LESSONS LESSON 1 The focus of Lesson 1 is a review of systems of linear equations To begin, turn to Chapter One, Investigating Equations in 3-Space on page 1 in your Mathematical Modeling textbook and read down to and including item number 7. This will give you direction for the first chapter. Each of the numbered items on page 1 of the text are connected to a specific curriculum outcome which defines what you are responsible for learning. A review section in the text (see pages 54-65) is available to you to keep track of important mathematical concepts and terms. You should read the review section before beginning the work in this lesson. In lesson 1 you will review systems of linear equations which have already been studied in the Mathematics 10 course. Remember, from grade 10, you have learned two ways to solve a linear system of equations: ALGEBRAICALLY, BY SUBSTITUTION The method of substitution is most often used when one equation is solved for a variable or can easily be solved. FINDING THE INTERSECTION POINT ON THE GRAPH Graphing is the most visual method, allowing you to see the "whole picture" before and after the intersection point. It is also the least accurate for several reasons, one of which is graph quality. LEARNING OUTCOMES Upon completion of Lesson 1 students will be expected to: 11B15 11C12 11C19 solve systems of "m" equations in "n" variables interpret geometrically the relationships between equations in systems solve problems involving systems of equations Remember, you are finding the point of intersection any time you solve a system of equations, whether the method is graphing or algebra. You will extend this understanding into larger systems. 1. a) Begin by reading about Mark s family received a flyer... the situation at the top of page 2 in the text. b) Complete Investigation 1, Choosing a Phone Plan. Procedure B: After answering parts i), and ii) answer this... How could you find the answers to part i) without using the table? Procedure C: Mark is trying to get the equation by translating the situation from words to algebra. He says that Option C will cost a flat rate of $15 and 5 cents for every minute. This gives him the equation: Cost = 15 + 0.05 m. CORRESPONDENCE STUDY PROGRAM PAGE 17
MATHEMATICS 11 i) Did you get this equation for Option C? ii) Was your strategy different? Explain. c)* Do Check Your Understanding questions 2, 3, and 4 on page 3. 2. a) Complete Investigation 2, Choosing a Rental Plan, Procedure A and B on page 4. b)* Complete Procedure steps C, D, and E. c)* Complete Investigation Questions 5 and 6 on page 5. d)* Complete Check Your Understanding questions 9, 10, 2, 14, and 16 on pages 6 and 7. 3. Read and develop a response to the Chapter Project, Student Awards on page 8. PF YOU HAVE FINISHED LESSON 1 Make sure you have completed the assignment questions and the portfolio questions. Send the assignment questions (*) to your marker now. Keep the portfolio items (PF) until after the end of the 10th lesson, then send them to your marker. PAGE 18 CORRESPONDENCE STUDY PROGRAM
LESSONS LESSON 2 The focus of Lesson 2 is the threedimensional coordinate system -4 x z 4 3 2-3 -1 1-1 -2-3 -4 1 2 3 4 View the standard coordinate grid displayed above. Note the placement of the x, y, and z axes. This is the standard format for three dimensional graphs and is used throughout the world. Read 1.2 Visualization in Three Dimensions and Focus A, Labelling Axes in Three Dimensions. 1 4 y LEARNING OUTCOMES Upon completion of Lesson 2 students will be expected to: 11C8 demonstrate an understanding of real-world relationships by translating between graphs, tables, and written descriptions 11C12 interpret geometrically the relationships between equations in systems 11C13 demonstrate an understanding that an equation in three variables describes a plane 11E1 demonstrate an understanding of the position of axes in 3-space 11E2 locate and identify points and planes in 3-space 1. a) Use a photocopy of the 2 cm graph paper at the end of this lesson for the horizontal x-y-z plane. It is set up for you to use as shown in the diagram on textbook page 10. Use a photocopy of the cube-a-links at the end of this lesson or similar blocks to build the two towers to represent the two points discussed at the bottom of page 9. Place the towers appropriately on the on the graph paper. b)* Write a sentence or two to explain how your towers represent the two points in the discussion. 2. a)* Complete Focus Questions 1 and 2 on page 10. 3. a)* Complete Investigation 3, Visualizing the Phone Charges on pages 10 to 12 as well as the following questions Procedure A: Why do you think the diagram suggests that you use one cube to represent $10? Why are the towers 4 cubes high? Hint: in Procedure B the piece of paper would represent the plane. CORRESPONDENCE STUDY PROGRAM PAGE 19
MATHEMATICS 11 Procedure C: Billie said that the plane is flat, and that each point on the plane has the f-value of 40, so the equation for the plane must be f = 40. What does Billie mean that the plane is "flat"? Do you agree with Billie's equation for Marv's plan? Explain. Procedure D: Why would a cube value of $5 make sense in this step? Remember, you should have 20 co-ordinates when you are finished step D. Procedure E: The piece of paper should be lying on 20 towers. Describe the graph of Leila s Plan in many ways... as u gets larger, f gets larger and this results in a plane that slopes upwards towards me... Procedure F: Why is there no c-value in the equation that describes Leila s plan? What effect does this no c-value have on the graph? How can you tell from the graph that there should be a constant value of 20 in the equation? Procedure G: Why are there so many fewer cube towers needed to represent this situation? b)* Complete Investigation Questions 3 to 8 on pages 12, and 13. c)* Do Check Your Understanding questions 9, 10 and 11 on page 13. 