Grammar: Gerund: Gerundial Constructions

Objectives: By the end of this unit, students should be able to use active vocabulary of this theme in different forms of speech exercises.

Students should be better at discussing on theme “What is mathematics?”.

Students should know the rule of non-finite form of the verb: Gerund and fulfill grammar exercises.

 

 

Methodical instructions: This theme must be worked out during two lessons a week according to timetable.

Lexical material: Introduce and fix new vocabulary on theme “What is mathematics?”. Students should be better at thinking about their future and what they are to do.

Students should know Gerund: Gerundial Constructions till the end of this course.

Grammar: Introduce and practice the Gerund and Gerundial Constructions. Revise the use of Infinitive Constructions.

What is mathematics?

The students of mathematics may wonder where the word "mathematics "comes from. Mathematics is a Greek word, and, by origin or etymologically, it means "something that must be learnt or understood", perhaps “acquired knowledge" or "knowledge acquirable by learning" or “general knowledge". The word "mathematics'' is a contraction of all these phrases. The celebrated Pythagorean school in ancient Greece had both regular and incidental members. The incidental members were called "auditors"; the regular members were named "mathematicians" as a general class and not because they specialized in mathematics; for them mathematics was a mental discipline of science of learning. What is ma­thematics in the modern sense of the term, its implications and connota­tions? There is no neat, simple, general and unique answer to this question.

Mathematics as a science, viewed as a whole, is a collection of branches. The largest branch is that which builds on the ordinary whole num­bers, fractions, and irrational numbers, or what, collectively, is called the real number system. Arithmetic, algebra, the study of functions, the cal­culus differential, equations, and various other subjects which follow the calculus in logical order, are all developments of the real number sys­tem. This part of mathematics is termed the mathematics of number. A second branch is geometry consisting of several geometries. Mathematics contains many more divisions. Each branch has the same logical structure: it begins with certain concepts, such as the whole numbers or integers in the mathematics of number, and such as point, line and tri­angle in geometry. These concepts must verify explicitly stated axioms. Some of the axioms of the mathematics of number are the associative, commutative, and distributive properties and the axioms about equalities. Some of the axioms of geometry are that two points determine a line, all right angles are equal, etc. From the concepts and axioms theorems are deduced. Hence, from the standpoint of structure, the concepts, axioms and theorems are the essential components of any compartment of ma­thematics. We must break down mathematics into separately taught sub­jects, but this compartmentalization taken as a necessity, must be com­pensated for as much as possible. Students must see the interrelation­ships of the various areas and the importance of mathematics for other domains. Knowledge is not additive but an organic whole and mathema­tics is an inseparable part of that whole. The full significance of mathe­matics can be seen and taught only in terms of its intimate relationships to other fields of knowledge. If mathematics is isolated-from other provinces, it loses importance.

Ex.1 Match the columns:

А	В
1. fraction 2. whole numbers 3. irrational numbers 4. differential equations 5. concept 6. point 7. line 8. triangle 9. equality 10. axiom	a) аксиома b) иррациональные числа c) дробь d) целые числа e) концепция f) точка g) дифференциальные уравнения h) равенство i) линия j) треугольник

Ex.2. Choose a,b,c or d.

1. Where does the word “mathematics” come from?

a) Greece b) England

c) Russia d) Alexandria

2. What does the word “mathematics” mean by origin or etymologically?

a) “acquired knowledge” b) “logical construction”

c) “scientific knowledge” d)“knowledge about nature”

3. What is mathematics as a science?

a) a real number system b) a collection of branches

c) a calculus in logical order

d) a calculus, differential equations, and functions

4. What is the largest branch of mathematics?

a) geometry b) differential equations

c) the whole number system d) the real number system

5. What is the certain concept of mathematics of number?

a) whole numbers or integers b) points, lines, and triangles

c) differential equations d)fractions and irrational numbers

6. What is deduced from the concepts and axioms?

a) structures b) theorems

c) calculus d) equations

Ex.3. Choose rhe title of the text according to summary.

a. Geometry

b. Mathematics of number

c. Mathematics as a science

d. The Pythagorean school

Ex.4. Translate the highlighted word correctly.

