State Autonomous Professional

educational institution

"Orsk Medical College"

Methodological development by discipline

ODB.06 Mathematics

Subject:

COMPILER REVIEWED

at the CMC meeting

Mathematics teacher: general humanities,

I.V. Abroskina mathematical and

natural sciences

Protocol No. ____

dated _____________ 2016

CMC Chairman:

T.V. Gubskaya

Orsk, 2016

EXPLANATORY NOTE

At the heart of the Federal State educational standard is a system-activity approach. GEF sets new tasks for teachers.

development and education of the individual in accordance with the requirements of the modern information society;

development of students' ability to independently receive and process information on educational issues;

individual approach to students;

development of communication skills among students;

orientation to the application of a creative approach in the implementation of pedagogical activities.

The system-activity approach as the basis of the Federal State Educational Standard helps to effectively implement these tasks. The main condition for the implementation of the standard is the inclusion of students in such activities, when they independently carry out an algorithm of actions aimed at obtaining knowledge and solving the learning tasks assigned to them. The system-activity approach as the basis of the Federal State Educational Standard helps to develop children's abilities for self-education.

Within the framework of this approach, the theme "Trigonometric functions, their properties and graphs".

Methodological development is based on work program(FSES, specialty 34.02.01 Nursing, 31.02.03 Laboratory diagnostics), for which 2 hours are allotted for studying the topic "Trigonometric functions, their properties and graphs" practical session. Within the framework of the topic, the main properties of trigonometric functions and their graphs are considered, the connection of these functions with medicine and other fields of knowledge, the importance of this topic is emphasized.

In the course of mastering the topic "Trigonometric functions, their properties and graphs", students are aware of the role of mathematics and trigonometry in medicine, namely, by deciphering the cardiogram of the heart, they learn to calculate heart rate (heart rate), recognize sinus rhythm (normal, tachycardia, bradycardia).

When studying this topic, there is a connection with medicine, biology, anatomy, which certainly motivates students to study this topic, and allows them to further deepen their knowledge of the subject.

In the process of studying the topic "Trigonometric functions, their properties and graphs" students will be able to real life and in Soviet professional activity determine the heart rate from the cardiogram of the heart and draw a conclusion about the nature of the sinus rhythm.

Topic: Trigonometric functions, their properties and graphs

Tutorials:

Know all the properties of trigonometric functions, be able to plot trigonometric functions. To be able to make a conclusion on the cardiogram of the heart about sinusoidal rhythm and heart rate.

Developing:

yfromx

Educational:

Cultivate accuracy, purposefulness, discipline.

to continue the education of activity, mutual assistance, creative attitude to business.

Teaching aids, equipment

Plan-outline, computer, projector, presentation.

View training session

Theoretical and practical

Applied technologies

System-activity approach, information Technology, problem-based learning technology.

Lesson structure

Stage 1.

Organizing time / 1-2 minutes

Student activities

Preparing for the lesson

Teacher activity

Checking those present, setting the mood for the lesson

Stage 2.

Motivational moment / 2 minutes

Student activities

Formulation of the purpose of the lesson

Teacher activity

1. Formulates the topic of the lesson

2. Leads students to formulate the purpose of the lesson

3. Causes interest in the material being studied by various methods 4. Creates motivation

Stage 3.

Frontal survey / up to 8 minutes

Student activities

Answer questions

Teacher activity

Stage 4.

Learning new material /50 minutes

Student activities

1. Work with notes, writing in a notebook of the main points indicated by the teacher

2. Independent description of the properties of trigonometric functions according to the schedule

3. Trigonometry in human life; Relationship of trigonometry with medicine, research work (presentations) - 2 groups of students

Teacher activity

Explanation of the new material:

1. Statement of the problem question:

What is the significance of trigonometry for medicine?

2. View function (definition, graph)

3. Function of the form (definition, graph

4. Showing the video "ECG is within the power of everyone"

Stage 5.

