Experiences of 15 years using graphic calculators in Saxonia

1. Experiences of 15 years using graphic calculators in Saxonia Dr. Rainer Heinrich Dresden, Germany Dr. Rainer Heinrich Dresden, Germany

2. "No, Sir, the Americans need the telephone perhaps, we don´t. We have many messengers." (Sir William Preece, chief engineer of the British mail, 1896) Dr. Rainer Heinrich Dresden, Germany

3. Harry M. Warner (1927) for the suggestion to produce a sound film anyway: "Who to the devil wants to hear actors talking?" Dr. Rainer Heinrich Dresden, Germany

4. "The television set won't gain acceptance on the market. The people will be tired to stare at a plywood box every evening very soon." (Darryl F. Zanuck, boss of 20th Century Fox, 1946) Dr. Rainer Heinrich Dresden, Germany

5. Situation in Saxony • In Saxony, one state of the Federal Republic of Germany, the use of graphic calculators is obligatory from the 8th form on in the gymnasium (high school). It is also necessary in the central school leaving examinations. Dr. Rainer Heinrich Dresden, Germany

6. Situation in Saxony • Results of the „Abitur“ • with CAS with GC • 2,4 2,4 • 2045 students 4858 • 2,6 2,6 • 1922 4421 Dr. Rainer Heinrich Dresden, Germany

7. Reasons for using CAS/Graphic Calculators • methodical and didactical reasons: • explorative learning – experimentation • visualisation • motivation • calculator help • change of assignment culture • cross-curricular teaching and learning. Dr. Rainer Heinrich Dresden, Germany

8. Phases of the cognition way • developing fundamental interest • discovering CAS as a hand tool for oneself • use as a demonstration equipment in the hand of the teacher • use as a calculation aid (most at well known problems) • use as experimenting tool •  at recognizing each the necessity of change of the mathematics lesson Dr. Rainer Heinrich Dresden, Germany

9. Dr. Rainer Heinrich Dresden, Germany

10. The sum of the squares of three natural numbers, succeeding one another, is 590. Find the three numbers. • Teacher´s expected solution: Dr. Rainer Heinrich Dresden, Germany

11. Student´s solution Dr. Rainer Heinrich Dresden, Germany

12. Classical Way • 1. The teacher explain the new method with one or more examples. • 2. The students practise it without technology on many examples. • 3. If the students have solve enough examples, the teacher demonstrate the way with technology. Dr. Rainer Heinrich Dresden, Germany

13. Set and resolve of brackets The variable a shall stand for your age, the variable m for your weight. =100a+m-115 You should find rules to resolve brackets by experimenting with the CAS orders "expand" or "factor". After the working phase in groups You should give a presentation for the class. Dr. Rainer Heinrich Dresden, Germany

14. Set and resolve of brackets • Discovery phase with technology • Exercise phase without technology • Application phase with technology Dr. Rainer Heinrich Dresden, Germany

15. Example 1: Streetcar-Company Visualizing, motivating, experimenting Dr. Rainer Heinrich Dresden, Germany

16. The streetcar company in Dresden is merciless!  • institute 5000 court proceedings against fare dodgers • Passenger without ticket: fare dodger •  Every passenger should be checked average at least once in three month. Dr. Rainer Heinrich Dresden, Germany

17. FactsEveryday 480 000 passengers use the streetcar. With a honest passenger the company earns 0,40€, with a fare dodger 40€ fine (punishment). The 456 000 honest passengers pay every day 182 400€, the 24000 fare dodgers had to pay 960 000€, if a streetcar-company guard would catch them all.But:If all 480 000 passengers would pay honestly, the streetcar-company would only earn 480 000 • 0,40€ = 192 000€. Dr. Rainer Heinrich Dresden, Germany

18. The Problem is: The company would have to check the tickets exactly that often, that the company gets sufficient money and the fare dodgers would be preserved. Which number of inspectors is necessary for the biggest profit of the company? Dr. Rainer Heinrich Dresden, Germany

19. We look at a period of time of three month, nearly 180 trips for a student. • Assumptions: • At the beginning there are 5% fare dodgers • If a fare dodger would be caught at least two times in this period, he will change into a honest passenger • If a honest passenger would never be controlled in this period, he will change into a fare dodger within the next time. Dr. Rainer Heinrich Dresden, Germany

20. X is the number of checks in the period for any passenger. Distribution of X is binomial with n=180 and unknown probability p. p is the probability, that a passenger would be checked. Dr. Rainer Heinrich Dresden, Germany

21. Dr. Rainer Heinrich Dresden, Germany

22. 0,40€ 40,00€ 0,00€ y2(p) Z: profit of the streetcar-company in the period for any passenger in dependence of the probability p For p = 1,6% earn the company the highest profit. Dr. Rainer Heinrich Dresden, Germany

23. model reality number of inspectors 13 30 number of checks (600 per inspector per day ) 7800 18000 check-probability p (480000 passengers) 0,01625 0,0375 average earn of the company per passenger (Z) 0,415€ 0,40€ daily earn of the company ( 480 000 passengers) 199 200€ 192000€ Compare the model and the reality: Dr. Rainer Heinrich Dresden, Germany

24. result: If the streetcar-company would dismiss 17 inspectors, they would earn 7200€ more per day. Dr. Rainer Heinrich Dresden, Germany

25. Example 2: How safe are airplanes? • Would you rather fly with an airplane with two jet engines or with four jet engines over the Atlantic ocean? Dr. Rainer Heinrich Dresden, Germany

26. Acceptances • Breakdowns of the jet engines are independent of each other • An airplane is still maneuverable if the half of the jet engines works. Dr. Rainer Heinrich Dresden, Germany

27. Solution Dr. Rainer Heinrich Dresden, Germany