What is magnitude? Relative comparison value

2/ If quantities a and b are measured using the unit of quantity e, then to find the numerical value of the sum a + b it is enough to add

numerical values ​​of quantities a and b. a+b= c m (a+b) = m (a) + m (b). For example, if a = 15 kg, b = 12 kg, then a + b = 15 kg + 12 kg = (15 + 12) kg = 27 kg

3/ If the quantities a and b are such that b = x a, where x is a positive real number, and the quantity a is measured using the unit of the quantity e, then to find the numerical value of the quantity b with the unit e, it is enough to multiply the number x by the number m (a):b=x a m (b)=x m (a).

For example, if mass a is 3 times greater than mass b, i.e. b = For and a = 2 kg, then b = For = 3 ∙ (2 kg) = (3∙2) kg = 6 kg.

The concepts considered - object, subject, phenomenon, process, its magnitude, numerical value of a value, unit of value - must be able to be identified in texts and tasks.

For example, the mathematical content of the sentence “We bought 3 kilograms of apples” can be described as follows: the sentence considers an object such as apples, and its property is mass; to measure mass, the unit of mass was used - kilogram; As a result of the measurement, we obtained the number 3 - the numerical value of the mass of apples with a unit of mass - kilogram.

Let's look at the definitions of some quantities and their measurements.

Topic: QUANTITIES AND THEIR MEASUREMENTS

Target: Give the concept of quantity and its measurement. Introduce the history of the development of the system of units of quantities. Summarize knowledge about quantities that preschoolers become familiar with.

Plan:

The concept of quantities, their properties. The concept of measuring a quantity. From the history of the development of the system of units of quantities. International system of units. Quantities that preschoolers become familiar with and their characteristics.

1. The concept of quantities, their properties

Quantity is one of the basic mathematical concepts that arose in ancient times and underwent a number of generalizations in the process of long-term development.

The initial idea of ​​size is associated with the creation of a sensory basis, the formation of ideas about the size of objects: show and name length, width, height.

Quantity refers to the special properties of real objects or phenomena of the surrounding world. The size of an object is its relative characteristic, emphasizing the extent of individual parts and determining its place among homogeneous ones.

Quantities characterized only by numerical value are called scalar(length, mass, time, volume, area, etc.). In addition to scalar quantities, mathematics also considers vector quantities, which are characterized not only by number, but also by direction (force, acceleration, electric field strength, etc.).

Scalar quantities can be homogeneous or heterogeneous. Homogeneous quantities express the same property of objects of a certain set. Heterogeneous quantities express different properties of objects (length and area)

Properties of scalar quantities:

§ any two quantities of the same kind are comparable, either they are equal, or one of them is less (greater) than the other: 4t5ts…4t 50kgÞ 4t5ts=4t500kg Þ 4t500kg>4t50kg, because 500kg>50kg, which means

4t5ts >4t 50kg;

§ quantities of the same kind can be added, the result is a quantity of the same kind:

2km921m+17km387mÞ 2km921m=2921m, 17km387m=17387m Þ 17387m+2921m=20308m; Means

2km921m+17km387m=20km308m

§ a quantity can be multiplied by a real number, resulting in a quantity of the same kind:

12m24cm× 9 Þ 12m24m=1224cm, 1224cm×9=110m16cm, that means

12m24cm× 9=110m16cm;

4kg283g-2kg605gÞ 4kg283g=4283g, 2kg605g=2605g Þ 4283g-2605g=1678g, which means

4kg283g-2kg605g=1kg678g;

§ quantities of the same kind can be divided, resulting in a real number:

8h25min: 5 Þ 8h25min=8×60min+25min=480min+25min=505min, 505min : 5=101min, 101min=1h41min, that means 8h25min: 5=1h41min.

Magnitude is a property of an object, perceived by different analyzers: visual, tactile and motor. In this case, most often the value is perceived simultaneously by several analyzers: visual-motor, tactile-motor, etc.

The perception of magnitude depends on:

§ the distance from which the object is perceived;

§ the size of the object with which it is compared;

§ its location in space.

Basic properties of the quantity:

§ Comparability– determination of a value is possible only on the basis of comparison (directly or by comparison with a certain image).

§ Relativity– the characteristic of size is relative and depends on the objects chosen for comparison; one and the same object can be defined by us as larger or smaller depending on the size of the object with which it is compared. For example, a bunny is smaller than a bear, but larger than a mouse.

§ Variability– the variability of quantities is characterized by the fact that they can be added, subtracted, multiplied by a number.

§ Measurability– measurement makes it possible to characterize a quantity by comparing numbers.

