Learn mathematics in English
Read the text and do the tasks below it.
Mixed numbers
A mixed
number
is the sum of a whole number and a proper fraction. This sum is
implied without the use of any visible operator such as "+";
for example, in referring to two entire cakes and three quarters of
another cake, the whole and fractional parts of the number are
written next to each other:
.
An
improper fraction can be thought of as another way to write a mixed
number; consider the
example below.
We can
imagine that the two entire cakes are each divided into quarters, so
that the denominator for the whole cakes is the same as the
denominator for the parts. Then each whole cake contributes
to the total, so
is another way of writing
.
A mixed number can be converted to an improper fraction in three steps:
1. Multiply the whole part by the denominator of the fractional part.
2. Add the numerator of the fractional part to that product.
3. The resulting sum is the numerator of the new (improper) fraction, with the 'new' denominator remaining precisely the same as for the original fractional part of the mixed number.
Similarly, an improper fraction can be converted to a mixed number:
1. Divide the numerator by the denominator.
2. The quotient (without remainder) becomes the whole part and the remainder becomes the numerator of the fractional part.
3. The new denominator is the same as that of the original improper fraction.
Task 1. Convert these mixed numbers into improper fractions and comment on your steps of conversion
|
|
|
|
Task 2. Convert these improper fractions into mixed numbers. Comment your steps of conversion
|
|
|
|
Unit 6
Mathematics is the queen of scienses
Vocabulary
1. Guess the meaning of these international words. Check with your teacher or a dictionary
Theorem, poet, poetry, idea, logical, structure, model, elegant, abstract, intelligence, method, psychology, pedagogics.
Key words:
Human spirit |
Людський дух |
Complicated |
Складний |
Require |
Вимагати |
Core |
Серцевина, глибинна частина, сутність |
Appreciate |
Оцінювати |
Completeness |
Повнота, завершеність |
Rigour |
Суворість, точність |
Consistency |
Узгодженість, несуперечливість |
Distaste for… |
Відраза, неприязнь до … |
Failure |
Невдача, провал |
To shake off something |
визволятися, звільнятися від будь-чого |
To take a liking to … |
Полюбити когось, щось |
Capacity |
Здібність |
To encourage |
Заохочувати |
Profound knowledge |
Глибокі знання |
Eventually |
Зрештою, з часом |
Reading
Task 1. Read the text and match the paragraphs 1-5 with the
headings A-E
Maths is everywhere.
What’s needed to become a good maths teacher.
Why mathematics?
There are no bad pupils – there are bad teachers.
The subject matter of mathematics.
1. I take a course in mathematics. It has been my favorite subject since I started school. I agree with Stefan Banach that ‘mathematics is the most beautiful and most powerful creation of the human spirit’. I feel a real pleasure when I manage to solve a difficult problem or prove a complicated theorem, and I quite agree with the people who compare maths with poetry. Sofia Kovalevsakaya believed that it was impossible to be a mathematician without being a poet in your soul. She thought that mathematics required a great amount of imagination. Albert Einstein said, ‘Pure mathematics is, in its way, the poetry of logical ideas’.
2. At present mathematics means the combination of pure and applied maths. Pure maths deals with logical structures, it studies them as structures themselves. Applied maths is the study of real objects by means of mathematical methods, that is, the study of structures modeling real phenomena. The core of mathematics is pure maths, its theories are beautiful and elegant abstract mathematical systems expressing not words, but thoughts and ideas. The results in pure mathematics are appreciated not by their practical applications, but by their completeness, rigour, consistency, and beauty. So pure maths is an abstract science but its applications can be concrete and practical.
3. Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine and social sciences. You can hardly find an area of human activities where mathematics is not used. No wonder that the great German mathematician Carl Gauss called mathematics the queen of all sciences.
4. Unfortunately, a lot of people have a deep distaste for mathematics and even fear it often as a result of early failure and poor teaching. I want them to shake off this fear and take a liking to maths, and who can help them do this? – A good teacher of mathematics! I think almost all of us have, in our natural intelligence, a capacity for mathematical thinking which may, sadly, just have been ‘blocked off’ by unhappy early experience, and the task of a mathematics teacher is to develop this capacity, find the way to encourage schoolchildren to do maths.
5. To become a good teacher of mathematics you should know the subject you teach very well, be well-educated and broad-minded, and of course, you should know the methods of maths teaching. Our university course of mathematics gives us profound knowledge of the subject. Besides, I’m going to take a course in psychology and pedagogics in my second and third year and I hope to eventually become a good maths teacher.
Task 2. Answer the questions on the text
1. What can mathematics be compared with according to the text?
What does pure maths deal with?
What is applied maths?
How are the results in pure maths appreciated?
Why is maths the queen of sciences?
Why do many people dislike mathematics?
What is the main task of a mathematics teacher?
What makes a good teacher of mathematics?
Speaking
Task 1. Do you agree with the following statements? Give your reasoning
Mathematics requires a great deal of imagination.
Mathematics is like poetry.
c. Many people don’t like maths because they were not encouraged at school.
d. All people have a capacity for mathematical thinking.
Task 2. Here are some quotes about mathematics and mathematicians. How do you understand them? Do you agree with them? Give your reasons
“The essence of mathematics is not to make simple things complicated, but to make complicated things simple”. /S.Gudder/
“But mathematics is the sister, as well as the servant, of the arts and is touched with the same madness and genius”. /Harold Marston Morse/
”Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe, and not make messes in the house”. /Robert Heinlein, Time Enough for Love/
”Although he may not always recognize his bondage(зависимость, подчинение), modern man lives under a tyranny of numbers”. /Nicholas Eberstadt, The Tyranny of Numbers?/
“I used to love mathematics for its own sake, and I still do, because it allows for no hypocrisy and no vagueness (неопределённость, неясность)...”. /Stendhal (Henri Beyle), The Life of Henri Brulard/
”As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality”. /Albert Einstein, Sidelights on Relativity/
”One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers...”. /Heinrich Hertz/
”Pure mathematics is the world's best game. It is more absorbing than chess, more of a gamble(азартная игра) than poker, and lasts longer than Monopoly. It's free. It can be played anywhere - Archimedes did it in a bathtub.”. /Richard J. Trudeau, Dots and Lines/
“The man ignorant of mathematics will be increasingly limited in his grasp (способность быстрого восприятия) of the main forces of civilization.”. /John Kemeny/