"user manual"
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1. 5 3 1 1 3 5 6 Let s see if this works with more systems Each system below has only one equation given Assign each system to one person in the group and create a second equation that will give the solution 1 2 when solved System A System B System C Equation 1 x 6y 11 Equation 2 Equation 1 14x 9y 4 Equation 2 Equation 1 9x 7y 2 5 Equation 2 7 Share your equation with everyone in the group and copy down the equations from the rest of the group Review the equations created and make sure they follow the rule that you noted in 5 You will solve these three systems as a part of your home activity TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 38 4 4 4 The Sub Steps Solving using the method of substitution requires five steps The steps are given below in the text boxes Discuss with your partner what you think the correct order is for the steps and then write the steps in the space provided Solve the system in the chart as model of solving by substitution Check using your handheld State the point of intersection Solve the resulting equation Substitute the isolated expression into Substitute your solution into an original the other equation equation to solve for the other variable Isolate for a variable The easiest variable to isolate for has a coefficient of 1 Example Solve Steps for Solving by S2. 12 Record the line that results once you have substituted and then solve the equation using your CAS handheld Resulting equation and my solution Don t forget to solve for both variables Check your solution using the CAS handheld Compare your solution to someone else in the class TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 29 4 3 4 The Sub Way e n groups of three have each person in the group solve one of the systems below e Use your CAS handheld to help you solve and check your system e Share your solutions with each person in the group System A System B y 24x 24andy 5x 12 13x y 4and 5x y 4 0 System C Challenge y X 8 and y 5x CHALLENGE Plot each of the POI s from Systems A B and C and find the equation of the line that connects the three points Equation of Line TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 30 4 3 5 What s My Equation Part 2 Problem A Yasser is renting a car Zeno Car Rental charges 45 for the rental equations of the car and 0 15 per kilometre driven Erdos Car Rental charges 45 0 15x 35 for the rental of the same car and 0 25 per kilometre driven y For what distance do the two rental companies charge the same Pub P 9 y 35 0 25x Problem B The school council is trying to determine where to hold the athletic Equatio
3. Finalize their three linear system problems making necessary changes based on input from peers CGE 2c 4f 5e 5g 10 Final Evaluation This is a one day paper and pencil test that evaluates student ML3 01 ML3 02 learning for the remaining overall expectations and math ML3 03 This is available from processes the members only Students will apply their knowledge to generic situations as well section of the OAME as real world contexts Web site Students will generate graphical and algebraic models from the AUW Oe OTL ca data calculated in tables of value 11 Jazz Day TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 2 Nspire CAS Handheld Manual Getting Started When you turn on the handheld press Q0 You will be asked whether you want to save the Do you want to save this document document Select No To do this use the large circular navpad to move to the right then press the Yes amp button in the middle of the navpad 1 Add Calculator Next select 1 Add Calculator To do this press the button again 2 Add Graphs amp Geometry 3 Add Lists amp Spreadsheet AA ddiRibres You are now ready to use CAS on the handheld 5 Add Data amp Statistics Some Helpful Shortcuts If you make a mistake at any point that you want to undo press C If you undo something that you want back again press C v How to Solve for a Variable Example One Say that you wish to
4. Support your answer with an example that includes a graph TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 8 4 1 1 Working on Commission Nahid works at Euclid s Electronics She is paid a salary of 200 per week plus a commission of 5 of her sales during the week The equation P 0 055 200 represents Nahid s pay for the week where P represents the total pay for the week and s represents her total sales If Nahid earned 290 in a week use the Euclid s Electronics Wages equation to algebraically determine how 500 much she sold Use your handheld to help you solve Refer to the user manual if you need to review how to solve using the handheld Week s Pay 1000 1500 2000 2500 Total Value of Sales Nahid is offered another job at Fermat s Footwear where the pay is a salary of 100 per week and 10 commission on all sales The graph below represents the Pay vs Sales for this job Which of the following equations do you think represents pay for one week at Fermat s Footwear Fermat s Footwear Wages 500 eL LLL LLLI b P 20 105 100 E LE tt d ft d c P 1005 10 ss 21 d 3 ot tt tt tt ee Reese are a P 2 0 015 100 d P 0 05s 200 xeu rr the equation for Euclid s Electronics for hints B eee i TTELECELHE TET i 500 1000 1500 2000 2500 Provide a reason or justify why you selected the equation that you chose Refer back to Week s Pay Tot
5. Using BLM 4 3 3 display the first system on an overhead projector Ask the class to identify what is different about the system they see on the overhead projector from the ones that they were able to solve on BLM 4 3 2 Stress student responses that highlight the fact that both equations now have two unknowns Note The first system is one of the systems they solved in the Minds On BLM 4 3 1 System B To solve the system lead the class in demonstrating the method of substitution by cutting the overhead along the dotted lines and physically replacing the y in the second equation with the expression 2x 7 from the first equation Ask students why this is correct mathematically Distribute BLM 4 3 3 Student Worksheet and ask students to solve the resulting equation Repeat this process using the second system on BLM 4 3 3 Note that in the second system we could isolate either x or y Small Groups gt Worksheet Using BLM 4 3 4 in groups of three each student solves one of three systems The point of intersection for each is System A 4 8 System B 0 4 System C 2 10 The three points are collinear The students are to determine the equation of the line connecting all three The correct equation is y 3x 4 gt CAS to solve their equations using an explicit method not using the SOLVE command If an overhead projector is not available use scrap paper with the individual pieces of both equations
6. zy 200 8 LL tt tt tt HRE HHHH 350 350 900 400 400 600 450 450 Pt TT ey yy yy 300 309 309 ELA TETTE 50 100 150 200 250 300 350 400 450 500 Number of Books How can the flat fee and the cost per book be used to draw the graph N What is the point of intersection of the two lines What does it represent e Under what conditions is it best to go with Descartes Publishing AR Under what conditions is it best to go with School Memories TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 14 4 1 2 What s My Equation Continued Problem D The school is putting on the play Algebra The Musical Adult tickets were sold at a cost of 8 and student tickets were sold at a cost of 5 A total of 220 tickets were sold to the premiere and a total of 1460 was collected from ticket sales How many adult and student tickets were sold to the premiere of the musical To solve the question complete the table of values and the graph Let x represent the of student tickets sold Let y represent the of adult tickets sold Algebra The Musical D x y x y x 0 0 40 40 E 80 80 E 120 120 5 160 160 D 200 200 t 2 z 20 40 60 80 100 120 140 160 180 200 Number of Student Tickets 1 What is the approximate point of intersection of the two lines What does it represent 2 Does the rest of the graph ot
7. Point of intersection The student council has 1 800 in their budget for the lunch They prefer Let s Do Lunch what is the greatest number of grade 10 students they can have at the lunch What does the ordered pair 25 750 mean on the Lunch Express line TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 67
