coordinate line. Points on the coordinate line. How to draw a coordinate line How to draw a coordinate line

So the 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 hit, but to which we can approach arbitrarily close, using smaller and smaller ones up to an infinitesimal fraction of a unit segment. These points correspond to infinite periodic and non-periodic decimal fractions. Let's give some 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 of its tenths, 1 hundredth, 1 thousandth, 7 ten-thousandths, 1 hundred-thousandth, 1 millionth of a unit segment, and so on. And one more point of 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 above information in this paragraph allows us to assert that we have assigned a specific real number to each point of the coordinate line, while it is clear that different points correspond to different real numbers.

It is also quite obvious that this correspondence is one-to-one. That is, we can associate a given point on the coordinate line with a real number, but we can also use a given real number to indicate a specific point on the coordinate line to which this real number corresponds. To do this, we will have to postpone a certain number of unit segments, as well as tenths, hundredths, and so on, of a single segment from the origin in the right direction. For example, the number 703.405 corresponds to a point on the coordinate line, which can be reached from the origin by setting aside 703 unit segments in the positive direction, 4 segments that make up a tenth of a unit, and 5 segments that make up a thousandth of a unit.

So, each point on the coordinate line corresponds to a real number, and each real number has its place in the form of a point on the coordinate line. That is why the coordinate line is often called number line.

Coordinates of points on the coordinate line

The number corresponding to a point on the coordinate line is called the 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 the 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.

It remains to say only about the accepted notation. The coordinate of the point is written in parentheses to the right of the letter that denotes the point. For example, if the point M has a coordinate of -6, then you can write M(-6) , and the notation of the form means that the point M on the coordinate line has a coordinate.

Bibliography.

• Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics: textbook for 5 cells. educational institutions.
• Vilenkin N.Ya. etc. Mathematics. Grade 6: textbook for educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8 cells. educational institutions.

It is impossible to claim that you know mathematics if you do not know how to build graphs, depict inequalities on a coordinate line, and work with coordinate axes. The visual component in science is vital, because without visual examples in formulas and calculations, sometimes you can get very confused. In this article, we will see how to work with coordinate axes and learn how to build simple function graphs.

Application

The coordinate line is the basis of the simplest types of graphs that a student encounters on his educational path. It is used in almost every mathematical topic: when calculating speed and time, projecting the size of objects and calculating their area, in trigonometry when working with sines and cosines.

The main value of such a direct line is visibility. Because mathematics is a science that requires a high level of abstract thinking, graphs help in representing an object in the real world. How does he behave? At what point in space will it be in a few seconds, minutes, hours? What can be said about it in comparison with other objects? What is its speed at a randomly selected time? How to characterize his movement?

And we are talking about speed for a reason - it is often the function graphs that display it. And they can also display changes in temperature or pressure inside the object, its size, orientation relative to the horizon. Thus, constructing a coordinate line is often required in physics as well.

1D Graph

There is a concept of multidimensionality. In just one number is enough to determine the location of the point. This is exactly the case with the use of the coordinate line. If the space is two-dimensional, then two numbers are required. Charts of this type are used much more often, and we will definitely consider them a little further in the article.

What can be seen with the help of points on the axis, if it is only one? You can see the size of the object, its position in space relative to some "zero", i.e., the point chosen as the origin.

It will not be possible to see the change in parameters over time, since all readings will be displayed for one specific moment. However, you have to start somewhere! So let's get started.

How to build a coordinate axis

First you need to draw a horizontal line - this will be our axis. On the right side, “sharpen” it so that it looks like an arrow. Thus, we indicate the direction in which the numbers will increase. In the downward direction, the arrow is usually not placed. Traditionally, the axis is directed to the right, so we will simply follow this rule.

Let's put a zero mark, which will display the origin of coordinates. This is the very place from which the countdown is taken, whether it be size, weight, speed, or anything else. In addition to zero, we must necessarily designate the so-called division price, i.e., introduce a unit standard, in accordance with which we will plot certain quantities on the axis. This must be done in order to be able to find the length of the segment on the coordinate line.

Through an equal distance from each other, we put dots or “notches” on the line, and under them we write 1,2,3, respectively, and so on. And now, everything is ready. But with the resulting schedule, you still need to learn how to work.

