Subject: "Coordinate beam".
Goals:
teach to determine the coordinates of points on a number line, navigate on a coordinate line, repeat the concept of “coordinate line”;
consolidate the ability to independently analyze and solve problems of various types;
develop skills in oral and written calculations, logical thinking, spatial representation.
DURING THE CLASSES
I. Organizational moment
II. Updating knowledge
A ray is drawn on the board with its origin at a pointABOUT .
Conversation on questions:
What's on the board? (Ray)
Is this ray a coordinate ray? (No. )
Why? (No single segment selected. )
How is a unit segment designated? (the student goes to the board and marks a unit segment )
Why is it called that?
How to understand the entry:IN (3)?
What is the name of the number 3?
How many pointsIN (3) can be marked on the coordinate ray? (One. )
Points C(7), E(4), M(8), T(10) are marked. Name the coordinates of points C, E, M, T.
At this time, 6 students work using cards
Option I
Option II
1. Write the coordinates of the pointsD , E , T AndTO
A (8), TO (12), R (1), M (9), N (6), S (3).
1. Write the coordinates of the pointsM , N , WITH AndR , marked on the coordinate ray.
2. Draw a coordinate ray and mark points on itA (6), IN (5), WITH (3), D (10), E (2), F (1).
III. Fastening the ZUN.
Exercise 1
Construct a coordinate ray in your notebook with a unit segment of 1 cell. On your beam, write the letters corresponding to the numbers of this key and read the resulting word.
21
9
27
3
0
24
15
12
6
18
A
R
A
O
To
T
And
d
O
n
The concept “coordinate” appears.
Task 2
What point on OM has coordinate 5? 7? What coordinate is the origin of the ray? Define other points in the figure.
Task 3
Name the coordinates of the points where the following are located: telephone, medical aid station, canteen, gas station.
b) Let one unit on the ray be equal to 5 km.
Which from the dining room to the telephone?
From a gas station to a medical aid station?
Task 4
Draw points A (1) and B (7) on the coordinate ray if: a) e = 2 cm; b) e = 5 mm. Find the distance between points A and B in unit segments, centimeters, millimeters.
Name three numbers whose images are located on the coordinate ray:
a) to the right of point A (25);b) to the left of point B (118);c) to the right of point C (2), but to the left of point D (15);d) to the right of point E (7), but to the left of point F (8).
Task 5
The ant crawled along the coordinate ray from point A (9) three units to the right. Where did he end up? He then crawled 5 units to the left. Where is he now? How many units and in what direction did the ant have to crawl to immediately get to this point?
b) The ant left point B (4) of the coordinate ray, made two movements along the ray and ended up at point C (7). What kind of movements could these be?
IV. Lesson summary
– Students name the key words of the lesson and comment on what they learned during the lesson.
.– The work of the class during the lesson is assessed.
V. Homework.
Task 6
The car traveled from some point A of the coordinate ray 6 units to the right and ended up at point B (17). Where did he leave from? How should he move to get from point A to point C(8)?
Task 7
How many units and in what direction must one shift in order to get from point M (16) to the point with coordinate: a) 14; b) 22; at 12; d) 6; e)21; e) 0; g)16?
So a unit segment and its tenth, hundredth, and so on parts allow us to get to the points of the coordinate line, which will correspond to the final decimal fractions (as in the previous example). However, there are points on the coordinate line that we cannot get to, but to which we can get as close as we like, using smaller and smaller ones down to an infinitesimal fraction of a unit segment. These points correspond to infinite periodic and non-periodic decimal fractions. Let's give a few examples. One of these points on the coordinate line corresponds to the number 3.711711711...=3,(711) . To approach this point, you need to set aside 3 unit segments, 7 tenths, 1 hundredth, 1 thousandth, 7 ten-thousandths, 1 hundred thousandth, 1 millionth of a unit segment, and so on. And another point on the coordinate line corresponds to pi (π=3.141592...).
Since the elements of the set of real numbers are all numbers that can be written in the form of finite and infinite decimal fractions, then all the information presented above in this paragraph allows us to state that we have assigned a specific real number to each point of the coordinate line, and it is clear that different the points correspond to different real numbers.
It is also quite obvious that this correspondence is one-to-one. That is, we can assign a real number to a specified point on a coordinate line, but we can also, using a given real number, indicate a specific point on a coordinate line to which a given real number corresponds. To do this, we will have to set aside a certain number of unit segments, as well as tenths, hundredths, and so on, of fractions of a unit segment from the beginning of the countdown in the desired direction. For example, the number 703.405 corresponds to a point on the coordinate line, which can be reached from the origin by plotting in the positive direction 703 unit segments, 4 segments constituting a tenth of a unit, and 5 segments constituting a thousandth of a unit.
So, to each point on the coordinate line there is a real number, and each real number has its place in the form of a point on the coordinate line. This is why the coordinate line is often called number line.
Coordinates of points on a coordinate line
The number corresponding to a point on a coordinate line is called coordinate of this point.
