Методические указания по английскому языку для студентов 1 курса математического факультета. 2004 (стр. 1 из 5)
Министерство образования Российской Федерации
Часть I
Методические указания по английскому языку для студентов 1 курса математического факультета.
2004
От составителя
Данные методические указания предназначены для студентов 1 курса математического факультета. Целью работы является развитие умений чтения литературы по специальности, а также оформления монологического высказывания по проблемам прочитанного. Учебные материалы данных методических указаний объединены в 3 блока:
- треугольники;
- простые числа;
- теория игр.
Тематика разработки охватывает не основные, базовые понятия математической науки, так как достаточная обеспеченность математического факультета учебными материалами снимает необходимость обращаться к ним. В методических указаниях затрагиваются мало освещенные в научной литературе на английском языке, доступной студентам, но довольно-таки актуальные проблемы современной высшей, а также прикладной математики.
Перед текстами и после них даются упражнения, которые различны по содержанию, целевой направленности и форме выполнения. В том числе упражнения на отработку вокабуляра, включающего специальную и общенаучную лексику, упражнения на проверку понимания содержания текста, упражнения по некоторым аспектам грамматики.
Блоки содержат творческие задания (кроссворды, головоломки), задачи, требующие логического осмысления, а также математические шутки. Тексты некоторых блоков сопровождаются иллюстративным материалом. Диаграммы, таблицы, схемы и чертежи призваны облегчить понимание тех или иных текстов, а также разнообразить комплекс послетекстовых упражнений.
В приложение включены правила чтения и названия некоторых чисел и цифр на английском языке; словарь математических символов и ответы к задачам, встречающимся в блоках.
Настоящая работа предназначена как для аудиторной работы под руководством преподавателя, так и для самостоятельной работы студентов.
Unit I.
Triangles
Text 1
Before you read:
Can you give a definition of a triangle? What features of a humble triangle can you remind yourself of from your geometry classes? Read the following text and find out whether you are right or not.
What is a triangle?
Everyone is familiar with the simplest mathematical objects, such as straight lines and circles and squares, and the counting numbers. To be a mathematician all you have to do is to learn to look at these objects with some insight and imagination, maybe do a few experiments too, and be able to draw reasonable conclusions.
The result of these activities – which are also quite familiar to you from everyday life – is that you soon see the square as more than something with four equal sides and four right angles; a circle as more than just a plain circle; and the number 8 as much more than merely the next number after 7. We are going to start this mathematical adventure by looking at another very simple and common mathematical object, the humble triangle.
What features does it have? The three sides could be any length at all – except that the two shorter sides together must be longer than the longest side, or the triangle would not close. You cannot make a triangle out of three strips of wood of length 3, 5 and 12 meters.
The three angles cannot be chosen as freely as the three sides. In fact, when we know the size of two of them, the other one can be calculated, because their sum is constant and is equal to 180°.
What other properties does the original triangle have? None, until we use our imagination and start asking some searching questions. As soon as we start to pose problems, and to solve them, we inevitably find ourselves discovering more of its many features. One natural question is: how big is this triangle? What is its area? The simplest and traditional way to find the area is to divide the triangle into two right-angled triangles, by drawing an altitude, as in Fig. 1.1, and then drawing a horizontal line which bisects the altitude at right angles. This dissects each right-angled triangle into a rectangle. The original triangle has been transformed into a rectangle of the same height, and half the width. This trick can be performed in three different ways, if all three angles of the triangle are less than a right angle, by starting with each of the three sides. This at once tells us something about the lengths of the altitudes and the sides:
BC × AD = CA × BE = AB × CF.
Fig. 1.1
Assignments:
1. Answer the following questions:
1. What is a humble triangle?
2. What main features of a humble triangle can you name?
3. What is the sum of all angles in a triangle equal to?
4. How can the area of a humble triangle be found?
2. Find Russian equivalents to English words and expressions:
Insight, to draw reasonable conclusions, to pose problems, to bisect, to dissect, to be performed.
3. Give definitions of the following geometrical figures:
A square, a circle, a triangle, a rectangle.
4. Comment on the following (use 2-3 sentences):
a) properties of a humble triangle
b) ways of finding the area of a triangle.
5. Make up a plan of the text “What is a triangle?” and retell the text, according to your plan, and using the following words and expressions:
It is well known, hence, consequently, therefore, so, it is obvious, it is evident, apparently, manifestly.
Text 2
Before you read:
What types of triangles do you know? Try to give a definition of each type. Read and translate the definitions of different types of triangles:
Types of Triangles
Equilateral triangle – a triangle with all three sides of the same length.
Isosceles triangle – a triangle with two of the three sides of the same length.
Right-angled triangle – a triangle with one angle equal to 90º.
Acute-angled triangle – a triangle with all the angles less than 90º.
Obtuse-angled triangle – a triangle with one of the angles greater than 90º.
Assignments:
1. Insert the missing letters:
Equ.lateral, isos.eles, acute-ang.ed, eq.al, rig.t-an.led, tri.ngle.
2. Match the words and their definitions:
| Right angle | a triangle with all three sides of the same length |
| Obtuse-angled triangle | a triangle with two of the three sides of the same length |
| Isosceles triangle | angle equal to 90º |
| Equilateral triangle | a triangle with one of the angles greater than 90º |
3. Name the type of each of the following triangles:
1. 2. 3. 