2. Concept of quantity measurement

The need to measure all kinds of quantities, as well as the need to count objects, arose in the practical activities of man at the dawn of human civilization. Just as to determine the number of sets, people compared different sets, different homogeneous quantities, determining first of all which of the compared quantities was larger or smaller. These comparisons were not yet measurements. Subsequently, the procedure for comparing values ​​was improved. One value was taken as a standard, and other values ​​of the same kind were compared with the standard. When people acquired knowledge about numbers and their properties, magnitude, the number 1 was assigned to the standard and this standard began to be called a unit of measurement. The purpose of measurement has become more specific – to evaluate. How many units are contained in the measured quantity. the measurement result began to be expressed as a number.

The essence of measurement is the quantitative division of measured objects and establishing the value of a given object in relation to the adopted measure. Through the measurement operation, the numerical relationship of the object is established between the measured quantity and a pre-selected unit of measurement, scale or standard.

The measurement includes two logical operations:

the first is the process of separation, which allows the child to understand that the whole can be split into parts;

the second is a substitution operation consisting of connecting individual parts (represented by the number of measures).

The measurement activity is quite complex. It requires certain knowledge, specific skills, knowledge of the generally accepted system of measures, and the use of measuring instruments.

In the process of developing measurement activities in preschoolers using conventional measures, children must understand that:

§ measurement gives an accurate quantitative description of a quantity;

§ for measurement it is necessary to choose an adequate standard;

§ the number of measurements depends on the quantity being measured (the larger the quantity, the greater its numerical value and vice versa);

§ the measurement result depends on the selected measure (the larger the measure, the smaller the numerical value and vice versa);

§ to compare quantities, they must be measured with the same standards.

3. From the history of the development of the system of units of quantities

Man has long realized the need to measure different quantities, and to measure as accurately as possible. The basis for accurate measurements are convenient, clearly defined units of quantities and accurately reproducible standards (samples) of these units. In turn, the accuracy of the standards reflects the level of development of science, technology and industry of the country and speaks of its scientific and technical potential.

In the history of the development of units of quantities, several periods can be distinguished.

The most ancient period is when units of length were identified with the names of parts of the human body. Thus, the palm (the width of four fingers without the thumb), the cubit (the length of the elbow), the foot (the length of the foot), the inch (the length of the joint of the thumb), etc. were used as units of length. The units of area during this period were: well (area , which can be watered from one well), plow or plow (average area processed per day by plow or plow), etc.

In the XIV-XVI centuries. In connection with the development of trade, so-called objective units of measurement of quantities appear. In England, for example, an inch (the length of three barley grains placed side by side), a foot (the width of 64 barley grains placed side by side).

Gran (weight of grain) and carat (weight of seed of one type of bean) were introduced as units of mass.

The next period in the development of units of quantities is the introduction of units interconnected with each other. In Russia, for example, these were the units of length: mile, verst, fathom and arshin; 3 arshins was a fathom, 500 fathoms was a verst, 7 versts was a mile.

However, the connections between units of quantities were arbitrary; not only individual states, but also individual regions within the same state used their own measures of length, area, and mass. Particular disparity was observed in France, where each feudal lord had the right to establish his own measures within the boundaries of his possessions. Such a variety of units of quantities hampered the development of production, hindered scientific progress and the development of trade relations.

The new system of units, which later became the basis for the international system, was created in France at the end of the 18th century, during the era of the French Revolution. The basic unit of length in this system was meter- one forty millionth of the length of the earth's meridian passing through Paris.

In addition to the meter, the following units were installed:

§ ar- the area of ​​a square whose side length is 10 m;

§ liter- volume and capacity of liquids and bulk solids, equal to the volume of a cube with an edge length of 0.1 m;

§ gram- the mass of pure water occupying the volume of a cube with an edge length of 0.01 m.

Decimal multiples and submultiples were also introduced, formed using prefixes: miria (104), kilo (103), hecto (102), deca (101), deci, centi, milli

The unit of mass, kilogram, was defined as the mass of 1 dm3 of water at a temperature of 4 °C.

Since all units of quantities turned out to be closely related to the unit of length meter, the new system of quantities was called metric system.

In accordance with accepted definitions, platinum standards of the meter and kilogram were made:

§ the meter was represented by a ruler with strokes applied to its ends;

§ kilogram - a cylindrical weight.

These standards were transferred to the National Archives of France for storage, and therefore they received the names “archival meter” and “archival kilogram”.

The creation of the metric system of measures was a great scientific achievement - for the first time in history, measures appeared that formed a coherent system, based on a model taken from nature, and closely related to the decimal number system.

But soon changes had to be made to this system.

It turned out that the length of the meridian was not determined accurately enough. Moreover, it became clear that as science and technology develop, the value of this quantity will become more precise. Therefore, the unit of length taken from nature had to be abandoned. The meter began to be considered the distance between the strokes marked on the ends of the archival meter, and the kilogram the mass of the standard archival kilogram.

