See the Unit 10 from the Oxford Practice Grammar by George Yule and follow the instructions.

Do the grammar exercises: Unit 10/10.17-10.19 p. 60 Oxford Practice Grammar. Supplementary exercises, 2007 by George Yule.

Self work tasks

Discussion What is mathematics?

http://www.english-easy.info/topics/topics_Mathematics.php#axzz2B47MFA98

Doing grammar exercises Unit 87/87.1-87.3 p. 175 Advanced Grammar in Use, 2006 by Martin Hewings

Unit 7 Theme: Counting. Natural Numbers. Notations

Grammar: Review

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 Counting. Natural Numbers. Notations and its functions.

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 “Counting. Natural Numbers. Notations”. Define the basic peculiarities of internet functions in our life.

Grammar: Revision

Ex 1 Read and translate the text. Sum the main ideas. Be ready to ask and answer all topical questions. Work in pairs.

Counting. Natural Numbers. Notations

What are numbers, after all? In view of the fact that most four year olds are able to manage finger – reckoning and can count at least to 10, the question posed may seem meaningless and irrelevant. Nevertheless, to answer it is by no means an easy task, as number is both an everyday word and a scientific term. Ordinary people take numbers for granted as a means in counting and measurements and never think about their origin and evolution. This is not the case with mathematicians. Numbers for them are not only basic concepts but the most mysterious math entities as well.

Generally, mathematicians try to clarify vague notions by means of precise definitions. The formal explicit (verbal) and rigorous definition of the term “number” has been unavailable so far. Even though we may know intuitively what natural or cardinal numbers -3, -2, 0, 1, 2, 3, ... are, they are not easy to define. The scientific definition demands a lot, indeed: the knowledge of maths, erudition, rich and productive imagination and lucky insight. It is customary nowadays that mathematicians do not give a single formal definition of a number concept, possibly because numbers have so many varied properties and interrelations which mathematicians fail to compress into one explicit definition. There might be other reason, of course.

No mathematician knows the name of the man who was the first to say 1, 2, 3, … Unfortunately, the origin of many fundamental math concepts is wholly anonymous. Obviously, this still unidentified person was abstract-minded, with number sense, had a lot of experience (practice) in counting and creative insight. Maths owes a lot of this generator of abstract mental idea of a number concept.

The existence of abstract math entities lies in their math properties and relations in terms of which mathematicians interconnect and group them. These relations and properties are the only possible aspects under which an object can enter the realm of math activity. As our primitive math objects we may take natural numbers or their positive integers both in cardinal (1, 2, 3, …) and ordinal ( the first 1^st, the second 2^nd, the third 3^rd, …) sense. Their practical uses are numerous. They are a concrete aspect of the physical universe in the form of the number of fingers and toes on the human hand and leg. The positive integers are the numbers a child learns to count with and the world nations symbolize them by different numerals and notations.

If we look at the names of natural and cardinal numbers, it becomes clear that the number 10 (ten) plays a very special role in numeration. When we study the historical development of maths, we find that even in the hieroglyphic symbols of ancient Egypt (2000 B.C.) the number 10 was exceptionally prominent. With hieroglyphics, however, it is quite a difficult problem to check up even the simplest operations of arithmetic. Most people are more familiar with the Roman numerals. We still use and see them on faces of some clocks, on official documents for chapters of the book. Their base is also the number 10 and though they may seem simpler than hieroglyphics, their arithmetic is just as bad. The advantage lies in the fact that even in simple calculations we must count how many times each symbol appears in the number.

In daily life we need not only to figure out (to get the result of counting) and enumerate individual objects but also to measure quantities such as lengths, weight, time, volume, etc. for such measurements we should employ the other number systems. By further abstraction, generalization and construction, mathematicians had been progressing from the primitive positive integers to real and complex numbers to render infinity which are several levels of abstraction higher than the positive integers.

Ex 2 Answer the following questions

1. Numbers are mental abstract concepts, aren’t they?

2. When do numbers become abstract concepts?

3. How do numbers originate and emerge?

4. How do mathematicians disclose the content of abstract number concepts?

5. Aren’t numbers private property and personal belongings of the mathematicians alone?

6. What is the difference between the two math terms: “natural numbers” and “cardinal numbers”? is the number 5 cardinal or ordinal?

7. It’s more or less clear. But about “power” –what can you say?

Grammar: Grammar Review.

Do the tests to repeat the previous grammar rules from the Oxford Practice Grammar by George Yule on page 194.

Self work tasks

Share your opinion How does the internet influence on the youth nowadays?

http://www.english-easy.info/topics/topics_Internet.php#axzz2B47MFA98

Grammar review p. 219 Advanced Grammar in Use, 2006 by Martin Hewings.

UNIT 8

Theme: Geometry as a science

Grammar: Complex Modals

Objectives: By the end of this unit, students should be able to use active vocabulary of this theme in different forms of speech 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 “Geometry as a science”. Define the basic peculiarities of its functions and its role in our life.

Grammar: Introduce and practice the Complex Modals.

Ex.1 Read and translate the text