8. and Music Mine are two online music providers Each company charges a monthly membership fee and then a per song download rate iTones charges 10 per month and 1 per song C 2 x 10 Music Mine charges 7 per month and 1 50 per song C 15x 7 1 Find the number of songs that would need to download where the costs are the same for the two music providers 2 Paris solved the problem and then made the following conclusion If you download less than 6 songs per month than choose iTones since the cost per song is less If you download more than 6 songs per month than choose Music Mine since the fixed cost is less If you download exactly 6 songs per month choose either Is the conclusion that Paris made correct If not underline the part s of her conclusion that are incorrect and then rewrite it so that it is correct TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 34 4 4 2 Putting the Pieces Together Solve the system y 10 2x and x 2y 10 The solution to this system is given in the pieces below Cut the pieces out and glue them in the correct order on a separate piece of paper Use the handheld to help you Hint There are 10 steps in the complete solution 2 lt TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 35 4 4 2 Putting the Pieces Together Solution Teacher Solve the system y 10 2x and x 2y 10 The solution to this system is given in the piec
9. choose If the job lasts more 30 w I b ee than hours choose l If the job lasts hours 20 choose either company and the cost is 10 IS 12 3 4 656 6 78 9 10 Time hours Car Rental Companies Point of intersection Interpretation of the point If the kilometers driven is less than choose If the kilometers driven is more than choose If the kilometers driven is choose either company and the cost is 10 20 30 40 50 60 70 80 90 100 Kilometers Driven TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 21 4 2 3 Where Do We Meet Continued For each of the following situations find the point of intersection and describe the meaning of this point Refer back to the template provided for the first situations Snowboard Rental LL T3 T LE e ELI Point of intersection Interpretation of the point Cost M QJ LA un e T e e Time hours Long Distance Plans B CI Point of intersection LLL LLL lel LE AA td Interpretation of the point LET OL WT TL IBS LLL LM Lu i Cost LLL MEE E LLLI Jr tal 157 4 E tT tt Yi i tt tt EE i i a 10 15 20 25 30 35 40 45 50 Talking Time hours TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 22 4 2 4 Does This Line Cross From the list of relations below determine which lines cross through the point 2 3 You ma
10. each equation that you re arranged What do you notice about how the two lines visually relate to each other Is there a solution to this linear system If so how many TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 53 Unit 4 Day 7 Which One is Best Grade 10 Applied Minds On 20 Min Math Learning Goals e Identify necessary conditions or look fors in a system of equations that make it best suited to be solved by graphing substitution or elimination Materials e BLM 4 7 1 4 7 5 e Computer Algebra Systems CAS Action 35 Min e Solve a system of equations using two of the three methods Consolidate e Solve a system of linear equations using the most appropriate method Debrief 20 Min Total 75 Min Assessment Opportunities Minds On Whole Class Kinesthetic Activity Teacher explains the 4 Corners activity and indicates that signs are in each corner representing the methods of Graphing Substitution Elimination and Not Sure BLM 4 7 1 The teacher then puts up the overhead BLM 4 7 2 displaying the first set of equations Students must go to the corner with the method that they believe is the best method for solving Each corner must justify their method choice Work through the three sets of equations Whole Class Discussion The overhead displaying all of the 4 Corners questions is displayed Students are asked to match the three sets up with
11. linear system by elimination Ax 2y 12 8x 4y 32 2 Did you encounter any results that are unusual Explain what is different compared to questions you have already solved 3 Re arrange each equation from the linear system into y mx b form then graph For example here is how the first equation can be re arranged Ax 2y 12 2y 4Ax 12 2 4 12 Se qur 2 2 2 y 2x 6 TT TT TT TTT eer yet yt tT et Te Ltt TE ETT LT TT TTT TTP ey tt LT TTT TT te eee te eT Tt BERR EEE See BERR eee BERR CE PT TT TTT eT EI eee eee LTT TTT TTT ee TTT Tt tT tT 4 What do you notice about the slope in each equation that you re arranged What do you notice about how the two lines visually relate to each other Is there a solution to this linear system TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 52 4 6 6 Here s To The Crazy Ones Continued 5 8 Solve the following linear system by elimination 6x 2y 2 12x 4y 4 Did you encounter any results that are unusual Explain what is different compared to questions you have already solved Re arrange each equation from the linear system into y mx b form then graph LITIIIIIIIPREITITIIITIL LLLI EEE eee LT TT TTT Teer eT tt LT TTT TT tT ee te eT TT BEER EEE eee BERR eee BERR Cee PT TT TTT eT TT Eli eee eee SL ZR pTIUPIpip m ppp DIT LTT TT TTT Tee TT TT tT tT TT What do you notice about the slope and y intercept in
12. of the problem BLM 4 2 3 Consolidate Debrief Pairs Investigation Through a series of graphing activities BLM 4 2 4 student explore the point of intersection from different perspectives Whole Class Discussion As a class discuss the strategies used Left Side Right Side verification is discussed as an alternative tool to graphing all of the lines to solve Questions and 2 Students brainstorm as a class about Questions 3 and 5 Concept Practice Exploration Home Activity or Further Classroom Consolidation Solve the system of equations problems BLM 4 2 5 TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 18 4 2 1 A Visual Cell Phone Problem Two cell phone companies charge a monthly flat fee plus an additional cost for each minute of time used The graph below shows the Time vs Cost relationship for one month Cell Phone Costs Talk More We Talk Cost 200 400 600 800 1000 1200 Time minutes 1 What is the Point of Intersection POI and what is the meaning of the POI in relation to the cell phone plans 2 Under what conditions is it best to use the Talk More cell phone plan 3 Under what conditions is it best to use the We Talk cell phone plan 4 How does the graph help you to determine which cell phone plan is the most appropriate at any given time TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 19 4 2 2 Music is
13. one of the methods and they can only use each method once E g System 1 is best suited to elimination System 3 could be graphing or substitution but substitution is best used for System 2 therefore System 3 is graphing Discuss what aspects make a system of equations best suited to a certain method e E Action Pairs Worksheet Utilizing Think Pair Share students must solve the three sets of equations choosing one of them to solve using two methods BLM 4 7 3 Whole Class Guided Exploration Take up the three sets of questions by having students put their solutions on the board using as many methods as are appropriate Discuss the key aspects again for selecting a method Consolidate Debrief Small Groups Model Making Students work in groups of three to complete a Frayer Model BLM 4 7 4 on one of the Graphing Substitution or Elimination methods Assign Frayer model as an assignment to be submitted next class Application Concept Practice Home Activity or Further Classroom Consolidation Students practice solving using multiple methods and identifying the best method BLM 4 7 5 Remind students to start thinking about ideas for summative where they must create their own real life question with systems of equations TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 54 4 7 1 Four Corners NOT SURE TIPS4RM Grade 10 Applied Unit 4 Linear Sy
14. page If you need help with this see the Getting Started section from earlier in this manual Press the following keys HPD DOOOPOOPVOODOD What you have typed should look like this RAD AUTO REAL a Press the key The handheld will 3 2ct3 Sy 3 7 display the result RAD AUTO REAL Now we have a system that we 6x 15y 21 TERET Ee A can begin to solve by subtracting x ay mj the equations TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 6 Installing Tl Nspire CAS Key Fonts Teacher In order to have the keys shown in your Word documents you need to download and install the Nspire Keys font on your computer Here s how Download the fonts 1 Goto www education ti com 2 Select Downloads from the top of the screen 3 Go to the Key Fonts section and select TI Nspire Key Fonts from the drop down box you may have to scroll to the bottom 4 Click on the Download button 5 Click on TI Nspire tm Key fonts for Windows r Make sure you know where you are saving the fonts i e Desktop My Documents etc Install the fonts 1 Go to the Control Panel on your computer Most computers just require you to click on the start button and the control panel folder should be visible from the start menu 2 Open the Fonts folder by double clicking on it 3 Go to the File Menu and Install New Font 4 You will now need to find the T NspireKeys font file Once you find it select it from th