4) a)* Read Focus B, Sketching Planes on Paper, pages 13-17 and reproduce all the exercises on a photocopy of the isometric dot paper at the end of this lesson. The diagram below demonstrates how to use the dot paper for drawing the axes. x z When you have finished drawing the diagram at the bottom of page 17, you have a shaded triangle that represents the plane in the 3rd octant (negative, negative, positive). To represent it in the first octant (positive, positive, positive) extend the two sides of the triangle that intersect at f = 10 upwards, then join the extended sides with a third side that is parallel to the side of the shaded triangle that was not extended. This new triangle lies entirely within the first octant. Explain. 4. b)* Complete Focus Questions 12 to 14 on page 18. c)* Do Check Your Understanding questions 19 to 25 on pages 19 and 20. y PAGE 20 CORRESPONDENCE STUDY PROGRAM
LESSONS 5. Read and think about Focus C, Visualizing and Describing Intersecting Planes on pages 21 and 22. a)* Complete Focus Questions 30 to 35 on pages 22 and 23. b)* Complete a response to the Chapter Project, Student Awards on page 24. PF YOU HAVE FINISHED LESSON 2. Make sure you have completed the assignment questions and the portfolio questions. Send the assignment questions (*) to your marker now. Keep the portfolio items (PF) until after the end of the 10th lesson, then send them to your marker. CORRESPONDENCE STUDY PROGRAM PAGE 21
ISOMETRIC DOT PAPER
CUBE-A-LINKS
2 CM GRAPH PAPER
LESSONS LESSON 3 The focus of Lesson 3 is methods for solving systems by graphing and algebraically, by substitution This lesson will review methods for solving systems of equations you have already learned in grade 10. The methods are: by reading the intersection points on a graph or algebraically, by substitution. This lesson will also extend the understanding of solving systems by examining new strategies and concepts. This lesson is intended to help you see that equation manipulation is possible, and that educated choices should be made when selecting appropriate strategies. 1. a) Read Focus D, Combining Information From Different Equations on page 25 and complete questions 1 and 2. Note: substitution is the process that allows one to take the value for t from the second equation, example, "25c - 500", and substitute it into the first equation where the t is, giving: c = 18 + 0.06 (25c - 500) SAMPLE SUBSTITUTION PROBLEM Geraldine has a choice of two cleaning services for her garage. Squeeky Cleaners charges a set fee of $60 and then $1 per hour. Grease Getters charges a set fee of $30 and $2 per hour. Find the cost when both cleaners would charge the same amount. LEARNING OUTCOMES Upon completion of Lesson 3 students will be expected to: 111B15 solve systems of "m" equations in "n" variables with and without technology 11C14 demonstrate an understanding of the relationships between equivalent systems of equations 11C19 solve problems involving systems of equations c = 30 + 2h c = 60 + h substitute for h h = c - 60 c = 30 + 2(c - 60) c = 30 + 2c - 120 c = 2c - 90 subtract 2c from both sides c - 2c = 2c - 2c - 90 -c = -90 multiply by -1 c = 90 The cost when both cleaners would charge the same is $90. b)* Complete Focus Questions 3 and 4 and Check Your Understanding questions 5-7 and 10 on pages 25 and 26. c) Do Check Your Understanding question 8 on page 26 if you feel you need more practice. CORRESPONDENCE STUDY PROGRAM PAGE 25
MATHEMATICS 11 2. Read Investigation 4, Graphing Equivalent Systems of Equations on pages 27 and 28 and do procedures A and B. adding the terms together the x-values are cancelled out leaving... a)* Complete Procedures C, D, and E, and question 12 on page 28. b) Complete Investigation Questions 13 and 14 on page 28. c)* Do Check Your Understanding questions 15, 16, 17, 19, and 20 on pages 28 to 29. 3. a) Read Investigation 5, Solving Equations by Elimination on pages 30-31 and develop a response to question 22. (PF) The method of elimination is used when graphing is difficult or substitution would require complex algebraic manipulation to solve for one variable (i.e. fractions). b)* Complete Investigation Questions 23 to 27 on page 31. c)* Do Check Your Understanding questions 28, 29 and 31, on pages 31 and 32. SAMPLE ELIMINATION PROBLEM 4x - 3y = 23 and 2x + 5y = 5 First, decide which variable to eliminate. In this case, eliminate x. When choosing the variable to eliminate, consider the following: is the numerical coefficient of one variable a multiple of the other are the variables opposite in sign In the example either variable could have been eliminated. However, eliminating the x-values involved less work and reduced the chance of error. To subtract the two equations, make one variable opposite in sign and add the two equations. If neither equations contain multiples or coefficients opposite in sign you must change both equations to a common multiple. If a system of equations looks complex (i.e. rational coefficients, series of brackets) simplify the complex equation(s) before solving the system. 4x - 3y = 23 2x + 5y = 5 6 Multiply by -2 PAGE 26 CORRESPONDENCE STUDY PROGRAM
LESSONS d) Do Check Your Understanding question 30 (if you feel you need more practice). 4.* Complete Investigation 6, Comparing Solution Methods and Investigation Questions 33 to 35. Do Check Your Understanding questions 39 to 41 on pages 32-34. 5. Work on the Chapter Project, Student Awards on page 35. PF YOU HAVE FINISHED LESSON 3. Make sure you have completed the assignment questions and the portfolio questions. Send the assignment questions (*) to your marker now. Keep the portfolio items (PF) until after the end of the 10th lesson, then send them to your marker. CORRESPONDENCE STUDY PROGRAM PAGE 27