How to find coordinates on a ray. Coordinate line (number line), coordinate ray
Using a flat wooden lath, two points A and B can be connected by a segment ( fig. 46). However, this primitive tool will not be able to measure the length of segment AB. It can be improved.
On the rail through each centimeter we will apply strokes. Under the first stroke we put the number 0, under the second - 1, the third - 2, etc. (Fig. 47). In such cases, they say that the rail is applied graduation scale 1 cm. This rail with the school looks like a ruler. But most often a scale with a division value of 1 mm is applied to the ruler ( fig. 48).
From Everyday life You are well aware of other measuring instruments that have scales of various shapes. For example: a watch dial with a division scale of 1 min ( fig. 49 ), a car speedometer with a division scale of 10 km / h ( fig. 50 ), a room thermometer with a division scale of 1 ° C ( fig. 51 ), a scale with a division scale of 50 g (Fig. 52).
The constructor creates measuring instruments, the scales of which are finite, that is, among the numbers marked on the scale there is always the largest. But a mathematician with the help of imagination can build an infinite scale.
Draw a ray OX. We mark some point E on this ray. Let's write the number 0 above the point O, and the number 1 under the point E (Fig. 53).
We will say that the point O depicts the number 0, and the point E is the number 1 . It is also customary to say that the point O corresponds the number 0, and the point E − the number 1 .
Set aside to the right of the point E a segment equal to the segment OE. Let's get the point M, which depicts the number 2 (see Fig. 53). In the same way, mark the point N, representing the number 3 . So, step by step, we get the points that correspond to the numbers 4, 5, 6, .... Mentally, this process can be continued as long as you like.
The resulting infinite scale is called coordinate beam, point O − reference point, and the segment OE − single segment coordinate beam.
In Figure 53, point K represents the number 5 . They say that the number 5 is coordinate points K, and write K(5 ). Similarly, we can write O(0 ); E(1 ); M(2); N(3 ).
Often instead of the words "mark a point with a coordinate equal to ..." they say "mark the number ...".
For a convenient representation of a fraction on a coordinate ray, it is important to correctly choose the length of a unit segment.
The most convenient option to mark fractions on the coordinate ray is to take a single segment from as many cells as the denominator of the fractions. For example, if you want to depict fractions with a denominator of 5 on the coordinate ray, it is better to take a single segment with a length of 5 cells:
In this case, the image of fractions on the coordinate beam will not cause difficulties: 1/5 - one cell, 2/5 - two, 3/5 - three, 4/5 - four.
If it is required to mark fractions with different denominators on the coordinate ray, it is desirable that the number of cells in a single segment be divisible by all denominators. For example, for the image on the coordinate ray of fractions with denominators 8, 4 and 2, it is convenient to take a single segment eight cells long. To mark on the coordinate line desired fraction, 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 portray 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 six-cell segment as a unit:
Subject: Coordinates on a beam.
Lesson Objectives:
- form the ability to determine the coordinates on number beam with a given unit segment;
- form the ability to record the coordinates of any points;
- train the skill of competent construction of coordinate rays.
During the classes
I. Self-determination to activity.
Children work standing up.
- Let's get to work. Close your eyes. Stroke your head, your face, wish yourself to think clearly, memorize firmly and be attentive, like scouts. Hug tight and love yourself. Open your eyes and repeat after me:
I really want to study!
I'm ready for success!
I'm doing great!
- What did you learn about previous lessons? (Scales. Number beam.)
We will continue this interesting work today.
– We have to climb one more step of the Ladder of Knowledge in order to learn a new concept related to the number ray.
II. Updating knowledge and motivation.
a) - At home, you had to build a numerical ray and mark on it the results of measuring the lengths of the sides of a similar polygon, arranging them in ascending order.
For example: the sides of a polygon are equal:
3cm, 6cm, 9cm, 12cm, 15cm, 18cm, 21cm, 24cm, 27cm.
- Show me what you did.
Who was having trouble?
(The children show the worksheets.)
- What interesting things did you notice? (Numbers that are multiples of 3.)
- What knowledge did you use when constructing a number beam?
