GeoGebra Construction Steps :-ICT in Mathematics Education

ICT in Mathematics Education

                Part I: Geo-Gebra

                 Construction on GeoGebra

             

Q. No.1. Square Construction

Construction Steps:

1.      Draw the segment AB between points A and B.

2.      Construct a perpendicular line g to segment AB thorough point B.

3.      Construct a circle c with center B through point A.

4.      Intersect the perpendicular line g with the circle c to get the intersection point C.

5.      Construct a perpendicular line h to segment AB through point A.

6.      Construct a circle d with center A through point B.

7.      Intersect the perpendicular line h with the circle d to get the intersection point D.

8.      Create parallel line i to segment AB through point C and create the polygon ABCD.

9.      Hide circles and perpendicular lines.

10.  Perform the drag test to check if your construction is correct and Save the construction.

Q. No.2. Equilateral Triangle Construction

Construction Steps:

1.      Create segment AB.

2.      Construct a circle with center A through B.

3.      Construct a circle with center B through A.

4.      Intersect both circles to get point C.

5.      Create the polygon ABC in counter-clockwise direction and Hide the two circles.

6.      Show the interior angles of the triangle.

7.      Save the construction.

Q. No.3. Isosceles Triangle Construction

Construction Steps:

1.      Create a line segment between two points AB.

2.      Create a midpoint or center on AB, Say C.

3.      Create a perpendicular line g to segment AB through point C.

4.      Create a New point on a perpendicular line g, say D.

5.      Create the polygon ABD.

6.      Show the all sides and interior angle of a triangle.

7.      Save the construction.

Q. No.4. Regular Hexagon Construction

Construction Steps:

1.      Draw a circle c with center A through point B.

2.      Construct a new circle d with center B through point A.

3.      Intersect the circles c and d to get the hexagon’s vertices C and D.

4.      Construct a new circle e with center C through point A.

5.      Intersect the new circle e with circle c in order to get vertex E.

6.      Construct a new circles f with center D through point A.

7.      Intersect the new circle f with circle c in order to get vertex F.

8.      Construct a new circle g with center E through point A.

9.      Intersect the new circle g with circle c in order to get vertex G

10.  Create polygon BDFGEC and Hide circles.

11.  Display the interior angles of the hexagon.

12.  Save the construction.

Q. No.5. Circumcircle of a Triangle Construction

Construction Steps:

1.      Active a polygon tool and create an arbitrary triangle ABC.

2.      Create perpendicular bisector d, e and f for all sides of the triangle.

3.      Construct intersection point D of the two of the line bisectors.

4.      Create a circle with center D through one of the vertices of triangle ABC.

5.      Rename point D to Circumcircle.

6.      Save the construction.

Q. No.6. Centroid of a Triangle Construction

Construction Steps:

1.      Active a polygon tool and create an arbitrary triangle ABC.

2.      Create midpoint D, E and F of the triangle sides.

3.      Connect each midpoint with the opposite vertex using segment d, e and f.

4.      Create intersection point G of two of two of the segments.

5.      Rename point G to Centroid.

6.      Save the construction.

Q. No.7. Centroid of a Triangle angle sum

Construction Steps:

1.      Active a polygon tool and create an arbitrary triangle ABC.

2.      Create midpoint D, E and F of the triangle sides.

3.      Connect each midpoint with the opposite vertex using segment d, e and f.

4.      Create intersection point G of two of two of the segments.

5.   Create interior angles α, β and  of triangles ABC.

6.      Rename point G to Centroid.

7.      Save the construction.

Q. No.8. Orthocenter of a Triangle Construction

Construction Steps:

1.      Create an arbitrary triangle ABC.

2.      Create perpendicular line f, g and h to each side through the opposite vertex of the triangle.

3.      Construct intersection point D of two of the perpendicular lines.

4.      Rename point D to orthocenter.

5.      Save the construction.

Q. No.9. Visualize the Theorem of Thales

Construction Steps:

