Coordinate line (number line), coordinate ray. Notes on mathematics "reconstruction of the origin of a coordinate ray and a unit segment from coordinates" Draw a coordinate ray

Topic: Coordinates on the beam.

Lesson objectives:

• to develop the ability to determine coordinates on a numerical line with a given unit segment;
• develop the ability to record the coordinates of any points;
• train the skill to competently construct coordinate rays.

During the classes

I. Self-determination for activity.

Children work while standing.

- Let's get ready for work. Close your eyes. Pat yourself on the head, on the face, wish yourself to think clearly, remember firmly and be attentive, like intelligence officers. Give yourself a big hug and love. Open your eyes and repeat after me:

I really want to study!
I'm ready for successful work!
I'm doing a great job!

– What did you learn in previous lessons? (Scales. Numerical beam.)

– Today we will continue this interesting work.

– We have to climb one more step of the Ladder of Knowledge in order to learn a new concept associated with the number ray.

II. Updating knowledge and motivation.

a) – At home, you should have built a number line and on it noted 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:

3 cm, 6 cm, 9 cm, 12 cm, 15 cm, 18 cm, 21 cm, 24 cm, 27 cm.

– Show me: what did you do?

Who had any difficulties?

(Children show sheets of paper with the task.)

– What interesting things did you notice? (Numbers that are multiples of 3.)

– What knowledge did you use when constructing the number beam?

(1. The number 0 is the beginning of the ray. 2. Equal unit segments were laid out on the number ray. 3. The distance from each point of the number ray to the beginning of the count is equal to the number corresponding to this point.)

– What actions does the number beam allow you to perform?

(Draw 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 unit segments, and divide the 16th into 3 equal parts, but take only 1 of the three.)

b) – And now I will give you the “key” to find out a new concept that stands on the next step of the ladder of Knowledge.

– To do this, put the letters on your number line that correspond to the numbers in 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 of which we must now find out the meaning of. scale

c) – I suggest you complete the following task on individual pieces of paper:

“In 1 minute, determine and write down the coordinates of points A, B, C, D in a given rectangular window.” You can invent your own recording method...

- Whoever completed the task - stand up!

What kind of recordings did you make? Show on the board...

(Several students show their options.)

– How is it possible: there was one task, but the recording options turned out to be different?

What knowledge did you use when recording?

III. Setting a learning task.

(Children work standing.)

– How does this task differ from the previous one, when you marked different numbers on the number line? (It was not necessary to determine and record the coordinates of the points.)

– So what exactly was the problem? Why did the recordings turn out different?

(They didn’t understand the meaning of the word “coordinate”; they didn’t know how to write it down 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 write down the coordinates of any points).

- Formulate the topic of the lesson... (a note appears on the board): Coordinates on the beam.

- 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 assistant in case of difficulties?

(Dictionary, textbook, teacher, knowledge from previous lessons...)

– Have you heard the phrase: “Leave your coordinates”? What does it mean?

(Leave your address. Give your phone number.)

– So, we are talking about...what?...( About the location.)

– What is used to record an address? (Number).

– So what is the “coordinate” of a point?

(This is a number indicating the location of a point on the number line, i.e. the “address” of the point.)

– So, we found out the meaning of the word “coordinate”. Those who wish can check the explanatory dictionary during the break! (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.”

– Whoever completed the task correctly, help those who made mistakes in it: explain to them what helped you complete this work correctly? (Students' statements).

– Indeed, in mathematics there are strict rules, there are symbols.

– Look carefully at the support: How is the coordinate of point A written here?

(In parentheses, next to the point designation.)

– What does the number in brackets show?

(Number of unit segments from the origin to point A.)

- Attention! The letter designation of the point is above the ray, and the corresponding number is below it!

– Correct the mistakes in your records by those who made them.

(Students’ choral response using a support.)

(Children sit down and continue working while sitting.)

c) – Test yourself using the textbook: p. 61 – reading the conclusion to yourself...

– So what is “point coordinate”?

– 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 conclusion in the textbook?

(It is also called a coordinate ray).

- 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 Ladder of Knowledge has been replenished with one more addition:

Physical exercise! (Standing.)

- Well done! You are doing a wonderful job. 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 consolidation.

Task 4, p. 62

a) Performed frontally on the board with commentary. If there are those who wish, it will be done “in a chain”.

b) Performed on the board “in a chain”, with commentary:

c) Performed in conjunction with mutual verification (1 pair works at the board):

Task 2 (b), p. 61 – performed orally, frontally.

