Video tutorial “Coordinate ray. Determination of a single segment and coordinates of a point on a scale. Coordinate beam. coordinates Coordinates of points on the ray

Single segment. ? A single segment can have different lengths. For example, we need to construct a coordinate ray with a unit segment equal to two cells. To do this, you need to: construct a ray (according to the rules discussed above), count two cells from point O, mark the point and give it coordinate 1, the distance from 0 to 1, equal to two cells, is a unit segment. O. 0. 1. Below is a coordinate ray with a unit segment equal to five cells. O. 0. 1.

Slide 6 from the presentation "Coordinate beam". The size of the archive with the presentation is 107 KB.

Mathematics 5th grade

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This article is devoted to the analysis of such concepts as a coordinate ray and a coordinate line. We will dwell on each concept and look at examples in detail. Thanks to this article, you can refresh your knowledge or become familiar with the topic without the help of a teacher.

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In order to define the concept of a coordinate ray, you should have an idea of what a ray is.

Definition 1

Ray- this is a geometric figure that has an origin of the coordinate ray and a direction of movement. The straight line is usually depicted horizontally, indicating the direction to the right.

In the example we see that O is the beginning of the ray.

Example 1

The coordinate ray is depicted according to the same scheme, but is significantly different. We set a starting point and measure a single segment.

Example 2

Definition 2

Unit segment is the distance from 0 to the point chosen for measurement.

Example 3

From the end of a single segment you need to put a few strokes and make markings.

Thanks to the manipulations that we did with the beam, it became coordinate. Label the strokes with natural numbers in sequence from 1 - for example, 2, 3, 4, 5...

Example 4

Definition 3

is a scale that can last indefinitely.

It is often depicted as a ray starting at point O, and a single unit segment is plotted. An example is shown in the figure.

Example 5

In any case, we will be able to continue the scale to the number we need. You can write numbers as convenient as possible - under the beam or above it.

Example 6

Both uppercase and lowercase letters can be used to display ray coordinates.

The principle of depicting a coordinate line is practically no different from depicting a ray. It's simple - draw a ray and add it to a straight line, giving it a positive direction, which is indicated by an arrow.

Example 7

Draw the beam in the opposite direction, extending it to a straight line

Example 8

Set aside single segments according to the example above

On the left side write down the natural numbers 1, 2, 3, 4, 5... with the opposite sign. Pay attention to the example.

Example 9

You can only mark the origin and single segments. See the example of how it will look.

Example 10

Definition 4

- this is a straight line, which is depicted with a certain reference point, which is taken as 0, a unit segment and a given direction of movement.

Correspondence between points on a coordinate line and real numbers

A coordinate line can contain many points. They are directly related to real numbers. This can be defined as a one-to-one correspondence.

Definition 5

Each point on the coordinate line corresponds to a single real number, and each real number corresponds to a single point on the coordinate line.

In order to better understand the rule, you should mark a point on the coordinate line and see what natural number corresponds to the mark. If this point coincides with the origin, it will be marked zero. If the point does not coincide with the starting point, we postpone the required number of unit segments until we reach the specified mark. The number written under it will correspond to this point. Using the example below, we will show you this rule clearly.

Example 11

If we cannot find a point by plotting unit segments, we should also mark points that make up one tenth, hundredth or thousandth of a unit segment. An example can be used to examine this rule in detail.

By setting aside several similar segments, we can obtain not only an integer, but also a fractional number - both positive and negative.

The marked segments will help us find the required point on the coordinate line. These can be either whole or fractional numbers. However, there are points on a straight line that are very difficult to find using single segments. These points correspond to decimal fractions. In order to look for such a point, you will have to set aside a unit segment, a tenth, a hundredth, a thousandth, ten-thousandths and other parts of it. One point on the coordinate line corresponds to the irrational number π (= 3, 141592...).

The set of real numbers includes all numbers that can be written as a fraction. This allows you to identify the rule.

Definition 6

Each point on the coordinate line corresponds to a specific real number. Different points define different real numbers.

This correspondence is unique - each point corresponds to a certain real number. But this also works in the opposite direction. We can also specify a specific point on the coordinate line that will relate to a specific real number. If the number is not an integer, then we need to mark several unit segments, as well as tenths and hundredths in a given direction. For example, the number 400350 corresponds to a point on the coordinate line, which can be reached from the origin by plotting in the positive direction 400 unit segments, 3 segments constituting a tenth of a unit, and 5 segments constituting a thousandth.

Natural numbers can be depicted on a ray. Let's construct a ray with the beginning at point O, directing it from left to right, marking the direction with an arrow.

