Contextualization

Hello, young mathematicians! Let's embark on an adventure through the world of angles and proportions. But first, have you ever stopped to think about how mathematics is present in our daily lives? Whether it's calculating change, preparing a recipe, or even in the games we play, mathematics is there, helping us understand and solve problems.

A simple example of how mathematics is present in our lives is the study of angles. Have you ever noticed that clocks, clock hands, eyeglass lenses, the shape of windows, doors, and furniture, all have angles? And most interestingly, these angles are very important for these objects to function correctly.

Angles are formed by two lines that meet at a point, which we call a vertex. They can be open, when they are less than 180 degrees, and closed, when they are greater than 180 degrees. In addition, angles can be classified as acute (less than 90 degrees), right (equal to 90 degrees), obtuse (greater than 90 degrees and less than 180 degrees), and straight (equal to 180 degrees).

Introduction

Now that we know what angles are, let's learn a very important concept: the congruence of angles. Just as two line segments are congruent when they have the same length, two angles are congruent when they have the same measure. This means that, for example, an angle of 30 degrees and another of 30 degrees are congruent, as they have the same measure.

The congruence of angles is a very useful property in solving mathematical problems. It helps us compare angles and better understand the relationship between them. Furthermore, the congruence of angles is present in various elements of our daily lives, such as in furniture assembly, drawing house plans, building bridges, among others.

Another important concept we will explore is proportionality. Did you know that proportionality is present in many situations in our daily lives? For example, when cooking a recipe, if we double the amount of ingredients, the cooking time will also double. This is because there is a proportion between the amount of ingredients and the cooking time.

Proportionality is the relationship between two quantities that vary together. For example, if a person's height is proportional to the time they live, we can say that as the person ages, their height also increases. Proportionality is a very important tool in mathematics, as it helps us solve problems more efficiently and accurately.

Now that we know the concepts of angles and proportions, let's move on to practice! Get ready, because the challenge will be to explore these concepts in a fun and creative way.

Practical Activity

Activity Title: "Adventure of Measures and Proportions"

Project Objective:

This project aims to develop students' ability to identify, create, and manipulate congruent angles, as well as explore situations of proportionality, both among angles and quantities. In addition, the activity aims to stimulate teamwork, creativity, communication, and proactivity.

Detailed Project Description:

Students, divided into groups of 3 to 5, will be challenged to build a "Pocket Sundial" and a "Handmade Compass". To build these objects, they must apply the concepts of congruent angles and proportionality in a practical and playful way. This activity will involve several steps, from planning, research, material collection, object construction, to the final project presentation.

Required Materials:

• Cardboard
• Ruler
• Compass
• Pencil
• Blunt scissors
• Glue
• Popsicle stick
• Aluminum foil
• Magnet
• Water
• Bottle caps (two of different sizes)
• Small needle or nail
• Small bowl

Step by Step:

1. Study and Planning: Each group should research the construction of sundials and compasses. They must understand how these objects work and which angles need to be considered in each of them. Based on this study, they should plan how they will build their own sundials and compasses.

2. Pocket Sundial: Using the cardboard, students will draw a circle of approximately 15 cm in diameter. In the center of the circle, they should mark a point that will be the center of the sundial. Then, with the help of the ruler and compass, students should divide the circumference into 12 equal parts, like a traditional clock. Finally, they should write the numbers from 1 to 12 in each of these divisions.

3. Handmade Compass: With a small bowl of water, students will conduct an experiment to understand how the compass works. They should place the bottle cap with the needle in the center of the water bowl. The bottle cap with the needle should float in the water. Then, students should observe what happens to the needle. It should always point in the same direction. This is the direction of the magnetic north.

4. Application of Concepts: Now comes the most challenging part! Students must apply the concepts of congruent angles and proportionality to enhance their sundials and compasses. For example, they can use a protractor to measure the angles between the numbers on the sundial and ensure they are congruent. In the compass, they can use the proportion between the needle's position and the magnetic north to determine other cardinal points.

5. Project Presentation: In the end, each group must present their "Pocket Sundial" and their "Handmade Compass" to the class. In the presentation, they should explain how they built the objects, which mathematical concepts they applied, and how these objects work.

Delivery Format:

The project will be delivered in two parts: the first will be the "Pocket Sundial" and the second the "Handmade Compass". Each part of the project will count points towards the final grade. In addition, students must submit a written report (it can be handwritten) describing all the project steps, the materials used, the challenges faced, and the solutions found. This report will also count points towards the final grade. Remember, the most important thing is not to have perfect sundials and compasses, but the learning process and the fun of building them together! Good luck, and let the adventure begin!