In Russia, the metric system of measures began to be used on a par with Russian national measures since 1899, when a special law was adopted, the draft of which was developed by an outstanding Russian scientist. Special decrees of the Soviet state legitimized the transition to the metric system of measures, first in the RSFSR (1918), and then in the entire USSR (1925).

4. International system of units

International System of Units (SI) is a single universal practical system of units for all branches of science, technology, national economy and teaching. Since the need for such a system of units, which is uniform for the whole world, was great, in a short time it received wide international recognition and distribution throughout the world.

This system has seven basic units (meter, kilogram, second, ampere, kelvin, mole and candela) and two additional units (radian and steradian).

As is known, the unit of length meter and unit of mass kilogram were also included in the metric system of measures. What changes did they undergo when they entered the new system? A new definition of the meter has been introduced - it is considered as the distance that a plane electromagnetic wave travels in a vacuum in a fraction of a second. The transition to this definition of the meter is caused by increasing requirements for measurement accuracy, as well as the desire to have a unit of magnitude that exists in nature and remains unchanged under any conditions.

The definition of the kilogram unit of mass has not changed; the kilogram is still the mass of a platinum-iridium alloy cylinder manufactured in 1889. This standard is stored at the International Bureau of Weights and Measures in Sevres (France).

The third basic unit of the International System is the time unit, the second. She is much older than a meter.

Before 1960, the second was defined as 0 " style="border-collapse:collapse;border:none">

Prefix names

Prefix designation

Factor

Prefix names

Prefix designation

Factor

For example, a kilometer is a multiple of a unit, 1 km = 103×1 m = 1000 m;

A millimeter is a submultiple unit, 1 mm = 10-3 × 1 m = 0.001 m.

In general, for length, the multiple units are kilometer (km), and the subunit are centimeter (cm), millimeter (mm), micrometer (µm), nanometer (nm). For mass, the multiple unit is megagram (Mg), and the subunit is gram (g), milligram (mg), microgram (mcg). For time, the multiple unit is the kilosecond (ks), and the subunit is the millisecond (ms), microsecond (µs), nanosecond (not).

5. Quantities that preschoolers become familiar with and their characteristics

The goal of preschool education is to introduce children to the properties of objects, teach them to differentiate them, highlighting those properties that are commonly called quantities, and introduce them to the very idea of ​​measurement through intermediate measures and the principle of measuring quantities.

Length- this is a characteristic of the linear dimensions of an object. In preschool methods of forming elementary mathematical concepts, it is customary to consider “length” and “width” as two different qualities of an object. However, in school, both linear dimensions of a flat figure are more often called “side length”; the same name is used when working with a three-dimensional body that has three dimensions.

The lengths of any objects can be compared:

§ approximately;

§ application or overlay (combination).

In this case, it is always possible to either approximately or accurately determine “how much one length is greater (smaller) than another.”

Weight is a physical property of an object measured by weighing. It is necessary to distinguish between the mass and weight of an object. With the concept item weight children meet in the 7th grade in a physics course, since weight is the product of mass and the acceleration of gravity. The terminological incorrectness that adults allow themselves in everyday life often confuses a child, since we sometimes, without thinking, say: “The weight of an object is 4 kg.” The very word “weighing” encourages the use of the word “weight” in speech. However, in physics, these quantities differ: the mass of an object is always constant - this is a property of the object itself, and its weight changes if the force of attraction (acceleration of free fall) changes.

To prevent your child from learning incorrect terminology, which will confuse him later in elementary school, you should always say: object mass.

In addition to weighing, the mass can be approximately determined by an estimate on the hand (“baric feeling”). Mass is a difficult category from a methodological point of view for organizing classes with preschoolers: it cannot be compared by eye, by application, or measured by an intermediate measure. However, any person has a “baric feeling”, and using it you can build a number of tasks that are useful for a child, leading him to understand the meaning of the concept of mass.

Basic unit of mass – kilogram. From this basic unit other units of mass are formed: gram, ton, etc.

Square- this is a quantitative characteristic of a figure, indicating its dimensions on a plane. The area is usually determined for flat closed figures. To measure the area, you can use any flat shape that fits tightly into the given figure (without gaps) as an intermediate measure. In elementary school, children are introduced to palette - a piece of transparent plastic with a grid of squares of equal size applied to it (usually 1 cm2 in size). Laying the palette on a flat figure makes it possible to count the approximate number of squares that fit in it to determine its area.

In preschool age, children compare the areas of objects, without naming this term, by superimposing objects or visually, by comparing the space they occupy on the table or ground. Area is a convenient quantity from a methodological point of view, since it allows the organization of various productive exercises in comparing and equalizing areas, determining the area by laying down intermediate measures and through a system of tasks for equal composition. For example:

1) comparison of the areas of figures by the superposition method:

The area of ​​a triangle is less than the area of ​​a circle, and the area of ​​a circle is greater than the area of ​​a triangle;

2) comparison of the areas of figures by the number of equal squares (or any other measurements);

The areas of all figures are equal, since the figures consist of 4 equal squares.

