Workshop 5: Transformations in the Plane with Geogebra
The workshop 5 for the subject Mathematical Bases in Primary Education for the First Course in the degree for Primary Teacher at the University of Granada.
Table of Contents
- 4.1. Getting familiar with GeoGebra
- 4.1. Getting familiar with Geogebra
- 4.2. Dynamic Geometry with GeoGebra
- 4.2. Dynamic Geometry with GeoGebra
- 4.3. Translations with GeoGebra
- 4.3. Translations with GeoGebra
- 4.4. Symmetries with GeoGebra
- 4.4. Symmetries with GeoGebra
- 4.5. Rotations with GeoGebra
- 4.5. Rotations with GeoGebra
- 5.1. Assessment Activities: Triangle Inequality
- Assessment activities: 5.1. Triangle Inequality
- 5.2. Assessment Activities: The Orthocentre
- Assessment activities: 5.2. The Orthocentre
- 5.3. Assessment Activities: Mosaics through Transformations
- Assessment activities: 5.3. Mosaics through transformations
4.1. Getting familiar with Geogebra
Find where you can add or remove the axes and the grid, and adjust the view to what is most comfortable for you. Then, carry out the following activities by searching for and using the tools you consider appropriate.
a) Draw any straight line and two others related to it: one parallel and one perpendicular.[br]
b) A non-convex polygon.[br]
c) A regular enneagon.[br]
d) An isosceles triangle.[br]
e) Two [i]externally tangent[/i] circles. Two[i] internally tangent [/i]circles.
4.2. Dynamic Geometry with GeoGebra
We say that a [b]construction is well done[/b] with GeoGebra when the properties that characterize the construction are maintained, [b]even when we drag[/b] (pull on the vertices) the figures with the pointer.[br][br]Practice creating well-made constructions with simple elements:
a) [i]Line parallel to a segment[/i]. Draw a segment and a line parallel to it that passes through a point not on the segment. If you draw the parallel line by eye, when you modify the segment, the line will no longer be parallel to it.[br][br]To create the parallel line correctly, use the “[i]Segment between two points[/i]” tool and the “Parallel line” tool. Once you do this, drag one of the ends of the segment with the pointer. What happens to the line?
b) [i]Right triangle[/i]. Draw a right triangle by eye using the “Segment between two points” tool three times, or using the “Polygon” tool. Then, drag one of its vertices with the pointer and observe that the triangle is no longer a right triangle.[br][br]Repeat the construction in such a way that the triangle you have built is always a right triangle. To do this, you must use the “Perpendicular line” tool, then use the “Polygon” tool and click on the ends of the segment and on a point on the perpendicular line. Now drag one of the ends of the segment. What happens now with the triangle? Does it stop being a right triangle?[br][br]Hide the perpendicular line so that only the triangle remains (use the button “Show/Hide Object”).[br]
c) Find the appropriate tools to construct a [i]regular polygon[/i] and the [i]perpendicular bisector of a segment[/i] so that they remain correctly constructed.
4.3. Translations with GeoGebra
Construct a square ABCD and take v=(3,−4). Use the “Vector” tool to tell the program that you want to define that vector.[br]Note: [color=#00000a]It is recommended to show the grid to perform this activity.[/color]
a) Perform the translation of the square ABCD according to the vector v using the "Translation" tool in GeoGebra and analyze if the result makes sense to you. Does it fulfill the definition of “translation” discussed in class?[br]b) Explore the dynamic nature of the program by modifying the initial square and observing what happens. You can also modify the vector and see what effect these changes have on the results.
4.4. Symmetries with GeoGebra
Choose any line [b]r[/b] in the plane; this exercise works on symmetry with respect to the line [b]r[/b].
a) Draw a non-convex heptagon that stays within one of the half-planes defined by the line and use the “axial symmetry” tool to find the symmetric polygon of the drawn heptagon with respect to [b]r[/b], then analyze if the result makes sense to you.[br][br]b) We are going to verify that the definition of “symmetry” discussed in class holds. To do this, choose any vertex of the heptagon (let's call it C) and use the “Perpendicular” and “Compass” tools to find C' following the steps discussed in class:[br][list=1][*]Draw the line s, which is perpendicular to [b]r[/b] and passes through C (using the “Perpendicular” tool).[/*][*]Select the point where [b]r[/b] and [b]s[/b] intersect, let's call it D.[/*][*]Use the “Compass” tool to find another point on [b]s[/b] that is equidistant from [b]r[/b], just like C. This point is C'.[/*][/list][br]c) Move the line [b]r[/b] in different directions and directions, and observe what happens to the heptagon. Similarly, move the original heptagon and see what happens. Perform the same exercise by moving only one vertex of the heptagon.[br][br]Recommendation: Throughout the work process, you will obtain a large number of elements that are useful but may become distracting. It might be a good idea to hide the elements that will not be interacted with again, as this will allow you to visualize the results of your work more easily.[br][br]
4.5. Rotations with GeoGebra
Choose any point A in the plane (it is recommended to center it on the screen). This exercise will work on rotating by an angle of 100° (clockwise) around point A.
