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PROBLEM-SOLVING with the TI-NSPIRE CAS CALCULATOR ICTMT 9 Metz 2009

PROBLEM-SOLVING with the TI-NSPIRE CAS CALCULATOR ICTMT 9 Metz 2009. Presenters Pauline Holland Shirly Griffith. Background. Education in Australia is state based Victorian Curriculum and Assessment Authority (VCAA) set the examinations for the State of Victoria

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PROBLEM-SOLVING with the TI-NSPIRE CAS CALCULATOR ICTMT 9 Metz 2009

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  1. PROBLEM-SOLVING with the TI-NSPIRE CAS CALCULATORICTMT 9Metz 2009 Presenters Pauline Holland Shirly Griffith

  2. Background • Education in Australia is state based • Victorian Curriculum and Assessment Authority (VCAA) set the examinations for the State of Victoria • 2002 - VCAA implemented a pilot mathematics subject in a small number of schools, called Mathematical Methods CAS. • 2006 - The subject Mathematical Methods CAS was open to all students wishing to study calculus based mathematics • 2009 - Mathematical Methods CAS is a prerequisite for most science based university courses.

  3. From 2010, students undertaking Specialist Mathematics will be assumed to be using CAS technology. • Mathematical Methods CAS and Specialist Mathematics are assessed as follows: 1. School Assessed Coursework (34%) 2. Examination 1 Technology Free (22%) 3. Examination 2 Technology Active (44%)

  4. Getting Started • When the TI-Nspire CAS is first turned on, it starts with the HOME c screen as shown. • You can return to this screen at any time by pressing the HOME c icon. • The System Info folder contains tools that will allow the user to change the settings on the calculator.

  5. Checking the Operating System To check which operating system is on your calculator, press: • 8: System Info 8 • 5: About 5 The calculator being used here has OS 1.7.2733 installed. To select OK, press: • Tab e • Enter ·

  6. System Settings To change System settings, press: • HOME c • 8: System Info 8 • 3: Graphs & Geometry 3 Use Tab e to move between fields. Press Click x to select options.

  7. Question 1 A debt of $450 is to be shared equally among the members of the Crackerjack club. When five of the members refuse to pay, each of the other members will have to pay an additional $3. How many members does the Crackerjack club have? Solution  Let the number of members be n. Let the value of the individual debt be s.  So

  8. On a Calculator page, press: • Menu b • 3: Algebra 3 • 1: Solve 1 Complete the entry line as: Then press Enter ·.

  9. The Crackerjack club has 30 members. Check If all members paid, then each member would pay . If only 25 members paid then each paying member will have to pay an additional $3. i.e. $18

  10. Question 2 Instead of walking along 2 sides of a rectangular field, Patrick took a short cut along the diagonal, thus saving distance equal to half the length of the shorter side. Find the length of the long side of the field given that the length of the short side is x metres.

  11. Solution  Let the length of the diagonal be d metres. Let the length of the longer side be y metres. Given the length of the shorter side is x metres.

  12. Complete the entry line as: Then press Enter · The longer side is equal to metres.

  13. Question 3 Suppose that the annual dues of a union are as follows: If d is the annual dues and s is the salary, graph the relationship and determine the annual union dues of an employee earning $62 000.

  14. Solution The relation between the annual dues d and the salary s can be expressed as:

  15. To graph this hybrid function On a Calculator page, select the hybrid function template from the Maths expression template by pressing • Ctrl / • × r Use the NavPad to move across to the selection highlighted.

  16. Then press Enter and select 3 function pieces. Press Enter ·. Type the expressions as shown. Tab e to move between fields. Do not press Enter ·.

  17. To define the function, press: • Tab e • Ctrl / • Var h Type f1(x), then press Enter ·.

  18. To draw the graph of the hybrid function, open a Graphs & Geometry page, press: • Menu b • 4: Window 4 • 1: Window Settings 1 Complete the table as shown.

  19. Press Enter ·. To determine the annual union dues of an employee earning $62 000, press: • MENU b • 6: Points & Lines 6 • 1: Point 1 Move the cursor to the line and when a point appears press Click x twice.

  20. Use the NavPad to move the cursor over the x-coordinate of the point, when an open hand appears, press Click twice and change the x- coordinate to 62 000. Press Enter ·. The annual dues of an employee earning $62 000 is $640.

  21. Question 4Lauren and Matt are travelling off road in the desert in Morocco. They are 10 km from a long, straight road. On the road, their 4WD can do 60 km/hr, but off road, it can only manage 40 km/hr. It is getting dark and Lauren and Matt are keen to reach the town where they are staying for the night. The town is 25 km down the road (from the nearest point P on the road). (a) How many minutes will it take for Lauren and Matt to drive to the townthrough the desert? (b)Would it be faster if they first drove to P and then used the road to town (T)? (c) Find an even faster route for Lauren and Matt to follow.

  22. Draw a diagram A DESERT 10 ROAD T 25 P

  23. Solution (a) 40 km/hr = ⅔km/min Time taken =

  24. 10 km at ⅔ km/min will take 15 minutes. 25 km at 1 km/min will take 25 minutes. A total time of 40 minutes.  This would save 0.39 minutes so it would be faster if Lauren and Matt drove to the road and then to the town.

  25. c To find a faster route, chose a point M, x metres from P on the road and construct a function F(x) for the total time it would take for Lauren and Matt to drive to M and then along the road T. Find the value of x which would make F a minimum. A DESERT 10 ROAD x T 25 M P

  26. Time taken = minutes Time taken = (25 − x) minutes The total time taken is:

  27. A graphical method of finding a solution On a Graphs & Geometry page, complete the Function entry line as: Then press Enter ·. Adjust the Window Settings as shown. (the maximum x could be is 25 and we know that the total time, y, is under one hour or 60 minutes)

  28. To locate the minimum, press: • Menu b • 5: Trace 5 • 1: Graph Trace 1 Move the cursor along the curve until the minimum is located or Press ? for hints and then n for minimum. The minimum time is 36.2 minutes when x is 8.94 km.

