some centuries before – бірнеше ғасыр бұрын
in some of the most complicated – кейбір күрделірек
as shown below – как показано ниже
Two cubed or two to the third power
two
squared
one
multiplied by two the third power
is the simplest – ең қарапайым
well familiar with – ... онымен жақсы таныс
the system has the advantage (disadvantage) of having – система обладает тем преимуществом что она имеет
Notes II
and so on – сонымен ары қарай
numeration system – есептеу жүйесі
in a special way – ерекше тәсілмен
Hindi–Arabic system – араб жүйесі
over and over again – көпеселі, жиірек
place-value system – разрядтардың позициялы жүйесі
one and the same – біреу және тағы да
a way of thinking of – өзіңе елестету әдісі
from the above – жоғарыда айтылғаннан
a whole number of times – (целое число раз)
a part left over – қалдық, қалған бөлік
check division by using multiplication – көбейтуді қолдана отырып, бөлуді тексеру
№ 3 practical work
Professional terminology of the school mathematics
Notes
1. and so on – сонымен ары қарай
2. numeration system – есептеу жүйесі
3. in a special way – ерекше тәсілмен
4. Hindi–Arabic system – араб жүйесі
5. over and over again – көпеселі, жиірек
6. place-value system – разрядтардың позициялы жүйесі
7. one and the same – біреу және тағы да
8. a way of thinking of – өзіңе елестету әдісі
9. from the above – жоғарыда айтылғаннан
10. a whole number of times – (целое число раз)
11. a part left over – қалдық, қалған бөлік
12. check division by using multiplication – көбейтуді қолдана отырып, бөлуді тексеру
Lecture №4
Topic: What is mathematics?
№ 4 practical work
Listen to the recording of the text «What is mathematics?». Analyze each paragraph and its Russian translation. Practice back translation and equations. Work in pairs.
Lecture №5
Topic: The subject matter of mathematics
Read and translate the text at home. Be ready a) to illustrate different meaning of the words bold-faced in the text with examples of your own.
№ 5 practical work
Self-Study
Lecture №6
Topic: Mathematics – the language of science
Тапсырма.
№ 6 practical work
Lecture №7
Topic: The Meaning of Geometry. Points and Lines
Contents of lectures:
Problems 101 to 107 are taken from a branch of mathematics called projective geometry. We are concerned here only with plane projective geometry, which deals with the properties of plane figures that are unchanged by protection from one plane onto another (fig.1.).
There are
two kinds of protection, central
and
parallel.
A central projection is obtained by choosing two planes
,
and a center O which is on neither of them. Given any point P of
,
we draw OP and extend it until it intersects
in a point
as in fig.1a. The mapping
is called a central projection. (Of course if OP is parallel to
there will be no point
of intersection. In this case P can be thought of as being mapped to
a «point a infinity» by the projection). A parallel projection from
to
is obtained by drawing a family of parallel lines and mapping each
point A of
onto the point
of
such that
is one of these lines (fig. 1b).
A
projection sends any figure F drawn in the plane
to a figure
in
.
We call
the image of
.
The image of a straight line is itself a straight line, but the
distance from a point P to a line
l,
or the angle between two lines
and
,
may be changed by protection (fig.1a). Similarly, the image of a
circle need not be a circle (fig. 1b). projective geometry deals only
with the properties of figures which are unchanged by projections,
and is therefore not concerned with such things as distances, angles,
and circles. However, concepts involving only the incidence of points
and lines (such as collinearity or concurrence) are preserved under
projection and therefore belong to projective geometry. Each
configuration of straight lines and curves in problems 101 to 107
would be transformed under projection into another configuration
having the same properties, and thus the results proved in these
problems are theorems of projective geometry.
For further reading see [4] and [10].
№ 7 practical work
Topic: The Meaning of Geometry. Points and Lines
A certain city has 10 bus routes. Is it possible to arrange the roués and the bus stops so that if one route is closed, it is still possible to get from any one stop to any other (possibly changing along the way), but if any two routes are closed, there are at least two stops such that it is impossible to get from one to the other?
Show that it is possible to set up a system of bus routes (more than one) such that every route has exactly three stops , any two routes have a stop in common, and it is possible to get from any one stop to any other without changing.
Consider a system of at least two bus lines with the following properties:
Every line has at least three stops.
Given any two stops there is at least one bus line joining them.
Any two distinct lines have exactly one stop in common.
Show that all the lines have the same number of stops. Calling this number n+1, show that every stop lies on n+1 different lines.
Prove that there are altogether
stops and
lines in the system.
a. Arrange nine points and nine straight lines in the plane in such a way that exactly three lines pass through each point, and exactly three points lie on each line.
b. Show that such an arrangement is impossible with seven points and seven straight lines.
Let S be a finite set of straight lines in the plane, arranged in such a way that thrjugh the point of intersection of any two lines of S there passes a third line of S. Prove that the lines of S are either all parallel or all concurrent.
Lecture №8
Topic: Degrees, measurement of the degree
Contents of lectures:
№ 8 practical work
Degrees, measurement of the degree
.
Lecture №9
Topic: Length of circle
Contents of lectures:
№ 9 practical work
Area of the shape
Lecture №10
Topic: Area of the circle
Contents of lectures:
№ 10 practical work
Area of circle. Basic equations of a circle.
Lecture №11
Topic: Percent
Contents of lectures:
Lecture №12
Topic: Correlation. Proportion
Contents of lectures:
№ 11, 12 practical work
Topic: Percent
Lecture №13
Topic: Duplication of the Cube
Тапсырма
№ 13 practical work
Lecture №14
Topic: Abstract (Precis) Writing Practice. Compass and Straightedge Constructions
Omit the unnecessary and extra information, compress and transform the following abstract into a) 8-sentences; b) 3-sentences long abstracts.
Compass and Straightedge Constructions
№ 14 practical work
Lecture №15
Topic: Lesson plans in mathematics in English
Class: 5 Date:______ _
Theme: The notion of ordinary fractions.
The purpose of the lesson:
To acquaint students with the basic concepts: common fraction, the numerator of the fraction, the denominator of a fraction; form the ability to read and write ordinary fractions,
Create in pupils a positive motivation to implement mental and practical exercises
Nurture a sense of satisfaction from being able to show their knowledge in the classroom.
The course of the lesson.