The stage of consolidation and generalization of knowledge / 20 minutes

Student activities

1. Work in groups. Creation of a "council" of physicians and statement of the conclusion on the cardiogram of the heart on sinusoidal rhythm and heart rate (HR)

2. Summing up, writing conclusions in a notebook

Teacher activity

1. Help in formulating conclusions

2. Control and correction of knowledge, providing the opportunity to identify the causes of errors and correct them.

Stage 6.

Reflection /6 minutes

Student activities

.

2. Work with abstracts

Marginal notes:

"+" - knew

«!» - new material(learned)

"?" - I want to know

Teacher activity

Result control learning activities, Assessment of knowledge.

Stage 7.

Homework / 2 minutes

Homework content

You can't understand the basics without knowing mathematics.

modern technology, nor how scientists study

natural and social phenomena.

A.N. Kolmagorov

Related lesson : Trigonometric functions, their properties and graphs.

organizational information

Lesson topic: Trigonometric functions, their properties and graphs

Thing: Mathematics

Teacher: Abroskina Irina Vladimirovna

Educational institution: GAPOU "Orsk Medical College"

Methodical base:

1. Lukankin A.G. - Mathematics: textbook. for students Wednesdays. prof. education / A.G. Lukankin. - M.: GEOTAR - Media, 2012. - 320 p.

2. Mordkovich A.G. - Algebra and the beginning of analysis. 10-11 cells: Proc. for general education institutions. - M.: Mnemosyne, 2012. - 336 p.

3. Studies.en

4. Math. en"library"

5. History of mathematics from ancient times to early XIX century in 3 volumes// ed. A.P. Yushkevich. Moscow, 1970 - volume 1-3 E. T. Bell Creators of mathematics.

6. Predecessors of modern mathematics / / ed. S. N. Niro. Moscow, 1983 A. N. Tikhonov, D. P. Kostomarov.

7. Stories about applied mathematics//Moscow, 1979. A. V. Voloshinov. Mathematics and Art // Moscow, 1992. Newspaper Mathematics. Supplement to the newspaper dated 1.09.98.

Lesson type: combined

Duration: 2 teaching hours

The purpose of the lesson: The study of trigonometric functions, their properties and graphs.

Definition of the role of trigonometry for medicine.

Lesson objectives:

Educational : Know all the properties of trigonometric functions, be able to plot trigonometric functions. To be able to make a conclusion on the cardiogram of the heart about sinusoidal rhythm and heart rate.

Developing: Continue building skills and abilities to build charts using dependencyyfromx. Show the importance of trigonometry for medicine.

Educational: Cultivate accuracy, purposefulness, discipline. Pcontinueeducation of activity, mutual assistance, creative attitude to business.

Used technologies: system-activity approach, developing training, group technology, elements research activities, ICT.

Equipment and materials for the lesson: computer, projector, student presentations, video "ECG for everyone"

Lesson plan:

1. Organizational moment - 1-2 minutes.

2. Motivational moment - 2 min.

3. Frontal survey - 8 min.

4. Learning new material - 50 min.

5. Consolidation and generalization of knowledge - 20 min

6. Reflection - 6 min.

7. Homework - 2 min.

During the classes

1. Organizational moment

Checking those present, set the mood for the lesson.

2. Motivational moment

Lesson topic message

Leading students to independently formulate the goal of the lesson

Emphasizing the importance of this topic for medicine and the world around.

3. Frontal survey

Answers to questions on homework (analysis of unsolved problems)

Students' answers to teacher's questions ( At this stage, the students' knowledge is updated, which is necessary for further work in the lesson):

1. What is trigonometric functions numeric argument?

2. What is the meaning of trigonometric functions in the first quarter (table of values)?

3. Which features are even and which are odd?

4. What is the symmetry of the graphs of even and odd functions?

5. Which of the trigonometric functions are even (odd)?

4. Learning new material

1) I would like to start studying the topic with the words of the great mathematician Nikolai Ivanovich Lobachevsky: "There is not a single area of ​​mathematics that will not someday be applicable to the phenomena of the real world.

