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HOW TO “DOMESTICATE” LOGARITHMS IN SCHOOL?

HOW TO “DOMESTICATE” LOGARITHMS IN SCHOOL?. Dr. Marina Rugelj SLOVENIA. T he logarithmic function is introduced as the inverse function of the exponential one:. log a x = y  x = a y a > 0 and x > 0. Rules of calculations with logs. log a 1 = 0 log a a = 1

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HOW TO “DOMESTICATE” LOGARITHMS IN SCHOOL?

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  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: logax = y x = ay a > 0 and x > 0

  3. Rules of calculations with logs • loga 1 = 0 • loga a = 1 • loga (x y) = logax + logay • loga (x/y) = logax - logay • logaxr = r logax

  4. Students can: • Repeat the definition Put down the rules of calculation with logs Draw the graph of function: f(x) = log3 (x +3) - 1 Solve the equation: log3 (x – 2) – log3 (x + 1) = 3

  5. Students have: • Repeat the definition KNOWLEDGE Put down the rules of calculation with logs Draw the graph of function: f(x) = log3 (x +3) - 1 SKILLS Solve the equation: log3 (x – 2) – log3 (x + 1) = 3

  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. Teaching for understanding (Perkins, 1993)

  7. The spreading activation model Earthquake, Richtar scale … Exponential function Sound, dBel… Logarithmic function pH, acidity … Distances in Universe… 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 1Loudness of sound

  10. Human is equiped with very sensitive ears: threshold of hearing: 10-12 W/m2 threshold of pain: 10W/m2 10-12 W/m2…1 W/m2 … 200 km/h 1.8 mm/year

  11. Coversation is 1 000 000 times more intenstive than TOH … 10-11 10-12 W/m2 10-10 10-6 1km Conversation TOH Rustling leaves Whisper

  12. Loudness: I …. Intensity of sound Io …. Intensity of threshold Alexander Graham Bell 1847 - 1922

  13. The graph of f(x) = logax

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

  15. Example 2Map of the Universe

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

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

  18. Moon 1 cm Sun 5 m Saturn 19 m Proxima Centauri 1422 km Center of Milky Way 8 600 727 km Adromeda 727 753 846 km

  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 3pH measurement

  21. pH = -log[0H3+] 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 4Earthquakes

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

  26. 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)?

  27. Observations

  28. 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.

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

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

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

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

  33. Thank you!

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