Types of points on the coordinate line

At first glance at the drawings proposed in the textbooks, it becomes clear: the points on the axis can be filled or not filled. Do you think it's a coincidence? Not at all! A "solid" dot is used for non-strict inequality - one that reads "greater than or equal to". If we need to strictly limit the interval (for example, "x" can take values ​​from zero to one, but does not include it), we will use a "hollow" point, that is, in fact, a small circle on the axis. It should be noted that students do not really like strict inequalities, because they are more difficult to work with.

Depending on what points you use on the chart, the constructed intervals will be named as well. If the inequality on both sides is not strict, then we get a segment. If on the one hand it turns out to be “open”, then it will be called a half-interval. Finally, if a part of a line is bounded on both sides by hollow points, it will be called an interval.

Plane

When constructing two lines on we can already consider the graphs of functions. Let's say the horizontal line is the time axis and the vertical line is the distance. And now we are able to determine what distance the object will overcome in a minute or an hour of travel. Thus, working with a plane makes it possible to monitor the change in the state of an object. This is much more interesting than exploring a static state.

The simplest graph on such a plane is a straight line; it reflects the function Y(X) = aX + b. Does the line bend? This means that the object changes its characteristics in the process of research.

Imagine you are standing on the roof of a building holding a stone in your outstretched hand. When you release it, it will fly down, starting its movement from zero speed. But in a second he will overcome 36 kilometers per hour. The stone will continue to accelerate further, and in order to draw its movement on the chart, you will need to measure its speed at several points in time by setting points on the axis in the appropriate places.

Marks on the horizontal coordinate line by default are named X1, X2,X3, and on the vertical - Y1, Y2,Y3, respectively. By projecting them onto a plane and finding intersections, we find fragments of the resulting pattern. Connecting them with one line, we get a graph of the function. In the case of a falling stone, the quadratic function will look like: Y(X) = aX * X + bX + c.

Scale

Of course, it is not necessary to set integer values ​​next to divisions by a straight line. If you are considering the movement of a snail that crawls at a speed of 0.03 meters per minute, set as values ​​on the coordinate straight line. In this case, set the division value to 0.01 meters.

It is especially convenient to carry out such drawings in a notebook in a cage - here you can immediately see whether there is enough space on the sheet for your schedule, whether you will go beyond the margins. It is not difficult to calculate your strength, because the width of the cell in such a notebook is 0.5 centimeters. It took - reduced the picture. By changing the scale of the graph, it will not lose or change its properties.

Point and line coordinates

When a mathematical problem is given in a lesson, it may contain the parameters of various geometric shapes, both in the form of side lengths, perimeter, area, and in the form of coordinates. In this case, you may need to both build a shape and get some data associated with it. The question arises: how to find the required information on the coordinate line? And how to build a figure?

For example, we are talking about a point. Then a capital letter will appear in the condition of the problem, and several numbers will appear in brackets, most often two (this means we will count in two-dimensional space). If there are three numbers in brackets, separated by a semicolon or a comma, then this is a three-dimensional space. Each of the values ​​is a coordinate on the corresponding axis: first along the horizontal (X), then along the vertical (Y).

Remember how to draw a segment? You passed it on geometry. If there are two points, then a line can be drawn between them. Their coordinates are indicated in brackets if a segment appears in the problem. For example: A(15, 13) - B(1, 4). To build such a line, you need to find and mark points on the coordinate plane, and then connect them. That's all!

And any polygons, as you know, can be drawn using segments. Problem solved.

Calculations

Suppose there is some object whose position along the X axis is characterized by two numbers: it starts at the point with coordinate (-3) and ends at (+2). If we want to know the length of this object, then we must subtract the smaller number from the larger number. Note that a negative number absorbs the sign of the subtraction, because "a minus times a minus equals a plus." So we add (2+3) and get 5. This is the required result.

Another example: we are given the end point and length of the object, but not the start point (and we need to find it). Let the position of the known point be (6), and the size of the object under study be (4). By subtracting the length from the final coordinate, we get the answer. Total: (6 - 4) = 2.

Negative numbers

Often it is required in practice to work with negative values. In this case, we will move along the coordinate axis to the left. For example, an object 3 centimeters high floats in water. One-third of it is immersed in liquid, two-thirds is in air. Then, choosing the water surface as an axis, we get two numbers using the simplest arithmetic calculations: the top point of the object has a coordinate (+2), and the bottom one - (-1) centimeter.

It is easy to see that in the case of a plane, we have four quarters of the coordinate line. Each of them has its own number. In the first (upper right) part there will be points that have two positive coordinates, in the second - from the top left - the values ​​\u200b\u200bof the X axis will be negative, and along the Y axis - positive. The third and fourth are counted further counterclockwise.

