5. Prepare the GeoGebra window for the export of the graphics view as a picture (e.g. rearrange the squares, reduce the size of the GeoGebra window).

6. Export the graphics view as a picture and save your picture file.
Note Hint: Since your students are supposed to measure the side length of the squares you should not change the scale of the picture!

7. Open a word processing document and type in the heading and tasks of the worksheet.

8. Insert the picture of the squares into the worksheet.
Note Hint: Make a printout of the worksheet and try it out by measuring the side lengths of the squares.

Challenge 1

Create similar examples for different geometric shapes (e.g. circle with given radius, equilateral triangle, rectangle). For which of these shapes do the same relations between the length of the given side (radius) and the resulting area apply? Try to find an explanation for this relationship between the given length and the area of the figure.

Challenge 2

Create a dynamic worksheet based on your construction that helps your students to generalize their conjecture about the relationship between the side length and the area of such geometric shapes (for example, see Area_Circles.html).

Visualizing the Angle Sum in a Triangle

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

Tool Polygon.gif Polygon
Tool Angle.gif Angle
Tool Slider.gif Slider
Tool Midpoint or Center.gif Midpoint
Tool Rotate Object around Point by Angle.gif Rotate Object around Point by Angle
Tool Move.gif Move
Tool Insert Text.gif Insert Text

Construction Steps

1. Create a triangle ABC.
Note Hint: Use counterclockwise orientation.

2. Create the angles α, β, and γ of triangle ABC.

3. Set the number of decimal places to 0 (Options Menu).

4. Create sliders δ and ε with the settings angle (type); 0° to 180° (interval); 10° (increment).

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 to show 180° before you create the angles ζ (A’C’B’) and η (C'1B'1A'1).

9. Enhance your construction using the Properties Dialog.
Note Hint: Congruent angles should have the same color.

13 anglesum.PNG

Challenge 1

Insert dynamic text showing that the interior angles add up to 180°.
Note Hint: reate dynamic text for the interior angles (e.g. α = and select α from Objects), calculate the angle sum using sum = α + β + γ and insert the sum as a dynamic text.

Match colors of corresponding angles and text. Fix the text in the Graphics View.

Challenge 2

Export the figure to a dynamic worksheet. Come up with instructions that guide your students towards discovering the angle sum in a triangle. Have them check their conjecture using the provided worksheet.

Visualizing Integer Addition on the Number Line

In this activity you can either use the following tools or corresponding commands. Make sure you know how to use them before you begin.

Tool Slider.gif Slider
Tool New Point.gif New Point
Tool Vector between Two Points.gif Vector
Tool Move.gif Move
Tool Segment between Two Points.gif Segment Between Two Points
Tool Insert Text.gif Insert Text
Tool Check Box to Show Hide Objects.gif Checkbox to Show/Hide Objects

Construction Steps

1. Open a new GeoGebra window and hide the Algebra View. Set the labeling option to All new objects (Options Menu).

2. Open the Properties dialog for the Graphics View. On tab yAxis, uncheck Show yAxis. On tab xAxis, set the distance of tick marks to 1 by checking the box Distance and entering 1 into the text field. On tab Basic set the minimum of the x-Axis to -21 and the maximum to 21.

3. Create sliders a and b (interval -10 to 10; increment 1). Show the value of the sliders instead of their names (Properties dialog).

4. Create points A = (0 , 1) and B = A + (a , 0).

5. Create vector u = Vector[A, B] which has the length a.

6. Create points C = B + (0 , 1) and D = C + (b , 0) as well as vector v = Vector[C , D] which has the length b.

7. Create point R = (x(D) , 0).
Note Hint: x(D) gives you the x-coordinate of point D. Thus, point R shows the result of the addition.

8. Create point Z = (0, 0) as well as the following segments: g = Segment[Z, A], h = Segment[B, C], i = Segment[D, R].

9. Use the Properties Dialog to enhance your construction (e.g. change color, line style, fix sliders, hide labels).

13 integer.PNG

Challenge 1

Enhance your interactive figure by inserting dynamic text that displays the corresponding addition problem.

10. Calculate the result of the addition problem: r = a + b

11. In order to display the parts of the addition problem in different colors you need to insert the dynamic text step by step. a. Insert text1: Select a from Objects b. Insert text2: + c. Insert text3: Select b from Objects d. Insert text4: = e. Insert text5: Select r from Objects

12. Match the color of text1, text3, and text5 with the color of the corresponding sliders and point R. Hide the labels of the sliders and fix the text (Properties Dialog).

13. Export the interactive figure as a dynamic worksheet.

13 integer2.PNG

Challenge 2

Insert a checkbox into the Graphics View that allows you to show or hide the result of the addition problem (text5, point R, and segment i).

14. Activate tool Checkbox to show and hide objects.

15. Click on the graphics view next to the result of the addition problem.

16. Enter Show result into the Caption text field.

17. From the drop down menu successively select all objects whose visibility should be controlled by the checkbox (text5, point R, and segment i).

18. Click Apply to create the checkbox.

19. In Move mode check and uncheck the checkbox to try out if all three objects can be hidden / shown.

20. Fix the checkbox so it can’t be moved accidentally any more (Properties dialog).

21. Export this new interactive figure as a dynamic worksheet.
Note Hint: You might want to use a different name for this worksheet.

13 integer3.PNG

Creating a "Tangram" Puzzle

In this activity you will create the "Tangram" puzzle. It consists of 7 geometric shapes which can all be constructed using the side length a (see Tangram_puzzle.html).

13 tangram.PNG

For these constructions you will need a selection of geometry tools. Please read through the following hints before you actually start to create the geometric shapes.

1. Enter the number a = 6. It will provide a basis for the construction of all triangles and quadrilaterals necessary for a "Tangram" puzzle.

2. Try to figure out the side lengths of the geometric shapes.
Note Hint: In some cases you might want to look at the diagonals or height. Their lengths can be expressed more easily using the variable a than the lengths of the corresponding sides.

3. Begin each of the geometric figures using a segment with given length. This will allow you to drag and rotate the figure later on.

4. Construction hints:

a. If the height of a right triangle is half the length of the hypotenuse you might want to use the theorem of Thales for the construction (see practice block 1).

b. If you know the legs of a right triangle you might want to construct it similar to a square construction.

c. For constructing a square using its diagonals, it is helpful to know that they are perpendicular and bisect each other.

d. For constructing the parallelogram it is helpful to know the size of the acute angle.

5. Check your construction by trying out if you can manage to create a square with side length a using all figures.

Challenge 1

Arrange the geometric shapes arbitrarily around the edges of the interactive applet. Export the figure to a dynamic worksheet and add an explanation for your students.

Challenge 2

With these geometric shapes other figures than a square can be created as well. Search the Internet for a "Tangram" figure other than a square and import this figure into the Graphics View. Export the GeoGebra construction again using a different name and different instructions.

13 cat.PNG
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