Topicm6030930c7ef54b4e_1528449000663_0Topic

Application of the exponential functionexponential functionexponential function

Levelm6030930c7ef54b4e_1528449084556_0Level

Third

Core curriculumm6030930c7ef54b4e_1528449076687_0Core curriculum

V. Functions. The student:

14) uses the exponential and logarithmic function, including their graphs, to describe and interpret concepts connected with practical applications.

Timingm6030930c7ef54b4e_1528449068082_0Timing

45 minutes

General objectivem6030930c7ef54b4e_1528449523725_0General objective

Interpretation and the use of information presented both in a mathematical and popular science texts also using graphs, diagrams and tables.

Specific objectivesm6030930c7ef54b4e_1528449552113_0Specific objectives

1. Communication in English, developing mathematical, IT and basic scientific and technical competence, developing learning skills.

2. Getting to know the application of exponential function in realistic context.

Learning outcomesm6030930c7ef54b4e_1528450430307_0Learning outcomes

The student:

- gets to know the application of exponential function in realistic context.

Methodsm6030930c7ef54b4e_1528449534267_0Methods

1. Chain of association.

2. Situational analysis.

Forms of workm6030930c7ef54b4e_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm6030930c7ef54b4e_1528450127855_0Introduction

Students work in groups. The teacher gives every group a “chain” of empty links, which the students have to complete with already introduced concepts or other associations connected with exponential function.

During the presentation of the “chains”, the students pay attention to the elements repeating in the “chains”.

The teacher verifies the students’ answers and explains the doubts.

Procedurem6030930c7ef54b4e_1528446435040_0Procedure

The teacher informs the students that the aim of the class is getting to know the application of the exponential functionexponential functionexponential function in a realistic context.

The students work individually. They analyse the graph presenting the changes in the number of inhabitants in two towns.

Task 1

In the beginning there were 25 000 inhabitants in Town A. Every year the number of inhabitants grew by 4%.

In the beginning there were 10 000 inhabitants in Town B. Every year the number of inhabitants grew by 10%. Observe how the number of inhabitants of the tow towns has been changing.

[Illustration 1]

After how many years the number of inhabitants of Town B will be bigger than Town A? What is the shape of the curved line representing the change of the number of inhabitants? Formulate your conclusion.

The conclusion

The curved line representing the change of the number of inhabitants has the shape of the exponential functionexponential functionexponential function.

The students work in groups analyzing the material presented in the applet.

[Geogebra applet]

The students solve the tasks.

Task 2

Changing the position of the slider, observe how the number of the population of micro‑organismspopulation of micro‑organismspopulation of micro‑organisms changes. Complete the table for each initial number of micro‑organisms.

The initial number of micro‑organisms - 300, 500, 700, 1000.

The percentage increase of the micro‑organisms in every hour - 4%, 5%, 10%, 20%.

Example table.

[Table 1]

What is the shape of the curved line illustrating the change in the number of the population of micro‑organismspopulation of micro‑organismspopulation of micro‑organisms?

Task 3

Radioactive isotopes undergo spontaneous degradation. Look for the following ideas in the available sources: half‑life and the relationship describing the change of the sample mass over time. Write down an appropriate conclusion.m6030930c7ef54b4e_1527752263647_0Radioactive isotopes undergo spontaneous degradation. Look for the following ideas in the available sources: half‑life and the relationship describing the change of the sample mass over time. Write down an appropriate conclusion.

The conclusion:

Half‑life is the term used for the time a sample of radionuclide requires to decrease by half.  

$m_0$ - the initial sample mass,

$m_{\mathrm{t}}$ - the mass of the sample after time $t$,

$T$ - the half‑life.

Task 4

How many milligrams of 131 iodine isotope will remain from the sample of 40 mg after 48 days? What percentage of the isotope will decay in this time?m6030930c7ef54b4e_1527752256679_0How many milligrams of 131 iodine isotope will remain from the sample of 40 mg after 48 days? What percentage of the isotope will decay in this time?

[Table 2]

Having solved the tasks, the students present their results. The teacher explains the doubts and assesses the students’ work.

An extra task:

The mass of a radioactive substance is 100 grams and the decay causes the decrease of its mass by 25% every year.

a) Write the formula of a function m(t) describing the mass of this substance after a time t (t‑time given in years)

b) Calculate after how long time the mass of the substance will equal 45,5 grams.

Lesson summarym6030930c7ef54b4e_1528450119332_0Lesson summary

The students do the consolidation tasks. They formulate the conclusions to memorize.

The properties of the exponential functionexponential functionexponential function are, among others, used to describe:

- The change of the number of inhabitants of a given town,
- The change of the number of micro‑organisms,
- The change of the mass of radioactive isotopes.

Selected words and expressions used in the lesson plan

age of findsage of findsage of finds

decay of radioactive isotopesdecay of radioactive isotopesdecay of radioactive isotopes

exponential functionexponential functionexponential function

number of inhabitants of a townnumber of inhabitants of a townnumber of inhabitants of a town

population of micro‑organismspopulation of micro‑organismspopulation of micro‑organisms