(1. The number 0 is the beginning of the beam. 2. Equal unit segments were plotted on the numerical beam. 3. The distance from each point of the numerical beam to the origin is equal to the number corresponding to this point.)
- What actions does the number beam allow you to perform?
(Depict any number; add, subtract and compare numbers).
- Then draw a mixed number on your number line.
(Children sit down, 1 student shows on the board or on a demonstration sample.)
– What is needed for this?
(Take 15 whole single segments, and divide the 16th into 3 equal parts, but take only 1 of the three.)
b) - And now I will give you a “key” to learn a new concept that is on the next step of the Knowledge ladder.
- To do this, on your numerical beam, put down the letters corresponding to the numbers of this table and read the resulting word:
- So, on the next step of the Ladder of Knowledge, a new concept “appears” - “coordinate”, the numerical ray, the meaning of which we must now find out. scale
c) - I suggest that you complete the following task on individual pieces of paper:
“For 1 minute, determine and write down the coordinates of points A, B, C, D in a given rectangular window.” You can invent your own way of recording ...
- Who completed the task - stand up!
What records did you get? Show on board...
(Several students show their options.)
- How is it that the task was one, but the versions of the records turned out to be different?
What knowledge did you take into account when recording?
III. Statement of the educational task.
(Children work standing.)
- How is this task different from the previous one, when you noted different numbers on the number line? (It was not necessary to determine and record the coordinates of the points.)
“So what exactly is the problem?” Why are the records different?
(They didn’t understand the meaning of the word “coordinate”; they didn’t know how to write correctly; they didn’t have time ...)
What is the purpose of our lesson? (Or what should we learn?)
(Clarify the meaning of the concept of “coordinate” of a point; learn to determine and record the coordinates of any points).
- Formulate the topic of the lesson ... (Writing appears on the board): Beam coordinates.
- Well done!
- And at the next stage of our lesson, we will clarify the meaning of the concept of “coordinate” and learn how to correctly write down the coordinates of any points.
IV. “Discovery” of new knowledge by children.
a) - So, who or what is your first mate in times of trouble?
(Dictionary, textbook, teacher, knowledge from previous lessons ...)
– Have you heard the phrase: “Leave your coordinates”? What does she mean?
(Leave your address. Give phone number.)
- Means, we are talking…about what?…( about the location.)
What is used to write an address? (Number).
– So what is the “coordinate” of a point?
(This is a number indicating the location of the point on the number line, i.e. the "address" of the point.)
- So, with the meaning of the word "coordinate" found out. Those who wish can check on the break explanatory dictionary! (The explanatory dictionary is on the teacher's desk).
b) - Let's return to our task: "Determine and write down the coordinates of points A, B, C, D".
- Who coped with the task correctly, help those who made mistakes in it: explain to them what helped you to accurately complete this work? (Statements of students).
- Indeed, in mathematics there are strict rules, there are conventions.
- Look carefully at the support: How is the coordinate of point A written here?
(In parentheses, next to the point symbol.)
What does the number in brackets indicate?
(The number of unit segments from the origin to point A.)
- Attention! The letter designation of the point is above the beam, and the corresponding number is below it!
- Correct in your records the mistakes of those who made them.
(Choral response of students with the help of a support.)
(Children sit down and continue to work while sitting.)
c) - Check yourself according to the textbook: p. 61 - reading the conclusion to yourself ...
– So what is a “point coordinate”?
- And why is the coordinate of your point B equal to (8)?
(It is this number that shows the distance from point B to the beginning of the beam.)
- What new did you learn about the number ray from the output in the textbook?
(It is also called a coordinate beam).
Why is it still called that?
(Since each point of the numerical ray corresponds to a number equal to the coordinate of this point).
– The Knowledge Ladder has been replenished with one more addition:
Fizminutka! (Standing.)
- Well done! You are doing wonderful work. And to cheer yourself up a little more - again a little auto-training - close your eyes, repeat after me:
I am healthy and strong in spirit!
I am a magnet for success!
I trust myself and life!
I deserve all the best!
V. Primary fastening.
Task 4, p. 62
a) Performed frontally on the board with commentary. If there are those who wish, “along the chain”.
b) It is carried out on the board “in a chain”, with commenting:
c) It is carried out in tandem with mutual verification (1 pair works at the board):
Task 2 (b), p. 61 - performed orally, frontally.