Important property

You know that a line can be represented as an infinite number of points. We can view as carefully as we like any number of values ​​in each direction of the axis, but we will not meet repeating ones. It seems naive and understandable, but that statement stems from an important fact: each number corresponds to one and only one point on the coordinate line.

Conclusion

Remember that any axes, figures and, if possible, graphics must be built on a ruler. Units of measurement were not invented by man by chance - if you make an error when drawing, you run the risk of seeing an image that was not the one that should have been.

Be careful and accurate in plotting graphs and calculations. Like any science studied in school, mathematics loves accuracy. Put in a little effort and good grades won't take long.

Lesson topic:

« Coordinates on a straight line»

The purpose of the lesson:

introduce students to the coordinate line and negative numbers.

Lesson objectives:

Training: introduce students to the coordinate line and negative numbers.

Developing: development of logical thinking, broadening one's horizons.

Educational: development of cognitive interest, education of information culture.

Lesson plan:

Organizational moment. Checking students and their readiness for the lesson.

Updating of basic knowledge. Oral survey of students on the topic covered.

Explanation of new material.

4. Consolidation of the studied material.

5. Summarizing. A summary of what was learned in the lesson. Questions from students.

6. Conclusions. Summarizing the main points of the lesson. Knowledge assessment. Putting marks.

7. Homework. Independent work students with learning material.

Equipment: chalk, board, slides.

Expanded outline plan

Stage name and content

Activity

Activity

students

I stage

Organizational moment. Greetings.

Filling out the journal.

greets the class, the head of the class gives a list of absentees.

say hello to

teacher

II stage

Updating of basic knowledge.

The ancient Greek scientist Pythagoras said: "Numbers rule the world." We live in this world of numbers, and in our school years we learn to work with different numbers.

1 What numbers do we already know for today's lesson?

2 What problems do these numbers help us solve?

Today we are moving on to the study of the second chapter of our textbook "Rational Numbers", where we will expand our knowledge about numbers, and after studying the entire chapter "Rational Numbers" we will learn how to perform all the actions you know with them and start with the topic coordinate line.

1. natural, common fractions, decimal fractions

2.addition, subtraction, multiplication, division, finding a fraction from a number and a number from its fraction, solve various equations and problems

Stage III

Explanation of new material.

Let's take the line AB and divide it with the point O into two additional rays - OA and OB. We select a single segment on a straight line and take the point O as the origin and direction.

Definitions:

A straight line with a reference point chosen on it, a unit segment and a direction is called a coordinate line.

The number showing the position of a point on a straight line is called the coordinate of this point.

How to construct a coordinate line?

draw a direct

set a single segment

indicate the direction

The coordinate line can be drawn in different ways: horizontally, vertically and at any other angle to the horizon, and has a beginning, but no end.

Exercise 1. Which of the following lines are not coordinate? (slide)

Let's draw a coordinate line, mark the origin of coordinates, a unit segment and set aside points 1,2,3,4 and so on to the left and right.

Let's look at the resulting coordinate line. Why is such a straight line inconvenient?

The direction to the right from the origin is called positive, and the direction on the straight line is indicated by an arrow. Numbers located to the right of the point O are called positive. Negative numbers are located to the left of the point O, and the direction to the left of the point O is called negative (negative direction is not indicated). If the coordinate line is located vertically, then above from the origin - positive numbers, below from the origin - negative. Negative numbers are written with a “-” sign. They read: “Minus one”, “Minus two”, “Minus three”, etc. The number 0 - the origin is neither positive nor negative. It separates positive from negative numbers.

The solution of equations and the concept of "debt" in trading calculations led to the emergence of negative numbers.

Negative numbers appeared much later than natural numbers and ordinary fractions. The first information about negative numbers is found among Chinese mathematicians in the 2nd century BC. BC e. Positive numbers were then interpreted as property, and negative numbers as debt, shortage. In Europe, recognition came a thousand years later, and even then for a long time negative numbers were called “false”, “imaginary” or “absurd”. In the 17th century, negative numbers received a visual geometric representation on the number line.

You can also give examples of a coordinate line: a thermometer, a comparison of mountain peaks and depressions (sea level is taken as zero), distance on a map, an elevator shaft, houses, cranes.

Think do you know any other examples of coordinate line?

Tasks.