15. y 0 25x 35 14 y 2 4 50x 475 15 y 2 85x 4 0 25 16 5x 8y 1460 TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 11 4 1 2 What s My Equation Continued Problem A Yasser is renting a car Zeno Car Rental charges 45 for the rental of the car and 0 10 per kilometre driven Erdos Car Rental charges 35 for the rental of the same car and 0 25 per kilometre driven Which company should Yasser choose to rent the car from To solve the question complete the table of values and the graph Zeno vs Erdos Zeno Erdos Distance Cost Distance Cost km km 0 0 10 10 20 20 30 30 ea 40 40 z 50 50 amp 60 60 70 70 80 80 90 90 100 100 10 20 30 40 50 60 70 80 90 100 Kilometers Driven 1 How can the car rental cost and the cost per kilometre be used to draw the graph 2 What is the point of intersection of the two lines What does it represent 3 Under what conditions is it best to rent from Zeno Car Rental 4 Under what conditions is it best to rent from Erdos Car Rental TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 12 4 1 2 What s My Equation Continued Problem B The school council is trying to determine where to hold the athletic banquet The Algebra Ballroom charges an 800 flat fee and 60 per person The Geometry Hall charges a 1000 flat fee and 55 per person Which location shoul
16. 16 5x 2y 8 You probably agree that if we add these two equations the y variable will be eliminated Here is how to do this on a handheld First be certain that you are on a Calculator page If you need help with this see the Getting Started section from earlier in this manual When you type in the equations be certain to enclose them in brackets Remember that we decided to add the equations Here are the keys you should press OVOOOVOVOO OO POOPUOG It will look like this on your handheld RAD AUTO REAL a Press the key The handheld will display the result RAD AUTO REAL a You have eliminated the y variable Sct2 y 16 5 x 2 y 8 8x 24 i Subtracting two equations works the same way as adding two equations The key is to remember that you must enclose each equation in brackets when you type it into the handheld How to Multiply to Find an Equivalent Equation Consider the following linear system 2x 5y 7 6x 3y 3 If we immediately add or subtract the equations neither the x or the y variable is eliminated Instead we must multiply one of the equations by an integer so that the coefficients match Then if we subtract the equations a variable will be eliminated Let s multiply the first equation by 3 Remember that all terms on both sides of the equation must be multiplied by 3 so that the equation stays balanced Here is how to do this on a handheld First be certain that you are on a Calculator
17. My Best Friend iTones and Music Mine are two online music providers Each company charges a monthly membership fee and then a per song download rate iTones charges 10 per month and 1 per song C n 10 Music Mine charges 7 per month and 1 50 per song C 1 5n 7 Where C represents the total cost for one month and n represents the number of songs purchased Create a table of values showing the total charges for up to 8 songs purchased Graph the lines on the same graph below iTones Music Mine Online Music Purchasing 20 C n n 0 0 1 1 Cost 23 45 6 78 89 e Number of Songs 1 If Lulu plans to purchase 7 songs this month which is the best plan for her Explain 2 Which plan is cheaper if you only plan to buy 4 songs per month How do you know from the graph 3 Which cell phone plan would you choose and why Relate your answer back to the POI TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 20 4 2 3 Where Do We Meet For each of the following situations find the point of intersection and describe the meaning of this point Describe which company or service you would choose under what circumstances A template has been provided for the first situations Lawn Service 1074 7 TF YO ggi_ 1A Lo ud an Point of intersection 70 Interpretation of the point 60 D z 50 If the job lasts less than hours amp 40 A
18. Unit 4 Grade 10 Applied Linear Systems Lesson Outline BIG PICTURE Students will e solve systems of two linear equations and solve related problems that arise from realistic situations A summative task and paper and pencil assessment are included as a part of the unit Teachers can decide to use one or both assessments for the unit If both assessments are used then the unit will last ten days The unit was written so that the use of CAS is integrated into every lesson Although there are no examples showing specific screen shots it is expected that students will have it available to use for solving and checking whenever needed However the materials in this unit can be used without CAS A CAS user manual is included as well as video tutorials for the skills outlined in the manual The manual and videos include keystrokes for the commonly used skill sets needed for linear systems such as solving for one variable or adding subtracting two equations This manual should be available to students as they work through the concepts in the unit To be able to see the keystrokes the TI Nspire key fonts must be installed on your computer A guide to install the key fonts is included Day Lesson Title Math Learning Goals Expectations 1 Where Do We Meet e Identify Solve problems that arise from realistic situations that ML1 02 ML 2 05 can be modelled by two linear relations ML3 01 ML3 03 e Determine graphically the point of intersection
19. al Value of Sales TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 9 4 1 1 Working on Commission Continued Which One is Better Nahid needs help determining which job she should keep She decides to look at them as a system of equations when she creates a graph comparing the two equations at the same time Analyze the graph and complete the questions below Euclid s Electronics Fermat s Footwear 500 1000 1500 2000 2500 Total Value of Sales 1 Where the two lines cross is called the point on intersection or the solution to the system At what coordinates do the two lines cross 2 What does this coordinate represent in terms of Nahid s sales and pay for the week 3 If Nahid usually makes 1500 worth of sales per week which job should she take Explain 4 How does the graph help Nahid determine which is the better job 5 What does the point 1000 250 represent in the graph TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 10 4 1 2 What s My Equation You are given four problems below Each problem will require two equations to solve it The equations that are needed to solve each problem appear at the bottom of the handout Match the equations with the problems and compare your answers with another student Note There are more equations than problems and all the equations use x for the independent variable and y for the dependent variable Problem A Equat
20. class Whole Class Discussion Ensure that all of the students have the scenarios and equations matched properly Review the parts of y mx b and how the words in the scenario matched the components of the equations E g flat fee corresponds with b the y intercept Small Groups Practice Assign students a letter A through D and then have them work in their letter group attempting the problem associated with their letter Students practice analyzing systems of linear equations creating a table of values plotting points finding the point of intersection and interpreting the meaning in the larger context of the problem BLM 4 1 2 Afterwards they must recreate the graph on a sheet of chart paper and summarize their findings along with the questions I Consolidate Debrief Whole Class Presentation Each of the letter groups presents their findings and displays their chart paper graph Whole Class Discussion Discuss the similarities in the types of questions for Problems A B and C and what makes Problem D different e Chart paper presentations should be evaluated to be aware of misguided understanding Reflection Skill Drill Home Activity or Further Classroom Consolidation Assign BLM 4 1 3 as home activity and ask students to complete the following journal entry Journal The equation of a vertical line is x 2 a and the equation of a horizontal line is y b Where will these lines intersect
