GeoGebra and its applications - especially for teaching of geometrical optics

The aim of this paper is to create two interactive electronic materials for teaching geometrical optics, especially imaging on thin lenses. Geogebra mathematical dynamic geometry software is employed for preparing the interactive constructions. In this software it is possible to work both in online mode and after installation in offline mode. There are constructs created for the software for mathematics and physics that can be used by the teacher. The paper does not describe a detailed procedure for preparing both constructions, and as an appendix we attach the original procedure for the construction as exported by the software. The reader is also introduced to the basics of geometrical optics, especially imaging on thin lenses. The values of parameters a, á and f that are measured directly by the created construction will be compared in the results with the results of these parameters, which are obtained from the derived rendering equation. In this way, we will confirm the correct function of the interactive constructions that have been created. The conclusion of the paper summarises the advantages and disadvantages of the created teaching materials.


Introduction
This paper is primarily intended for secondary school teachers.Looking at the Framework Educational Programme (in the Czech Republic), it can be seen that the field of geometrical optics is represented in secondary schools by the following topics: reflection on plane and spherical mirrors; refraction on thin lenses; angle of view; the eye as an optical system; magnifying glass.The guide (in the Czech Republic) [1] states that the optimal hourly allocation for the area of geometric imaging (including the complete area of geometric optics) for all variants is 4+2 lessons.We certainly do not consider these recommendations to be binding, and they are often reduced, especially when students do not study physics in their 4th year of grammar school.However, the frequently low hour allocation of geometrical optics is not the only problem in teaching optics at secondary school.An equally important problem is the very poor mathematical, especially geometric, knowledge and skills of pupils.Paradoxically, this poor knowledge and skills place higher demands on the quality of the study materials prepared.The aim of this paper is to create a dynamic interactive construction of a thin converging and diverging lens in GeoGebra and to show the possibilities of using this software in teaching geometrical optics at secondary school (for review see [2]).Similar issues have been addressed in studies [3][4][5].However, our solution offers teachers a ready-to-use online construction that can be freely modified and further used in teaching.Thus, a basic overview of dynamic geometry software is needed at the outset.This type of software is very popular among teachers nowadays, especially in conjunction with interactive whiteboards.For inspiration, this is the most widely used paid software, and two variants of the GNU GPL-licensed software.So, in the first place we present Cabri Geometry, specifically Cabri II Plus with the official website https://cabri.com/,where all information is available including licences, guides, purchase options, etc.The exclusive supplier for the Cabri II plus Czech version, AKERMANN ELECTRONIC PRAHA, spol.s r. o., is also listed here.(https://akermann.cz/).The second software in our list is the GeonexT software.In this case it is a software that is distributed under the GNU General Public Licence.The official website of the software manufacturer is http://geonext.uni-bayreuth.de/.The Czech version of the software can also be downloaded at these websites.The undeniable advantage is, of course, that the software is completely free for schools.As a disadvantage we can mention the fact that the software requires Java to be installed.The second software licensed under the GNU GPL that we will present here is GeoGebra.We will look at the use of this software in more detail in the article.For completeness, however, we offer the official website of the software manufacturer https://www.geogebra.org/.On this site you can get all the information, tutorials, installation packages, but also work online (https://www.geogebra.org/classic?lang=cs).To GeoGebra is a computer software for interactive dynamic geometry, algebra and analysis.It is primarily intended for teachers and students of all types of schools.The software is licensed under the GNU General Public Licence, and most versions of GeoGebra are available to users for free.On the developers' page https://www.geogebra.org/you can choose the Czech environment to start with.If we want to install the application, it is possible to do so in the "Applications to download" menu, where we select the "Geometry" application for our case.The downloaded .exefile directly starts the software.Considering the limited possibilities when working with .exefiles on school PCs, we decided to continue working in the online version of the software.This can be done by ticking the blue box "RUN THE CALCULATOR".Then select "Geometry" from the drop-down menu in the top middle.The second way to perform the construction online is via the "Applications for download" drop-down menu, but this time select the "Geometry" software, specifically the "Start" item.This online tool also offers us a number of interesting configuration options.For example, it is possible to display both the main and secondary grid.It is also possible to switch the attachment to the grid on.Another very useful option is to save the work file or export it to an image format.It is also worth mentioning the option to save the design as a "html dynamic design", but this option is only available after creating a Geogebra account and logging in.If you create this registration for free, you will also get other options, such as sharing your designs.We will also expand the export options (e.g. the construction procedure), etc.At this point it should be noted that if we are working in different versions, or in different variants of the software, then its environment may be slightly different.

