Setting the Stage

Introduction

Inequalities are mathematical statements that, instead of being defined by the equals sign (=), are represented by one of the four inequality symbols (> , < , ≤ , ≥). These express a relationship of inequality between two quantities or values. Like equations, inequalities are a fundamental part of mathematics and have applications in many fields, including engineering, economics, and physics, just to name a few.

The main ideas that you need to understand to work with inequalities are a) the meaning of the inequality symbols, b) how to solve one-step inequalities, and c) the rules for manipulating both sides of an inequality. For instance, if provided with the inequality 2x + 3 > 7, one would need to isolate x to find the solution, or solution set.

However, it’s important to remember that when solving an inequality if you multiply or divide both sides of the inequality by a negative number, you must reverse the inequality symbol. Understanding these concepts will be essential for completing this project.

Real-World Context

Inequalities are a tool that aids in making everyday decisions. For example, when you are buying something and are on a budget, you are essentially solving an inequality: the price of the item must be less than or equal to the amount of money that you have.

Inequalities are also used in many professions in more complex ways. Engineers use inequalities to determine the safety limits of a structure. Economists use them to model economic behavior and analyze profit and loss.

Therefore, understanding and being able to solve inequalities is not just a mathematical skill. It’s a skill that can be useful in many aspects of life, both personal and professional.

Lesson Plan

Title of the Activity: Challenging Inequalities

Learning Objectives:

The main objective of this lesson is to provide a practical and meaningful understanding of the concept of inequalities, how to solve them, and some real world applications of inequalities while promoting teamwork, effective communication, and time management.

Detailed Description of the Lesson:

Students will be placed in groups of 3-5 and each group will be given a unique word problem or scenario that involves solving one-step inequalities. Each group will work together to solve their individual word problem and create a final report that describes the theory needed, the method of solution, the solution itself, and how this ties into a real world example.

Groups will be encouraged to create and utilize a variety of manipulatives to further explain their findings, including graphs, charts, drawings, or any visual aid that will help communicate their findings.

Materials Needed:

• Scrap paper and pencils
• Calculators
• Computer with internet access
• Software for creating a digital report (Word, Google Docs, etc.)
• Graphing software or app (Excel, Google Sheets, Desmos, etc.) if desired

Step-by-Step Procedure:

1. Organize students into groups of 3-5.
2. Present students with their assigned word problem and encourage a group discussion to make sure they understand the problem.
3. Have students research the theory of inequalities, how to solve them, and applications for using them. Resources are suggested below to help complete this step.
4. Once complete, have the group solve the inequality within the context of their word problem, discussing as a group the best strategy to do so.
5. After solving the inequality, have the group discuss what the solution means in terms of the word problem they were given.
6. With the inequality solved and discussed, have the group work to create a digital report that will include: Introduction to the concept of inequalities and their usage; Theory of how to solve inequalities; Method used by the group to solve the inequality; Solution and its real world application within the word problem; Group reflection and list of resources and/or references used.
7. Depending on the outcome, the group may also choose to represent the solution graphically (not required, but a bonus).
8. Finalize the report to ensure all parts of the project are addressed correctly and thoroughly.
9. Turn in the final report within one week.

Deliverables:

By the end of the week, each group should turn in:

• Digital report including all of the components listed in step 6 of the "Step-by-Step Procedure";
• Final presentations on their solutions to the class, explaining the word problem, the inequality that was solved, the solution, and how it addresses the problem. Students should be prepared to answer questions and talk about their work with the class.

Remember: Written reports should be well-organized, well-written, and use clear and precise language. They should thoroughly explain the group’s thinking and methodology so that someone unfamiliar with the problem will be able to follow and understand the question and the answer. The final presentation is the group’s chance to show what they have learned and how they worked together to solve a complex, real-world problem using inequalities.