This assignment will prepare us for the next topic.
1) 15-1=14 (single segments) distance from the canteen to the telephone;
2) 14 5 km = 70 (km) distance from the dining room to the phone.
(If a single segment is 5 km, then the distance from the canteen to the phone is 14 single segments, or 70 km.)
VI. Independent work with self-examination according to the model.
Task 3 (a, b), p. 62 - according to options, independently:
- Who finished, stand up! Let's check the example.
a) Sample on the board:
- Who made a mistake, explains what exactly (where?) And why?
What else needs to be worked on?
Children who make mistakes work independently at the next stage of the lesson, performing a similar task, for example, task 4 (c), p. 62.
VII. Inclusion in the system of knowledge and repetition.
Students who make mistakes in independent work work on their own (task 4 (c), p. 62),
performing a similar task. Then they compare according to the standard, or according to the sample (on individual sheets). Having completed their task, they are connected to the work of the class.
At this time, the whole class is doing frontal work.
- Let's solve the problem for the specific application of new knowledge about the coordinate beam:
Task 7, p. 62 - orally, frontally, or in pairs. Reading the problem aloud by 1 student.
What is known about the problem? Where was the car going? (From left to right.)
– What do you need to know? How? (Point of departure. From the end point B (17) subtract 6 units of segments.)
So where did the car leave from? (From point A (11.)
Answer the 2nd question of the task. (Right to left at 3 e.)
Task 9 (b, c, d, e), p. 63 - group work:
- Let's repeat the solution of problems using the formulas of the path, cost, work.
The team captains will write the letter on the board and prove their choice.
1g: b) (x + x3): 7;
2g: c) (y:5)12;
3g: d) (s:20)d;
4gr.: e) c-(a4 + c).
VIII. Reflection of activity.
(Children work standing.)
- Name the key words of the lesson ...
- Where in life can you use the knowledge of today's lesson?
(When solving problems, determining the address of something, someone, etc.)
- And our lesson prepared you for the next one, in which you will learn how to find the distance
between the points of the numerical beam by their known coordinates.
* Well done! Wonderful!
*Good, but could be better!
* Try! Be careful!
Close with your finger that snowflake with the statement opposite to which you agree.
How would you rate the work of the entire class?
(“Shock” - hands up “to the castle”, “It could have been better” - hands behind the back).
Homework: Assignment 5, p. 62 - creative nature (oral);
Task 8, p. 62; Task 12 (a) or 13, p. 63-64 (1 optional).
Think to everyone: what else should he work on?
Natural numbers can be represented on a ray. Let's build a ray with the beginning at point O, directing it from left to right, mark the direction with an arrow.
The beginning of the beam (point O) is assigned the number 0 (zero). Let us put off from the point O the segment OA of arbitrary length. Point A will be assigned the number 1 (one). The length of the segment OA will be considered equal to 1 (one). The segment AB = 1 is called single segment. Let us set aside the segment AB = OA from point A in the direction of the beam. Let's put point B in correspondence with the number 2. Note that point B is located at a distance from point O at a distance twice as large as point A. Hence, the length of the segment OB is 2 (two units). Continuing to postpone segments equal to one in the direction of the beam, we will get points corresponding to the numbers 3, 4, 5, etc. These points are removed from point O, respectively, by 3, 4, 5, etc. units.
A ray 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 the 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 the coordinate ray, while the point P corresponding to it will be removed from the point O by n units. They write: OP = n and P( n) - point P (read: "pe") with coordinate n(read: "en"). For example, in order to mark the point K(107) on the numerical ray, it is necessary to set aside 107 segments from the point O, equal to one. As a unit, you can choose a segment of any length. Often the length of a single segment is chosen such that it is possible to depict the necessary integers. Consider an example
5.2. Scale
An important application of the number line is in scales and charts. They are used in measuring instruments and devices that measure various quantities. 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 numerical beam. Each stroke on a straight or circular scale is assigned a certain 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 a pointer. 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.). Like a shifting arrow, the boundary of a column of mercury or tinted alcohol in a thermometer (Figure 3.1). In some devices, it is not the arrow that moves along the scale, but the scale moves relative to the fixed arrow (mark, stroke), for example, in floor scales. In some tools (ruler, tape measure), the pointer is the boundaries of the measured object itself.