21. d the school council select for the athletic banquet To solve the question complete the table of values and the graph Algebra Ballroom Geometry Hall Algebra Ballroom vs Geometry Hall Number Number of Cost of Cost People People 0 0 10 10 30 30 3 40 40 gt 50 50 60 60 70 70 80 80 90 90 100 100 G 20 30 40 50 60 70 80 90 100 Number of People 1 How can the flat fee and the per person cost be used to draw the graph 2 What is the point of intersection of the two lines What does it represent 3 Under what conditions is it best to go with Algebra Ballroom 4 Under what conditions is it best to go with Geometry Hall TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 13 4 1 2 What s My Equation Continued Problem C The yearbook club is considering two different companies to print the yearbook The Descartes Publishing Company charges a flat fee of 475 plus 4 50 per book School Memories charges a flat fee of 550 plus 4 25 per book Which company should the yearbook club select to print this year s yearbook To solve the question complete the table of values and the graph Descartes School Memories Descartes vs School Memories Number Number 3000 Hime cost Formoe Cost TTT TTT 0 0 50 55 uw LE LL LA 100 100 2100 LLL LLL LLLLJI e 150 150 9 1800
22. e CAS manual included at the beginning of this unit file The purpose of this exercise is for students to understand that adding and subtracting equations can eliminate a variable After completing the exercise students should be able to identify what conditions are necessary for elimination to occur when adding and what conditions are necessary for elimination to occur when subtracting The use of the CAS on the handheld is encouraged to allow a student to complete this exercise even if their algebra skills are not at grade level 4 5 3 Solving a Linear System by Elimination Teacher Notes Guide students to a solution for the system Use of CAS is encouraged so that students with algebra skills that are below grade level are still able to learn the elimination algorithm Students to describe in their own words the steps required to solve a linear system by elimination after guided example Show students how to check a solution to a linear system using CAS on the handheld see CAS manual at the beginning of the unit file TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 41 4 5 1 A Trip to Jim Hortons It s summer vacation Ah sweet freedom The only problem is that you re the designated coffee gopher at the office where you have a summer job On Monday you were sent out to Jimmies to pick up five small coffees and seven extra large coffees You remember that the total cost was 14 95 including tax On Tuesday
23. e List of Fonts box and select OK 5 An easier way Minimize all windows except the Fonts folder window Reduce the size of this window so you can see the TINspireKeys font file on your desktop or in a folder and drag and drop it in the Fonts folder TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 7 Grade 10 Introduce the idea of systems of equations by building on linear application problems from the previous unit BLM 4 1 1 Whole Class gt Discussion Summarize the idea of a Point of Intersection or Solution to a System Unit 4 Day 1 Where Do We Meet Applied Math Learning Goals Materials Minds On 15 Min dentify Solve problems that arise from realistic situations that can be rh ie modelled by two linear relations C a ia Algebra Action 45 Min Determine graphically the point of intersection of two linear equations Systems CAS Consolidate e Interpret the story the graph tells including point of intersection and what Debrief 15 Min occurs before and after the intersection point Total 75 Min Assessment Opportunities B Minds On Small Groups gt Guided Exploration Divide class into homogeneous groups of at most 4 students per group Action Pairs gt Worksheet Utilizing Think Pair Share students interpret the system of equations application scenarios BLM 4 1 2 connecting them to the appropriate equations then discuss with the
24. e the system using each of the three methods you have learned and if you can then solve y 2x 7 y 4x 5 Graphing Justify why you can or cannot Justification solve using this method Substitution Elimination Justification Justification TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 62 4 7 4 The Frayer Model Facts Characteristics Non Examples TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 63 4 7 5 3 Ways Two catering companies provide food and the banquet hall for weddings proms and anniversaries Nick and Heather are getting married in September and they have two catering companies to choose from Cookie s Frugal Gourmet Catering Minestrone Cesaer Salad Soup Chicken Picatta Mixed green Roasted salad Potatoes Prime Rib Steamed Garlic Mashed Vegetables Potatoes Sherbert Asparagus Coffee or Tea Apple Pie Coffee or Tea The cost C for the different menu options includes the cost of the hall rental and price per person n Cookie s Catering C 40n 500 Frugal Gourmet C 45n 350 Cookie s Catering Frugal Gourmet Solve the system using the graphing method 10 20 30 40 50 60 70 80 90 100 Point of intersection TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 64 4 7 5 3 Ways Continued Solve the system using subst
25. elop a context include a linear system that describes the context and then solve the linear system by graphing and by the students choice of substitution or elimination Make five copies of BLM 4 6 1 on an overhead transparency Select constants from the group 2 4 8 12 and 20 Concept Practice Skill Drill Home Activity or Further Classroom Consolidation Using BLM 4 6 6 provide practice with questions that have parallel and coincidental solutions TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 46 4 6 1 What s the difference x 2y 6 Original equation The new equation is Multiply the equation by the constant Complete the table of values for your equation and plot the values on the grid provided ptt tt tt eiii 4 47 TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 6 2 Elimination Preparation 1 Consider the following linear system 3k 4 15m 15 k m 1 If you add or subtract the equations will a variable be eliminated Explain 2 What could be done to create the conditions necessary for elimination TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 48 4 6 3 Algebra the Musical Redux Recall that in the first lesson of this unit you solved the following problem by graphing The school is putting on the play Algebra The Musical Adult tickets were sold at a cost of 8 and student ticket
26. equation You probably see that to finally isolate the y variable it is necessary to divide the equation by 2 on both sides Again just start typing the operation you want to perform Press i The handheld will insert Ans for you Press to calculate the result As you can see the handheld reports that RAD AUTO REAL y 3 x 3 6 x 2 y 18 6x 2 y 18 A Is this the result you expected E meS Your teacher will discuss this with your class 2 y 18 6 x y 3 x 3 After the discussion use the space below to write your 2 own explanation of what this means TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 4 Nspire CAS Handheld Manual Continued How to Check a Solution to a Linear System Say that you have solved the following linear system AX 2y 24 8x 6y 18 and you believe the solution is E 3 This would be tedious to check by pencil and paper but it is quick to check with the handheld Here is how to do it First be certain that you are on a Calculator page If you need help with this see the Getting Started section from earlier in this manual You are going to press the following keys to check the solution against the first equation QUO 09 Y C ty 9 0 9X Qt This means Check this equation such that x z and y 3 Don t forget to press the key lower right corner of keypad before and after the ANO Here is what it will l
