Contextualization

Understanding and solving equations involving absolute value is an essential skill in algebra. A step towards more complex problems, absolute value equations are a foundational component in the study of mathematics, leading to more advanced concepts such as inequalities, graphs, calculus, etc. The absolute value of a number refers to the distance from that number to zero on the number line, regardless of the direction. An important detail to remember with absolute values is that they are always non-negative.

Absolute value equations often require a unique strategy for solving: splitting the equation into two separate cases. This is based on the fundamental concept that any absolute value equation |X|=A has two solutions: X=A and X=-A. Taking this into consideration allows for a systematic method of solving absolute value equations.

Relevance

Now, one might wonder, where is this useful in the real world? The concepts of absolute value come into play in various real-world scenarios. In finance, absolute values are used to calculate absolute differences, which can measure the size of errors or deviations. In engineering, they can be used to indicate magnitude without specifying direction, like in the case of sound levels, magnitude of earthquakes or wind speed. In computer science, absolute values help in algorithms where only the magnitude of difference matters and not the direction.

Resources

For a more in-depth understanding of absolute values and their use in solving equations, the following resources can be beneficial:

1. Khan Academy's Course on Absolute Value – This course gives a detailed explanation of the concept and various examples to practice.
2. MathBitsNotebook's Module on Solving Absolute Value Equations – This is a great resource for understanding the splitting technique.
3. Math is Fun's Section on Absolute Value – It provides an easy-to-understand introduction to the concept.
4. YouTube's playlist by The Organic Chemistry Tutor – This playlist includes numerous videos solving absolute value equations step-by-step.

With these resources, you should be ready to dive into the world of absolute value equations. Remember, practice is key. Have fun learning!

Practical Activity

Activity Title: "Solving and Applying Absolute Value Equations"

Objective of the Project

The goal of this project is to deepen understanding of absolute value equations and develop an ability to apply these concepts in real-world scenarios. By the end of this project, students will become proficient in solving absolute value equations and understand how these mathematical principles apply to real-world situations.

Detailed Description of the Project

The project will involve two primary activities: The first part will be solving absolute value equations, and the second part will involve creating real-world problems that can be solved using absolute value equations.

Each group will be given a set of absolute value equations of varying difficulties to solve. Here they will apply the understanding they have gained from their learning resources, showcasing their problem-solving skills.

For the second part, each group must think creatively to come up with real-life scenarios where absolute value equations would be required. They will present the problem and solution in the class.

Necessary Materials

• Pencil and paper for solving equations.
• Classroom blackboard or digital whiteboard for presenting their work.
• Access to the internet to research real-world applications of absolute value equations.

Detailed Step-by-Step for Carrying Out the Activity

1. Split into groups of three to five students.
2. Each group will be given a set of absolute value equations. These will be of varying difficulties to cater to different learning levels within the group.
3. Each student within the group will solve the equations individually.
4. Afterward, the team will discuss and compare their solutions. If there are discrepancies, they will have to solve them as a group and reach a consensus on the correct solution.
5. Now, the group will brainstorm and come up with real-world scenarios where absolute value equations would apply. They will form an absolute value equation based on the scenario and solve it.
6. Each group will then present their real-world scenarios and equation solutions to the class.
7. After the presentations, each group will compile their work into a written document following the "Introduction, Development, Conclusions, and Used Bibliography" format.

Project Deliveries

At the end of this project, each group will need to submit the following:

1. A written document containing:

   • The solved absolute value equations.
   • The real-world scenarios and their solutions.
   • A documentation of the discussions they had in the group while comparing solutions, in the Development section.

2. The presentations of real-world problems: video or slides.

The written report should be organized as follows:

• Introduction: Explanation of the theme, its relevance, and real-world application. This should also include the main objective of this project.
• Development: Detailed explanation of the theory behind absolute value equations, the methodology used in solving the exercises, and the presentation of real-world scenarios. Also document the discussion and problem-solving process within the group.
• Conclusions: Revisit the main points and explicitly state the learning obtained and conclusions about the project. Each group member should state what they have learned from this project.
• Used Bibliography: Indicate the sources they relied on to work on the project such as books, web pages, videos, etc.

Remember the goal of this project is not only to understand absolute value equations but also to foster teamwork, time management, and problem-solving skills. Have fun and learn together!