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Applications of Math 11 Linear Programming Unit Outline Chapter 4 General Prescribed Learning Outcomes: Represent and analyze situations that involve expressions, equations and inequalities. Use linear programming to solve optimization problems. Specific Prescribed Learning Outcomes: Graph linear inequalities, in two variables. Notes: Students were introduced to inequalities with one variable in Grade 9 mathematics, where they solved inequalities and graphed inequalities on a number line. Coordinate geometry skills are essential for this specific outcome. The concepts of plotting points and intercepts, line graphing, and the use of calculators are the more important concepts to review. Ax + By + C = 0 can be sketched using intercepts. Conversion from the Ax + By + C = 0 form to any y = form is a necessary preliminary step to the use of a graphing calculator. Window settings, on the graphing calculator, are useful to replace the plotting of horizontal and vertical lines. For example, x < 5 could be entered into window settings as an x-max of 5. Students should first graph manually—to solve for and sketch the solution region—and then graph using the graphing calculator. An even balance between both approaches is recommended Acceptable Standard: graph the boundary line between two half planes use a test point, usually (0, 0), to determine the solution region that satisfies the inequality, given a boundary line graph a linear inequality expressed in the form y = mx + b, using <, >, £, ³ rewrite any inequality expressed in the Ax + By = C form in the y = mx + b form, where A, B, C are integral and B > 0 Standard of Excellence: distinguish between the use of solid and broken lines in solution regions graph any linear inequality in two variables rewrite any inequality expressed in the Ax + By = C form in the y = mx + b form, and graph explain why the shaded half plane represents the solution region of the inequality Mark Healy mbhealy@telus.net Page 1 West Vancouver Secondary School Solve systems of linear inequalities in two variables using technology. Notes: Many constraint inequalities are of the form Ax + By = C, rather than Ax + By + C = 0. It is important for students to first sketch the system of linear inequalities, because the graphing calculator may need adjustments in window settings or use of the zoom function to show the intersection point or points of the inequalities. With multiple inequalities, students may find it easier to see the intersection by graphing the opposite inequalities (reverse shading), so the intersection region is unshaded. Examples should include both open solution regions and closed polygon solution regions Acceptable Standard: graph the boundary line between two half planes use a test point, usually (0, 0), to determine the solution region that satisfies the inequality, given a boundary line graph a linear inequality expressed in the form y = mx + b, using <, >, ≤, or ≥ rewrite any inequality expressed in the Ax + By = C form in the y = mx + b form, where A, B, C are integral and B > 0 explain why the solution region represents the solution to the problem Standard of Excellence: distinguish between the use of solid and broken lines in solution regions graph any linear inequality in two variables rewrite any inequality expressed in the Ax + By = C form in the y = mx + b form, and graph explain why the shaded half plane represents the solution region of the inequality explain how the solution region is a combination of different half planes Design and solve linear and nonlinear systems, in two variables, to model problem situations. Notes: Technology is a key mathematical process for this outcome. Use window settings for horizontal and vertical lines. For problems where the intersection of the inequalities is difficult to distinguish on the graphing calculator, students could reverse the inequality signs and view the intersection as the unshaded region. Most problem situations have solutions in the first quadrant. Price and profit problems can lead to quadratic inequalities. Area and perimeter problems often lead to non-linear inequalities. Intersection points are often found using technology. All nonlinear inequalities should be solved using technology Students may find this outcome challenging as a whole; it may be useful to guide students into structured solutions. Mark Healy mbhealy@telus.net Page 2 West Vancouver Secondary School Apply linear programming to find optimal solutions to decision-making problems. Notes: The optimal solution is found by substituting the coordinates of the vertices into the expression for the objective function. The optimal solution must lie within the domain of the relation; e.g., if x represents the number of people, then x must be a whole number. In addition to the stated restrictions, other inequalities may be implicit and must be considered to find the solution; e.g., if x represents the length of a rectangle, then x > 0 The optimal solution may involve either real numbers or natural numbers, depending on the context of the problem. Vertices should be part of the domain for all assessment problems. Acceptable Standard: Write the system of inequalities corresponding to a problem context Graph inequalities Find vertices Write an expression for the objective function Substitute vertices into an expression for the objective function to find the optimal solution Standard of Excellence: Integrate the elements of the solution process to find the optimal solution to a decision-making problem Summarize and explain why the solution is correct Mark Healy mbhealy@telus.net Page 3 West Vancouver Secondary School Lesson Sequence Day 1 – Tutorial 4.1 – The Graph of a Linear Inequality Discuss preamble at beginning of tutorial Use attached handout to discuss lesson or go through Investigation 1 and 2 as a class Go through ―Discussing the Ideas‖ #3 – 6 Assignment: Attached Worksheet Day 2 – Tutorial 4.2 – Graphing a Linear Inequality in Two Variables http://faculty.stcc.mass.edu/zee/newpage128.htm provides good examples for this lesson Discuss steps outlined at beginning of Tutorial. Utility 21 on page 417 discusses use of graphing calculator to graph inequalities. Use steps to complete examples 1 and 2 using both graphing by hand and graphing using technology. Go through ―Discussing the Ideas‖ #1 – 4 Assignment: Page 178 #1, 2, 3aceg, 4ac, 6ac, 7, 9 Day 3 and 4 – Tutorial 4.3 – The Solution of a System of Linear Inequalities Day 3: Discuss the steps to graphing a system of linear inequalities by hand o http://faculty.stcc.mass.edu/zee/newpage210.htm provides good examples for this lesson o Complete example 1 (need different color pencil crayons) o Assignment: Page 187 #2, 3, 4ab, 6 Day 4: Discuss the steps to graphing a system of linear inequalities using technology (reverse shading) o Discuss using window settings for horizontal and vertical lines. o Use reverse shading so solution area becomes the only part non-shaded. o Go through examples 2 and 3 o Go through ―Discussing the Ideas‖ #1-5 o Assignment: Page 187 #1, 3(using TI-83), 4c, 5, 7(either way), 8 (either way) Day 5 – Review and Quiz Day 6 and 7 – Tutorial 4.4 – Modelling a Problem Situation Day 6: Modelling linear system situations o Go through Examples 1 and 2 o Go through Discussing the Ideas #1-4 o Assignment: Page 195 #1 – 3 Day 7: Modelling nonlinear system situations o Go through Example 3 o Assignment: Page 196 #4 – 7 Mark Healy mbhealy@telus.net Page 4 West Vancouver Secondary School Day 8 – Project & Introduction to Tutorial 4.5 – Optimization Problems Read through Project Introduction on page 168 Complete #1 – 3 of project on page 198 (~50 minutes) using the attached handout. Introduce Optimization problems by completing Investigation on Page 200. Day 9 – Tutorial 4.5 continued Go through attached lesson to discuss optimization. Go through Examples 1 and 2 for more examples. Assignment: Page 207 #1 – 5 ***If time, add an extra day to complete Tutorial 4.5!!! Use attached handout for more examples*** Day 10 – Review and Project Completion Complete project requirements on page 210 Review Unit using Unit Review on Pages 213 – 218 Day 11 – Unit Exam Mark Healy mbhealy@telus.net Page 5 West Vancouver Secondary School