Contextualization

Pyramids are geometric solids that have inhabited the human imagination for millennia, from the ancient Egyptians to modern architects. Understanding the mathematics behind pyramids - such as calculating their area, volume, and the distances between their points - is a crucial step in understanding three-dimensional geometry, a key component in disciplines such as architecture, engineering, and graphic design.

However, pyramids do not exist only in the form of imposing architectural monuments. They can also be found in natural structures, such as mountains, and even in everyday phenomena, such as the path of the sun in the sky.

The mathematics behind pyramids is a field of study that establishes connections with other disciplines and practical applications. By mastering the calculation of pyramid areas, for example, you will be able to transform simple two-dimensional shapes into complex three-dimensional structures. This has direct applications in areas such as civil engineering, architecture, and product design, where this knowledge is used to design and build real-world objects.

Furthermore, this theme allows for the exploration of other fundamental mathematical concepts, such as angles, trigonometry, and ratios, showing how they relate to the study of pyramids and three-dimensional geometry.

Practical Activity: Construction and Modeling of Pyramids

Project Objective

The objective of the project is to build a scale model pyramid, apply the concepts of calculating the lateral and total area of the pyramid in this model, and finally, relate them to a real-world problem in another discipline such as physics, where pyramids can be used to address phenomena like light refraction.

Detailed Project Description

Divided into groups of 4 to 5 students, the project should be developed over three weeks, with 4 hours of work per week recommended for each student.

The activity will involve the following steps:

1. Advanced Studies: Students will familiarize themselves with the key theoretical concepts involved in the area of pyramids.

2. Planning: Students will decide on the dimensions of their pyramid and the material to be used, planning all the necessary details for the construction of the model.

3. Construction: Students will build a scale model pyramid.

4. Calculations and Measurements: Students will make the necessary measurements and calculate the lateral and total area of the pyramid.

5. Relating the pyramid to an external discipline: They will apply the mathematical concepts learned to a practical situation linked to another discipline, for example, how light refraction occurs in a glass pyramid.

Required Materials

• Cardboard, tape, ruler, protractor, pen, tracing paper (for drawing the faces of the pyramid).
• Calculator.
• Various materials for the light refraction experiment (depending on the experiment chosen by the group).

Detailed Step-by-Step

1. Read the provided material and watch the videos to understand the calculation of a pyramid's area.
2. As a group, decide on the dimensions of the pyramid you will build.
3. Draw the faces of the pyramid on cardboard, using the ruler and protractor to ensure accuracy, and cut them out.
4. Assemble the pyramid using tape to secure the edges. Make sure the pyramid is stable and all measurements are correct.
5. Calculate the lateral and total area of the pyramid and verify if the results obtained are as expected.
6. Choose an experiment that relates the pyramid to another discipline. For example, in physics, you can explore how light reflects and refracts in a glass pyramid. Plan and conduct the experiment, carefully observing the results.
7. Document the construction process, calculations, and experiments in detail to include them in the final report.

Project Deliverables

1. Pyramid Model: The constructed pyramid will be part of the project delivery.

2. Written Report: The report should be structured into four main parts: Introduction, Development, Conclusions, and Bibliography.

• In the Introduction, students should contextualize the topic, explaining the importance of calculating a pyramid's area and how it applies to the real world.

• In the Development, students should describe in detail all the steps they followed in carrying out the project, explaining the methods used to build the pyramid, calculate its area, and perform the experiment related to another discipline. They should present and discuss the results obtained.

• In the Conclusion, students should summarize the main points addressed in the work, explain the learnings obtained, and draw conclusions about the project.

• In the Bibliography, students should indicate the sources of information used for the project development.

The written report should be submitted along with the pyramid model on the date defined for the project evaluation.