LWB Level 3 Linear Programming 3.2 Learning Workbook

The LWB Level 3 Linear Programming 3.2 Learning Workbook is a targeted, write-on resource designed for the NCEA Level 3 Internal Assessment: Apply linear programming methods in solving problems. This workbook focuses on the mathematical art of "finding the best outcome"—whether that is maximizing profit or minimizing costs—within a set of real-world limitations.

Key Features

• Defining Constraints: Learning how to translate written "real-life" limitations (e.g., "we only have 40 hours of labor available" or "we must produce at least 10 units") into mathematical linear inequalities.

• The Feasible Region: Extensive practice in graphing multiple inequalities on a single Cartesian plane to identify the "feasible region"—the area where all conditions are met.

  • Boundary Lines: Distinguishing between solid lines ($\le, \ge$) and dashed lines ($<, >$).

  • Shading: Mastering the convention of shading out the "unwanted" regions to leave the feasible solution clear.

• The Objective Function: Formulating the goal of the problem (e.g., $P = 5x + 8y$) and understanding how to move this "search line" across the graph to find the optimal point.

• Vertex Theory: Learning that the optimal solution in a linear programming problem always occurs at one of the vertices (corners) of the feasible region. Students practice calculating the exact coordinates of these vertices using simultaneous equations.

• Integer Constraints: A critical real-world check. Learning how to adjust solutions when the answer must be a whole number (e.g., you can't sell 4.7 bicycles).

Achievement, Merit, and Excellence Scaffolding

• Standard Mastery (Achievement): Focused on forming and graphing simple inequalities and identifying a basic feasible region.

• Relational Application (Merit): Transitioning to problems where the objective function must be optimized, and the student must clearly communicate the solution in the context of the story.

• Complex Reasoning (Excellence): High-level tasks that involve "what-if" scenarios. For example, investigating how the optimal solution changes if the profit per item changes or if a constraint is loosened (Sensitivity Analysis).

Workbook Highlights

• Internal Assessment Preparation: Provides a clear report-writing structure that mirrors the assessment tasks used in New Zealand schools, ensuring students know how to present their graphs and findings.

• Real-World Contexts: Exercises are built around authentic business and logistics scenarios, such as manufacturing mixes, dietary planning, or transport scheduling.

• Graphics Calculator Integration: Specific guides on using Casio or TI calculators to graph inequalities and find the intersection points of boundary lines quickly.

• Step-by-Step Worked Examples: Provides templates for translating complex word problems into a mathematical system of equations and inequalities.

• Full Answer Appendix: Every question includes a fully worked solution at the back of the book, allowing for independent study and immediate verification.

• Glossary of Terms: A guide to essential vocabulary—such as Bounded vs. Unbounded Regions, Objective Function, and Constraint—to ensure students use precise mathematical language.