Read more about the purpose of the MAP *Classroom Challenges*…

## Optimization Problems: 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 the data generated 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 scenarios where limited resources must be used 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 about the same problem.
- 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 the alternative solution strategies they have seen and used.
- Finally, students revise their individual solutions, and comment on what they have learned.

### Materials required

- Each individual student will need a calculator, and a copy of the assessment task
*Boomerangs.* - 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.
- There are also some projector resources to help you with whole-class discussions.

### Time needed

15 minutes before the lesson, a one-hour lesson, and 10 minutes in the next lesson (or for homework). Timings are approximate and will depend on the needs of the class.

### 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

### Mathematical Content

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

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

### Resources

#### Lesson (complete)

- boomerangs_complete.pdf ( 3179.8K PDF/Acrobat 19 Mar 2012)

#### Projector Resources

- boomerangs_slides.ppt ( 1151K MS PowerPoint 19 Mar 2012)

A draft Brief Guide for teachers and administrators (PDF) is now available, and is recommended for anybody using the MAP *Classroom Challenges* for the first time.

We have assigned lessons to grades based on the Common Core State Standards for Mathematical Content. During this transition period, you should use your judgement as to where they fit in your current course.

The Beta versions of the MAP Lesson Units may be distributed, unmodified, under the Creative Commons Attribution, Non-commercial, No Derivatives License 3.0. All other rights reserved. Please send any enquiries about commercial use or derived works to map.info@mathshell.org.

Can you help us by sending samples of students' work on the *Classroom Challenges?*