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

Tartu, 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)

Tartu, 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?"

Tartu, 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)

Tartu, 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.

Tartu, Dr. Rainer Heinrich Dresden, Germany 6 Situation in Saxony Results of the „Abitur“ with CAS with GC ,4 2, students ,6 2,

Tartu, 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.

Tartu, 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

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Tartu, 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:

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

Tartu, 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.

Tartu, 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.

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

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

Tartu, 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.

Tartu, Dr. Rainer Heinrich Dresden, Germany 17 Facts Everyday passengers use the streetcar. With a honest passenger the company earns 0,40€, with a fare dodger 40€ fine (punishment). The honest passengers pay every day €, the fare dodgers had to pay €, if a streetcar-company guard would catch them all. But: If all passengers would pay honestly, the streetcar-company would only earn ,40€ = €.

Tartu, 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?

Tartu, 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.

Tartu, 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.

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

Tartu, Dr. Rainer Heinrich Dresden, Germany 23 Compare the model and the reality: modelreality number of inspectors1330 number of checks (600 per inspector per day ) check-probability p ( passengers) 0,016250,0375 average earn of the company per passenger (Z) 0,415 € 0,40 € daily earn of the company ( passengers) € €

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

Tartu, 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?

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

Tartu, Dr. Rainer Heinrich Dresden, Germany 27 Solution