Task2. Name the coordinates of the points.

Task3. Plot points on a coordinate line

Task 4 . Draw a horizontal line and mark point O on it. Mark points A, B, C, K on this line if it is known that:

A is 9 cells to the right of O;

B is 6.5 cells to the left of O;

C is 3½ spaces to the right of O;

K is 3 spaces to the left of O .

Recorded in base notes.

Listen, complement.

Complete the task in your notebook and then explain your answers aloud.

Draw, mark the origin of coordinates a single segment

Such a straight line is inconvenient in that the same number corresponds to 2 points on the straight line.

History before our era and our era.

IV stage

Consolidation of the studied material.

1. What is a coordinate line?

2. How to build a coordinate line?

1. A straight line with a reference point chosen on it, a unit segment and a direction is called a coordinate line

2) draw a straight line

mark the beginning of the countdown

set a single segment

indicate the direction

Stage V

Summarizing

What new did we learn today?

Coordinate line and negative numbers.

VI stage

Knowledge assessment. Putting marks.

Homework.

Make up questions on the topic covered (know the answers to them)

coordinate line.

Let's take a straight line. Let's call it a straight line x (Fig. 1). We choose a reference point O on this line, and also indicate the positive direction of this line with an arrow (Fig. 2). Thus, to the right of the point O we will have positive numbers, and to the left - negative. We choose the scale, that is, the size of the straight line segment, equal to one. We got it coordinate line(Fig. 3). Each number corresponds to a specific single point on this line. Moreover, this number is called the coordinate of this point. Therefore, the line is called the coordinate line. And the reference point O is called the origin.

For example, in fig. 4 point B is at a distance of 2 to the right of the origin. Point D is at a distance 4 to the left of the origin. Accordingly, point B has a coordinate of 2, and point D has a coordinate of -4. The point O itself, being a reference point, has a coordinate of 0 (zero). It is usually written like this: O(0), B(2), D(-4). And in order not to constantly say “point D with coordinate such and such”, they say more simply: “point 0, point 2, point -4”. And in this case, it is enough to designate the point itself with its coordinate (Fig. 5).

Knowing the coordinates of two points of the coordinate line, we can always calculate the distance between them. Let's say we have two points A and B with coordinates a and b respectively. Then the distance between them will be |a - b|. Record |a - b| read as "a minus b modulo" or "the modulus of the difference between the numbers a and b".

What is a module?

Algebraically, the modulus of x is a non-negative number. Denoted as |x|. Moreover, if x > 0, then |x| = x. If x< 0, то |x| = -x. Если x = 0, то |x| = 0.

Geometrically, the modulus of the number x is the distance between the point and the origin. And if there are two points with coordinates x1 and x2, then |x1 - x2| is the distance between these points.

The module is also called absolute value.

What else can we say when it comes to the coordinate line? Certainly about numerical intervals.

Types of numerical intervals.

Let's say we have two numbers a and b. Moreover, b > a (b is greater than a). On the coordinate line, this means that point b is to the right of point a. Let us replace b in our inequality with the variable x. That is x > a. Then x is all numbers greater than a. On the coordinate line, these are, respectively, all points to the right of the point a. This part of the line is shaded (Fig. 6). Such a set of points is called open beam, and this numerical interval is denoted by (a; +∞), where the +∞ sign is read as “plus infinity”. Note that the point a itself is not included in this interval and is indicated by a light circle.

Consider also the case when x ≥ a. Then x is all numbers greater than or equal to a. On the coordinate line, these are all points to the right of a, as well as the point a itself (in Fig. 7, point a is already indicated by a dark circle). Such a set of points is called closed beam(or just a ray), and this numerical interval is denoted by .

The coordinate line is also called coordinate axis. Or just the x-axis.

At the end of Chapter 1, we said that in the course of algebra, you and I need to learn to describe real situations in words (verbal model), algebraically (algebraic or, as mathematicians often say, analytical model), graphically (graphic or geometric model). The entire first section textbook(chapters 1-5) was devoted to the study of the mathematical language with which analytical models are described.

Starting from Chapter 6, we will study not only new analytical, but also graphical (geometric) models. They are built using a coordinate line, coordinate plane. These concepts are a little familiar to you from the mathematics course in grades 5-6.

Straight line /, on which the initial dot O (reference point), scale (single line segment, i.e., a segment, the length of which is considered equal to 1) and the positive direction, is called the coordinate line, or the coordinate axis (Fig. 7); The term "x-axis" is also used.