Let us assign the number 0 (zero) to the beginning of the ray (point O). Let us lay off a segment OA of arbitrary length from point O. Let us associate point A with the number 1 (one). The length of the segment OA will be considered equal to 1 (unit). The segment AB = 1 is called single segment. Let us lay off the segment AB = OA from point A in the direction of the ray. Let us assign the number 2 to point B. Note that point B is located from point O at a distance twice as great as point A. This means that the length of the segment OB is equal to 2 (two units). Continuing to plot segments equal to one in the direction of the ray, we will obtain points that correspond to the numbers 3, 4, 5, etc. These points are removed from point O by 3, 4, 5, etc., respectively. units.

A beam constructed in this way is called coordinate or numerical. The beginning of the number line, point O, is called starting point. The numbers assigned to points on this ray are called coordinates these points (hence: coordinate ray). They write: O(0), A(1), B(2), read: “ point O with coordinate 0 (zero), point A with coordinate 1 (one), point B with coordinate 2 (two)" etc.

Any natural number n can be depicted on a coordinate ray, and the corresponding point P will be removed from point O by n units. They write: OP = n and P( n) - point P (read: "pe") with coordinate n(read: "en"). For example, to mark point K(107) on a number line, it is necessary to plot 107 segments equal to one from point O. You can select a segment of any length as a single segment. Often the length of a unit segment is chosen such that it is possible to depict the necessary natural numbers on a number line within the limits of the picture. Consider an example

5.2. Scale

An important application of the number beam is in scales and charts. They are used in measuring instruments and devices with which various quantities are measured. One of the main elements of measuring instruments is the scale. It is a numerical beam applied to a metal, wood, plastic, glass or other base. Often the scale is made in the form of a circle or part of a circle, which are divided by strokes into equal parts (divisions-arcs) like a number line. Each stroke on a straight or circular scale is assigned a specific number. This is the value of the measured quantity. For example, the number 0 on the thermometer scale corresponds to a temperature of 0 0 C, read: “ zero degrees Celsius" This is the temperature at which ice begins to melt (or water begins to freeze).

Using measuring instruments and instruments with scales, determine the value of the measured quantity by position pointer on the scale. Most often, arrows serve as indicators. They can move along the scale, marking the value of the measured value (for example, a clock hand, a scale hand, a speedometer hand - a device for measuring speed, Figure 3.1.). The boundary of a column of mercury or tinted alcohol in a thermometer is similar to a moving arrow (Figure 3.1). In some instruments, it is not the arrow that moves along the scale, but the scale that moves relative to the stationary arrow (mark, line), for example, in floor scales. In some instruments (ruler, tape measure), the pointer is the boundaries of the object being measured.

The spaces (parts of the scale) between adjacent scale strokes are called divisions. The distance between adjacent strokes, expressed in units of the measured value, is called the division price(the difference in numbers that correspond to adjacent scale strokes.) For example, the price of the speedometer division in Figure 3.1. is equal to 20 km/h (twenty kilometers per hour), and the division price of the room thermometer in Figure 3.1. equal to 1 0 C (one degree Celsius).

Diagram

To visually display quantities, line, column or pie charts are used. The diagram consists of a numerical ray-scale directed from left to right or from bottom to top. In addition, the diagram contains segments or rectangles (columns) depicting the compared values. In this case, the length of segments or columns in scale units is equal to the corresponding values. On the diagram, near the numerical ray-scale, sign the name of the units of measurement in which the quantities are plotted. In Figure 3.2. shows a bar chart, and Figure 3.3 shows a line chart.

3.2.1. Quantities and instruments for measuring them

The table shows the names of some quantities, as well as devices and instruments designed to measure them. (Bold type indicates the basic units of the International System of Units.)

5.2.2. Thermometers. Temperature measurement

Figure 3.4 shows thermometers that use different temperature scales: Reaumur (°R), Celsius (°C) and Fahrenheit (°F). They use the same temperature range - the difference between the boiling temperatures of water and the melting temperatures of ice. This interval is divided into a different number of parts: in the Reaumur scale - into 80 parts, in the Celsius scale - into 100 parts, in the Fahrenheit scale - into 180 parts. Moreover, in the Reaumur and Celsius scales, the temperature of ice melting corresponds to the number 0 (zero), and in the Fahrenheit scale - to the number 32. The temperature units in these thermometers are: degree Reaumur, degree Celsius, degree Fahrenheit. Thermometers use the property of liquids (alcohol, mercury) to expand when heated. At the same time, different liquids expand differently when heated, as can be seen in Figure 3.5, where the strokes for a column of alcohol and mercury do not coincide at the same temperature.

