THE COLLECTION OF INTERACTIVE SOLID FIGURES
AND SPATIAL SITUATIONS IN THE CABRI - GEOMETRY
The planar nature of Cabri-geometry makes it possible to generate stereometrical figures of
descriptive geometry, but the creation of interactive figures of solids is more difficult.
Drawing, for example, an image of a polyhedron in an oblique projection, and wanting it to
rotate about an axis, one has to ensure that the relevant changes are made to both the form of
the faces and the visibility of the edges. In practice it is more convenient for a teacher to open
a file with a pre-prepared basic solid figure, rather than to create the solid figure himself
(herself). These figures can then simply be changed, completed and manipulated as required.
There are such files containing a range of interactive solid figures and spatial situations. The
files were developed by Karel Kabelka, a student of the Pedagogical Faculty University of
South Bohemia, in his diploma thesis "Applying of Cabri-geometry in high school
stereometry". The entire collection and the manuscript of Kabelka’s thesis are accessible
(only in Czech) from the Internet address
http://www.pf.jcu.cz/cabri/temata/kabelka
Download of the brief English version collection:
http://www.pf.jcu.cz/cabri/temata/leischner/collection.zip
The main purpose of this collection is to help teachers with teaching of stereometry in
secondary and elementary schools. The authors would like to know your experiences with an
exercise of this aid, especially in school lessons. Send it please to the authors at
leischne@pf.jcu.cz
The English version of collection contains three types of our files for use
with Cabri-geometry:
1. SOLID FIGURES
The first group of files consists of solid figures, to which one can apply additional
constructions. For example, it is possible to change their form and measurements and to rotate
them about the vertical axis by means of control-diagrams, which are located on the left hand
side of each figure. It is also possible to change the shape of the base of some of solid figures,
for example pyramids and prisms. All controls work in the same way: you grasp the point
which is marked on the line of the control by pressing the mouse button and then drag the
point.
The solid figures are drawn in two different forms of visibility. When first presented to the
youngest pupils it is better to work with those of the first type (full solids). Later, when we
want pupils to imagine even the back part of the solid, we use the other form, in which a
dashed line marks the invisible edges. Interactive objects can also be used to construct
sections of solids, making a decision about the mutual position of lines, which can be added
into the interactive polyhedron, etc.
When working on a task, one can complete the task with the figure as presented. If the
orientation of the figure is not suitable for the task, we can turn the solid into an optimal
position.
Taking measurements of diagonals and angles is clearer if we use colour to highlight the
triangles with which we are working. By rotating the object, pupils can better image the
situation in space.
When making a decision about the mutual position of lines in a figure, pupils can add the
required lines into interactive polyhedron (see examples 1.6.1.1 – 1.6.1.7). By rotating the
object, pupil can better image the situation in space. We can use the same method to convince
a pupil that he or she is wrong to consider the illusive point of intersection of skew lines to be
a real point.
Interactive objects can also be used to construct of section of solids (examples 1.6.2.1 –
1.6.2.4, 1.6.3).
The generation of a cone or cylinder by rotating a triangle or rectangle (see 1.5) can be
demonstrated to the class as a whole.
2. PROJECTIONS
The second group of teaching files are used to illustrate vertical, oblique and central
projections. We can study the features of a particular projection by applying the projection to
a cube. Parameters of each projection are variable. In this way we can compare oblique
projections, which differ by the angle between the x, y axes and by the ratio of units on those
axes. Later we can experimentally verify whether a given projection preserves parallelism,
perpedincularity or length.
We recommend showing the illustration of central projections because of their similarity to
our visual perception of the world. Pupils who use only parallel projection in maths lessons
would certainly be interested in it because of an attractive link between mathematics and art.
(The teacher should emphasize these facts as well with the presentation of single adjustable
parameters of projection.)
3. NETS OF POLYHEDRA
Our third group of files can be used to show the development of the boundaries of polyhedra
into planar nets. The creation of nets of polyhedra is a useful task for helping students
develops spatial imagery. Each of our pictures presents the boundary of a polyhedron.
Colourful edges are cut lengthwise. The pupils have to imagine the movement (revolving
round edges which are not cut), whose result is a net of the given polyhedron.
Faces of a polyhedron can be opened by using the control-diagram or by selecting one of the
vertices of the face and dragging it. We can even change the sequence in which we open the
faces. Later pupils can imagine this happening and then check their hypothesis by using
Cabri. Ultimately, they are asked to solve such problems on their own without using a
computer.
Interactive computer solid pictures join the possibility to manipulate with solid models and
the abstraction of a geometric projection. These Cabri files can help students, through
experimentation, develop the visualisation skills necessary to carry out complex geometric
manipulations and abstract projections mentally.