27. es below Cut the pieces out and glue them in the correct order on a separate piece of paper Use the handheld to help you Hint There are 10 steps in the complete solution x 2 10 2x 10 X 20 4 4x 2 10 5x 10 20 5x 30 Xx 30 5 x II o y 10 2 6 y 10 12 y 2 Point of Intersection 6 2 TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 36 4 4 3 An Interesting Problem Consider the following three systems of equations x 2y23and 4x 5y 6 2x y 4and 7x 10y 13 5x 3y 2 1 and x 3y 5 1 In order to solve these systems by substitution we need to first isolate one variable in one equation Circle the variable in each system that would require just one step to isolate Compare your choice with a neighbour 2 Using your handheld to help you isolate the variable you selected in 1 for each system 3 Assign each system to one person in your group and solve the system assigned to you in the space below Use your handheld to help solve and check am solving system 4 Compare your solution to the rest of the group What do you notice TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 37 4 4 3 An Interesting Problem Continued 5 The chart below lists out only the numbers that appear in each system What do you notice about the numbers in each system System A System B System C 1 2 3 4 5 6 2 1 4 7 10 13
28. ess to calculate the result As you can see the handheld reports that y 3x 8 TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 3 Nspire CAS Handheld Manual Continued How to Solve for a Variable Example Two Say that you wish to solve the equation 6x 2y 18 for the variable y To do this first be certain that you are on a Calculator page If you need help with this see the Getting Started section above First type in the equation that you want to solve Use the number pad and the green letter keys the operations x are located on the right and the equals sign is in the top left corner of the keypad When you have typed in the equation press the key found in the bottom right corner RAD AUTO REAL a The top of your screen will look something like this 6 2 y 18 6 x 2 y 18 A Now decide how you would start solving for y Perhaps you ve decided that subtracting 6x from both RAD AUTO REAL m sides of the equation is a good start Wonderful To 6202 18 6 x 2 y 18 e do this immediately press Ce o Notice that the handheld automatically inserts Ans What is this Ans 6x Ans stands for the last answer you found If you now RAD AUTO REAL Ll press the key the handheld will subtract 6x from A 6 x 2 y 18 the left side and the right side of 6x 2y 18 You will see this result 6 x 2 y 18 6 x 6 xt 2 y 18 2 y 18 6 x Continue solving the
29. fficient for isolating and then solve linear systems using the method substitution Applied Math Learning Goals Materials Minds On 10 Determine the conditions for which a linear relation application is preferred to BLM 4 4 1 4 4 4 the algebraic models by comparing the rate of change and initial value of two dt ue Action 55 linear relations making up a system of equations 5 Capa Alpe Systems CAS Assessment Opportunities ie Minds On Individual Worksheet Distribute BLM 4 4 1 to each student Students are to complete the solution to the contextual problem as a review of solving by substitution Optional Discuss the home activity assigned in yesterday s class to consolidate the algorithm for solving by substitution e E Action Pairs Kinesthetic Activity Distribute BLM 4 4 2 and a blank piece of paper to each pair of students Students are to cut the boxes necessary for the solution of the system at the top of the handout and glue them on the blank piece of paper Once they have completed their solution they are to compare their solution with another pair of students Whole Class Discussion Ask students what is different in solving this system from the systems they were solving the previous day Highlight the need to distribute once they have substituted in the expression Small Groups gt Investigation Distribute BLM 4 4 3 to each student After students respond to 1 highlig
30. hen solve the system together and check the solution using CAS on the handheld Solutions for teacher reference included at right Each equation works only with it s partner as shown in the table below Once they have found their partner have them work together to solve They may use CAS to check the solution pn 85 85 58 5 ds 6x 4y 12 5x 2y 11 7x 13y 72 18x lly 11x l5y 204 60 36 48 6x 2X O3Xy x Ay 7X y 10x 20y 12x 12y lex 16y TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 5 ec 4 3 9 18 6 18 37 32 4 45 Unit 4 Day 6 There Is Another Way Part Il Grade 10 Applied Minds On 15 Min Math Learning Goals e Determine that an equation multiplied by a constant is equivalent to the original equation Materials BLM 4 6 1 4 6 6 e Computer Algebra Systems CAS Action 45 Min e Develop a complete algorithm for the algebraic method of elimination to e Six overhead Consolidate solve a linear system marker multiple Debrief 15Min Solve problems using the method of elimination that arise from realistic colours suggested situations described in words Total 75 Min Assessment Opportunities mI Minds On Small Groups Guided Exploration Divide class into groups of four students Display graph of p
31. her than the POI give us any information about the number of tickets sold TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 15 4 1 3 Meaning of the Point of Intersection 1 Your family wants to rent a car for a weekend trip Cars R Us charges 60 00 per weekend for a midsize car plus 0 20 per km Travel With Us charges 0 50 per km Renting a Car a Graph both options on the grid and determine the number of kilometres where both companies will cost the same amount 200 b Explain what this means for your weekend trip 150 Cost 100 50 100 200 300 400 Number of kms 2 Anthony and Anne are bicycling at a Provincial Park Anthony travels at the rate of 10 km hr and begins 2 km from the park entrance Anne begins at the park entrance and travels at a rate of 15 km hr They both travel at a constant rate towards the Outdoor Education Centre Bicycling in the Park Graph both routes on the grid and determine the meaning of the point of intersection Distance From Park Entrance km Time hours TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 16 4 1 3 Meaning of the Point of Intersection continued 3 Foracar wash fundraiser Team A washes 2 cars per hour starting a 7 00 a m Team B begins washing cars at 9 00 a m and washes 3 cars per hour Graph the car washing progress of each team on the grid and determine the meaning of the point of intersection as well as the meani
32. ht the importance of identifying which variable it is easiest to isolate for when using substitution Students will complete the solution to one of the three systems provided The solution to all three systems is 1 2 Check to see that students understand that as long as a system has values that follow an arithmetic progression the solution to the system will be 1 2 Discuss as a class the second equations created in question 7 to ensure that everyone has the correct systems as these will be assigned for students to complete for their home activity e Consolidate Debrief Pairs gt Worksheet Distribute BLM 4 4 4 to each student Students are to write the algorithm for solving by substitution using the text boxes provided and then solve the system as model of solving by substitution Whole Class Discussion Discuss correct order of steps for solving by substitution Refer back to the graphical solution of this problem and connect the appropriate region of the graph with the appropriate part of the solution Students will need scissors and glue for this portion of the lesson Skill Practice Home Activity or Further Classroom Consolidation Complete the systems created in question 7 from BLM 4 4 3 TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 33 4 4 1 The Lowdown on Downloads Let s return to our music downloading problem Here s the problem again iTones
33. ion is not ML3 02 ML3 03 efficient Identify necessary conditions or look fors that make solving by CGE 4b 7b elimination possible Develop the algebraic method of solving a system by elimination 6 There Is Another Way Determine that an equation multiplied by a constant is equivalent ML3 02 ML3 03 Part II to the original equation Develop a complete algorithm for the algebraic method of elimination to solve a linear system Solve problems using the method of elimination that arise from realistic situations described in words 7 Which One is Best Identify necessary conditions or look fors in a system of ML1 01 ML1 03 equations that make it best suited to be solved by graphing ML3 01 ML3 02 substitution or elimination ML3 03 Solve a system of equations using two of the three methods Solve a system of linear equations using the most appropriate CGE 3c 4f 5a 5e method 8 Summative Task Create contextual linear systems problems complete with ML1 01 ML1 03 Evaluation Day 1 solutions and graphs and present them in a visual format e g ML3 01 ML3 02 poster project or PowerPoint slideshow ML3 03 This is available from the members only CGE 5b section of the OAME web site WWW oame on c 9 Summative Task Exchange problems created amongst them for others to solve ML1 01 ML1 03 Evaluation Day 2 verify their solutions and justify the selection of preferred ML3 01 ML3 02 method of solution ML3 03
