Applications of Mathematics 11 -
Patterns and Relations (Variables and Equations)

This sub-organizer contains the following sections:
Prescribed Learning Outcomes
Suggested Instructional Strategies
Suggested Assessment Strategies
Recommended Learning Resources

PRESCRIBED LEARNING OUTCOMES

It is expected that students will :

• represent and analyse situations that involve expressions, equations and inequalities.
• use linear programming to solve optimization problems.

It is expected that students will:

• graph linear inequalities, in two variables
• solve systems of linear equations, in two variables:
  • algebraically (elimination and substitution)
  • graphically
• solve nonlinear equations, using a graphing tool.
• solve systems of linear inequalities in two variables using graphing technology
• design and solve linear and nonlinear systems, in two variables, to model problem situations
• apply linear programming to find optimal solutions to decision-making problems

SUGGESTED INSTRUCTIONAL STRATEGIES

When students create, identify, and interpret linear and non-linear function graphs that represent real-world situations or events, they connect their knowledge of linear systems and graphs with their technology and problem-solving skills.

• Consult with science teachers to obtain laboratory data that represents real-world events and that can be modeled with linear or non-linear relations. Have students use a graphing utility to generate a line of best-fit and to analyse the function in context. Technology such as a Calculator Based Laboratory (CBL) could also be used to generate data.
• Model the process of creating and analysing graphs on a variety of topics (e.g., population growth, car value depreciation, radioactive decay), paying special attention to the patterns and characteristics of each graph.
• Emphasize the relationship between two variables by encouraging students to use relational terminology (e.g., the distance an object falls depends on or is a function of, the time that it falls).
• Ask students to compare various company rates (e.g., vehicle towing, cellular phones, automobile rentals) to conduct a break-even analysis.
• Emphasize that relations between two variables give a line on the Cartesian plane, and that problems of multiple relations can be solved graphically and algebraically.
• Use graphing calculators or computer software to introduce the concepts of linear inequalities and systems of linear inequalities. Have students:
  • shade the solution planes and mark the boundaries of graphs of linear inequalities
  • shade and find areas of overlap for systems of linear inequalities
  • determine vertices and possible restrictions to the solution of systems of linear inequalities
  • determine maximum and minimum values
• Present students with models of problem situations and have them determine the linear equations, inequalities, and/or systems that define the problem. Have students discuss the problems and ways of solving them, analyse constraints on the problems, and discuss feasibility of the constraints.

SUGGESTED ASSESSMENT STRATEGIES

Linear and non-linear functions occur in real-life situations (e.g., maximizing profit and productivity, exponential growth of bacteria, earthquakes, radiocarbon dating). It is important to observe students’ abilities to recognize patterns and functional relationships as well as their abilities to select and use techniques to solve various types of problems. Assessment of students’ linear programming and graphing skills should focus on the extent to which students apply the concepts to model and solve optimization problems.

Observe

• While students are designing and solving linear systems, note the extent to which they:
  • use linear programming to accurately model various problems
  • use correct terminology such as optimization, objective functions, constraints
  • draw graphs accurately
  • correctly use graphing calculators with shade and trace capabilities
  • draw appropriate conclusions and interpretations
  • recognize discrete or continuous data

Collect

• Assign a series of problems that require students to:
  • use graphing calculators
  • identify the appropriate graph with its associated function
    
    
• Check students’ work for evidence that they:
• clearly understand the requirements of the problem
• use efficient strategies and procedures to solve the problem
• can verify the accuracy and reasonableness of their answers

Presentation

• Have students work in small groups to complete a project to design and analyse a profit and productivity problem using linear programming. Look for evidence that they:
  • interpreted and modeled the problem correctly
  • arrived at logical conclusions or solutions
  • logically and clearly presented the work

RECOMMENDED LEARNING RESOURCES

Print Materials

• Applied Mathematics 11, Western Canada Edition
• Explore Quadratic Functions with the TI-83 or TI-82
• Exploring Functions with the TI-82 Graphics Calculator
• An Introduction to the TI-82 Graphing Calculator
• Modeling Motion: High School Math Activities with the CBR

© 2000 Copyright. All Rights Reserved. Curriculum Branch.
Maintained by: Mathematics Coordinator
Revised: November 22, 2000

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