# Maximizing Profits: Selling Boomerangs

## Mathematical goals

This lesson unit is intended to help you assess how well students are able to:

- Interpret a situation and represent the constraints and variables mathematically.
- Select appropriate mathematical methods to use.
- Explore the effects of systematically varying the constraints.
- Interpret and evaluate generated data and identify the optimum case, checking it for confirmation.
- Communicate their reasoning clearly.

## Introduction

This lesson is designed to help students develop strategies for solving optimization problems. Such problems typically involve using limited resources to greatest effect, as in, for example, the allocation of time and materials to maximize profit.

- Before the lesson, students attempt the problem individually. You then review their work and formulate questions for students to answer in order to improve their solutions.
- At the start of the lesson, students work alone answering your questions.
- Students are then grouped and engage in a collaborative discussion of the same task. In the same small groups, students are given sample solutions to comment on and evaluate.
- In a whole-class discussion, students explain and compare solution strategies seen and used.
- Finally, students revise their individual solutions and comment on what they have learned.

## Materials required

- Each individual student will need a copy of the
*Boomerangs*task, some plain paper, a calculator, and a copy of the*How Did You Work?*questionnaire. - Each small group of students will need copies of the
*Sample Responses to Discuss.* - Graph paper should be kept in reserve and used only when necessary or requested.

## Time needed

Approximately 15 minutes before the lesson, a 1-hour lesson, and 10 minutes in a follow-up lesson.

## Lesson Type

## Mathematical Practices

This lesson involves a range of *mathematical practices* from the standards, with emphasis on:

- MP1: Make sense of problems and persevere in solving them
- MP2: Reason abstractly and quantitatively
- MP3: Construct viable arguments and critique the reasoning of others
- MP4: Model with mathematics
- MP5: Use appropriate tools strategically
- MP6: Attend to precision
- MP7: Look for and make use of structure
- MP8: Look for and express regularity in repeated reasoning

## Mathematical Content Standards

This lesson asks students to select and apply mathematical content from across the grades, including the *content standards:*

- H.A-CED: Create equations that describe numbers or relationships
- H.A-REI: Solve equations and inequalities in one variable
- H.A-REI: Represent and solve equations and inequalities graphically.