5.2.3. Air humidity measurement

Air humidity depends on the amount of water vapor in it. For example, in the summer in the desert the air is dry and its humidity is low, since it contains little water vapor. In the subtropics, for example, in Sochi, the humidity is high and there is a lot of water vapor in the air. You can measure humidity using two thermometers. One of them is a regular one (dry bulb). The second has a ball wrapped in a damp cloth (wet thermometer). It is known that when water evaporates, body temperature decreases. (Remember the chill when you come out of the sea after swimming). Therefore, the wet bulb thermometer shows a lower temperature. The drier the air, the greater the difference between the readings of the two thermometers. If the thermometer readings are the same (the difference is zero), then the air humidity is 100%. In this case, dew falls. A device that measures air humidity is called psychrometer (Figure 3.6 ). It is equipped with a table that shows: dry bulb readings, the difference between the readings of two thermometers, and air humidity as a percentage. The closer the humidity is to 100%, the more humid the air. Normal indoor humidity should be about 60%.

Block 3.3. Self-preparation

5.3.1. Fill the table

When answering the questions in the table, fill in the empty column (“Answer”). In this case, use the pictures of devices in the “Additional” block.

760 mm. rt. Art. considered normal. Figure 3.11 shows the change in atmospheric pressure when climbing the highest mountain, Everest.

Construct a linear diagram of pressure changes, plotting height above sea level on the vertical ray and pressure along the horizontal ray.

Block 5.4. Problem

Construction of a numerical ray with a unit segment of a given length

To solve this educational problem, work according to the plan given in the left column of the table, while it is recommended to cover the right column with a sheet of paper. After answering all the questions, compare your conclusions with the solutions given.

Block 5.5. Facet test

Number beam, scale, chart

The facet test tasks used pictures from the table. All tasks begin like this: “ IF the number ray is represented in the figure...., then...»

IF: the number ray is represented in the figure... Table

The number of units between adjacent strokes of a number line.
Coordinates of points A, B, C, D.
Length (in centimeters) of segments AB, BC, AD, BD, respectively.
Length (in meters) of segments AB, BC, AD, BD, respectively.
Natural numbers located on the number line to the left of point D.
Natural numbers located on the number line between points A and C.
The number of natural numbers lying on the number line between points A and D.
The number of natural numbers lying on the number line between points B and C.
Instrument scale division price.
Vehicle speed in km/h if the speedometer needle points to points A, B, C, D, respectively.
The amount (in km/h) by which the speed of the car increased if the speedometer needle moved from point B to point C.
The speed of the car after the driver reduced the speed by 84 km/h (before reducing the speed, the speedometer needle pointed to point D).
The mass of the load on the scales in centners, if the arrow - the scale indicator - is located opposite points A, B, C, respectively.
The mass of the load on the scales in kilograms, if the arrow - the scale pointer - is located opposite points A, B, C, respectively.
The mass of the load on the scales in grams, if the arrow - the scale pointer - is located opposite points A, B, C, respectively.
Number of students in 5th grade.
The difference between the number of students achieving “4” and the number of students achieving “3”.
The ratio of the number of students achieving grades “4” and “5” to the number of students achieving grades “3”.

EQUAL (equal, equal, this):

a) 10 b) 6,12,3,3 c) 1 d) 99,102,106,104 d) 2 f) 201,202 g) 49 h) 3500,3000,8000,4500

i) 5,2,1,4 k) 599 l) 6,3,3,9 m) 10,4,16,7 n) 100 o) 4 km/h p) 65,85,105,115 p) 7,2, 4,6 c) 20,20,50,30 t) 0 y) 700,600,1600,900 f) 1,2,3,4,5,6 x) 25,10,5,20 c) 3,4, 5.2 h) 203,197,200,206 w) 15,20,25,10 w) 1599 s) 11,12,13,14,15 e) 30,60,15,15 y) 0,700,1300,1600 i) 100,100,250,150 aa) 30,15,15,45 bb) 4 vv) 1,2,3,4,5 y) 17 dd) 500 kg ee) 19 zh) 80 zz) 100,101,102,103,104,105 ii)5,6 kk) 28,64,100,164 ll) 1500000 ,3000000,4500000 mm) 11 nn) 36 oo) 1500,3000,4500 pp) 7 rr) 24 ss) 15,30,45

Block 5.6. Educational mosaic

The mosaic tasks used devices from the “Additional” block. Below is the mosaic field. The names of the devices are indicated on it. In addition, for each device the following are indicated: the measured value (V), the unit of measurement of the value (E), the instrument reading (P), the scale division value (C). Next are the mosaic cells. After reading a cell, you must first identify the device to which it belongs and put the device number in the circle of the cell. Then you need to guess what this cell is about. If we are talking about a measured quantity, you need to add a letter to the number IN. If this is a unit of measurement, put a letter E, if the instrument reading is a letter P, if the division price is a letter C. In this way, you need to designate all the cells of the mosaic. If the cells are cut out and arranged as on the field, then you can systematize information about the device. In the computer version of the mosaic, with the correct arrangement of cells, a pattern is created.

To conveniently display a fraction on a coordinate ray, it is important to choose the correct length of a unit segment.

