Contextualization

Understanding equations involving absolute values, often referred to as absolute value equations, is an integral component in algebra, a branch of mathematics that deals with symbols and the rules for manipulating these symbols. The absolute value of a quantity refers to its value without considering its sign; it is the "distance" of the number from zero on a number line. Absolute value equations therefore are equations where the variable is inside an absolute value expression.

The primary focus of this project is to study absolute value equations and unravel the beauty and intricacy of mathematics. The aim is not just to memorize procedures and rules to solve these types of equations, but to comprehend their underlying principles. This understanding will be valuable as you progress further in your mathematical education, enhancing your ability to solve a broader range of complex problems.

Absolute value equations often require splitting. This is because an absolute value equation can have two possible solutions. The numbers -3 and 3, for instance, both have an absolute value of 3. Therefore, when solving an equation like |x| = 3, we have to consider both x = 3 and x = -3. Splitting the equation is hence an essential skill to master when dealing with absolute value equations.

Importance

In the real world, absolute value can be found in various fields, from engineering and physics to economics and finance. It's a tool used to represent quantities that are always positive. For instance, in finance, the absolute value can represent the absolute change in a stock's price, irrespective of the direction of change.

Moreover, learning about absolute value equations contributes to the development of logical and critical thinking skills. It promotes a deep understanding of mathematical concepts, fostering the ability to apply these concepts in different contexts and to find solutions to practical problems.

Resources

1. Khan Academy: Absolute Value Equations
2. Purple Math: Solving Absolute Value Equations
3. YouTube: Absolute Value Equations
4. Ronnie R., Absolute Value Equations and Inequalities, a book to dig deeper into the subject.
5. Wolfram MathWorld: Absolute Value

These resources provide comprehensive overviews, solved examples, practice problems and more detailed insight into the topic. The first three resources are particularly good starting points, offering beginner-friendly explanations and guides.

Practical Activity

Activity Title: "Exploring Real-world Scenarios with Absolute Value Equations"

Objective of the Project

The primary objective of this project is to develop an in-depth understanding of absolute value equations and their real-world applications. The project aims to encourage students to collaborate and apply the theoretical knowledge they have acquired on absolute value equations to analyse and solve problems they might encounter in real-world scenarios.

Detailed Description of the Project

Each group will be given a real-world scenario where absolute value equations are applicable. The teams will be required to create a mathematical model of the scenario, identify the absolute value equations in the model, solve them, and interpret their solutions in the context of the given scenario. The scenarios could include:

1. Understanding stock market fluctuations.
2. Predicting and interpreting temperature changes (weather forecasting).
3. Understanding population growth and decline trends.

The scenarios will be complex enough to necessitate the use of at least four absolute value equations. The project will integrate mathematics and social sciences, requiring students to exercise both their mathematical and contextual reasoning skills.

Necessary Materials

1. A notebook or a digital document for maths calculations, project work, and final report writing.
2. Calculator for solving equations.
3. Online resources to research the topic and gather information about the chosen scenario.
4. Graph paper or online graphing tool to visually represent the solutions.

Detailed Step-by-step for carrying out the activity

1. Divide into groups and choose a scenario (1-2 hours): Split the class into groups of 3-5 students. Each group will choose one of the provided scenarios.

2. Research and Contextual Understanding (3-4 hours): The group will research their chosen scenario, gather necessary information, draw context, and build the appropriate mathematical model.

3. Modeling and Equation Formation (3-4 hours): Using the mathematical model, the group will create at least four absolute value equations that correlate with the context of their scenario.

4. Solving the Equations (2-3 hours): The students will solve the absolute value equations, demonstrating their knowledge of splitting the equations and handling multiple solutions.

5. Interpretation and Presentation (2-3 hours): Finally, the students will interpret their solutions in the context of their scenario. They will prepare a presentation summarizing their process and findings.

6. Writing the Report (6-8 hours): The group will collaborate to write a detailed report on their project.

Project Deliverables

1. A mathematical model of the real-world scenario.
2. Four absolute value equations derived from the model.
3. The solutions to these equations, showing a clear understanding of how to solve absolute value equations, including the process of splitting them.
4. A presentation summarizing the group's findings and how they arrived at them.
5. A comprehensive report following the format specified: Introduction, Development, Conclusions, and Used Bibliography.

Writing the Report

The report should contain the following sections:

• Introduction: Contextualize the chosen scenario, explaining its relevance, real-world application, and the objective of the project from the group's perspective.
• Development: Detail the theoretical concept of absolute value equations, explain the activity, the methodology used, present and discuss the results. Include the equations formed and solved, with explanations for each step of the solution.
• Conclusion: Conclude the work by revisiting the main points, explicitly stating the learnings obtained, and the conclusions drawn from the project.
• Bibliography: Indicate the sources, such as books, web pages, videos, etc. used for the project.

At the end of the project, students should be able to understand the concept of absolute value, how to model real-world situations using absolute value equations, the method to solve these equations, and the ability to interpret the solutions in the context of the real-world scenario. Further, students will also improve their ability to work effectively as a team, manage time, and solve problems creatively.