34. ions Yasser is renting a car Zeno Car Rental charges 45 for the rental of the car and 0 15 per kilometre driven Erdos Car Rental charges 35 for the rental of the same car and 0 25 per kilometre driven Which company should Yasser choose to rent the car from Problem B Equations The school council is trying to determine where to hold the athletic banquet The Algebra Ballroom charges an 800 flat fee and 60 per person The Geometry Hall charges a 1000 flat fee and 55 per person Which location should the school council select for the athletic banquet Problem C Equations The yearbook club is considering two different companies to print the yearbook The Descartes Publishing Company charges a flat fee of 475 plus 4 50 per book School Memories charges a flat fee of 550 plus 4 25 per book Which company should the yearbook club select to print this year s yearbook Problem D Equations The school is putting on the play Algebra The Musical Adult tickets were sold at a cost of 8 and student tickets were sold at a cost of 5 A total of 220 tickets were sold to the premiere and a total of 1460 was collected from ticket sales How many adult and student tickets were sold to the premiere of the musical EQUATIONS 1 y 4 50 475x 2 60 800x y 3 y 1000 55x 4 x 45 0 15x 5 y 2 1000x 55 6 y 245 0 15x 7T X y 220 8 5x 4 8y 220 9 y 2 4 25x 550 10 y 2 550x 4 25 11 y 2 800 60x 12 x y 1460 13
35. itution Point of intersection Solve the system using elimination Point of intersection What does the point of intersection mean in this catering problem You used 3 different methods to solve the system what did you notice about the points of intersection Does this surprise you Nick and Heather have invited 80 people to their wedding How much will it cost for each menu Heather prefers the Frugal Gourmet menu to Cookie s Catering How much more will she pay for her preference TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 65 4 7 5 3 Ways Continued The student council is providing lunch and music for the grade 10 class They have two quotes from Lunch Express and Let s Do Lunch The costs for each were given as follow Lunch Express If 100 students attend it will cost 1 000 If 200 students attend it will cost 1 500 Let s Do Lunch If 50 students attend it will cost 700 If 150 students attend it will cost 1 350 Solve the system using the three different methods Equations for the companies Lunch Express Let s Do Lunch Lunch Express Let s Do Lunch Graphing Method Substitution Method 50 100 150 200 250 300 Point of intersection Point of intersection TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 66 4 7 5 3 Ways Continued Elimination Method
36. ng apar Mi e Develop understanding of facility of solving systems using the oa i substitution method to improve the accuracy of a solution Total 75 Min Assessment Opportunities EE Minds On Pairs Investigation Distribute BLM 4 3 1 to each student Students are to work in pairs One student in each pair is to solve one of the problems Students are to then exchange solutions and provide feedback to each other regarding their solution Write the heading System A and System B on the board chart paper or overhead transparency Students are to write their solution to each system under the appropriate heading Whole Class Discussion Discuss the solutions provided to each system Ask for reasons as to why their solutions are not exactly the same The goal is to draw out the approximate nature of the solutions and that this is one of the disadvantages of solving systems graphically Students are encouraged to use CAS to check their solutions They will probably get a false statement returned which should reinforce the approximate nature of their solution Students can use E Action Individual Worksheet Distribute BLM 4 3 2 to each student Students are to solve these simple systems algebraically and note which equation they solve first in each equation and why they chose that equation The goal is to stress that they can only solve equations that are expressed in a single unknown Whole Class Demonstration
37. ng of the points before and after the point of intersection Fund Raiser Number of Cars Hours Elapsed TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 17 Grade 10 Utilizing Think Pair Share students interpret the linear system BLM 4 2 1 then discuss with the class Stress the point that the solution depends on the conditions under consideration Unit 4 Day 2 What s the Point Applied Math Learning Goals Materials Minds On 15 Min Determine the coordinates of the points of intersection for pairs of linear d ot T relations given their graphs and interpret the meaning of the point of ius C AS Action 35 Min intersection Consolidate e Solve a problem that arises from a realistic situation represented by two Debrief 25 Min linear relations Total 75 Min Assessment Opportunities B Minds On Pairs gt Worksheet Action Whole Class Guided Exploration Represent graphically a situation that is modelled by a linear system Students should understand the meaning of specific parts of the graph e g point of intersection Demonstrate the graphical method of solving a linear system BLM 4 2 2 Discuss the questions emphasize the meaning of the point of intersection in terms of the problem Individual Practice Students practice analyzing systems of linear equations finding the point of intersection and interpreting the meaning in the larger context
38. ns banquet The Algebra Ballroom charges an 800 flat fee and 60 60x 800 per person The Geometry Hall charges a 1000 flat fee and 55 y per person For what amount of guests do the two banquet halls charge the same amount a Problem C The yearbook club is considering two different companies to print equations the yearbook The Descartes Publishing Company charges a flat fee 475 4 50x of 475 plus 4 50 per book School Memories charges a flat fee of y 550 plus 4 25 per book For what amount of books do the two E companies charge the same amount a mex am solving problem TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 31 4 3 5 What s My Equation Part 2 continued 1 Looking at your problem how can you tell from the equation which company is cheaper before the point of intersection where the costs are equal 2 Looking at your problem how can you tell from the equation which company is cheaper after the point of intersection where the costs are equal 3 Is this true for all problems 4 Now that you ve solved the problems using two different methods which method do you prefer Why 5 When do you think solving by substitution would be preferable to solving by graphing TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 32 Unit 4 Day 4 The SubMathiner Grade 10 Consolidate 10 Total 75 min e Identify which variable is the most e
39. ns and check your solutions 1 Afitness club charges an annual fee and an hourly fee In a single year member A worked out for 76 hours and paid 277 in total Member B worked out for 49 hours and paid 223 in total What is the annual fee What is the hourly fee HINT Start by writing let statements to define the variables you will use For example Let a represent the amount of the annual fee Let h represent the amount of the hourly fee 2 This past summer you ran a food booth at a local festival You sold hotdogs for 1 each and samosas for 2 50 each From 205 purchases you made 400 in total To help plan purchases for next year s festival you d like to know how many hotdogs and samosas were sold Unfortunately you forgot to keep track of this when selling the food Can you determine how many hotdogs and samosas were sold NOTE Assume one hotdog or one samosa per purchase TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 50 4 6 5 Help an Absent Friend Consider the following linear system 2x 3y 1 3x y 7 How would you solve it Write in words a description of the steps you would take To help you understand what to write pretend for a moment that you are writing the instructions for a friend who is not in class today What steps would you need to describe TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 51 4 6 6 Here s To The Crazy Ones 1 Solve the following