In the previous paragraph, we said that each real number corresponds to a single point on the coordinate line, therefore, the coordinate of a point uniquely determines the position of this point on the coordinate line. In other words, the coordinate of a point uniquely defines this point on the coordinate line. On the other hand, each point on the coordinate line corresponds to a single real number - the coordinate of this point.
All that remains to be said is about the accepted notation. The coordinate of the point is written in parentheses to the right of the letter that represents the point. For example, if point M has coordinate -6, then you can write M(-6), and notation of the form means that point M on the coordinate line has coordinate.
Bibliography.
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics: textbook for 5th grade. educational institutions.
- Vilenkin N.Ya. and others. Mathematics. 6th grade: textbook for general education institutions.
- Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8th grade. educational institutions.
A ray is a part of a straight line that has a beginning and no end (a ray of the sun, a ray of light from a flashlight). Look at the drawing and determine which figures are depicted, how they are similar, how they differ, and what they can be called. http://bit.ly/2DusaQv
The figure shows parts of a straight line that have a beginning and no end; these are rays that can be called “o x”.
- one ray is designated by large letters OX, and in the name of the second one letter is large and the second is small Ox;
- the first ray is clean, and the second looks like a ruler, since numbers are marked on it;
- on the second ray the letter E is marked, and below it is the number 1;
- there is an arrow at the right end of this beam;
- perhaps it could be called a number beam.
The second ray can be called the numerical ray Ox:
- O is the origin and has coordinate zero;
- written O(0); point O with coordinate zero is read;
- It is customary to write the number zero (0) under the point marked with the letter O;
- segment OE - unit segment;
- point E has coordinate 1 (marked with a dash in the drawing);
- E (1) is written; read point E with coordinate one;
- the arrow at the right end of the beam indicates the direction in which the count is being taken;
- we introduced new concepts of coordinates, which means that the ray can be called coordinate;
- Since the coordinates of various points are plotted on the ray, we write a small letter x in the name of the ray on the right.
Construction of a coordinate ray
We have revealed the concept of a coordinate ray and the terminology associated with it, which means we must learn how to build it:
- we construct a ray and denote Ox;
- indicate the direction with an arrow;
- We mark the beginning of the countdown with the number 0;
- We mark a single segment OE (it can be of different lengths);
- mark the coordinate of point E with the number 1;
- the remaining points will be at the same distance from each other, but it is not customary to put them on the coordinate beam, so as not to clutter the drawing.
To visually represent numbers, it is customary to use a coordinate ray, on which the numbers are arranged in ascending order from left to right. Thus, the number located to the right is always greater than the number located to the left on the straight line.
The construction of a coordinate ray begins from point O, which is called the origin of coordinates. From this point we draw a ray to the right and draw an arrow to the right at its end. Point O has coordinate 0. From it on the ray we lay a unit segment, the end of which has coordinate 1. From the end of the unit segment we lay off one rot that is equal in length, at the end of which we put coordinate 2, etc.
§ 1 Coordinate ray
In this lesson you will learn how to build a coordinate ray, as well as determine the coordinates of points located on it.
To build a coordinate beam, we first need, of course, the beam itself.
Let's denote it OX, point O is the beginning of the ray.
Looking ahead, let's say that point O is called the origin of the coordinate ray.
The beam can be drawn in any direction, but in many cases the beam is drawn horizontally and to the right of its origin.
So, let's draw the ray OX horizontally from left to right and denote its direction with an arrow. Let's mark point E on the ray.
We write 0 above the beginning of the ray (point O), and the number 1 above point E.
The segment OE is called unit.
So, step by step, putting aside single segments, we get an infinite scale.
The numbers 0, 1, 2 are called the coordinates of points O, E and A. Write point O and in brackets indicate its coordinate zero - O (o), point E and in brackets its coordinate one - E (1), point A and in brackets its coordinate two is A(2).
Thus, to construct a coordinate ray it is necessary:
1. draw a ray OX horizontally from left to right and indicate its direction with an arrow, write the number 0 above the point O;
2. you need to set the so-called unit segment. To do this, you need to mark some point on the ray other than point O (at this place it is customary to put not a dot, but a stroke), and write the number 1 above the stroke;
3. on the ray from the end of a unit segment, you need to set aside another unit segment, equal to the unit one, and also put a stroke, then from the end of this segment, you need to set aside another unit segment, also mark it with a stroke, and so on;
4. In order for the coordinate ray to take its finished form, it remains to write down numbers from the natural series of numbers above the strokes from left to right: 2, 3, 4, and so on.
§ 2 Determining the coordinates of a point
Let's complete the task:
The following points should be marked on the coordinate ray: point M with coordinate 1, point P with coordinate 3 and point A with coordinate 7.
Let's construct a coordinate ray with the beginning at point O. We will choose a unit segment of this ray of 1 cm, that is, 2 cells (2 cells from zero we will put a prime and the number 1, then after another two cells - a prime and the number 2; then 3; 4; 5 ; 6; 7 and so on).
Point M will be located to the right of zero by two cells, point P will be located to the right of zero by 6 cells, since 3 multiplied by 2 will be 6, and point A will be located to the right of zero by 14 cells, since 7 multiplied by 2 will be 14.
