Activity 1: Solving a System of Linear Equations

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1 John Neral Teaching Notes and Solutions Objective: Students will be able to solve a system of linear equations by using the Casio fx-5es and interpret the solutions. Getting Started: When solving a system of linear equations, it is important to remember that when the equations are graphed on a coordinate plane, they will either intersect at a particular point, be parallel indicating that there is "no solution," or be identical lines indicating that there are "infinitely many solutions." Students should achieve a solid understanding of how to solve these systems by graphing, substitution, and elimination. However, once the student has obtained mastery, incorporating technology to assist in obtaining the answer will allow the student the opportunity to interpret the solution and begin to solve more complex problems. Answers to Scenario Problems:. Michael must cut lawns in order to break-even. Theoretically, Michael will break-even once he has cut /9 lawns. Since, it is unacceptable to cut only /9 of a lawn, he must complete the lawn bringing the total to lawns.. Todd and Betty must clean for 37 /7 hours in order to earn $, A "dog day" is defined as taking care of one dog for one day. Since vacations typically run for more than one day, we will use the "dog days" to indicate $.50 earned per dog, per day. Therefore, Denny and Jeri will have to work 400 "dog days" in order to earn $5,000. Answers to Problems:. Y = 0,000 ; Y = 50X This system is best solved by substitution. Michael will need to cut 00 lawns in order to earn $0,000.. N + Q = 5 ; 0.05N + 0.5Y =.5 Let N = Nickels and Q = Quarters This system is best solved by elimination. There are 8 nickels and 7 quarters. 3. (4, ) 4. (0.75, -0.5) 5. While the calculator displays the solution as a "Math Error", we can see from the equations that the second equation is double the first equation. We can interpret this system as having infinitely many solutions. Extension: Provide a detailed business plan for any of the three scenarios. In your business plan, you should include three different fee schedules and a rationale as to which fee schedule is best. Keep in mind that charging more does not always equate to earning more. There is always a balance between supply and demand and careful attention to detail is essential to any successful business.

2 Student Worksheet Activity You have been hired as a business consultant to work with three new business owners. The owners are opening their businesses in three months and are looking for some sound financial advice from you. Your job is to determine the break-even point for each of the businesses and provide each owner with alternatives for reaching their individual break-even points. Examine each of the scenarios and answer the questions that follow. Scenario : Michael is opening a landscaping business. He believes that the best way to begin his business is to start small and simply cut lawns. Because he is limiting his business to the small town of Oakville and the lawns are relatively the same size, he is able to charge the same price per lawn. Michael has bought a new lawn mower and supplies totaling $950. This amount equals his "start-up" costs for his business. Michael will charge $45 to cut each lawn. How many lawns must Michael cut in order to break-even? Scenario : Todd and Betty have established a housecleaning business. They will charge $8 per hour for their combined cleaning services. They purchased $40 worth of cleaning supplies as a start-up cost, but have informed all homeowners that they are responsible for the purchase of any additional cleaning supplies. Todd and Betty believe that by making the owners responsible for the supplies, it will save them money. How many hours must Todd and Betty clean in order to make a profit of $,000? Scenario 3: Denny and Jeri are creating a dog walking service. They believe this is business will thrive because people who don t want to put their dogs in a kennel will hire them to care for their dogs when they go away on vacations. Denny and Jeri will charge $.50 per dog, per day and their services include visiting the animal twice per day along with daily walks and feedings. How many days must they care for dogs in order to profit $5,000? Calculator Notes:. Turn the fx-5es ON.. Press MODE. 3. Press 5 for EQN (Equation). 4. Press for anx+bny=cn. 5. Input the values for a, b, and c for each equation into the matrix. (Note: All linear equations must be expressed in standard form in order to enter the values into the fx-5es.)

3 Student Worksheet Activity (continued) 6. Once you have entered all of the values, press = to obtain the value for X. 7. Press = again to obtain the value for "Y". 8. Press = again to return to the equation matrix. 9. Press AC to clean the values contained in the matrix. Problems:. With the cost of oil increasing, Michael is forced to raise his cost per lawn. If Michael charges $50 per lawn, how long will it take him to earn $0,000? Write a system of equations, state whether it is best to solve by substitution or elimination and state your answer.. There are 5 coins in a jar. Some of the coins are quarters and some are nickels. Their total value is $.5. How many quarters and how many nickels are there in the jar? Write a system of equations, state whether it is best to solve by substitution or elimination and state your answer. 3. Solve the following system. X Y = 6 3X + 5Y = 4. Solve the following system. 3X 7Y = 4 6X + 4Y = 5. Solve the following system. 9X + Y = 37 8X + 4Y = 74 3