Gaps (parts of the scale) between adjacent strokes of the scale are called divisions. The distance between adjacent strokes, expressed in units of the measured value, is called the division price(the difference between the numbers that correspond to adjacent strokes of the scale.) For example, the price of a division of a speedometer in Figure 3.1. is equal to 20 km/h (twenty kilometers per hour), and the division value of a room thermometer in Figure 3.1. equals 1 0 C (one degree Celsius).
Diagram
For a visible display of quantities, line, column or pie charts are used. The diagram consists of a numerical beam-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, the name of the units of measurement in which the values are plotted is signed. Figure 3.2. a bar chart is shown, and in Figure 3.3 a line chart.
3.2.1. Quantities and instruments for their measurement
The table shows the names of some quantities, as well as devices and tools designed to measure them. (The main units of the International System of Units are in bold type).
5.2.2. Thermometers. Temperature measurement
Figure 3.4 shows thermometers that use different temperature scales: Réaumur (°R), Celsius (°C) and Fahrenheit (°F). They use the same temperature interval - the difference between the temperatures of boiling water and melting ice. This interval is divided into a different number of parts: in the Réaumur scale - into 80 parts, the Celsius scale - into 100 parts, in the Fahrenheit scale - into 180 parts. At the same time, in the Reaumur and Celsius scales, the melting temperature of ice corresponds to the number 0 (zero), and in the Fahrenheit scale - the number 32. The temperature units in these thermometers are degrees Reaumur, degrees Celsius, degrees Fahrenheit. The device of thermometers uses the property of liquids (alcohol, mercury) to expand when heated. Wherein various 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 match at the same temperature.
5.2.3. Humidity measurement
The humidity of the air depends on the amount of water vapor in it. For example, in the summer in the desert, the air is dry, its humidity is low, since it contains little water vapor. In the subtropics, for example, in Sochi, the humidity is high, there is a lot of water vapor in the air. Humidity can be measured using two thermometers. One of them is ordinary (dry thermometer). The second ball is wrapped in a damp cloth (wet bulb). It is known that when water evaporates, the temperature of the body decreases. (Recall the chills coming out of the sea after swimming.) Therefore, a wet bulb thermometer shows a lower temperature. The drier the air, the greater the difference in 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 the humidity of the air is called psychrometer (Figure 3.6 ). It is equipped with a table that shows: the readings of a dry thermometer, the difference in the readings of two thermometers, air humidity in percent. The closer the humidity is to 100%, the more humid the air. Normal indoor humidity should be around 60%.
Block 3.3. Self-training
5.3.1. Fill the table
When answering the questions in the table, fill in the free column (“Answer”). In this case, use the drawings 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.
Plot a line diagram of pressure change by plotting height above sea level on the vertical line and pressure on the horizontal line.
Block 5.4. Problem
Construction of a numerical ray with a unit segment of a given length
To solve this learning problem work according to the plan given in the left column of the table, while it is recommended to close 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, diagram
In the tasks of the facet test, figures from the table were used. All tasks start like this: IF the number beam is represented in the figure ...., then ...»
IF: the number line is shown in the figure… Table
- The number of units between adjacent strokes of the number line.
- Coordinates of points A, B, C, D.
- The length (in centimeters) of segments AB, BC, AD, BD, respectively.
- The length (in meters) of segments AB, BC, AD, BD, respectively.
- Natural numbers located on the number line to the left of the 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.
- The price of division of the scale of the device.
- 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 vehicle's speed increased if the speedometer needle moved from point B to point C.
- The speed of the car after the driver slowed down by 84 km/h (the speedometer needle was pointing at point D before the speed reduction).
- The mass of the load on the scales in centners, if the arrow - the pointer of the scales - is located opposite points A, B, C, respectively.
- The mass of the load on the scales in kilograms, if the arrow - the pointer of the scales - is located opposite points A, B, C, respectively.
- The mass of the load on the scales in grams, if the arrow - the pointer of the scales - is located opposite points A, B, C, respectively.