  29. An algebraic method of finding a solution It is possible to find the minimum time taken by using calculus.  Find x when f’(x) = 0 On a Calculator page, complete the entry line as: Then press Enter ·. The minimum time is 36.18 mins, when x is 8.94 km

  30. Question 5 A confectionery manufacturer wishes to market the latest chocolate sensation in an eye-catching pyramid shape package. The volume of the pyramid is to be 1000 cm3 and the base must be a square. Find the dimensions of the packaging if the manufacturer wishes to keep the surface area to a minimum.

  31. Solution Volume = ⅓x2h = 1000 To find the perpendicular height of a sloping side, complete the entry line as: Then press Enter ·.

  32. The surface area of the pyramid is made up of the square base and four triangular sloping sides. On a Calculator page, define SA as f1(x).

  33. To find the value of x, when the surface area is a minimum, complete the entry line as: Then press Enter ·. To find the height , complete the entry line as: Then press Enter ·. The surface area will be a minimum when the base is 12.85 cm and the height is 18.17cm.

  34. Question 6 Two Year 7 classes complete the same end-of-year mathematics test. The marks expressed as percentages are given in the following table. Is it possible to determine which class overall has achieved at a higher level?

  35. To compare the two classes, use parallel boxplots. On a Lists & spreadsheets page, enter the data for 7A into Column A and label the column ClassA, the data for 7B into Column B and label the column ClassB.

  36. To highlight Column A, with the cursor in the top cell, press the up arrow £ on the NavPad, then press: • Menub • 3: Data 3 • 6: Quick Graph 6 • Menu b • 1: Plot Type 1 • 2: Box Plot 2 Repeat for Class B The scales are different.

  37. To change the scale for Class B, press: • Ctrl / • Menu b • 5: Zoom 5 • 1: Window Settings 1 Change the Window Settings to match the scale for Class B.

  38. It is now possible to compare the results of the two classes. Class B has a smaller range and IQR. Class B has a substantially higher minimum, medium and maximum. Class B appears to have achieved better results overall than Class A.

  39. Question 7 The following beep test data was obtained from a Year 11 Physical Education student. The table below shows the heart rate recorded each minute. Use this data to predict the heart rate of a student at 6.5 minutes.

  40. To draw a scatterplot of the data, on a Lists & Spreadsheets page, enter the data in Columns A and B, labelling as shown. Then press: • HOME c • 5: Data & Statistics 5 Press Tab to enter the variables on the axes. The data could possibly be logarithmic in shape.

  41. To fit a logarithmic regression line to the data, return to the Lists & Spreadsheets page and delete the first row. ( loge0 is not possible) then press: • Menu b • 4: Analyze 4 • 6: Regression 6 • 9: Show Logarithmic 9 The regression line and it’s equation will be displayed.

  42. To predict the heart rate at 6.5 mins, return to the Lists & Spreadsheets page, press: • Menu b • 4: Statistics 4 • 1: Stat Calculations 1 • B: Logarithmic Regression B Complete the table as shown. Then select OK. Note: The equation has been saved as f1(x).

  43. To predict the heart rate, on a Calculator page, complete the entry line as: f1(6.5) Then press Enter ·. Using the logarithmic regression model, at a time of 6.5 mins the heart rate is predicted to be 187.55.

  44. Question 8 John rides a Ferris wheel for five minutes. The diameter of the wheel is 10 metres, and its centre is 6 metres above the ground. Each revolution of the wheel takes 30 seconds. Being more than 9 metres above the ground causes John to suffer an anxiety attack. For how many seconds does John feel uncomfortable? Anxiety 9 m 10 m 6 m

  45. Maximum height: 11 m • Minimum height: 1 m • Amplitude: 5 • Period: 30 s An initial model of this situation might be: To graph this function, on a Graphs & Geometry page complete the entry line as:

  46. Clearly this function is not quite right as it suggests that at the start of the ride, f(0) , John was 6 m off the ground. John would start the ride when the chair is at the lowest point. That is when it is 1 m off the ground. To discover when the model above is at the lowest point, press: • Menu b • 5: Trace 5 • 1: Graph Trace 1 Use the NavPad to trace to the minimum. The minimum point will be displayed on the graph screen. It is (22.5, 1).

  47. Use this information to modify the equation that models John’s situation. Sketch this graph.

  48. John is uncomfortable when the Ferris wheel is more than 9 metres above the ground. Complete the entry line as: f2(x) = 9 To find the points of intersection, press: • Menu b • 6: Points & Lines 6 • 3: Intersection Point(s) 3 Move the cursor to a point of intersection and press Click x twice. For one rotation, John will feel uncomfortable for 19.43 − 10.57 = 8.86 secs.

  49. Question 9 After Adam finishes his Multimedia Design course, his first job pays him a weekly salary of $600 after tax. He sees a second hand Peugeot for sale at $35 000. He has a deposit of $2000 and can get a personal loan of $33 000 at 13% p.a. compounded monthly for 7 years. How much money will Adam have to live on each week, after he makes a monthly repayment? How much money will he still owe after 5years?

  50. Solution To calculate the monthly repayments, on a Calculator page, press: • Menu b • 8: Finance 8 • 1: Finance Solver 1 Complete the table as shown. Return to the Pmt field and press Enter ·.

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