1. The incidental members were called “auditors”.

a) называют b) назывались

c) названный

2. This part of mathematics is termed the mathematics of number.

a) будет определена b) была определена

c) определяется

3. From the concepts and axioms theorems are deduced.

a) выводят b) выводятся

c) были выведены

4. Mathematics as a science, viewed as a whole, is a collection of branches.

a) рассматривается b) рассмотрела

c) рассматриваемая

5. A second branch is geometry consisting of several geometries.

a) состоящая b) состояла

c) состоит

6. These concepts must verify explicitly stated axioms.

a) установив b) установленные

c) устанавливающие

Ex.5 Listening

Follow the link: http://www.youtube.com/watch?v=r2HJcWg1Moo (Beauty and Truth in Mathematics and Science)

 

Grammar: Gerund: Gerundial Constructions

Do exercises from Units 53, p.106-107 ex.:53.1-53.4 and Unit 58 p.116-117 ex.:58.1-58.4 (Raymond Murphy “English Grammar in Use” A self-study reference and practice book for intermediate students of English Third Edition. Cambridge)

 

БӨЖ тапсырмалар:

Write an essay or be ready to speak about “What is mathematics?”.

Follow the link and pass the test for grammar:

http://www.study.ru/test/test.php?id=367

Unit 4

Theme: Probability of occurence.

Grammar: Gerundial Constructions

Objectives: By the end of this unit, students should be able to use active vocabulary of this theme in different forms of speech exercises.

Students should know the Gerundial Constructions till the end of this course.

 

Methodical instructions: This theme must be worked out during two lessons a week according to timetable.

Lexical material: Introduce and fix new vocabulary on theme “Probability of occurence”.

How long is the branch of science alive?. Discuss in groups.

Grammar: Introduce and practice the Gerundial Constructions. Revise the use of Gerundial Constructions

Probability of occurence

In mathematical language the choice, the probability of success is the ratio of the number of ways in which the trial can succeed to the total number of ways in which the trial can result. Here nothing favors the choice of any particular circle; they are all on the same page, and you are just as likely to cover one as another. The trial can result in five ways; there are five black circles. The trial can result in nine ways; there are nine circles in all (in exercise 1.1). If p represents the probability of success, then p = ,5-9..

Similarly, the probability of failure is the ratio of the number of ways in which the trial can fail to the total number of ways in which it can result. If q represents the probability of failure, in this case q = ,4-9.. Notice that the sum of probability of success and failure is 1. If you put your finger on a circle, it is certain to be either a black circle or a white one, for no other kind of circle is present. Thus p+q = ,5-9.+,4-9.=1. The probability that an event will occur can not be more than 1. When p =1, success is a certainty. When q =1, failure is sure. Let S represent the number of ways in which a trial can succeed. And let f represent the number of ways in which a trial can fail.

𝑝=,𝑆-𝑆+𝑓.;𝑞=,𝑓-𝑆+𝑓.;𝑝+𝑞=,𝑆-𝑆+𝑓.+,𝑓-𝑆+𝑓.=1

When S is greater than f, the odds are S to f in favor of success, thus the odds in favor of covering a black circle are 5 to 4. Similarly, when f is greater than S, the odds are f to S against success. And when S and f are equal, the chances are even; success and failure are equally likely. Tossing a coin illustrates a case in which S and f are equal. There are two sides to a coin, and there is no reason why a normal coin should fall one side up rather than the other. So if you toss a coin and call heads, the probability that it will fall heads is ,1-2.. Suppose you toss a coin a hundred times, for each of the hundred trials, the probability that the coin will come down heads is ,1-2.. You might expect fifty of the tosses to be heads. Of course, you may not get fifty heads.But the more times you toss a coin, the closer you come to the realization of what youexpect.

If p is the probability of success on one trial, and K is the number of trials, then the expected number is K p. Mathematical expectation in this case is defined as K p.

Ex.1. Answer the following questions.

a. What does the article deal with?

b. If you were shown 9 red circles and 6 black circles and were asked to choose one

of them which on these circles would you be likely to choose? Why?

c. Can you give the definition of the probability of failure? What is it?

d. What are the odds in case f > S?

e. What are the odds in case f < S?

f. Suppose S = f, what would the chances be?

g. Could you give some examples to illustrate a case when S and f are equal?

Ex.2. Are these statements true or false? Correct the false statements.

a. The trial can succeed in nine ways when you suppose that you have nine circles.

b. The sum of the probability of success and failure is equal to 1.

c. The probability that an event will occur can be more than 1.

d. In tossing two coins the fact that one fell heads would not affect the way the other

fell.

Ex.3. Fill in each gap using a word from the text.

a. There are differences of opinion among mathematicians and philoso– phers about ______ theory.

b. Suppose two dice are thrown. What are the chances that the ______of the faces is five?

c. Two coins are ______ simultaneous. Since a coin will come down ______ () or tail (T), each possible outcome is a member of A × A where A = { …, T}.

d. To describe this sample space ______ each situation in terms of events and discuss the chances of each event ______.

e. When we try to do something several times we say that we have had several ______.

Ex.4.Listening

Follow the link: http://www.youtube.com/watch?v=LSxqpaCCPvY (Mathematics Gives You Wings)