1. LOGARITHMIC FUNCTION Jasna KOS Gimnazija Bežigrad, Ljubljana
Matija LOKAR
Faculty of mathematics and physic
University of Ljubljana, SLOVENIA

2. Slovene school system Various types of secondary schools
Gymnasium (grammar school)
15 years +
4 years
Preparation for higher education (University)
Ends with Matura – central external examination
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

3. Logarithmic function Second year of gymnasium
10th year of schooling (soon 11th)
16 year old students
Inverse function of exponential function
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

4. Approaches to teaching
Different materials, styles
Classical lecturing/demonstrations/lab work
Videos
Computer prepared lectures
Internet
Self discovering
Active participation
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

5. Workshop Introduction to the materials
The Rules of Logarithmic Computation
Translation and Scaling
Solutions to Equations and Inequalities
Using logarithmic function
In Psychology
Music
Loudness of Sound
Fractals
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

6. Workshop WWW: Brief explanation of all worksheets
Brief explanation of all worksheets
Introduction to two worksheets (with necessary DERIVE command explained)
Translation and scaling
Using logarithmic function in psychology
Group work: choose a worksheet
? Discussion
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

7. Worksheets Teacher’s part: Student’s part: explanations,
what to expect,
sometimes solutions, hints
Attention to “technical” aspects
Student’s part:
Completely prepared worksheets
Teachers are encouraged to use prepared work only as a basis
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

8. The rules of logarithmic computation
Discovering three main rules
Addition, subtraction, powers
Lab exercise or as a demonstration
Quite short exercise
discussion
WEB
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

9. The rules of logarithmic computation
On history of logarithmic function:
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

10. The rules of logarithmic computation
On Eulers' number e and natural logarithm
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

11. The rules of logarithmic computation
On rules for calculating with logarithms
 and many more, especially on Ask Dr. Math:  
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

12. The rules of logarithmic computation
Log(x) in DERIVE denotes natural logarithm
Log(x, 10) is a common logarithm
Natural logarithms are used
Be careful about settings in DERIVE
No difficulties with math. notation of rules – but expressing rules with words?
Next lesson:
Summarizing the findings
Proofs
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

13. Translation and scaling
Families of functions:
Ln(x – a)
a Log5(x)
Loga(x)
How a logb(x – c) looks like
Next hour – summarize the findings: curriculum requires capability of drawing log functions by hand
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

14. Translation and scaling
Students are already familiar with the basic graph of logarithmic function
Brief explanation on DERIVE use
Log(x) is not a common logarithm
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

15. Log(x) = ln(x) Use DERIVE to graph the function f(x) = log x !
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

16. Log(x) = ln(x) This is easy: Author, Expression, log x, OK, Plot, Plot
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

17. Log(x) = ln(x) Are you sure you have drawn
the graph of the logarithmic
function with the basis 10 ?
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

18. Log(x) = ln(x) Of course. Something seems wrong.
No. If I look at the point
with ordinate 1 on the
graph,….
I think this is the graph
of the function ln x.
Something seems wrong.
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

19. Translation and scaling
Different settings
Observation of the teacher’s presentation
Fill in the worksheets
Home work
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

20. Solutions to equations and inequalities
Most present exercises: elementarily solvable
Most harder exercises marked and avoided
Graphic approach
Numerical solution with DERIVE
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

21. Using logarithmic function
Trying to avoid “classical examples”
Present examples:
Music (musical ladders)
Sound (loudness)
Curve of forgetting (psychology)
Fractals
Additional suggestions on WWW
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

22. Using logarithmic function
Rocket equations ( with calculating how high the rocket model will go (
Exercises in Math Readiness on with number of exercises where log function is needed
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

23. Using logarithmic function
Radioactive decay
On earthquakes and Richter Magnitude
Many different problems:
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

24. Curve of forgetting F(t) = A – B log(t + 1)
A, B: experimentally determined
Simple math model:
Weakness of the model
Still some results possible
We show why logarithmic function is needed
Why is it necessary to interpret the results (calculation when we forget everything)
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

25. Mathematics and music Frequencies, hearing, C-minor scale, octaves
Table of frequencies for the tones in the first octave
Regression curve
Due to interest of the students: suitable utility function
Introduction of the Least squares method
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

26. Loudness of sound Logarithmic scale
Connection between the intensity and loudness
“Real life” examples
Different questions about the noise
Can we add loudness (noise)?
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

27. Fractals Students hear about them in art, nature, ...
Mathematic concepts necessary
Fractal = fractioned dimension
Homework, cauliflower,
Dimension of a fractal
Introduction to programming (recursion, ...)
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

28. SAMPLE: Translation and Scaling
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

29. SAMPLE: Curve of forgetting
Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000

30. YOUR WORK Computer algebra in mathematics education
Liverpool, 12th – 15th July 2000