Perfect Tenses (Active, Passive). Numerals. Cleft sentences: it is (was)…that (who)

Конспект урока

Иностранные языки, филология и лингвистика

In our scientific ge there is generl belief tht ll science s it grows to perfection becomes mthemticl in its ides. It is generlly true tht in the development of lgebr three stges hve been pssed successively: verbl bbrevited nd symbolic. Verbl lgebr is chrcterized by the complete bsence of ny symbols except of course tht the words themselves re used in their symbolic sense.



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9 чел.

                                                             Unit 4


  1.  Perfect Tenses (Active, Passive)
  2.  Numerals
  3.  Cleft sentences: it is (was)…that (who)

                                                  Reading Exercises

  1.  Practise reading the two – syllable words:

      concise, because, conceal, consist, design, translate, precise, upon, pervade, include, belief, become, idea, complete, themselves, denote, itself, appear, displace, notation, survive, supply, extreme, permit, consider, involve, effect, subject, attach, transform.

  1.  Read observing the correct pronunciation of vowels in stressed and unstressed syllables:

      language, algebra, alphabet, practical, capable, particular, common, consonant, level, general, even, latter, except, develop.

     3.   Read the following many-syllable words:

       influence, reasoning, capacity, typical,  gradual,  eventual, superscript, century, magnitude, consonant, alphabet, geometry, quantity, difficult, literal, ambiguity, paraphrase, equivalent.

     4.   Memorize the spelling and pronunciation of the following words: permeate ['pq:mIeIt] - проникати, throughout[Tru:'aut] - повсюди, pervade [рq:'veid] - наповнювати, пронизувати, though ['Dou[- хоча, certain ['sq:tn[- певний, enough [I'nAf[– достатньо, доволі, medieval ["medi'Jvql[- середньовічний, technique [tek'ni:k]- техніка, метод, ambiguity ["xmbI'gjHItI[- двозначність, невизначеність, metamorphosis ["metq'mLfqsIs[ - метаморфоза.

  1.  Read observing the correct pronunciation of  the - ed  ending:

   abbreviated, illustrated, denoted, designated, calculated, generalized, symbolized, civilized, concealed, enabled, supplied, involved, solved, passed, expressed, developed, displaced.

  1.  Read paying  attention to the shift of stress in the following numerals:

      thirteen — thirteen years                    thirteen — thirty

      fourteen — fourteen languages          fourteen — forty

      fifteen — fifteen students          fifteen — fifty

      sixteen — sixteen centuries          sixteen — sixty

      seventeen — seventeen words          seventeen — seventy

      eighteen — eighteen sentences          eighteen — eighty

      nineteen — nineteen countries          nineteen — ninety.



       Human language is capable of precise statements because it is a system of symbols. But common language is a product of social development, customs and traditions. Even by the most careful choice of words the meaning concealed in them may influence our reasoning. Algebra, the language of mathematics, consists mostly of signs and symbols and is carefully and purposefully designed. It is precise, concise and universal, i. e. one and the same throughout the civilized world, though the people in each country translate it into their own spoken language.

Algebra in the broad sense of the term, deals with operations upon symbolic forms.

In this capacity it not only permeates all of mathematics, but pervades practically all

sciences   including formal logic, philosophy, and even linguistics, poetry and music.

In our scientific age there is a general belief that all science, as it grows to perfection, becomes mathematical in its ideas.

It is generally true that in the development of algebra three stages have been passed successively: verbal, abbreviated and symbolic. Verbal algebra is characterized by the complete absence of any symbols, except, of course, that the words themselves are used in their symbolic sense. To this day verbal algebra is used in such a statement as "the sum is independent of the order of the terms", which in symbols is designated by a + b = b + a.

     Abbreviated algebra of which the Egyptian is a typical example, is a further development of verbal one. Certain words of frequent use are gradually abbreviated. The history of the symbols "+" and "-" may illustrate the point. In medieval Europe the latter was denoted by the full word "minus", then by the first letter "m" duly superscribed. Eventually the letter itself was dropped, leaving the superscript only. The sign "plus" has passed through a similar metamorphosis. The abbreviation has become a symbol.