Theory
The result of the article will be a dynamic interactive geometric construction of an image on a thin converging lens and a thin diverging lens, including a check calculation based on the image equation.Therefore, the theoretical basis that will result in the thin lens image equation in secondary school physics should be presented here.For the geometric construction of the image on a thin lens the so-called 3 prominent rays are important (they start from point A, which is not on the lens axis).The first ray (1), going parallel to the optical axis; the second ray (2), going through the optical centre of the lens; the third ray (3), passing through the object focus.The situation is shown in figure 1 for a thin converging lens and a thin diverging lens.The derivation of the lens equation (as given in high school textbooks) can be based on Newtonian two image equations, but this is beyond the scope of the high school instruction (given in [7]).From the first Newtonian image equation, it is possible to derive the "secondary school" lens equation, given in the form [8][9][10]: This equation is important to verify our construction.However, it still needs to be supplemented by the so-called "high school" sign convention.We use here the convention as given in [8], i.e., distance a is positive in front of the lens (in object space) and negative behind the lens (in image space).For the distance a´ it is the other way around: if a´ > 0, the image is in image space and is real, if a´ < 0, the image is in object space and is unreal.
A similar sign convention is also used in the textbook [10] (table 16.3, page 502).As a side note, from the second Newtonian image equation it is possible to derive the so-called transverse magnification (the scale of optical imaging), which we denote by Z.This magnification is introduced as the ratio of the image size y´ to the object size y, so we can express it as [8,10]: The basic knowledge in this section should be the lens equation.

Method
We now turn to the actual construction of the thin double converging and double diverging lens.There are ready-made applications for teachers in the field of thin lens imaging, see [11-13] https://www.herramientasingenieria.com/onlinecalc/optics/thin-lenses.html,https://www.physicsclassroom.com/Physics-Interactives/Refraction-and-Lenses/Optics-Bench/Optics-Bench-Refraction-Interactive,https://phet.colorado.edu/sims/html/geometric-optics/latest/geometric-optics_all.html,etc.These online applications cannot be edited by teachers and are therefore not fully under their control.In our construction we name the endpoint of the displayed object P, and thus the endpoint of the created image P´.We will construct the point P as a free point, then we can choose any distance a (in the construction AO = OA), and any size of the object y.In the dynamic construction, a´ (in the construction A´O = OA´) will be calculated based on the similarity of the triangles (the software does not respect the sign convention, i.e., it does not calculate distances based on the image equation).To measure a, a´, f use the "distance measurement" option in the software.When creating the "image of thin converging lens" and ""image of thin diverging lens" constructs, we had the software automatically name all the construct elements.At the end, we hid the unnecessary elements of the constructions and renamed the basic points O, F, F´, A, A´, P, P´ manually.The software offers this option as well.The students are thus shown the above mentioned values of a, a´, and f in the interactive worksheet (which can be exported in .htmlformat), which can also be freely changed.We added a "Checkbox" element called "Hide a'" at the end of the design.This field will allow the teacher to hide the value of a' if necessary, i.e., at any arbitrarily chosen value and the student will have to calculate a' independently.
As an example, we insert here a part of the construction procedure for displaying on a thin converging lens, as generated by the software in .htmlformat (figure 2).

Name Definition Value
Point A1(-2, -3) A1 = (-2, -3) An example of the design procedure as generated by the software.Complete construction procedure for both situations is available in the appendix.

Results
Thus, the output of our paper is two interactive constructions, "image of thin converging lens" (figure 3) and "image of thin diverging lens" (figure 4).They will be presented as dynamic .htmlconstructions with the possibility of arbitrarily setting the value of the object distance a.Here in the paper, static images of both constructions with randomly chosen values of a and f are presented.The calculation confirms the validity of the image equation from the theory.