– This task will prepare us for studying the next topic.

1) 15-1=14 (single segments) distance from the dining room to the telephone;

2) 14 · 5 km=70 (km) distance from the dining room to the telephone.

(If a unit segment is 5 km, then the distance from the dining room to the telephone is 14 unit segments, or 70 km.)

VI. Independent work with self-test according to the sample.

Task 3 (a, b), p. 62 – according to options, independently:

- Whoever has finished, stand up! Let's check it using the sample.

A) Sample on the board:

– Who made the mistake, explains what exactly (where?) and why?

What else should you work on?

Children who made mistakes work independently at the next stage of the lesson, completing a similar task, for example, task 4(c), p. 62.

VII. Inclusion in the knowledge system and repetition.

Students who made mistakes in independent work work on their own (task 4 (c), p. 62),

performing a similar task. Then they are checked against a standard or a sample (on individual pieces of paper). Having completed their task, they join the work of the class.

And at this time the whole class is doing frontal work.

– Let’s solve a problem for the specific application of new knowledge about the coordinate ray:

Task 7, p. 62 – orally, frontally, or in pairs. Reading the problem aloud by 1 student.

– What is known in the problem? Where was the car going? (From left to right.)

– What do you need to know? How? (Departure point. Subtract 6 units of segments from end point B (17).

- So from what point did the car leave? (From point A (11.)

– Answer the 2nd question of the problem. (Right to left at 3rd.)

Task 9 (b, c, d, e), p. 63 – group work:

– Let’s repeat solving problems using formulas for path, cost, work.

– Team captains will write down a letter expression on the board and prove their choice.

1st group: b) (x+x3):7;

2nd group: c) (y:5)12;

3rd group: d) (p:20)d;

4th group: 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 has prepared you for the next one, in which you will learn to find distance

between points of a numerical ray according to their known coordinates.

* Well done! Amazing!
*Good, but could have been better!
*Try hard! Be careful!

Cover with your finger the snowflake with the statement opposite which you agree.

– How would you evaluate the work of the whole class?

(“Shock” – hands up “locked”, “It could have been better” – hands behind the back).

Homework: Task 5, p. 62 – creative nature (orally);

Task 8, p. 62; Task 12 (a) or 13, p. 63-64 (1 optional).

Everyone should think: what else should they work on?

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 the form the ray presented on the board is called a coordinate ray. It can be said 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 slide change in the pictures, confirming the 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.

Using a flat wooden strip, two points A and B can be connected with 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, we will apply strokes every centimeter. Under the first stroke we will put the number 0, under the second - 1, third - 2, etc. (Fig. 47). In such cases, they say that the rail is marked scale with division price 1 cm. This rod with a school is similar to 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 clock dial with a scale of 1 min (Fig. 49), a car speedometer with a scale of 10 km/h (Fig. 50), a room thermometer with a scale of 1 °C (Fig. 51), scales with a scale of 50 g (Fig. 52).

The designer creates measuring instruments whose scales are finite, that is, among the numbers marked on the scale there is always the largest. But a mathematician, with the help of his imagination, can construct an infinite scale.

Draw the ray OX. Let's 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 point O depicts the number is 0, and point E is the number 1. It is also customary to say that point O corresponds number 0, and point E is number 1.

Let us lay off a segment equal to segment OE to the right of point E. We obtain point M, which represents the number 2 (see Fig. 53). In the same way, mark 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 − starting point, and the segment OE − single segment coordinate ray.

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 saying “let’s mark a point with a coordinate equal to...” they say “let’s mark a number...”.

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 under it 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 number 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; point E with coordinate one is read;
• 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;
• 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 single 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:

1. Mathematics 5th grade. Vilenkin N.Ya., Zhokhov V.I. and others. 31st ed., erased. - M: 2013.
2. Didactic materials for mathematics grade 5. Author - Popov M.A. – 2013.
3. We calculate without errors. Work with self-test in mathematics grades 5-6. Author - Minaeva S.S. – 2014.
4. Didactic materials for mathematics grade 5. Authors: Dorofeev G.V., Kuznetsova L.V. – 2010.
5. Tests and independent work in mathematics grade 5. Authors - Popov M.A. - 2012.
6. Mathematics. 5th grade: educational. for general education students. institutions / I. I. Zubareva, A. G. Mordkovich. - 9th ed., erased. - M.: Mnemosyne, 2009.