2) Let's put the question: What is the meaning of trigonometry for medicine?

I hope, after studying our topic, each of you will be able to answer the question posed.

3) So, let's start studying trigonometric functions, consider their basic properties and build their graphs.

Trigonometric functions

The main trigonometric functions are the functions y=sin(x), y=cos(x), y=tg(x), y=ctg(x). Let's consider each of them separately.

Y = sin(x)

Graph of the function y=sin(x).

Basic properties:

3. The function is odd.

Y = cos(x)

Graph of the function y=cos(x).

Basic properties:

1. The area of ​​​​definition is the entire numerical axis.

2. The function is limited. The set of values ​​is the segment [-1;1].

3. The function is even.

4. The function is periodic with the smallest positive period equal to 2*π.

Y = tan(x)

Graph of the function y=tg(x).

Basic properties:

1. The domain of definition is the entire numerical axis, except for points of the form x=π/2 + π*k, where k is an integer.

3. The function is odd.

Y = ctg(x)

Graph of the function y=ctg(x).

Basic properties:

1. The domain of definition is the entire numerical axis, except for points of the form x=π*k, where k is an integer.

2. The function is unlimited. The set value is the entire number line.

3. The function is odd.

4. The function is periodic with the smallest positive period equal to π.

4) Why does a person need to know the properties of functions and the ability to read graphs in life?Any repetitive movement is calledVOCATIONS

The practice of studying oscillations has shown a useful and harmful role.

Every specialist needs to know the theory of oscillatory processes.

The theory of oscillations is a field of science related to mathematics, physics and medicine. Harmonic vibrations

Mechanical vibrations

Vibration. Harmful effects of vibration

Ultrasound

infrasound sound

Electromagnetic oscillations (used for radio, television,

communications with space objects)

Conclusion :

Oscillations occur according to the laws of sines and cosines

Properties of trigonometric functions show which parameters can be changed

Measurement results and calculations show how to avoid harmful effects fluctuations and how to apply them

5) Let us dwell in more detail on the theory of oscillations in medicine. Where do you meet fluctuations in your body -A HEART. What is the cardiogram of the heart called?SINUSOID. Therefore, the heart works according to trigonometric laws, and we just need to know and understand them.

Trigonometric laws are also found in the world around us:

In nature (biology)

In architecture (buildings, structures)

In music (harmonious melodies)

and in other areas.

Now to your attention, a group of students will present their research papers to you at this topic. Presentation of presentations by students on the topics:

- "Communication of trigonometric function and medicine"

- "Trigonometry in medicine"

- "Trigonometry in the world around us and human life"

6) Watching the educational video "ECG for everyone"

7) Acquaintance of students with the ECG of a healthy person, and with rhythm disturbance.

8) Formula for calculating heart rate (heart rate)

5. Consolidation and generalization of knowledge

1. Divide students into 2 groups.

2. Work in groups. Creation of a "consilium" of physicians and statement of the conclusion on the cardiogram of the heart on sinus rhythm and heart rate (HR)

3. Voicing their conclusions (one representative from the group)

4. Main conclusions, correction by the teacher of the main conclusions.

6. Reflection

1. Independent summing up the lesson, introspection and self-assessment.

2. Working with abstracts

Marginal notes:

"+" - knew

"!" - new material (learned)

"?" - I want to know

3. Assessment of knowledge.

7. Homework

1. Mathematics, Bashmakov M.I., 2012 - P.107 / P.165

2. Prepare (optional) a message: "Trigonometry in medicine and biology"

Appendix to the lesson

Student presentations

(research groups)

Lessons 25-26. Functions y \u003d tg x, y \u003d ctg x, their properties and graphs

09.07.2015 7626 0

Target: consider graphs and properties of functions y = tg x, y = ctg x.

I. Communication of the topic and objectives of the lessons

II. Repetition and consolidation of the material covered

1. Answers to questions on homework (analysis of unsolved problems).

2. Monitoring the assimilation of the material (written survey).

Option I

2. Graph the function:

Option 2

1. How to graph a function:

2. Graph the function:

III. Learning new material

Consider the two remaining trigonometric functions - tangent and cotangent.

