# Augmented Reality as a Didactic Resource for Teaching Mathematics
## 3 Results
### Classroom Sessions  
Next, we explain how we would teach polyhedra using the created materials as support, with which we intend to facilitate their study and visualization, as well as to improve their motivation and to encourage the use of new technologies. Whenever it is mentioned during the activities that the polyhedra are displayed, we will remind that they are visualized using the AR, through markers and a computer or mobile device with camera and the applications installed. It is also assumed that students have previously been taught the geometry of the plane, because without doing so it would not make sense to explain the most advanced geometric concepts related to polyhedra. Finally, work groups of 3-4 students will be formed to carry out the activities. Groups will be determined based on students that complement each other and are able to work well together. This is intended for some students to help others to understand the concepts and to work as a team. To structure the contents and the implementation of the application of AR in the classroom, different sessions are planned:  
Class session 1  
This session will be dedicated to teaching students how to use augmented reality technology with markers, to visualize and describe polyhedra and the elements that define them (faces, vertices, edges, diagonal). We start teaching how to use the application we have created. In groups, we distribute a marker so that students can visualize various types of polyhedra using the AR application. The AR models will facilitate visualization (they can move, rotate the marker and the polyhedron will also move with the marker) and understand concepts like faces, vertices, and edges, that will be in different colors. Class session 2  
The notions of concave and convex polyhedra are introduced. Eight different concave and convex polyhedra will be shown with AR (see an example in Figure 3). Groups will be asked to identify what differentiates concave from convex polyhedra. They are asked to count the number of edges, vertices, and faces of convex polyhedra and try to find a relationship between these numbers. [PERSON]'s theorem for convex polyhedra is explained. They are asked to verify with other convex polyhedra that [PERSON]'s theorem is fulfilled and verify that the theorem is not fulfilled in the case of concave polyhedra. Figure 3: Examples of concave and convex polyhedra with augmented reality (AR). (**a**) Concave polyhedron; (**b**) Convex polyhedron. Class session 3  
Different types of polyhedra are presented: orthohedron, parallelepipeds, prisms, and pyramids (straight and oblique) (see Figures 4 and 5). We teach the calculation of the diagonal of an orthohedron and the apothem of a pyramid as an application of the [PERSON] theorem with the help of AR. The flat developments of polyhedra are visualized and explained and the formulas of the area deduced. Class session 4  
The Cavalieri principle is explained, and the formulas of the area and volume of the polyhedra are deduced, helping us with the AR application. Class session 5  
The concept of a regular polyhedron is explained and the fact that there are only five regular convex polyhedra displayed within the AR application: tetrahedron, cube, octahedron, icosahedron, dodecahedron (see Figure 6). In addition, the notion of dual or conjugate polyhedron for the previous five, is introduced (see some of these conjugated polyhedra in Figure 7). Class session 6  
In this session, we visualize the flat developments of the platonic solids (see some examples in Figure 8) and how to calculate their area is explained. The formulas of the volume of the tetrahedron, cube, and octahedron are also deduced and the formulas that allow the calculation of the volume of the icosahedron and the dodecahedron are explained. Class session 7  
In this session, we explain the symmetry plane of a polyhedron. In addition, we visualize them for different polyhedra as regular prisms, regular pyramids, platonic solids, and parallelepipeds.