1. HOW TO “DOMESTICATE” LOGARITHMS IN SCHOOL? Dr. Marina Rugelj SLOVENIA

2. The logarithmic function is introduced as the inverse function of the exponential one: log a x = y  x = a y a > 0 and x > 0

3. Rules of calculations with logs log a 1 = 0 log a a = 1 log a (x y) = log a x + log a y log a (x/y) = log a x - log a y log a x r = r log a x

4. Students can: Repeat the definition Draw the graph of function: f(x) = log 3 (x +3) - 1 Solve the equation: log 3 (x – 2) – log 3 (x + 1) = 3 log 3 (x – 2) – log 3 (x + 1) = 3 Put down the rules of calculation with logs

5. Students have: Repeat the definition Draw the graph of function: f(x) = log 3 (x +3) - 1 Solve the equation: log 3 (x – 2) – log 3 (x + 1) = 3 log 3 (x – 2) – log 3 (x + 1) = 3 Put down the rules of calculation with logs KNOWLEDGE SKILLS

6. Knowledge and skills in themselves do not guarantee understanding. Rote knowledge generally defies active use, and routine skills often serve poorly because students do not understand when to use them. Rote knowledge generally defies active use, and routine skills often serve poorly because students do not understand when to use them. Teaching for understanding (Perkins, 1993)

7. Exponential function Logarithmic function Earthquake, Richtar scale … Sound, dBel… pH, acidity … Distances in Universe… The spreading activation model Spreading Activation Model of Semantic Memory (Collins & Loftus, 1975)

8. Applicability of logarithms Physic teacher: intensity of sound, loudness Chemistry teacher: definition of pH Geography teacher: earthquakes

9. Example 1 Loudness of sound

10. Human is equiped with very sensitive ears: threshold of hearing: 10 -12 W/m 2 threshold of pain: 10 W/m 2 10 -12 W/m 2 … 1 W/m 2 1.8 mm/year … 200 km/h

11. TOHTOH WhisperWhisper Rustling leavesRustling leaves 1km ConversationConversation 10 -12 10 -11 10 -10 10 -6 W/m 2 Coversation is 1 000 000 times more intenstive than TOH …

12. Loudness: I …. Intensity of sound I o …. Intensity of threshold Alexander Graham Bell 1847 - 1922 Sound intensity Loudness Threshold of hearing I o 10 -12 W/m 2 0Bel = 0 dB Whisper 10 -10 W/m 2 2Bel = 20 dB Conversation 10 -6 W/m 2 6Bel = 60 dB Threshold of pain 10 W/m 2 130dB

13. The graph of f(x) = log a x

14. Sound level meter La, la 50 La, la 60 If we like the loudness of 70 dB there should be 100 students (in an ideal condition).

15. Example 2 Map of the Universe

16. 286 000 km 149 000 000 km aprox. 1 light second 8,278 light minute Distance from the Earth to MOON SUN

17. The Sun is 520 times more far away from the Erath than the Moon.

18. Proxima Centauri1422 km Center of Milky Way8 600 727 km Adromeda727 753 846 km Sun5 m Moon1 cm Saturn19 m

19. Logaritmic scale The Moon 1.0 cm the Sun 2.7 cm, Saturn 3.3 cm, Proxima Centauri 8.2 cm, the center of the Milky Way 11.9 cm, Andromeda 13.9 cm.

20. Example 3 pH measurement

21. pH = -log[0H 3 + ] Soren Peter Lauritz Sorensen 1868 - 1939

22. 10 ml add water 100 ml 1000 ml HCl pH: 1 2 3

23. 1 liter of acid solution of pH 4 how much pure water do we need to get the solution of pH 9?

24. Example 4 Earthquakes

25. Charles F. Richter 1900 - 1985 Richter magnitude scale is Logaritmic scale: Earthquake of magnitude 6 is 100 times stronger than Earthquake of magnitude 4.

26. Richter Magnitudes Number of earthquakes per year Earthquake Effects < 3.5800 000Generally not felt, but recorded (detected by seismometers) 3.5 – 4.230 000Just about noticeable indoors 4.3 – 4.84 800Most people notice them, windows rattle 4.9 – 5.41 400Often felt (open doors swing), but rarely causes damage 5.5 – 6.1500Slight damage, to buildings, plaster cracks, bricks fall 6.2 – 6.9100Much damage to buildings, chimneys fall, houses move on foundations, can be destructive in areas up to about 100 kilometers across where people live. 7.0 – 7.315serious damage over larger areas, bridges twist, walls fracture, building may collapse 7.4 – 7.94Great damage, most buildings collapse > 8.0One every 5 to 10 years Total damage, surface waves seen, object thrown in the air

27. Find the strongest earthquakes in Slovenia! How many earthquakes were last three days? Find earthquakes where many people died! How much stronger was the earthquake in Slovenia in 1998 (5.6) than in 2004 (4.2)?

28. Observations

29. At first students were not so enthusiastic about my new approach of teaching. It was difficult for them to transfer concepts, ideas and procedures learned in mathematics to real life, to science. They had to use knowledge from other subjects, such as physics, chemistry, biology... Active participation brought them a feeling of success. They became motivated. They better understood the concept of logs.

30. Which part do you like the most? N=30

31. How often would you like such practical lesson? N=30

32. Have these lessons helped you to understand logarithms better? N=30

33. Do you like these type of tasks would appear in exams? N=30

34. Thank you!