1. Function y \u003d tg x

Let us dwell on the graphs of the tangent and cotangent functions. First, let's discuss plotting the function y = tg x on the interval Such a construction is similar to constructing a graph of the function y \u003d sin x described earlier. In this case, the value of the tangent function at a point is found using the line of tangents (see figure).

Taking into account the periodicity of the tangent function, we obtain its graph over the entire domain of definition by parallel translations along the abscissa axis (to the right and left) of the already constructed graph by π, 2π, etc. The graph of the tangent function is called the tangentoid.

We present the main properties of the function y = tg x:

1. The domain of definition is the set of all real numbers, with the exception of numbers of the form

y(x

3. The function increases on intervals of the formwhere k ∈ Z .

4. The function is not limited.

6. The function is continuous.

8. Periodic function with the smallest positive period T \u003d π, i.e. y (x + n k) = y(x).

9. Function graph has vertical asymptotes

Example 1

Set whether the function is even or odd:

It is easy to check that for the functions a, b the domain of definition is a symmetric set. Let us examine these functions for evenness or oddness. To do this, find y(-x) and compare the values ​​of y(x) and y(-x).

a) We get: Since the equality y(-x ) = y(x), then the function y(x) is even by definition.

b) We have:

Since the equality y(-x ) = -y(x), then the function y(x) is odd by definition.

c) The domain of this function is an asymmetric set. For example, the function is defined at the point x = π/4 and not defined at the symmetrical point x = -π/4. Therefore, this function has no definite parity.

Example 2

Find the main period of the function

This function y(x) is algebraic sum three trigonometric functions whose periods are equal: T 1 \u003d 2π, Write these numbers as fractions with the same denominators.Least common multiple of LCM coefficients (6; 2; 3). Therefore, the main period of this function

Example 3

Let's plot the function

Let's take into account the rules for transforming function graphs. According to them, the graph of the functionis obtained by shifting the graph of the function y = tg x by π/4 units to the right along the abscissa axis and stretching it 2 times along the ordinate axis.

Example 4

Let's plot the function

Using the definition and properties of the module, in the function argument, we will reveal the signs of the module, considering three cases. If x< 0, то имеем: For 0 ≤ x ≤ π /4 we have: For x > π /4 we have: Then it remains to build three parts of this graph. At x< 0 строим прямую у = -1. Для 0 ≤ x ≤ π /4 we build a tangentoidThis graph is obtained by shifting the graph of the function y = tg x by π/8 to the right along the abscissa axis and halving along this axis. For x > π/4 build a line y = 1.

2. Function y \u003d ctg x

Similarly to the graph of the function y \u003d tg x or using the reduction formulaa graph of the function y \u003d ctg x .

We list the main properties of the function y = ctg x :

1. Domain of definition - the set of all real numbers, with the exception of numbers of the form x = n k , k ∈ Z .

2. The function is odd (i.e., y(-x) = - y(x )), and its graph is symmetrical with respect to the origin.

3. The function is decreasing on intervals of the form (n k; n + p k ), k ∈ Z .

4. The function is not limited.

5. The function does not have the smallest and largest values.

6. The function is continuous.

7. Range of values ​​E(y) = (-∞; +∞).

8. Periodic function with the smallest positive period T \u003d n, i.e. y (x + n k) = y(x).

9. The graph of the function has vertical asymptotes x = n k .

Example 5

Let's find the domain of definition and the range of values ​​of the function

It is obvious that the domain of the function y(x ) coincides with the domain of the function z=ctg x, i.e., the domain of definition is the set of all real numbers, except for numbers of the form x = nk , k ∈ Z .

function y (x) complex. Therefore, we write it in the formParabola vertex coordinates y(z): zB = 1 and y in = 2 - 4 + 5 = 3. Then the range of values ​​of this function is Е(у) = )