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PROGETTO CLIL 2006 – 2007 EXPONENTIAL AND LOGARITHMIC FUNCTIONS

Prof.ssa Barbara di Majo in collaborazione con Prof.ssa Cinzia Calella Istituto Tecnico Commerciale “G. R. Carli” Trieste e Prof.ssa Patrizia Torelli Prof.ssa Barbara Fasano Istituto Tecnico Nautico “T. di Savoia” Trieste.

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PROGETTO CLIL 2006 – 2007 EXPONENTIAL AND LOGARITHMIC FUNCTIONS

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  1. Prof.ssa Barbara di Majo in collaborazione con Prof.ssa Cinzia Calella Istituto Tecnico Commerciale “G. R. Carli” Trieste e Prof.ssa Patrizia Torelli Prof.ssa Barbara Fasano Istituto Tecnico Nautico “T. di Savoia” Trieste PROGETTO CLIL2006 – 2007EXPONENTIAL AND LOGARITHMIC FUNCTIONS

  2. EXPONENTIALS AND LOGARITHMS • WHO: IV CLASS • WHEN: I TERM • TIME: 8-10 HOURS

  3. AIMS Students should : • Improve their abilities with numbers and symbols • Understand some practical applications of maths

  4. OBJECTIVES • To work with powers, roots, logarithms • To revise and use some topics studied in the I and II class • To draw exponential and logarithmic functions

  5. CONTENTS • Powers (revise) • Exponential and logarithmic functions • Logarithms • Tasks • Practical application

  6. PREREQUISITES Students should know: • Powers of numbers and their properties • Coordinates

  7. LESSON 1 Objective: to encourage students to read and speak Time: 1 h (task included)

  8. VOCABULARY potenza power base base a alla n a to the n esponente exponent/index per times più plus meno minus diviso divided

  9. radice root radice quadrata square root radice cubica cube root alla seconda squared alla terza cubed elevare to raise to inversa inverse logaritmo logarithm log in base 10 common logarithm log naturale natural/neperian log

  10. Task: singular work Every student should create a sentence using one of the words of the glossary (except for the last 3) and report to the whole class

  11. LESSON 2 Objective: to revise the powers (contents studied in the I and II class) Time: 1 h (task included)

  12. WHAT ARE POWERS? There are many particular multiplications in which all the factors are all the same For example: 2·2∙2·2·2 Not to write in a such long way, it has been introduced a new mathematical operation: the power So 2·2·2·2·2 is written as 25

  13. PROPERTIES

  14. PROPERTIESan · am ≡an+mam : an≡ am-n(am)n ≡ am·n

  15. PROPERTIES

  16. Task: individual work Each student should create an example of the previous properties and report to the whole class

  17. Indicate if every raltionship is true or false: 1. 53 = 15 T F 2. 24 = 16 T F 3. T F 4. (53)7 = 521 T F 5. 00 =1 T F 6. 83 : 83 = 0 T F 7. (73 : 72)0 = 1 T F 8. (142 : 72) = (14 : 7)2 T F 9. (153 : 33) = 125 T F 10. (24)3 : (24)2 = 16 T F 11. (34)2 3 = 39 T F 12. 23 25 : 22 : 28 = 2-2 T F

  18. 1. 2. 3. 4. Calculate the value of these expressions using powers properties:

  19. LESSON 3 Objective: • To draw “a raised to the power of x” • To understand the characteristics of these functions Time: 2h

  20. First task (individual or pair work) Calculate the values of the function giving to x positive and negative values

  21. Remember! Properties of the power of the numbers:

  22. Second task (individual work) Calculate the values of the function giving to x positive and negative values

  23. Third task (small groups) • Draw the two functions on the same Cartesian Plane • Compare your results • Describe their properties

  24. Plenary lesson The teamleaders of the groups report their conclusions to the whole class

  25. Blue:a>1; Red: 0<a<1

  26. Properties • Increasing • Continue • Asymptote: x axis • Positive

  27. LESSON 4 Objective: to work with logarithms Time: 2 h (task included)

  28. We call logarithm of bin base a logarithms the exponent x we give to the base a to obtain the number b

  29. x = loga b • ax = b • base

  30. Definition:

  31. PROPERTIES:

  32. Changing the base of a logarithm logac = x → c ≡ ax so logbc ≡ logbax ≡ x·logba

  33. Therefore or

  34. Task: (individual work):Calculate (remember the properties!) 1. log (3xy) 6. log 2. log (a2bc3) 7. log 3. log 8. log 4. log 9. log 5. log 10. log

  35. Semplify: 1. log x – log y – log z 2. 2log a + 3log b – 5log c 3. log x + log y - log z 4. 2log a –log a3 5. log 16 – 3log 2 + log 4 6. log 27 – log 5 + log 3 7. 8.

  36. LESSON 5 Objective: • to apply the definition of logarithm • to draw logarithmic functions • to understand their carachteristics Time: 2 h

  37. Inverse functions:

  38. First task (in pairs) Calculate the values of the functions and draw them on the same cartesian plane giving to x positive values only

  39. Second task (small groups) • Draw the two functions on the same cartesian plane • Compare your results • Describe their properties

  40. Red:a>1, Blue: 0<a<1

  41. Exponential equations • You are dealing with an exponential equation when the unknown is at the exponent. So, in general, an exponential equation is presented in the form: ax=b

  42. How to solve it? As you certainly remember, to solve an equation, you pass through inverse operations (in both member of the equation), in order to semplify it. • For example: • x+3=5 x=5-3 - is the inverse operation of + • 5x=15 x=15:5 : is the inverse operation of • • x=32 the power is the inverse operation of , the unknown is the base

  43. So, if the unknown is at the exponent…try to guess how to solve it: which is the inverse operation of the power when the unknown is at the exponent? ax=b so logax=logb and, using the logarithmic properties: xloga=logb so

  44. First task • Try to solve the following exponential equations using the logarithms and their properties:

  45. Can you guess how to solve simplier if you have just powers of the same base? And can you try to solve equations like these?

  46. LESSON 6 Objective: to connect maths with other disciplines Time: 1 h

  47. Applications of exponential equations in financial mathematics • Glossary • Linear capitalization law • Exponential capitalization law • Searching for the time in the capitalization law

  48. Glossary Matematica finanziaria = financial mathematics Interest rate = tasso di interesse Montante = total amount Valore attuale = current value

  49. Linear capitalization law • In the linear capitalization law, interests are always proportional to the time and the nominal value of the capital • I = i C t where: • I = interest • i = interest rate • C = nominal value of the capital • t = time • So M=C+Cit=C(1+It) Where M = final amount

  50. Exponential capitalization • In the esponential capitalization the total amount is calculated on the total amount of the previous year 0 1 2 M=c M1=C+Ci M2=M1+M1i M1=C(1+i) M2=M1(1+i) M2=c(1+i)(1+i) M2=C(1+i)2

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