TASK: To draw spheres and to draw spherical triangles on a Lénárt sphere on paper and on a blackboard.

TIME FRAME: 180 minutes

AIMS:
‒ Explore how to draw the sphere using the concept of projection.
‒ Explore the concept of righttriangle on the sphere.
‒ Increase the research by using three different scales, tools, and supports for three distinct purposes.
‒ Investigate the relevance and applicability of these teaching/learning/assessment methodologies within educational contexts.

MATERIALS AND PREPARATIONS:
The materials needed for the activity are Lénárt spheres, A4 white sheets of paper, pens, pencils, a room equipped with blackboards or whiteboards, chalk of various colours, and two flashlights. Although all materials are provided to participants and no prior preparation is required, participants could benefit if they are familiar with the Lénárt Sphere Construction Materials (instructions: http://lenartsphere.com/pdf/sik-es-gomb_fuzet-izelito_en.pdf).

INSTRUCTIONS:
Participants should work in groups discussing doubts and achievements in three interdependent stages: drawing on paper (done individually), drawing on a Lénárt sphere(for group discussion), and drawing on a board (for a broader discussion).

I. Introduction to the main theme of the workshop

II. Sphere Projection onto the Plane: From Circle to Ray
1. Experiment with sphere projections onto the plane using a flashlight, focusing on Desargues' conic studies.
2. Trace the sphere projections with chalk.
3. In phases IV and V, apply the knowledge and practices acquired in this phase to draw spheres.

III. Triangles on the plane
Represent the following four (equivalent) definitions of a right triangle on a plane visually, without using words, on a white A4 paper:
1. a triangle with a right angle
2. a triangle with an angle equal to the sum of the other two angles
3. a triangle obtained by bisecting a rectangle by means of its diagonal
4. a triangle inscribed in a semicircle

IV. Drawing basic concepts of spherical geometry on the Lénárt sphereand their projections on the plane
1. What is the shortest path between two points on the sphere? Draw an arc.
2. What is a straight line on the sphere? Draw a great circle.
3. How many great circles can you draw through two points on the sphere?
4. What is a right angle? Draw two perpendicular lines.
5. What is a circle? Draw a circle. How many centers has a circle?
6. What is a triangle? Draw a triangle and use the protractor to measure its three interior angles.
7. What is a rectangle? Draw a rectangle.

V. Drawing right-angled triangles on the Lénárt sphereand their projections on the plane

Consider the four previous definitions of a right triangle on the plane, transfer them to the sphere, and for each case, draw the respective right triangle on the Lénárt sphere. Discuss the equivalence of these definitions.

TIPS AND THINGS TO CONSIDER:
- For greater accuracy in stating the conjectures, the instructions for working with the Lénárt Sphere must be followed, specifically ensuring that both the spherical ruler and protractor are properly placed on the surface of the spheres;
- To better control the drawing of great circles, it is advisable to consider the points of intersection with the sphere’s surface of the axes of a three-dimensional Cartesian reference system, with origin at the center of the sphere;
- For drawing the sphere, it is advisable to consider the projection plane in a frontal position relative to the observer, in order to obtain a circular projection of the sphere;
-For obtaining a good dynamic during the activity, it should be ensured that mathematical speculation through drawing carefully usesthree different scales, tools, and supports in the three interdependent stages: drawing on paper (done individually), drawing on a Lénárt sphere (for group discussion), and drawing on a board (for a broader discussion).