Tartu, 11.07.2012 Dr. Rainer HeinrichDresden, Germany
1
Experiences of 15 years using graphic calculators in Saxonia
Dr. Rainer Heinrich
Dresden, Germany
Tartu, 11.07.2012 Dr. Rainer HeinrichDresden, 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, 11.07.2012 Dr. Rainer HeinrichDresden, 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, 11.07.2012 Dr. Rainer HeinrichDresden, 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, 11.07.2012 Dr. Rainer HeinrichDresden, 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, 11.07.2012 Dr. Rainer HeinrichDresden, Germany
6
Situation in SaxonyResults of the „Abitur“
with CAS with GC2010 2,4 2,4 2045 students 4858
2011 2,6 2,6 1922 4421
Tartu, 11.07.2012 Dr. Rainer HeinrichDresden, 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, 11.07.2012 Dr. Rainer HeinrichDresden, 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
Tartu, 11.07.2012 Dr. Rainer HeinrichDresden, Germany
9
•
Tartu, 11.07.2012 Dr. Rainer HeinrichDresden, Germany
10
• 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
Tartu, 11.07.2012 Dr. Rainer HeinrichDresden, Germany
11
• Student´s solution
Tartu, 11.07.2012 Dr. Rainer HeinrichDresden, 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, 11.07.2012 Dr. Rainer HeinrichDresden, Germany
13
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.
Tartu, 11.07.2012 Dr. Rainer HeinrichDresden, Germany
14
Set and resolve of brackets
• Discovery phase with technology
• Exercise phase without technology
• Application phase with technology
Tartu, 11.07.2012 Dr. Rainer HeinrichDresden, Germany
15
Example 1: Streetcar-Company
Visualizing, motivating, experimenting
Tartu, 11.07.2012 Dr. Rainer HeinrichDresden, 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, 11.07.2012 Dr. Rainer HeinrichDresden, Germany
17
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€.
Tartu, 11.07.2012 Dr. Rainer HeinrichDresden, 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, 11.07.2012 Dr. Rainer HeinrichDresden, 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, 11.07.2012 Dr. Rainer HeinrichDresden, 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.
B
H
B
HB
H
0,05
0,95
P(X=0)
P(X>1)
P(X<1)
P(X>1)
Tartu, 11.07.2012 Dr. Rainer HeinrichDresden, Germany
21
)).0,x,180(binomcdf1(95.0))1,x,180(binomcdf1(05.02y
und)0,x,180(binomcdf95.0)1,x,180(binomcdf05.01y
Tartu, 11.07.2012 Dr. Rainer HeinrichDresden, Germany
22
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.
Tartu, 11.07.2012 Dr. Rainer HeinrichDresden, Germany
23
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€
Tartu, 11.07.2012 Dr. Rainer HeinrichDresden, Germany
24
result: If the streetcar-company would dismiss 17 inspectors, they would earn 7200€ more per day.
Tartu, 11.07.2012 Dr. Rainer HeinrichDresden, 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, 11.07.2012 Dr. Rainer HeinrichDresden, 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, 11.07.2012 Dr. Rainer HeinrichDresden, Germany
27
Solution
22 q1)q(S
q1q4q1)q(S 344
Top Related