40. of two linear equations CGE 3c Interpret the story the graph tells including point of intersection and what occurs before and after the intersection point 2 What s the Point e Determine the coordinates of the points of intersection for pairs of ML2 05 ML3 01 linear relations given their graphs and interpret the meaning of the ML3 03 point of intersection e Solve a problem that arises from a realistic situation represented by two linear relations 3 Approximate or e Discuss accuracy of solutions obtained through graphical means MLI 01 ML1 03 Accurate e Establish the need to isolate a variable in order to use the ML3 02 substitution method and practice the skill of isolating e Develop understanding of facility of solving systems using the substitution method to improve the accuracy of a solution CGE 2c 5b 4 The SubMathiner e Determine the conditions for which a linear relation application is ML3 02 ML3 03 preferred to the algebraic models by comparing the rate of change and initial value of two linear relations making up a system of equations e Identify which variable is the most efficient for isolating and then solve linear systems using the method substitution TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 1 Day Lesson Title Math Learning Goals Expectations 5 ThereIs Another Way Identify situations where the solving by substitut
41. oints from the equation x 2y 6 on an overhead Each group is then given a constant to multiply this equation by Groups create a table of values for their new equation and plot the points on an overhead copy of BLM 4 6 1 Each group s sheet is then placed over the original plot shown Students will see that their new equations are equivalent to the original Ask students to comment on what they observe as each group places their graph over the original transparency e Action Whole Class Guided Example Distribute BLM 4 6 2 Students use what they have observed from the Minds On to modify a linear system such that a variable will be eliminated by addition or subtraction Pairs gt Worksheet Distribute BLM 4 6 3 Using Think Pair Share return to an earlier example from this unit and solve by elimination One half of class instructed to eliminate x variable the other half instructed to eliminate y variable Discuss why the decision to eliminate one variable first as opposed to another does not change the solution Individual Practice gt Worksheet Using BLM 4 6 4 students solve two linear systems based on contextual examples Consolidate Pairs gt Reflection Debrief Dctahue DIM AK Distribute BLM 4 6 5 Given a question students are asked to write a summary of the steps necessary to solve by elimination Whole Class gt Discussion Introduce the performance task to students Students will be asked to dev
42. ook like on your handheld screen RAD AUTO REAL Li To have the handheld check the solution press the 9 key If the solution is correct the handheld will I IDE return the result true If the solution is notcorrect the handheld will report false You should find that the handheld reports that 2 is a correct solution for the first equation Remember it is necessary to check the solution in the second equation as well It works the same way as before but you can save RAD AUTO REAL E some typing Use the circular navpad and press up twice The first command you entered is highlighted Now press the key The handheld copies the command down to the line you are working on You can now use the navpad to move left until the cursor is behind the symbol It looks like this Press the key to erase the first equation then type the second equation Then press Ge 9 As you can see E is a correct solution for the 9 8 xx 6 y und y 3 second equation as well Since the solution is correct for both equations from the linear system we know it must be the right answer TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 5 Nspire CAS Handheld Manual Continued How to Add or Subtract Two Equations The elimination method for solving a linear system involves adding or subtracting the given equations Say that you are considering the following linear system 3x 4 2y
43. re confident in the solutions to the systems post your point of intersection under the appropriate heading on the class list l y 2x 7 us yz 3x44 101 8 6 AT zee eee eee pProelTpl ep TT pre pT ols Point of Intersection Point of Intersection TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 26 4 3 2 A Better Way Solve the following systems of equations algebraically Use your CAS handheld to solve and check if needed 1 Equation 1 y 22 10 and Equation2 x y 12 Point of intersection ___ 2 Equation 1 3x 2y 33 and Equation2 2x x 7 Point of intersection ___ In each of the systems you solved above which equation did you choose to solve first Why did you select that equation in each case TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 27 4 3 3 Putting the Pieces Together Teacher Demonstration TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 28 4 3 3 Putting the Pieces Together Student Worksheet System 1 y 2x Zand y 3x 4 Record the line that results once you have substituted and then solve the equation using your CAS handheld Resulting equation and my solution Don t forget to solve for both variables Check your solution using the CAS handheld Compare your solution to someone else in the class System 2 2x y 0 and x y
44. s were sold at a cost of 5 A total of 220 tickets were sold to the premiere and a total of 1460 was collected from ticket sales How many adult and student tickets were sold to the premiere of the musical If x represents the number of student tickets sold and y represents the number of adult tickets sold then the equations that model this problem are from cost of tickets 5x 8y 1460 from number of tickets sold x y 220 You probably remember that this problem took a while to solve by graphing and the answer you found was not necessarily very accurate since you read the point of intersection off of the graph You will work with a partner now to solve this problem using the method of elimination 1 Since you have been asked to eliminate the x or y variable circle one first what will be the first step you take to create the conditions necessary for elimination 2 Solve the linear system now You may use CAS to help find the answer and check the solution Use the space below for rough work 3 Doesit matter which variable is eliminated first That is does it change the final answer 4 Think back to when you solved this problem by graphing Do you find the method of elimination easier or harder Explain TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 49 4 6 4 Two for You Try solving the following questions using the method of elimination You may use CAS on the handhelds to help solve the questio
45. solve the equation 6x 2y 16 for the variable y To do this first be certain that you are on a Calculator page If you need help with this see the Getting Started section above First type in the equation that you want to solve Use the number pad and the green letter keys the operations x are located on the right and the equals sign is in the top left corner of the keypad When you have typed in the equation press the key found in the bottom right corner RAD AUTO REAL Li The top of your screen will look something like this 6x 2y 16 TTE Now decide how you would start in solving for y RAD AUTO REAL U Perhaps you ve decided that subtracting 6x from both sides of the equation is a good start Wonderful To x 2 16 6 x 2 y 16 do this immediately press Ce 9 Notice that the Pc handheld automatically inserts Ans What is this Ans stands for the last answer you found If you now RAD AUTO REAL press the amp amp key the handheld will subtract 6x from SD 6x 2v 16 2 the left side and the right side of 6x 2y 16 You eee aaa will see this result 6 x 2 y 16 6 x 2 y 16 6 x 1 Continue solving the equation You probably see that to finally isolate the y variable it is necessary to divide the equation by 2 on both sides Again just start 2 y 16 6 x y 3 x 8 typing the operation you want to perform Press 2 DAL The handheld will insert Ans for you Pr
