Note the difference in translation between the Present Perfect Continuous and the Present Perfect Tenses.

Ex. 8. Choose the correct variant:

1. For how long (have they discussed, have they been discussing) the situation? 2. Why (have you repeated, have you been repeating) these English words over and over again? 3. The students (have taken, have been taking) the examination for more than 5 hours. 4. They (were discussing, have been discussing) the situation for three hours. 5. She (has been answering, has answered) the lesson already. 6. She (has worn, has been wearing) glasses for two years. 7. Peter’s English is getting much better. He (is practising, has practised, has been practising) a lot this year. 8. I (have written, am writing, have been writing) my course paper for three months, but I (am not finishing, haven’t been finishing, haven’t finished) it yet. 9. “… you (are defending, have defended, have been defending) your course paper?” – “No, I (haven’t done, am not doing, haven’t been doing) it yet.” 10. Tom (is having, has been having, has had) a toothache for nearly a week. He (is going, has been going, has gone) to the doctor today and I’m waiting for him. 11. What you (are doing, have been doing, have done) with my cassette-recorder? I can’t find it anywhere. 12. You look tired! – Yes, I (am dancing, have danced, have been dancing) and I (haven’t danced, am not dancing, haven’t been dancing) for years, so I’m not used to it. 13. Everybody (is looking, has looked, has been looking) forward to this holiday for months. 14. Recently this scientific theory (is being proved, has been proved, has been proving) to be false.

Ex. 9. Match the beginnings and the ends of the sentences.

1. Tom had been working for two hours a. as if he had been running for several hours without a rest.

2. “You look tired.” b. she will have been working at the department for 35 years.

3. “Aren’t you hungry?” c. I decided to have a cup of tea.

4. By the 1^st of August, 2009 d. because she has been painting the ceiling.

5. He was out of breath e. “I have been writing my course paper for more than a month”.

6. “Why are my books all over the floor?” f. because she had been cleaning the flat the whole day.

7. After I had been walking for an hour g. “No, I’ve been eating all day”.

8. Her hair is white h. “He has been walking in the rain”.

9. “Why is his coat wet?” i. when his brother came

10. She looked very dirty j. “Your little sister has been playing with them”.

Ex. 10. Translate into English.

1. Они учат эти правила больше года. 2. Как долго этот студент переводит эту статью? 3. Весь день идет снег. 4. Она преподаватель английского языка. Она преподает с тех пор, как закончила университет. 5. Ты выглядишь усталой. – Я стирала белье весь день. 6. Я вымыл свою машину. – Разве она не выглядит чудесно? 7. Сейчас она учит испанский язык, но она еще не очень говорит. 8. Он часами играет эту музыку на фортепьяно. Пусть он перестанет играть. 9. Студенты пишут этот тест уже 20 минут. Только один студент уже написал его. 10. На этой неделе я написала несколько писем своим друзьям. 11. Как долго вы будете писать контрольную работу перед тем, как сдадите ее преподавателю? 12. Мой друг ждет вас уже с двух часов. Почему вы не пришли вовремя?

Pre-reading activity

Guess the meaning of the following words.

generalization [GenqrqlaI`zeIS(q)n] n, arithmetic [q`rITmetik] n, procedure [prq`sJGq] n, symbol [`sImbql] n, formula [fLmjulq] n, characteristic [,kxrIktq`ristik] n, coefficient [,kouI`fISqnt] n, zero [`zIqrou] n

Read and learn the basic vocabulary terms.

Memorize the following word combinations.

1. let the number 20 be replaced – давайте заменим число 20

2. then the statement is true – тогда утверждение справедливо

3. no matter what – независимо от того, какие

4. for convenience – для удобства

5. serve to distinguish – служат для того, чтобы различить

6. both plus and minus – как плюс, так и минус

7. is to be treated as – следует рассматривать как

Reading Activity

The Nature of Algebra

Algebra is a generalization of arithmetic. Each statement of arithmetic has been dealing with particular numbers for years: the statement (20 + 4)² = 20² + 2 • 20 • 4 + 4² = 576 explains how the square of the sum of the two numbers, 20 and 4, may be computed. It can be shown that the same procedure applies if the numbers 20 and 4 are replaced by any two other numbers. In order to state the general rule, we write symbols, ordinary letters, instead of particular numbers. Let the number 20 be replaced by the symbol a, which may denote any number, and the number 4 by the symbol b. Then the statement is true that the square of the sum of any two numbers a and b can be computed by the rule (a + b)² = a² + 2a • b+b².

This is a general rule which remains true no matter what particular numbers may replace the symbols a and b. A rule of this kind is often called a formula.

Algebra is the system of rules concerning the operations with numbers. These rules can be most easily stated as formulas in terms of letters, like the rule given above for squaring the sum of two numbers.

The outstanding characteristic of algebra is the use of letters to represent numbers. Since the letters used represent numbers, all the laws of arithmetic hold for operations with letters.

In the same way, all the signs which have been introduced to denote relations between numbers and the operations with them are likewise used with letters.

For convenience the operation of multiplication is generally denoted by a dot as well as by placing the letters adjacent to each other. For example, a • b is written simply as ab.

The operations of addition, subtraction, multiplication, division, raising to a power and extracting roots are called algebraic expressions.

Algebraic expressions may be given a simpler form by combining similar terms. Two terms are called similar if they differ only in their numerical factor (called a coefficient).

Algebraic expressions consisting of more than one term are called multinomials. In particular, an expression of two terms is a binomial, an expression of three terms is a trinomial. In finding the product of multinomials we make use of the distributive law.

In algebra, the signs plus (+) and minus (-) have their ordinary meaning, indicating addition and subtraction and also serve to distinguish between opposite kinds of numbers, positive (+) and negative (-). In such an operation as + 10 – 10 = 0, the minus sign means that the minus 10 is combined with the plus 10 to give a zero result or that 10 is subtracted from 10 to give a zero remainder.

The so-called "double sign" (±), which is read "plus-or-minus", is sometimes used. It means that the number or symbol which it precedes may be "either plus or minus" or "both plus and minus".

As in arithmetic, the equality sign (=) means "equals" or "is equal to".

The multiplication sign (•) has the same meaning as in arithmetic. In many cases, however, it is omitted.

The division sign ( ) has the same meaning as in arithmetic. It is frequently replaced by the fraction line; thus means the same as 6 3 and in both cases the result or quotient is 2. The two dots above and below the line in the division sign ( ) indicate the position of the numerator and the denominator in a fraction, or the dividend and the divisor in division.

Parentheses ( ), brackets [ ], braces { }, and other enclosing signs are used to indicate that everything between the two signs is to be treated as a single quantity.

Another sign which is sometimes useful is the sign which means "greater than" or "less than". The sign (>) means "greater than" and the sign (<) means "less than". Thus, a > b means that "a is greater than b", and 3<5 means "3 is less than 5".

Post-Reading Activity