4. 5. 
Text 3
Similarity criteria of triangles.
Two triangles are similar, if:
1) all their sides are proportional;
2) all their corresponding angles are equal;
3) two sides of one triangle are proportional to two sides of another and the angles concluded between these sides are equal.
Two right-angled triangles are similar, if
1) their legs are proportional;
2) a leg and a hypotenuse of one triangle are proportional to a leg and a hypotenuse of another;
3) two angles of one triangle are equal to two angles of another.
Assignments:
1. Agree or disagree:
1. Two triangles are similar if only two of their sides are proportional.
2. Two triangles are not similar if only two their corresponding angles are equal.
3. Two right-angled triangles are similar if a leg and a hypotenuse of one triangle are proportional to a leg and a hypotenuse of another.
4. There exist only three similarity criteria of right-angled triangles.
5. If all corresponding angles of two triangles are equal they (triangles) are not similar.
2. Formulate similarity criteria of triangles, using the following expressions:
It is known that…, we are quite familiar with…, every mathematician is sure of…, it should be pointed out…, in fact, thus.
Text 4
Before you read:
Consider Fig. 1.2 and try to prove Pythagorean Theorem, well familiar to you from your school years yourself.
Pythagorean Theorem
Fig. 1.2
In a right-angled triangle a square of the hypotenuse length is equal to a sum of squares of legs lengths.
A proof of Pythagorean Theorem is clear from Fig.1.2. Consider a right-angled triangle ABC with legs a, b and a hypotenuse c. Build the square AKMB, using hypotenuse AB as its side. Then continue sides of the right-angled triangle ABC so, to receive the square CDEF, the side length of which is equal to a + b. Now it is clear, that an area of the square CDEF is equal to (a + b) ². On the other hand, this area is equal to a sum of areas of four right-angled triangles and a square AKMB, that is
c² + 4 (ab / 2) = c² + 2 ab,
hence,
c² + 2 ab = (a + b) ²,
and finally, we have:
c² = a² + b².
Assignments:
1. Give the proof of Pythagorean Theorem, using the following expressions:
It is clear that…, considering, using hypotenuse as a side, to receive the square, is equal to, on the one hand, on the other hand, hence, it is evident that…
Text 5
Medians of a triangle
David Wells in one of his books wrote: if the triangle were a real physical sheet, made of some uniform material, it would not only have an area, but also a centre of gravity; see Fig. 1.3. This is the point on which the triangular sheet would balance on a pinpoint. We understand that, if the triangle is suspended from a vertex, a vertical line through that vertex will also pass through the centre of gravity. If the triangular sheet were resting on the edge of a table, it would start to tip and fall over the edge, if its centre of gravity were over the edge.
Fig. 1.3
You’ll find it natural to ask where the centre of gravity is. Actually, if the triangle is divided into numerous narrow parallel strips, each strip will balance about its midpoint, and all these midpoints appear to lie on the straight line joining the vertex to the midpoint of the opposite side (Fig. 1.4), called the median from that vertex.
Fig. 1.4
The centre of gravity of all the strips together will lie somewhere on the same straight line. Bearing in mind that we found area of the triangle in three different ways by starting with each side as base in turn, it is natural to do the same for the centre of gravity of the triangle. Three constructions can be made, each with a line of midpoints. If the centre of gravity lies on each of these lines, then there must be one point where all three lines meet. To find it, we’ll join each vertex to the midpoint of the opposite side, and the three lines concur, at the centre of gravity (Fig. 1.5). We have a bonus in this case, also. Because we are confident that the three medians do indeed concur. We do not even need to draw a diagram to check this fact, whereas we only discovered that the altitudes concurred with the aid of a drawing.
Fig. 1.5
Assignments:
1. Active vocabulary (memorize the following expressions and use them in sentences of your own):
a centre of gravity, to be suspended from, to bear in mind, to concur, to have a bonus, to be confident, with the aid of.
2. Answer the questions:
1. Where is the centre of gravity of a triangle?
2. What is a median of a triangle?
3. Do the three medians of a triangle necessarily concur?
4. What is the point where the medians concur called?
3. Agree or disagree:
1. If the triangle were a real physical sheet, made of some uniform material, it would not only have an area, but also a centre of gravity.
2. The median from a vertex is a straight line joining this vertex to the midpoint of the opposite side.
3. We can find area of the triangle in three different ways by starting with each side as base in turn.
4. Three medians of a triangle never concur.
4. Recall everything you know of Subjunctive Mood and insert suitable auxiliary verbs:
1. If the triangle were a real physical sheet, made of some uniform material, it … not only have an area, but also a centre of gravity.
2. If the triangular sheet were resting on the edge of a table, it … start to tip and fall over the edge.
3. If the triangle is suspended from a vertex, a vertical line through that vertex … also pass through the centre of gravity.
4. If the triangle … divided into numerous narrow parallel strips, each strip will balance about its midpoint.
5. If the centre of gravity lies on each of these lines, then there … one point where all three lines meet.
5. Complete the sentences, according to the text:
1. If the triangle were a real physical sheet …
2. If the triangle is suspended from a vertex …
3. If the triangular sheet were resting on the edge of a table …
4. If the triangle is divided into numerous narrow parallel strips …