tartu, 11.07.2012dr. rainer heinrich dresden, germany 1 experiences of 15 years using graphic...

Tartu, 11.07.2012 Dr. Rainer HeinrichDresden, Germany

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Experiences of 15 years using graphic calculators in Saxonia

Dr. Rainer Heinrich

Dresden, Germany


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• "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)


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Harry M. Warner (1927) for the suggestion to produce a sound film anyway:

"Who to the devil wants to hear actors talking?"


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"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)


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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.


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Situation in SaxonyResults of the „Abitur“

with CAS with GC2010 2,4 2,4 2045 students 4858

2011 2,6 2,6 1922 4421


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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.


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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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•


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• The sum of the squares of three natural numbers, succeeding one another,

is 590. Find the three numbers.

• Teacher´s expected solution:

)entfällt(15x

)zutrifft(13x

1961

)195(11

q4

p

2

px

0195n2n

3I0585n6n3

590I5905n6n3

5904n4n1n2nn

590)2n()1n(n

2

1

2

2;1

2

2

2

222

222


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• Student´s solution


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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.


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Set and resolve of brackets

365m50)52a( The variable a shall stand for your age, the variable m for your weight.

=100a+m-115You 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.


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Set and resolve of brackets

• Discovery phase with technology

• Exercise phase without technology

• Application phase with technology


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Example 1: Streetcar-Company

Visualizing, motivating, experimenting


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• 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.


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Facts

Everyday 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€.


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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?


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•

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.


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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.

B

H

B

HB

H

0,05

0,95

P(X=0)

P(X>1)

P(X<1)

P(X>1)


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)).0,x,180(binomcdf1(95.0))1,x,180(binomcdf1(05.02y

und)0,x,180(binomcdf95.0)1,x,180(binomcdf05.01y


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Z: profit of the streetcar-company in the period for any passenger in dependence of the probability p

iz

)zZ(P i)p(1yp )p(2y))p(1yp(1

0,40€ 40,00€ 0,00€

y2(p)

For p = 1,6% earn the company the highest profit.


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Compare the model and the reality:

model reality

number of inspectors 13 30

number of checks(600 per inspector per day )

7800 18000

check-probability p (480 000 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€


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result: If the streetcar-company would dismiss 17 inspectors, they would earn 7200€ more per day.


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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?


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• Breakdowns of the jet engines are independent of each other

• An airplane is still maneuverable if the half of the jet engines works.

Acceptances


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Solution

22 q1)q(S

q1q4q1)q(S 344

tartu, 11.07.2012dr. rainer heinrich dresden, germany 1 experiences of 15 years using graphic...

Documents

rainer heinrichdresden

federal republic of

students solution

hand tool

use of graphic calculators

cas orders

experimenting tool

didactical reasons