The most convenient way to mark fractions on a coordinate ray is to take a single segment of as many cells as the denominator of the fractions. For example, if you want to depict fractions with a denominator of 5 on a coordinate ray, it is better to take a unit segment 5 cells long:

In this case, depicting fractions on a coordinate beam will not cause difficulties: 1/5 - one cell, 2/5 - two, 3/5 - three, 4/5 - four.

If you want to mark fractions with different denominators on a coordinate ray, it is desirable that the number of cells in a unit segment be divided by all denominators. For example, to depict fractions with denominators 8, 4 and 2 on a coordinate ray, it is convenient to take a unit segment eight cells long. To mark the desired fraction on the coordinate ray, we divide the unit segment into as many parts as the denominator, and take as many such parts as the numerator. To represent the fraction 1/8, we divide the unit segment into 8 parts and take 7 of them. To depict the mixed number 2 3/4, we count two whole unit segments from the origin, and divide the third into 4 parts and take three of them:

Another example: a coordinate ray with fractions whose denominators are 6, 2 and 3. In this case, it is convenient to take a segment six cells long as a unit:

MATHEMATICS
Lessons for 5th grade

LESSON 12

Subject. Coordinate beam

Goal: to form in students the concept of a coordinate ray, its elements and the method of constructing a given number on a coordinate ray and determining the coordinates of a point on a coordinate ray; consolidate knowledge of terminology (“coordinate ray”, “origin”, “unit segment”, “point coordinate”) and develop the ability to construct points with given coordinates on a coordinate ray and find the coordinates of points with numerical (complete and incomplete) drawings.

Lesson type: learning new knowledge.

During the classes

I. Updating of reference knowledge

Oral exercises

1. Perform the addition: a) 17 + 15; b) 170 + 150; c) 170 + 15; d) 17 + 150. Between which natural numbers in the natural series there are numbers, what did you get?

2. On ray Ox (Fig. 9) 8 equal segments 1 cm long were laid out. Find the distance from point B to points A, B, C, F, N.

3. The wooden slats must be divided into 16 equal parts. How many cuts do you need to make?

II. Formation of new knowledge

1. An explanation of the content of new material can be carried out close to the text of the textbook in the form of frontal practical work, making drawings and notes on the board during the explanations (students make the same notes and drawings in notebooks). At the end of the explanations, the following entries (approximately) should appear in the notebooks and on the board (Fig. 10).

(ray xO - coordinate ray, O - origin, OE - unit segment; point O represents the number 0, or O(0); point E represents the number 1, or E(1); point M represents the number 2, or M( 2); numbers that are represented by dots are the coordinates of the points)

2. To the material presented in the textbook, you should add information about the properties of points on the coordinate ray: the larger of the two natural numbers on the coordinate ray corresponds to the point on the right, and vice versa. In addition, if a number lies between two given numbers on a coordinate line, then it lies between these numbers in the natural series.

III. Consolidation of knowledge. Formation of skills

To consolidate the new terminology, it is appropriate to complete task 1.

Problem 1

1) Draw the ray Ox from left to right, place a segment OB on it and put a zero under point A, and the number 1 under point B. What is the name of the segment OB?

2) To denote the number 4, how many unit segments must be set aside from the beginning of the ray Ox?

3) If a unit segment is postponed from the beginning of the ray Ox six times, then what number will correspond to the end of the sixth segment?

4) Is it possible to plot a unit segment on the ray Ox a million times? Why?

5) Let the number 9 correspond to the number 9 at point M on the coordinate ray Ox. How many times is the segment OB delayed from the beginning of the ray and how to write this correspondence?

@ After completing the task, you should repeat once again with the students that a certain natural number n is constructed by setting aside n unit segments from the origin, and vice versa - the number of unit segments that are placed between the origin of the coordinate ray and a point on it is the coordinate of the point.

Task 2

1) Construct a coordinate ray Ox with a unit segment of 1 cm. Mark the points on it: A(2); AT 4); C(7); B(0). Find the length of segments AB, BC, AC.

2) Point D removed from point C(7) by 3 cm and lies to the right. What is the coordinate of the point D?

3) Lay off a unit segment CE from point C(7) to the left, then point E corresponds to a number - its coordinate. Write down the coordinates of point E. Find the midpoint of the segment O.D. and mark this point on the rays F. Which is the coordinate of the point F?

conclusions

· To construct a point, which is the image of a certain number n on a coordinate ray, you need to: specify a unit segment; postpone it n times from the beginning of the beam.

· To find the number n, which corresponds to a specific point on a coordinate ray, you need to know the distance from the beginning of the ray to this point in unit segments.

After completing and analyzing the solutions to problems 1 and 2, you can offer students No. 124, 127 (see textbook).

At the end of the lesson (if there is time left), exercises No. 140 are solved; 142 (pay attention to the different number of solutions to problems in cases 1 and 2, related to the limited nature of the coordinate ray).