  The turning point in the history of algebra was an essay written late in the sixteenth century by a Frenchman; it was Viète who denoted the unknown magnitudes by vowels. The given magnitudes were designated by consonants.

 Within half a century of Viète's death there appeared Descartes's Geometry. In it, the first letters of the alphabet were used for the given quantities, the last - for those unknown.

The Cartesian notation not only displaced the Viètan one, but has survived to this day.

It is symbols that permit of concise, clear representation of ideas which are sometimes quite complex. Consider, for example, how much is involved in the calculus symbol "Dy". Once we have grasped the meaning and use of a symbol there is no need to think through the origin and development of the idea symbolized, each time it is used. It is due to a powerful technique based upon the use of symbols that mathematics is so effective in problems which are insoluble by other methods.

It is convenient because the literal notation is free from all ambiguities of words. The letter is susceptible of operations and this enables one to transform literal expressions and thus to paraphrase any statement into a number of equivalent forms. It is this power of transformation that lifts algebra above the level of a convenient shorthand.

It is symbolic language that is one of the basic characteristics of modern mathematics. And modern mathematics supplies a language for the treatment of the qualitative problems of physical and social sciences.


  1.   ...is capable of (precise) statement - здатний (точно) передавати висловлювання
  2.   a product of social development - продукт суспільного розвитку
  3.   throughout the (civilized) world - у всьому (цивілізованому) світі.
  4.   spoken language - розмовна мова
  5.   in this capacity - в цій якості
  6.   calculus symbol - символ обчислення
  7.   Dy (derivative of у) - похідна від у
  8.   to superscribe - робити напис зверху
  9.   late in the sixteenth century - наприкінці XVI століття
  10.   to think through - додумувати до кінця, проникати в суть справи
  11.   a powerful technique - могутній спосіб (метод), засіб 
  12.   literal notation — буквене позначення, буквений запис
  13.   susceptible (of) який (що) допускає, піддається  чому-н,

     14.  to take the form (of) набирати вигляду

Answer the questions:

  1.  Why is it important to know mathematics? 2. What is the distinction between

common human language and the language of mathematics? 3. How is the language of mathematics designed? 4. Why is algebra called the language of mathematics? 5.What signs and symbols in mathematics do you know? 6. What three stages has algebra passed through in its development? 7. What event may be called the turning point in the history of algebra? 8. What sciences does mathematics embrace? 9. What can you say about the expression "mathematics is the language of science"?


     I. Learn the meaning and pronunciation of the following words:

   Mathematics, ratio, arithmetics, tangent, algebra, series, calculus, graph, variable, zero, derivative, designate, area, infinitesimal, increments, integral, subscript, superscript, magnitude, insoluble, addition, multiplication, division, raising to power, limits, subtraction.

  1.  Find in the text the words  which can form:

     a)  adverbs after the model A + -ly: full — fully;

 b)  adjectives after the model S + -ic: symbol — symbolic;

     c)  nouns after the model V + -ation: abbreviate — abbreviation.

  1.  Form words with the opposite meaning by adding prefixes un-, mis-, in-, im-:
dependent, possible, known, understand, soluble, interpretation, successful, favourable, productive, precise, complete, careful, frequent, limited, calculation, exact, convenient.

  IV.     Read and remember the Plural of the following Nouns:
radius — radii           datum — data            

   formula— formulae   index— indices

   V. Read, translate into Ukrainian and memorize the following pairs of

                                verbs and nouns:

to produce — product to    superscribe — superscription

to choose — choice    to calculate — calculus

to speak — speech     to add — addition

to use — use[ju:s]     to subtract — subtraction

to design — design     to multiply — multiplication

to place — place     to divide — division

VI. Read aloud and give the Ukrainian equivalents of the following words:

social, design, universal, operation, symbol (symbolic), sense, idea, illustrate,

problem, technique, interpretation, base, transform (transformation), equivalent.

