HOW TO HOW TO “DOMESTICATE” “DOMESTICATE” LOGARITHMS IN LOGARITHMS IN

SCHOOL?SCHOOL?

Presented at Presented at

5th International Colloquium 5th International Colloquium on the Didactics of Mathematics on the Didactics of Mathematics

University of Crete, GreeceUniversity of Crete, Greece

April 2008April 2008

Dr. Marina Rugelj, SLOVENIADr. Marina Rugelj, SLOVENIA

St. Stanislav Institution St. Stanislav Institution

Private high school

Students from 15 to 19 years old

Ljubljana, Slovenia

Combined lessonsCombined lessons

Dr. Marina Rugelj - maths teacherTine Golež, MSc. - physics teacher

European project: ScienceMath

Example: Logaritmic function

TThe logarithmic function is he logarithmic function is introduced as the inverse function introduced as the inverse function

of the exponential of the exponential one:one:

loglogaaxx = = yy xx = a = ayy

a > 0 and x > 0

The graph of The graph of ff((xx) = log) = logaa xx

Rules of calculations with logsRules of calculations with logs

loglogaa 1 = 0 1 = 0

loglogaa a = 1 a = 1

loglogaa ( (x yx y) = log) = logaa xx + log + logaa yy

loglogaa ( (x/yx/y) = log) = logaa x x - log- logaa yy

loglogaa xxrr = = rr log logaa xx

Students can:

Repeat the definitionRepeat the definition

Draw the graph of function:Draw the graph of function:

ff((xx) = log) = log33 ( (xx +3) - 1 +3) - 1

Solve the equation: Solve the equation:

loglog33 ( (xx – 2) – log – 2) – log33 ( (xx + 1) = 3 + 1) = 3

Put down the rules of calculation with logsPut down the rules of calculation with logs

Students have:Students have:

Repeat the definitionRepeat the definition

Draw the graph of function:Draw the graph of function:

ff((xx) = log) = log33 (x +3) - 1 (x +3) - 1

Solve the equation: Solve the equation:

loglog33 (x – 2) – log (x – 2) – log33 (x + 1) = 3 (x + 1) = 3

Put down the rules of calculation with logsPut down the rules of calculation with logsKNOWLEDGE

SKILLS

KKnowledge and skillnowledge and skillss in themselves do not in themselves do not guarantee understanding.guarantee understanding.

Rote knowledge generally defies active use, Rote knowledge generally defies active use, and routine skills often serve poorly and routine skills often serve poorly because students do not understand when because students do not understand when to use them. to use them.

Teaching for understanding (Perkins, 1993)Teaching for understanding (Perkins, 1993)

Exponential function

Logarithmic function

Earthquake, Richtar scale …

Sound, dBel…

pH, acidity …

Distances in Universe…

The spreading activation modelThe spreading activation model

Spreading Activation Model of Semantic Memory (Collins & Loftus, 1975)

Applicability of logarithmsApplicability of logarithms

Physic teacher: Physic teacher: intensity of sound, loudnessintensity of sound, loudness

Chemistry teacher:Chemistry teacher:definition of pHdefinition of pH

Geography teacher:Geography teacher:earthquakesearthquakes

Example 1Example 1

Loudness of soundLoudness of sound

Human is equiped with very sensitive ears:Human is equiped with very sensitive ears:

threshold of hearing: 10threshold of hearing: 10-12 -12 W/mW/m22

threshold of pain: 10threshold of pain: 10 W/mW/m22

10-12 W/m2 … 1 W/m2

1.8 mm/year … 200 km/h

TOH

Whisper

Rustling leaves

1km

Conversation

10-12 10-11

10-10 10-6 W/m2

Coversation is 1 000 000 times Coversation is 1 000 000 times more intenstive than TOH …more intenstive than TOH …

Loudness:Loudness:

II …. Intensity of sound …. Intensity of sound

II o o …. Intensity of threshold…. Intensity of threshold

oI

Ilog

Alexander Graham BellAlexander Graham Bell1847 - 19221847 - 1922

Sound intensity Sound intensity LoudnessLoudness

Threshold of hearing Threshold of hearing IIoo 1010-12 -12 W/mW/m22 0Bel = 0 dB0Bel = 0 dB

WhisperWhisper 1010-10 -10 W/mW/m22 2Bel = 20 dB2Bel = 20 dB

ConversationConversation 1010-6 -6 W/mW/m22 6Bel = 60 dB6Bel = 60 dB

Threshold of painThreshold of pain 10 10 W/mW/m22 130dB130dB

Sound level meterSound level meter

La, la

50

La, la

La, la La,

la

La, la

La, la

La, la 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).