- The 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 who are in time for "4" and "5" to the number of students who are in time for "3".
EQUAL (equal, equal, this):
a) 10 b) 6.12.3.3 c) 1 d) 99.102.106.104 e) 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 r) 7.2, 4.6 s) 20.20.50.30 t) 0 s) 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 s) 0.700.1300.1600 i) 100.100.250.150 aa) 30.15.15.45 bb) 4 cc) 1.2.3.4.5 y) 17 dd) 500 kg of her) 19 fj) 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
In the tasks of the mosaic, devices from the "Additional" block were used. Below is the mosaic box. It contains the names of the devices. In addition, for each device, the following are indicated: the measured value (V), the unit of measurement of the value (E), the indication of the device (P), the scale division value (C). Next are the cells of the mosaic. After reading the cell, you must first determine the device to which it refers, and put the number of the device in the circle of the cell. Then you have to guess what this cell is about. If we are talking about a measured value, it is necessary to assign a letter to the number AT. If it is a unit of measure, put a letter E, if the instrument reading is a letter P, if the division price is a letter C. Thus, it is necessary to designate all cells of the mosaic. If the cells are cut out and arranged as in the field, then information about the device can be systematized. In the computer version of the mosaic, with the correct arrangement of the cells, a pattern is created.
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 at the address.
More lessons on the site
Let's remember what a natural series is. These are all 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. We add 1 - and we get the next number, another 1 - and again the next. And, no matter what number from this series we take, there are neighboring natural numbers 1 to the right and 1 to the left of it. The only exception is the number 1: there is a natural number following it, but not the previous one. 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 easy to guess that this figure is a ray. Indeed, the beam has a beginning, but no end. And it would be possible to continue and continue it, but only the notebook or board will simply run out, and there is nowhere else to continue.
Using these similar properties, we correlate together the natural series of numbers and geometric figure- 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 also be written. Now each natural number that occurs in the natural series has two neighbors on the ray - a smaller one and a larger one. Taking just one step +1 from zero, you can get the number 1, and taking the next step +1 - the number 2 ... Stepping so on, we can get all the natural numbers one by one. In this form, the beam presented on the board is called the coordinate beam. It can be said more simply - a numerical beam. It has smallest number is the number 0, which is called reference point , each subsequent number is the same distance from the previous one, and largest number no, just as there is no end to either the ray or the natural series. I emphasize once again that the distance between the origin and the number 1 following it is the same as between any other two neighboring numbers of the numerical beam. This distance is called single segment . To mark any number on such a ray, exactly the same number of unit segments must be postponed from the origin.
For example, to mark the number 5 on the beam, we postpone 5 unit segments from the origin. To mark the number 14 on the beam, we set aside 14 unit segments from zero.
As you can see in these examples, in different drawings, unit segments can be different (), but on one beam, all unit segments () are equal to each other (). (maybe there will be a slide change in the pictures confirming the pauses)
As you know, in geometric drawings it is customary to give names to points capital letters Latin alphabet. Let's apply this rule to the drawing on the board. Each coordinate ray has an initial point, on the numerical ray this point corresponds to the number 0, and this point is usually called the letter O. In addition, we mark several points in places corresponding to some numbers of this ray. Now each point of the beam has its own specific address. A (3), ... (5-6 points on both rays). The number corresponding to a point on the beam (the so-called point address) is called coordinate points. And the ray itself is a coordinate ray. coordinate beam, or numerical - the meaning of this does not change.
Let's complete the task - mark the points on the numerical ray by their coordinates. I advise you to do this task yourself in a notebook. M(3), T(10), Y(7).
To do this, we first construct a coordinate ray. That is, a ray, the beginning of which is the point O (0). Now you need to select a single segment. He needs it 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 10 cm. Then we take a single segment of 1 cm, mark it on the beam, and the number 10 is 10 cm from the beginning of the beam. Point T corresponds to this number. (...)
But if you need to mark the point H (15) on the coordinate ray, you will need to select another unit segment. Indeed, as in the previous example, it will no longer work, because the beam of the required visible length will not fit in the notebook. You can choose a single segment with a length of 1 cell, and count 15 cells from zero to the required point.