Next task:
Find and write down the coordinates of points A; IN; and C marked on this coordinate ray
This coordinate ray has a unit segment equal to one cell, which means the coordinate of point A is 4, the coordinate of point B is 8, and the coordinate of point C is 12.
To summarize, the ray OX with its origin at point O, on which the unit segment and direction are indicated, is called a coordinate ray. The coordinate ray is nothing more than an infinite scale.
The number that corresponds to a point on a coordinate ray is called the coordinate of this point.
For example: A and in brackets 3.
Read: point A with coordinate 3.
It should be noted that very often the coordinate ray is depicted as a ray with a beginning at point O, and a single unit segment is laid off from its beginning, above the ends of which the numbers 0 and 1 are written. In this case, it is understood that, if necessary, we can easily continue constructing the scale, sequentially laying down single segments on the ray.
Thus, in this lesson you learned how to build a coordinate ray, as well as determine the coordinates of points located on the coordinate ray.
List of used literature:
- Mathematics 5th grade. Vilenkin N.Ya., Zhokhov V.I. and others. 31st ed., erased. - M: 2013.
- Didactic materials for mathematics grade 5. Author - Popov M.A. – 2013.
- We calculate without errors. Work with self-test in mathematics grades 5-6. Author - Minaeva S.S. – 2014.
- Didactic materials for mathematics grade 5. Authors: Dorofeev G.V., Kuznetsova L.V. – 2010.
- Tests and independent work in mathematics grade 5. Authors - Popov M.A. - 2012.
- Mathematics. 5th grade: educational. for general education students. institutions / I. I. Zubareva, A. G. Mordkovich. - 9th ed., erased. - M.: Mnemosyne, 2009.
The coordinate of a point is its “address” on the number line, and the number line is the “city” in which numbers live and any number can be found by address.
More lessons on the site
Let's remember what a natural series is. These are all the numbers that can be used to count objects, standing strictly in order, one after another, that is, in a row. This series of numbers begins with 1 and continues to infinity with equal intervals between adjacent numbers. Add 1 - and we get the next number, 1 more - and again the next one. And, no matter what number we take from this series, there are neighboring natural numbers on 1 to the right and 1 to the left of it. The only exception is the number 1: the next natural number is there, but the previous one is not. 1 is the smallest natural number.
There is one geometric figure that has a lot in common with the natural series. Looking at the topic of the lesson written on the board, it is not difficult to guess that this figure is a ray. And in fact, the ray has a beginning, but no end. And one could continue and continue it, but the notebook or board would simply run out, and there would be nowhere else to continue.
Using these similar properties, let us relate together the natural series of numbers and the geometric figure - the ray.
It is no coincidence that an empty space is left at the beginning of the ray: next to the natural numbers, the well-known number 0 should be written down. Now every natural number found in the natural series has two neighbors on the ray - a smaller one and a larger one. By taking just one step +1 from zero, you can get the number 1, and by taking the next step +1, you can get the number 2... Stepping so on, we can get all the natural numbers one by one. This is how the ray presented on the board is called a coordinate ray. You can say it more simply - by a numerical beam. It has the smallest number - number 0, which is called starting point , each subsequent number is the same distance from the previous one, but there is no largest number, just as neither a ray nor a natural series has an end. Let me emphasize once again that the distance between the beginning of the count and the following number 1 is the same as between any other two adjacent numbers of the numerical ray. This distance is called single segment . To mark any number on such a ray, you need to set aside exactly the same number of unit segments from the origin.
For example, to mark the number 5 on a ray, we set aside 5 unit segments from the starting point. To mark the number 14 on the ray, we set aside 14 unit segments from zero.
As you can see in these examples, in different drawings the unit segments may be different(), but on one ray all the unit segments() are equal to each other(). (perhaps there will be a change of slides in the pictures, confirming pauses)
As you know, in geometric drawings it is customary to name points in capital letters of the Latin alphabet. Let's apply this rule to the drawing on the board. Each coordinate ray has a starting point; on the numerical ray, this point corresponds to the number 0, and this point is usually called the letter O. In addition, we will mark several points in places corresponding to some numbers of this ray. Now each beam point has its own specific address. A(3), ... (5-6 points on both beams). The number corresponding to a point on the ray (the so-called point address) is called coordinate points. And the beam itself is a coordinate beam. A coordinate ray, or a numerical one - the meaning does not change.
Let's complete the task - mark the points on the number line according to their coordinates. I advise you to complete this task yourself in your notebook. M(3), T(10), U(7).
To do this, we first construct a coordinate ray. That is, a ray whose origin is point O(0). Now you need to select a single segment. This is exactly what we need choose so that all the required points fit on the drawing. The largest coordinate is now 10. If you place the beginning of the beam 1-2 cells from the left edge of the page, then it can be extended by more than 10cm. Then take a unit segment of 1 cm, mark it on the ray, and the number 10 is located 10 cm from the beginning of the ray. Point T corresponds to this number. (...)
But if you need to mark point H (15) on the coordinate ray, you will need to select another unit segment. After all, it will no longer work as in the previous example, because the notebook will not fit a beam of the required visible length. You can select a single segment 1 cell long, and count 15 cells from zero to the required point.