a) Draw a trapezoid far from the point and use the “Rotation” tool in GeoGebra to rotate the trapezoid by 100° (clockwise) around point A. Analyze if the result makes sense to you.[br][br]b) We are going to verify that the definition seen in class holds. To do this, take any vertex of the trapezoid (let's call it B). Use the “Line”, “Angle”, and “Compass” tools to manually find the point B' that results from rotating B around A. Follow the steps used in class:[list=1][*]Draw the line [b]r [/b]that connects A and B (using the “Line” tool).[/*][*]Draw a line [b]s[/b] that forms a 100° angle (clockwise) with the line that connects A and B (using the “Angle” and “Line” tools)[/*][*]Use the “Compass” tool to find B' on line s, such that |AB'| = |AB|.[/*][/list][br]c) Move point A in different directions and directions and observe what happens to the trapezoid. What happens when A coincides with the vertices? And what happens if it is inside the trapezoid?
Assessment activities: 5.1. Triangle Inequality
Construct, using the appropriate GeoGebra tools, a triangle with sides of 3, 4, and 5 [i]u[/i], where [i]u[/i] is the unit of length defined by the grid. [br][br]Then, do the same for the triangles with the data provided below.[br][br]Have you noticed anything unusual? Why do you think this happens? Formulate your own conjecture about what you observe.
a) 3, 4 and 8 [i]u[/i][br]
b) 3, 4 and 6 [i]u[/i][br][br]
c) 3, 4 and 7 [i]u[/i].[br][br]
d) 9, 5 and 3 [i]u[/i]
e) 7, 5 and 3[i] u.[/i]
f) 8, 5 and 3[i] u.[/i]
g) 6, 9 and 3 [i]u.[/i]
h) 6, 10 and 3[i] u.[/i]
i) 6, 8 and 3[i] u.[/i]
Assessment activities: 5.2. The Orthocentre
An [i]altitude[/i] of a triangle is a line that passes through a vertex of the triangle and is perpendicular to the opposite side. Therefore, [b]every triangle has three altitudes[/b], which all meet at a single point. This point is a candidate for the center of the triangle and is called the [b]orthocentre[/b].[br][br]a) Draw an arbitrary triangle in GeoGebra and use the appropriate tools to find its orthocentre in a reasoned manner.[br][br]b) With the construction already made, move the initial triangle and observe what happens to its orthocentre. Based on what you observe, propose a criterion (a rule) to determine, without drawing it, whether the orthocentre of a triangle will lie inside, outside, or on one of its sides (if possible).
Assessment activities: 5.3. Mosaics through transformations
A [b]mosaic[/b] is a composition of flat figures that exhibit certain regularity, such as invariance under symmetries or rotations. Follow these steps to construct mosaics using GeoGebra.[br][br][list=1][*]Choose any point A and draw two segments of equal length that meet at A, forming a 60° angle.[/*][*]Construct a non-convex trapezoid using the three points you already have. We will call it[b] T.[/b][br][/*][*]Perform two rotations of [b]T[/b] with respect to point A (it doesn't matter the direction, but both should go in the same direction): one of 60° (which we will call [b]T'[/b]) and another of 120° (which we will call [b]T*[/b]). We will call [b]S[/b] the union of [b]T[/b], [b]T'[/b], and [b]T*[/b] (a new polygon).[br][/*][*]Use the longest side of [b]S[/b] to perform the symmetry of the polygon with respect to that side, called [b]S'[/b]. You will obtain a new polygon [b]R[/b], which is the union of [b]S[/b] and [b]S'[/b].[br][/*][*]Check if [b]R[/b] is invariant under a 60° rotation and find its axes of symmetry (if any).[br][/*][*]Select a diagonal of maximum length (in the direction that is most appropriate for your screen) as a vector and perform [b]successive translations[/b] of the polygon according to that vector (successive means that if the result of translating [b]R[/b] is [b]R'[/b], you need to perform the translation again on [b]R'[/b] to obtain [b]R''[/b], then translate [b]R''[/b] to obtain [b]R'''[/b], and continue in this way until the translations go off the screen).[br][/*][*]Finally, select another diagonal that is different and appropriate, and repeat the procedure. This time, you must translate all the polygons you obtained in step 6).[br][/*][/list][br]To display the resulting mosaic, it is enough to make the names of the elements that are emerging invisible. I recommend doing this from the beginning to avoid complicating the view.[br]