VII. Read and learn by heart:

1 inch = 2.54 centimeters (cms)

12 inches = 1 foot

1 foot = 30.48 cms

3 feet = 1 yard

1 yard = 91.44 cms

1,760 yards = 1 mile

1 mile = 1609.3 metres

                  220 yards = 1 furlong

                  1 furlong = 201.16 metres

             8 furlongs = 1 mile

VIII. Translate the following words of the same root into English:

використовувати — використовування — користь — корисний —      некорисний;



IX. Arrange the following words according to:

  1.  similar meaning:  precise, identical,   exact, similar, careful, complicated, complex, attentive, universal, recurrent, global, frequent, brief, entire, complete, concise;
  2.  opposite meaning: simple, free, different, chief, precise, minor, similar,  dependent,  complex, long, ambiguous, brief.
    1.  Read the following:

    a) numerals: 30; 80; 100; 103; 3,200; 2,045,237;

  1.  mathematical expressions: З3, 43, 5б, 73, 1010;
  2.  fractions; ½ (km), 1/3 (ton), 0.5 (km), 3.152 (ton);
  3.  dates: in 1985; January 22, 1919; in 1995.
    1.   Translate into English using words and word-combinations from the text:
      1.  Математична мова — це мова знаків та символів. 2. Мова математики

проста і універсальна. 3. Люди різних країн перекладають знаки i 

символи мови математики на свою рідну мову. 4. Алгебра — це мова математики. 5. В своєму розвитку алгебра пройшла декілька ступенів. б. Сучасна алгебра об'еднує велику кількість самостійних дисциплін.

7. Метод аналізу математичних моделей посідає провідне мicцe серед інших  методів дослідження.

Text 4 B


  1.  Read and memorize:

to reward — винагороджувати, the game of chess — rpa в шахи,

chess-board — шахівниця, square — квадрат, a bag of wheat —мішок пшениці,

to double — подвоювати, to enjoy — насолоджуватися, радіти, treasure — скарб,

to order — наказувати, to count — рахувати, to account for — пояснювати,

to increase — збільшувати, to fulfil — виконувати, pretty big — досить великий,

assuming (that) — припускаючи, (що), to contain — вміщати, the amount 

requested — потрібна кількість, incessant flow (of latter's demands) — безпе-

рервний (безконечний) noтік (вимог останнього).

King Shirham of India, according to an old legend, wanted to reward his grand

vizier Sissa Ben Dahir for inventing and presenting to him the game of chess.

The desires of the clever vizier seemed very modest. "Majesty", he said kneeling in front of the king, "give me a grain of wheat to put on the first square of this chessboard, and two grains to put on the second square, and four grains to put on the third and eight grains to put on the fourth. And so, oh King, doubling the number for each succeeding  square, give me enough grains to cover all 64 squares of the board". "You do not ask for much, oh my faithful servant", exclaimed the king, silently enjoying the thought that his generous proposal of a gift to the inventor of the miraculous game would not cost him much of his treasure. "Your wish will certainly  be granted". And he ordered a bag of wheat to be brought to the throne. But when the counting began, with a grain for the first square, 2 for the second, 4 for the third and so forth, the bag was emptied before the twentieth square was accounted for. More bags of wheat were brought before the king but the number of grains needed for each succeeding square increased so rapidly that it soon became clear that with all the crop of India the king could not fulfil his promise to Sissa Ben. To do so would have required 18,446,744,073,709,551,615 grains... That's not so large a number as the total number of atoms in the universe, but it is pretty big anyway. Assuming that a bushel of wheat contains about 5,000,000 grains, one would need some 4,000 billion bushels to satisfy the demand of Sissa Ben. Since the world production of wheat averages about 2,000,000,000 bushels a year, the amount requested by the grand vizier was that of the world's wheat production for the period of some two thousand years.

    Thus King Shirham found himself deep in debt to his vizier and had either to  face the incessant flow of the latter's demands or to cut his head off. What alternative did he choose ?

 I. Read the text and answer the following questions:

1. Where did the story take place? 2. What is the legend about? 3. What is your opinion of the king's and the vizier's knowledge of mathematics? 4. Can you name the mathematical rule of doubling each succeeding number? 5.When did the king realize his mistake? 6. How many bushels of "wheat would be needed to satisfy the vizier's demand? 7. How many grains of wheat does a bushel contain? 8. Could the king satisfy the vizier's demand? 9. What is the author's suggestion as to the end of the legend? 10. Do you know any other stories about chess invention?

II. Make up a plan of the story and render its contents.


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