1.      Draw a segment AB.

2.      Construct a semicircle through points A and B.

3.      Create a new point C on the semicircles.

4.      Create the triangle ABC in counterclockwise direction for using polygon tool.

5.      Create the interior angles of triangle ABC.

6.      Drag point C to check for construction is correct and Save the construction.

Q. No.10. Drawings, Constructions, and Drag Test

Construction Steps:

1.      Create a Point A, B, C, D on graphics view.

2.      Create a segment AB, BC, CD and DA.

3.      Active a polygon tool and create the square ABCD.

4.      Hide the all labels.

5.      Similarly, Create a Point E, F, G, and H on graphics view.

6.      Create a polygon EFGH.

7.      Hide the all labels.

8.      Create a Point I, J, K and L on graphics view.

9.      Create a polygon IJKL.

10.  Repeat step 4.

11.  Create a Point M, N, O and P on graphics view.

12.  Create a polygon MNOP.

13.  Repeat step 4.

14.  Create a Point Q, R, S and T on graphics view.

15.  Create a polygon QRST.

16.  Repeat step 4.

17.  Create a Point U, V, W and X on graphics view.

18.  Create a polygon UVWX.

19.  Hide the all labels.

20.  Save the construction.

Q. No.11.Using Sliders to Modify Parameters

Construction Steps:

1.      Active a slider tool and click on Graphics view than Slider box appear and fill in the box. (e.g.:-click Number write name a, Min = 1, Max =20 and increment=1) and Apply. i.e., create a Sliders a.

2.      Display the variable a as a slider in the Graphics view.

3. Enter the quadratic polynomial f(x) = a * in input box.

               Note: - * means space between a and .

4. Save the construction.

Q. No. 12. Using Sliders to Modify Parameters

Construction Steps:

1.      Create a Slider m and c respectively.

2.   Enter the polynomial in input box.

3.      Create a new point A on line y.

4.      Active the slope tool and create the slope of a line y.

5.      Save the construction.

Q. No.13. Introducing Derivatives – The Slope Function

Construction Steps:

1.   Enter the polynomial in input box.

2.      Create a new point A on function f.

Note: - 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) in input box.

Note: x (A) gives you the x-coordinate of point A.

6.      Connect points A and S using a segment.

7.       Save the construction.

Q.No.14. Visualizing Multiplication of Natural Numbers

Construction Steps:

1.      Crate a horizontal slider Columns for number with Interval from 1 to 10, Increment 1 and Width 300.

2.      Create a new point A.

3.      Construct segment a with given length Columns from point A.

4.      Move slider Columns to check the segment with given length.

5.      Construct a perpendicular line b to segment a through point A.

6.      Construct a perpendicular line c to segment a through point B.

7.      Create a vertical slider Rows for number with Interval from 1 to 10, Increment 1 and Width 300.

8.      Create a circle d with center A and given radius Rows.

9.      Move slider Rows to check the circle with given radius.

10.  Intersect circle d with line c to get intersection point C.

11.  Create a parallel line e to segment a through intersection point C.

12.  Intersect lines c and e to get intersection point D.

13.  Construct a polygon ABDC.

14.  Hide all lines, circle d and segment a.

15.  Hide labels of segments.

16.  Set both sliders Columns and Rows to value 10.

17.  Create a list of vertical segments.

Write in input box: - Sequence[Segment[A+ i*(1, 0), D+ 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.

D + i*(1, 0) specifies a series of points starting at point C with distance 1 from each other.

Segment[A + i*(1, 0), D + 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.

Write input box :-  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:

22.  Result = Columns * Rows

23.  22. Insert dynamic text5: Select result from Objects.

24.  23. Hide points A, B, C and D.

25.  24. Enhance your construction using the Properties dialog.

26.  25. Save the construction.