4 Activity : Solving Matrices John Neral Teaching Notes and Solutions Objectives: Students will demonstrate the ability to input values into a matrix and determine its solution. Students will also demonstrate the ability to interpret the solution from a given matrix problem. Getting Started: Matrices are designed to organize information. Whether you are compiling statistical data or displaying data, matrices present a simple and detailed representation of large quantities of data. Some may choose to think of matrices like a spreadsheet where the cells are the locations of various pieces of data. It is important to realize when information can be operated within matrices. For example, students should clearly understand when two or more matrices could be added or subtracted. Furthermore, students should know when matrices can be multiplied together or how a scalar impacts the data within a spreadsheet. Answers: Orders of small fries were not as frequent in May as they were in April. 3. Projected Sales for June Projected Figures (June) - Actual Figures (June) = Answer Matrix A positive value signifies that your projection was greater than your actual sales figure meaning that you would have had enough products to meet your consumer demand. A negative value signifies that your projection was less than your actual sales figure. In both cases, this data can be useful in planning future projections, determining sales figures for a particular month, and determining trends in consumer buying preferences. Activity : Solving Matrices

5 Activity : Solving Matrices Student Worksheet Activity You have just been promoted to manager of a local fast food restaurant. You are responsible for all of the ordering as well as tracking sales records for your products. Examine the following sales records for April and May and answer the following questions. April Sales Totals May Sales Totals Calculator Notes: To Enter Data Into a Matrix: Turn Calculator ON. Press MODE and Enter 6 for MATRIX. Press for Matrix A. Select the matrix dimensions from the list. Enter the values for Matrix A into their appropriate locations. Once you have entered the last value, press AC. To enter values for another matrix, press MODE 6 for MATRIX and repeat the process. To Perform Matrix Calculations: Once you have entered all of the data for a given matrix and are ready to perform a particular operation, press AC to exit the Matrix editor and enter into the calculation mode. Make certain that MAT appears at the top of the screen as this indicates that you are in the Matrix Calculation Mode. Press SHIFT 4 to perform a Matrix Operation. To add Matrix A with Matrix B, press 3 for Matrix A + SHIFT 4 followed by 4 for Matrix B, then press =. Activity : Solving Matrices

6 Activity : Solving Matrices Student Worksheet Activity (continued) Problems:. Create a new matrix that shows the combined totals for April and May.. Which product was purchased less frequently in May than in April? 3. During the summer, sales typically more than double. Your supervisor suggests that you take the May figures and multiply them by.3 in order to project sales for June. Create a new matrix that shows your projected sales figures for June. Round all answers to the nearest whole number. 4. At the end of June, you compare the actual sales figures with your projections. Calculate the differences and describe how this information is useful in planning future projections. 5. Once you have compared the projected results with the actual results, what does a negative value represent? What does a positive value represent? Activity : Solving Matrices 3

7 Activity 3: Exploring Functions and Corresponding Tables of Values John Neral Teaching Notes and Solutions Objective: Students will demonstrate the ability to input a function, generate a table of values, and interpret those values. Getting Started: This activity reinforces the concepts of functions. Understanding the properties of linear, quadratic, and absolute value functions is essential to mastering Algebra. Students should be able to graph these functions on a coordinate plane, generate a table of values, and interpret those values. While the Casio fx-5es does not generate a visual representation of the graph, it does provide a table of values that can be easily used to interpret the results of any function. Once functions are transferred to various problem-solving scenarios, integrating this technology can be an extremely useful tool in the solving and analyzing process. Answers:.. X F(X)= X X F(X)= X + 6X X X F(X)=π X F(X)=x Activity 3: Exploring Functions and Corresponding Tables of Values

8 Activity 3: Exploring Functions and Corresponding Tables of Values Student Worksheet Activity 3 Problems: For each of the following problems, create a function, enter it into the Casio fx-5es, and generate a table of values to help you solve the problems.. You have opened a savings account at a local bank that gives you simple interest on your money. Based on the formula, I = Principal X Rate X Time, you are trying to determine how much interest you will generate over the next ten years. If you open the savings account with $50 and the money will earn.5% per year, how much interest will you earn each year for the next ten years?. Determine the area of a circle whose radius ranges from cm 0cm. Use a step increment of cm. Generate a table of values to determine your answers. 3. Create a table of values for the function, f(x) = x + 6x + 8, where 5 < x <. 4. Given the function, f(x) = x - 9, evaluate the function where -5 < X < 5. Extension: Write a linear function that models a business scenario where you earn a certain amount of money per job and must deduct some type of expense. Create a table of values that will show how much money you will profit after completing a certain number of jobs. Calculator Notes: To Enter a Function and Create a Table of Values: Press MODE. Press 7 for TABLE. Enter the function after f(x) =. Press =. Set a Start Value. Press =. Set an End Value. Press =. Set a Step Value. Press =. Activity 3: Exploring Functions and Corresponding Tables of Values