46. stems August 2008 4 55 4 7 1 Four Corners continued GRAPHING TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 56 4 7 1 Four Corners continued SUBSTITUTION TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 57 4 7 1 Four Corners continued ELIMINATION TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 58 4 7 2 Which Method 2x 3y 10 4x 5y 2 System A y x 2 x 5y 4 System B y 2x 7 y 4x 5 System C 4 7 3 Which Method Continued For System A determine if you can solve the system using each of the three methods you have learned and if you can then solve 2x 3y 10 4x 5y 2 Justify why you can or cannot solve using this method Graphing Justification Substitution Justification Elimination Justification TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 60 4 7 3 Which Method Continued For System B determine if you can solve the system using each of the three methods you have learned and if you can then solve y x 2 x 5y 4 Graphing Justify why you can or cannot Justification solve using this method Substitution Elimination Justification Justification TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 61 4 7 3 Which Method Continued For System C determine if you can solv
47. t possible to solve for x By subtracting the equations in the second linear system the x variable was eliminated there were 0x which makes it possible to solve for y 1 Work in pairs to consider the following linear systems Decide what operation addition or subtraction would result in the elimination of a variable You may use CAS on the handheld to help you decide 9x y 4 3x y 50 7x 6y 338 14x y 1 12x y 115 9x 6y 366 18x 5y 454 19x 2y 102 17x 8y 323 12x 5y 316 19x 2y 50 6x 8y 114 9x 4y 235 7x 16y 441 5x 3y 188 15x 2y 409 7x 17y 476 6x lly 344 2 What needs to be true about a linear system so that a variable is eliminated when the equations are added or subtracted TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 43 4 5 3 Solving a Linear System by Elimination 1 How would you begin solving this linear system Addition or Subtraction 5X 4y 7 3x Ay 17 2 Solve the system 3 In your own words describe what you must do to solve a linear system by elimination TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 44 4 5 4 Find Your Elimination Partner Set of Systems Teacher Give each student in your class an equation Matching equations are organized here beside each other Students to move around room and find their elimination partner t
48. ubstitution 4x y 6and 2x 3y 10 TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 39 Unit 4 Day 5 There Is Another Way Grade 10 Applied Math Learning Goals Minds On 15 Min e Identify situations where the solving by substitution is not efficient Action 45 Min e Identify necessary conditions or look fors that make solving by elimination possible Consolidate Debrief 15 Min e Develop the algebraic method of solving a system by elimination Total 75 Min Materials e BLM 4 5 1 4 5 4 e Computer Algebra Systems CAS Assessment Opportunities mI Minds On Pairs gt Investigation Students are given a contextual problem BLM 4 5 1 to be solved by substitution using Think Pair Share Discuss issues that occur when they try to solve the problem Lead students to the conclusion that we need a better way to solve this system _ Action Pairs gt Worksheet Distribute BLM 4 5 2 Students add and subtract equations in a given set of linear systems Identification of what operation will eliminate a variable Develop necessary conditions for elimination Whole Class gt Guided Example Reinforce the elimination algorithm for students using BLM 4 5 3 Whole Class gt Kinesthetic Activity Using BLM 4 5 4 students find their elimination partner who has an equation suitable for eliminating a
49. variable by addition or subtraction Each equation will have only one suitable match Checks for student understanding of the conditions necessary for elimination _ Consolidate Debrief Individual gt Practice Students return to the contextual problem discussed in the Minds On BLM 4 5 1 and solve by elimination Students to check their solution using CAS Whole Class gt Discussion Compare ideas on conditions necessary for elimination Reflection Home Activity or Further Classroom Consolidation Students create three sets of equations in which one variable could be eliminated TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 40 4 5 1 A Trip to Jim Hortons Teacher Notes In question 1 encourage students to develop equations to represent the context given This will help students prepare for the summative task in this unit The linear system that describes the context 5s 7e 14 95 3s 7e 12 75 Is designed to be problematic when solved by substitution but simple to solve by elimination When solving by substitution students will note that upon re arranging to solve for a variable fractional coefficients and or repeating decimals are encountered 4 5 2 An Elimination Introduction Notes to Teacher After discussing the introduction to this section you may wish to have your students go through the How to Add or Subtract Two Equations section of th
50. written on each piece of paper and post on the board or use the movie file Method of Sub Intro that demonstrates the process Note The process is narrated in the file Students are to identify which is cheaper before the point of intersection Consolidate Debrief Small Groups Worksheet Distribute BLM 4 3 5 to students Each group is to divide the three problems among the group members and solve them using substitution and then complete the questions on the reverse of the handout The goal is to identify for what types of problems the method of substitution is preferable to solving by graphing i e approximate solutions and or values for graphing are to large for practical solution lower flat fee and which is cheaper after lower rate of change Concept Practice Reflection Home Activity or Further Classroom Consolidation Journal Dillon was absent for today s lesson Explain to him how to find the point of intersection between 4x y 7 and 6x y 13 Provide a detailed explanation of each step Assign concept practice questions TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 25 4 3 1 What s my POI e Each one of you will solve one of the systems of equations given below e Once you have solved the system you were assigned trade with your partner and check their solution Share your feedback with your partner e Once you have shared your feedback and a
51. y use the graph to assist you 1 y 2x 3 2 y x l 3 y 2x 7 4 y 3 b x 2 6 y 2 Questions 1 Which of the lines passes through the point 2 3 2 Is there another way to determine if the line passes through the point other than graphing Explain 3 Without graphing how can you quickly determine if a horizontal or vertical line passes through a point 4 Other than the point 2 3 what are the other points of intersection on your graph 5 Is it possible for two lines to have more than one point of intersection with each other Discuss this with your partner TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 23 4 2 5 Is this Accurate 1 Find the point of intersection Solve the system using graphical method a y 22x 1 b y x 2 y 3x 2 y 2x 7 Point of intersection is Point of intersection is C y 22x 1 d y 5 y 4x 4 y 8x 2 D D P aq z 5 b B Point of intersection is Point of intersection is TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 24 Grade 10 Unit 4 Day 3 Approximate or Accurate Applied i Math Learning Goals oe ies Minds On 15 Min e Discuss accuracy of solutions obtained through graphical means C omput i Klei Action 50 Mi e Establish the need to isolate a variable in order to use the substitution Systems CAS E acid i method and practice the skill of isolati
52. you were sent out to get three small coffees and seven medium coffees You recall that the total came to 12 75 with tax Its Wednesday morning and your coffee crazy co workers are calling for their cup o joe Unfortunately since the morning fix hasn t arrived yet no one can remember how much a small or extra large coffee costs including yourself You need to find out how much each size costs to collect the correct amount of money for the Wednesday coffee run 1 Let s be the number of small coffees ordered on a single day Let e be the number of extra large coffees ordered on a single day As a class can we decide on an equation to represent the purchases made on Monday and an equation to represent the purchases made on Tuesday Monday s equation Tuesday s equation 2 Now we have a linear system Take a few minutes to solve the linear system using substitution in the space below Then pair with another student to discuss your solution 3 What problems if any did you encounter TIPS4RM Grade 10 Applied Unit 4 Linear Systems August 2008 4 42 4 5 2 An Elimination Introduction You know that two integers can be added or subtracted 15 5 7 6 12 9 In the same way equations can be added or subtracted 3x t 2y 19 10x 20y 80 t 5x 2y 5 10x 15y 25 8x 24 5y 55 Notice that by adding the equations in the first linear system the y variable was eliminated there were Oy which makes i
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