Q.No.15.Visualizing Integer Addition on the Number Line

Construction Steps:

1.      Open the Properties dialog for the Graphics view.

2.      On tab x-Axis 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 y-Axis uncheck Show y-Axis.

5.  Close the Properties dialog for the Graphics view.

6.  Create a slider for number a with Interval -10 to 10 and Increment

7.  Create a slider for number b with Interval -10 to 10 and Increment

8.  Show the value of the sliders instead of their names.

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.

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

12. Create point C = B + (0 , 1).

13. Create point D = C + (b , 0) .

14. Create vector v = Vector[C , D] which has the length b.

15. Create point R = (x(D) , 0).

Note: x(D) gives you the x-coordinate of point D. Thus, point shows the result of the addition on the number line.

16. Create point Z = (0, 0).

17. Create segment g = Segment [Z, A].

18. Create segment h = Segment [B, C].

19. Create segment i = Segment [D, R].

20. Use the Properties dialog to enhance your construction (e.g. match the color of sliders and vectors, line style, fix sliders, and hide labels).

Insert dynamic text

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

2.      Insert dynamic text1: a

3.      Insert static text2: +

4.      Insert dynamic text3: b

5.      Insert static text4: =

6.      Insert dynamic text5: r

7.       Match the color of text1, text3 and text5 with the color of the Corresponding sliders, vectors and point R.

8.      Line up the text on the Graphics view.

9.      Hide the labels of the sliders and fix the text (Properties dialog).

10.  Export your interactive figure as a dynamic worksheet.

11.  Save the construction.

Q.No.16 Circumcenter of a Triangle (Euler’s Discovery)

Construction Steps:

1.      Create an arbitrary triangle ABC.

2.      Create perpendicular bisectors d, e and f for all sides of the triangle.

      Note: The tool Perpendicular Bisector can be applied to an existing segment.

3.  Construct intersection point D of the two of the line bisectors.

4.  Create a circle with center D through one of the vertices of triangle ABC.

5.  Rename point D to Circumcenter.

6.  Use the drag test to check if construction is correct.

7.  Create a custom tool for the circumcenter of a triangle.

    Output objects: point Circumcenter

    Input objects: points A, B and C

    Name: Circumcenter

    Toolbar help: Click on three points

8. Save your custom tool as file circumcenter.ggt.

Q.No.17. Orthocenter of a Triangle (Euler’s Discovery)

Construction Steps:

1.      Create an arbitrary triangle ABC.

2.      Create perpendicular lines d, e and f to each side through the opposite vertex of the triangle.

3.      Construct intersection point D of two of the perpendicular lines.

4.      Rename point D to Orthocenter.

5.      Use the drag test to check if your construction is correct.

6.      Create a custom tool for the orthocenter of a triangle.

Output objects: point Orthocenter

Input objects: points A, B and C

Name: Orthocenter

Toolbar help: Click on three points

7.      Save your custom tool as file orthocenter.ggt.

Q.No.18. Centroid of a Triangle (Euler’s Discovery)

Construction Steps:

1.      Create an arbitrary triangle ABC.

2.      Create midpoints D, E and F of the triangle sides.

3.      Connect each midpoint with the opposite vertex using segments d, e and f.

4.      Create intersection point G of two of two of the segments.

5.      Rename point G to Centroid.

6.      Use the drag test to check if your construction is correct.

7.      Create a custom tool for the centroid of a triangle.

Output objects: point Centroid

Input objects: points A, B and C

Name: Centroid

Toolbar help: Click on three points

8.      Save your custom tool as file centroid.ggt.

Q.No.19. the Theorem of Pythagoras

Construction Steps:

1.      Create segment a with endpoints AB.

2.      Create semicircle c through points A and B.

3.      Create a new point C on the semicircle.

Note: Check if point C really lies on the arc by dragging it with the mouse.