Example 2Example 2

Map of the UniverseMap of the Universe

286 000 km286 000 km

149 000 000 km149 000 000 km

aprox. aprox. 1 light second1 light second

8,278 light minute8,278 light minute

Distance from the Earth to

MOON

SUN

The Sun is 520 times more far away

from the Eraththan the Moon.

Proxima Centauri 1422 km

Center of Milky Way 8 600 727 km

Adromeda 727 753 846 km

Sun 5 m

Moon 1 cm

Saturn 19 m

Logaritmic scaleLogaritmic scale

TheThe Moon Moon 11.0.0 cm cm the Sun the Sun 2.7 cm, 2.7 cm, Saturn Saturn 3.3 cm, 3.3 cm, Proxima Centauri Proxima Centauri 8.2 cm, 8.2 cm, the center of the center of the Milky Way the Milky Way 11.9 cm, 11.9 cm, Andromeda Andromeda 13.9 cm.13.9 cm.

Example 3Example 3

pH measurementpH measurement

pH = -log[pH = -log[00HH33++]]

Soren Peter Lauritz SorensenSoren Peter Lauritz Sorensen1868 - 19391868 - 1939

10 ml 10 ml add wateradd water 100 ml 1000 ml 100 ml 1000 mlHClHCl

pH: pH: 11 2 2 3 3

1 liter of acid solution of pH 41 liter of acid solution of pH 4

how much pure water do we needhow much pure water do we need

to get the solution of pH 9?to get the solution of pH 9?

Example 4Example 4

EarthquakesEarthquakes

Charles F. RichterCharles F. Richter1900 - 19851900 - 1985

Richter magnitude scale isLogaritmic scale:

Earthquake of magnitude 6

is 100 times stronger than

Earthquake of magnitude 4.

Richter Magnitudes

Number of earthquakes per year

Earthquake Effects

< 3.5 800 000 Generally not felt, but recorded (detected by seismometers)

3.5 – 4.2 30 000 Just about noticeable indoors

4.3 – 4.8 4 800 Most people notice them, windows rattle

4.9 – 5.4 1 400 Often felt (open doors swing), but rarely causes damage

5.5 – 6.1 500 Slight damage, to buildings, plaster cracks, bricks fall

6.2 – 6.9 100 Much 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 – 7.3 15 serious damage over larger areas, bridges twist, walls fracture, building may collapse

7.4 – 7.9 4 Great damage, most buildings collapse

> 8.0 One every 5 to 10

years

Total damage, surface waves seen, object thrown in the air

Find the strongest Find the strongest earthquakes in Slovenia!earthquakes in Slovenia!How many earthquakes How many earthquakes were last three days?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)?

Observations Observations

At first students were not so enthusiastic about At first students were not so enthusiastic about my new approach of teaching.my new approach of teaching.

It was difficult for them to transfer concepts, It was difficult for them to transfer concepts, ideas and procedures learned in mathematics to ideas and procedures learned in mathematics to real life, to sciencereal life, to science. .

They had to use knowledge from other subjects, They had to use knowledge from other subjects, such as physics, chemistry, biology ...such as physics, chemistry, biology ...

Active participation brought them a feeling of Active participation brought them a feeling of success.success.

They became motivated.They became motivated.

They better understood the concept of logs.They better understood the concept of logs.

Which part do you like the most?Which part do you like the most?

Loudness8%

Map of the Universe

42%pH25%

Earthquake25%

N=30

How often would you like such How often would you like such practical lesson?practical lesson?

Once per months

45%

Every lesson7%

Some lessons per year

7%

Once per w eek41%

N=30

Have these lessons helped you to Have these lessons helped you to understand logarithms better?understand logarithms better?

Yes63%

No15%

I do not know22%

N=30

Do you like these type of tasks Do you like these type of tasks would appear in exams?would appear in exams?

Yes15%

No 55%

I do not know30%

N=30

Next year we will revise all the Next year we will revise all the mathematical concepts that the mathematical concepts that the

students have learned in secondary students have learned in secondary school for the final exams. school for the final exams.

At that time I will check if my At that time I will check if my students have truly constructed a students have truly constructed a

knowledge network and if they knowledge network and if they know better the concept of the know better the concept of the

logarithm. logarithm.

Thank you!Thank you!