Each number corresponds to a single point on the line. For example, the number 3.5 corresponds to the point M (Fig. 8), which is removed from the origin, i.e., from the point O, at a distance equal to 3.5 (on a given scale), and postponed from the point O in a given ( positive) direction. The number -4 corresponds to the point P (see Fig. 8), which is removed from the point O at a distance equal to 4, and postponed from the point O in the negative direction, i.e., in the direction opposite to the given one.

The converse is also true: each point of the coordinate line corresponds to a single number.

For example, point K, which is 5.4 from point O in the positive (given) direction, corresponds to the number 5.4, and point N, which is 2.1 from point O in the negative direction, corresponds to the number - 2.1 (see fig. 8).

These numbers are called the coordinates of the corresponding points. So, in fig. 8 point K has a coordinate of 5.4; point P - coordinate -4; point M - coordinate 3.5; point N - coordinate -2.1; point O - coordinate 0 (zero). Hence the name - "coordinate line". Figuratively speaking, the coordinate line is a densely populated house, the residents of this house are points, and the coordinates of the points are the numbers of apartments in which the points-residents live.

Why do we need a coordinate line? Why characterize a point by a number, and a number by a point? Is there any benefit to this? Yes there is.
Let, for example, two points are given on the coordinate line: A - with the coordinate o and B - with the coordinate b (usually in such cases they write shorter:
A(a), B(b)). Suppose we need to find the distance d between points A and B. It turns out that instead of doing geometric measurements, just use the ready-made formula d \u003d (a - b) (you studied it in grade 6).
So, in figure 8 we have:

In an effort to conciseness of reasoning, mathematicians agreed instead of the long phrase “point A of the coordinate line, having coordinate a”, to use a short phrase: “point a”, and, accordingly, on the drawing, the point under consideration is denoted by its coordinate. So, figure 9 shows a coordinate line, on which points are marked - 4; - 2.1; 0; one; 3.5; 5.4.

The coordinate line gives us the opportunity to freely switch from algebraic to geometric language and vice versa. Let, for example, the number a be less than the number b. In algebraic language, this is written as: a< b; на геометрическом языке это означает, что точка а расположена на координатной прямой левее точки b.
However, both algebraic and geometric languages ​​are varieties of the same mathematical language that we are studying.

Let's get acquainted with several more elements of the mathematical language that are associated with the coordinate line.

1. Let a point a be marked on the coordinate line. Consider all the points that lie on the line to the right of the point a, and mark the corresponding part with a coordinate line hatching (Fig. 10). This set of points (numbers) is called an open ray and denoted by (a, + oo), where the + oo sign reads: “plus infinity”; it is characterized by the inequality x > a (by dz we mean any point of the beam).

Please note: point a does not belong to an open beam, but if this point needs to be attached to an open beam, then write x\u003e a or and, accordingly, paint over point b on the drawing (Fig. 13);

for (-oo, b) we will also use the term ray.

3. Let points a and b be marked on the coordinate line, and< b (т. е. точка а расположена на прямой левее точки b). Рассмотрим все точки, которые лежат правее точки а, но левее точки b отметим соответствующую часть координатной прямой штриховкой (рис. 14).

This set (of numbers) is called an interval and denoted by (a, b).

It is characterized by a strict double inequality a< х < b (под х понимается любая точка интервала).

Please note: the interval (a, b) is the intersection (common part) of two open rays (-oo, b) and (a, + oo) - this is clearly seen in Figure 15.

If we add its ends to the interval (a, b), i.e. points a and b, then we get the segment [a, b] (Fig. 16),

which is characterized by a non-strict double inequality a< х < b. Обратите внимание: в обозначении отрезка используют не круглые скобки, как это было в обозначении интервала, а квадратные; на чертеже точки а и b отмечены темными кружками, а не светлыми, как это было в случае интервала.

The segment [a, b] is the intersection (common part) of two rays (-oo, b] and and which is characterized by double inequalities: a< х < b - в первом случае, a < х < b - во втором случае.

So, we have introduced five new terms of the mathematical language: ray, open ray, interval, segment, half-interval. There is also a general term: numerical gaps.

The coordinate line itself is also considered a numerical interval; the notation (-oo, +oo) is used for it.

Mathematics for grade 7 free download, lesson plans, getting ready for school online

A. V. Pogorelov, Geometry for grades 7-11, Textbook for educational institutions