Image of thin converging lens:
At this point, we perform a control calculation for the given situation.So we start from the imaging equation given in the theory.In this case, a, a´ > 0 (in accordance with the sign convention mentioned above).The following therefore applies:

units
Calculated value a´ thus corresponds to the value measured by the software.

Image of thin diverging lens:
At this point, we perform a control calculation for the given situation.We start from the imaging equation given in the theory.In this case, a > 0, a´ < 0 (in accordance with the sign convention mentioned above).Also, f < 0, i.e., f = −4 units.Here it is necessary to note that the software works with distance as a positive number, so the calculation should give us the value a´ = −2.77units.The following therefore applies:

units
Absolute value of the calculated value a´ thus corresponds to the value measured by the software.

Conclusion
The results presented that both constructions can be implemented as dynamic interactive constructions.
The software calculates distances based on the similarity of triangles, and takes the distance of two points in the plane as a positive number.Therefore, when working with study material created in this way, it is necessary to explain the sign convention correctly to the pupils and work with it carefully, especially in the case of the thin converging lens, and specifically for all situations where the image on the thin converging lens appears in front of the lens, i.e., for all situations where the image is in subject space (a´ < 0).In these cases it is more convenient to prove the validity of the image equation by comparing both sides of the equation, with distance a' measured by the software marked with a "minus" sign.In the case of a thin diverging lens, students should also be reminded that f < 0 and the focal length in the image equation should be given a minus sign.We consider the main advantage of these interactive teaching materials the ability to continuously change the position of the P point (i.e. the object) both relative to the optical axis of the lens and to the lens itself.The pupils can also test for themselves what happens when point P approaches the focus F. The interactive design also clearly shows that the position of the image does not depend on the distance of point P from the optical axis.It is also possible to use the constructions in student examinations, where students can solve specific situations prepared by the teacher.For control it is then sufficient to reveal the already calculated values, which can be hidden with the created button.The second OSS software mentioned above is GeonexT.In this software it is possible to create analogous constructions of images of thin lenses [14].The use of GeonexT currently has a big disadvantage in the fact that the basis of the application is a Java applet.This fact used to be an undeniable advantage, because it was not necessary to install the application, it could work directly in the browser.Today, however, the situation around the use of the Java applet is completely different, so this fact must be seen as a disadvantage.Geogebra uses the HTML5 environment, which is clearly the main advantage of Geogebra over the otherwise comparable GeonexT.As the app was created just before the conference, it has not yet been tested on students and secondary school teachers.
We want teachers to be able to use and change our app.So in the appendix we present both complete construction procedures generated by Geogebra.Both constructions are public and available at the following links [15], [16]: https://www.geogebra.org/m/kfafhd43,https://www.geogebra.org/m/ncjpnzku.

Figure 1 .
Figure 1.Three prominent rays at a thin converging lens (a), thin diverging lens (b).Adapted according to [6].Point P indicates the object, point P´ indicates the image.The FO distance indicates the object focal length and the F´O distance indicates the image focal length.Let us further denote: a -object distance (distance of the object from the optical centre of the lens O), y -size of the object, a´-image distance (distance of the image from the optical centre of the lens O), y´ -image size, f -object focal length, f´ -image focal length.In the case of a thin lens (plus the similar environment in front of and behind the lens) we can say that f = f´ and use the common notation ffocal length of the lens.The derivation of the lens equation (as given in high school textbooks) can be based on Newtonian two image equations, but this is beyond the scope of the high school instruction (given in[7]).From the first Newtonian image equation, it is possible to derive the "secondary school" lens equation, given in the form [8-10]:

Figure 3 .
Figure 3.An example of the construction of an image of thin converging lens.For our chosen situation, a = 9 units, f = 4 units and a´ =7.19 units.The distance a' is verified by calculation.

Figure 4 .
Figure 4.An example of the construction of an image of thin diverging lens.For our chosen situation, a = 9 units, f = 4 units and a´ =2.77 units.The distance a' is verified by calculation.