4.      Hide the segment and the semicircle.

5.      Construct a triangle ABC in counterclockwise direction.

6.      Rename the triangle sides to a, b and c.

7.      Create interior angles of triangle ABC.

Note: Click in the middle of the polygon to create all angles.

8.      Drag point C to check if your construction is correct.

9.      Create a perpendicular line d to segment BC through point C.

10.  Create a perpendicular line e to segment BC through point B.

11.  Create a Circle f with center C through point B.

12.  Intersect the circle f and the perpendicular line d to get intersection point D.

13.  Create a parallel line g to segment BC through point D.

14.  Create intersection point E of lines e and g.

15.  Create the square CBED.

16.  Hide the auxiliary lines and circle.

17.  Repeat steps 8 to 15 for side AC of the triangle.

18.  Repeat steps 8 to 15 for side AB of the triangle.

19.  Drag the vertices of the right triangle to check if your squares are correct.

20.  Enhance your construction using the Properties dialog.

21.  Copy Visual Style, New!

Note: Click on an object to copy its visual style. Then, click on other objects to match their visual style with the first object.

22.  Create the midpoints of all three squares.

Note: Click on diagonal opposite vertices of each square.

23.  Insert static text1: a^2 and attach it to the midpoint of the corresponding square.

Note: Don’t forget to check the box LaTeX formula to get a^2.

24.  Insert static text2: b^2 and attach it to the midpoint of the corresponding square.

25.  Insert static text3: c^2 and attach it to the midpoint of the corresponding square.

26.  Hide the midpoints of the squares.

27.  Format the text to match the color of the corresponding squares.

28.  Insert text that describes the Pythagorean Theorem.

29.  Save the construction.

Q. No.20. Visualizing the Angle Sum in a Triangle

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 Properties dialog.

Note: 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

14.   

15. Insert the angle sum as a dynamic text: and select sum from Objects.

16.  Match colors of corresponding angles and text and Save the construction.

Q. No.21. Constructing a Slope Triangle

Construction Steps:

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

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

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

4.       Intersect perpendicular lines b and c to get intersection point C.

Note: You might want to hide the perpendicular lines.

5.       Create a triangle ACB.

6.       Hide the labels of the triangle sides.

7.    Calculate the rise:

Note: y(A) gives you the y-coordinate of point A.

8.   Calculate the run:

Note: 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.

14.  Save The Construction.

Q.No.22. Lower and Upper Sum

Construction Steps:

1.   Enter the cubic polynomial

2.      Create two points A and B on the x-axis.

Note: These points will determine the interval, which restricts the area between the function and the x-axis.

3.      Create slider for the number n with Interval 1 to 50 and Increment 1.

4.      Enter uppersum = UpperSum[f, x(A), x(B), n].

Note: x(A) gives you the x-coordinate of point A. Number n determines the number of rectangles used in order to calculate the lower and upper sum.

5.      Enter lowersum = LowerSum[f, x(A), x(B), n].

6.      Insert dynamic text Upper Sum = and select uppersum from Objects.

7.      Insert dynamic text Lower Sum = and select lowersum from Objects.

8.      Calculate the difference diff = uppersum – lowersum .

9.      Insert dynamic text Difference = and select diff from Objects.

10.  Enter integral = Integral[f, x(A), x(B)].

11.  Insert dynamic text Integral = and select integral from Objects.

12.  Fix slider and text using the Properties dialog.

13.  Save the construction.

Q.No.23. Rotating of polygons

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 new point to O.

Note: GeoGebra offers a ‘fast-renaming’ option. Activate Move mode and select the object. When you start typing the new name, GeoGebra opens the Rename dialog window.

4.      Create a slider for angle α.

Note: 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.

Note: Check counter clockwise rotation.

6.      Create segments AO and A’O.

7.      Create angle AOA’.

Note: Select the points in counter clockwise order. Hide the label of this angle.

8.      Move slider and check the image of the triangle.

9.      Save the Construction.