When performing such tasks, children indirectly become acquainted with some area properties:

§ The area of ​​a figure does not change when its position on the plane changes.

§ Part of an object is always smaller than the whole.

§ The area of ​​the whole is equal to the sum of the areas of its constituent parts.

These tasks also form in children the concept of area as number of measures contained in a geometric figure.

Capacity- this is a characteristic of liquid measures. At school, capacity is examined sporadically during one lesson in 1st grade. Children are introduced to the measure of capacity - the liter in order to later use the name of this measure when solving problems. The tradition is that capacity is not associated with the concept of volume in elementary school.

Time- this is the duration of the processes. The concept of time is more complex than the concept of length and mass. In everyday life, time is what separates one event from another. In mathematics and physics, time is considered as a scalar quantity, because time intervals have properties similar to the properties of length, area, mass:

§ Time periods can be compared. For example, a pedestrian will spend more time on the same path than a cyclist.

§ Time periods can be added together. Thus, a lecture in college lasts the same amount of time as two lessons in school.

§ Time intervals are measured. But the process of measuring time is different from measuring length. To measure length, you can use a ruler repeatedly, moving it from point to point. A period of time taken as a unit can be used only once. Therefore, the unit of time must be a regularly repeating process. Such a unit in the International System of Units is called second. Along with the second, others are also used. units of time: minute, hour, day, year, week, month, century.. Units such as year and day were taken from nature, and hour, minute, second were invented by man.

A year is the time it takes for the Earth to revolve around the Sun. A day is the time the Earth rotates around its axis. A year consists of approximately 365 days. But a year in a person’s life is made up of a whole number of days. Therefore, instead of adding 6 hours to each year, they add a whole day to every fourth year. This year consists of 366 days and is called a leap year.

A calendar with such an alternation of years was introduced in 46 BC. e. Roman Emperor Julius Caesar in order to streamline the very confusing calendar existing at that time. That's why the new calendar is called Julian. According to it, the new year begins on January 1 and consists of 12 months. It also preserved such a measure of time as a week, invented by Babylonian astronomers.

Time sweeps away both physical and philosophical meaning. Since the sense of time is subjective, it is difficult to rely on the senses in assessing and comparing it, as can be done to some extent with other quantities. In this regard, at school, almost immediately, children begin to become familiar with instruments that measure time objectively, that is, regardless of human sensations.

When introducing the concept of “time” at first, it is much more useful to use an hourglass than a clock with arrows or an electronic one, since the child sees the sand pouring in and can observe the “passage of time.” Hourglasses are also convenient to use as an intermediate measure when measuring time (in fact, this is exactly what they were invented for).

Working with the quantity “time” is complicated by the fact that time is a process that is not directly perceived by the child’s sensory system: unlike mass or length, it cannot be touched or seen. This process is perceived by a person indirectly, compared to the duration of other processes. At the same time, the usual stereotypes of comparisons: the course of the sun across the sky, the movement of hands on a clock, etc. - as a rule, are too long for a child of this age to really follow them.

In this regard, “Time” is one of the most difficult topics in both preschool mathematics teaching and elementary school.

The first ideas about time are formed in preschool age: the change of seasons, the change of day and night, children become familiar with the sequence of concepts: yesterday, today, tomorrow, the day after tomorrow.

By the beginning of school, children develop ideas about time as a result of practical activities related to taking into account the duration of processes: performing routine moments of the day, maintaining a weather calendar, becoming familiar with the days of the week, their sequence, children become familiar with the clock and orienting themselves by it in connection with a visit kindergarten. It is quite possible to introduce children to such units of time as year, month, week, day, to clarify the idea of ​​the hour and minute and their duration in comparison with other processes. The tools for measuring time are the calendar and the clock.

Speed- this is the path traveled by the body per unit of time.

Speed ​​is a physical quantity, its names contain two quantities - units of length and units of time: 3 km/h, 45 m/min, 20 cm/s, 8 m/s, etc.

It is very difficult to give a child a visual idea of ​​speed, since it is the ratio of path to time, and it is impossible to depict it or see it. Therefore, when getting acquainted with speed, we usually turn to comparing the time of movement of objects over an equal distance or the distances covered by them in the same time.

Named numbers are numbers with names of units of measurement of quantities. When solving problems at school, you have to perform arithmetic operations with them. Preschoolers are introduced to named numbers in the “School 2000” (“One is a step, two is a step...”) and “Rainbow” programs. In the School 2000 program, these are tasks of the form: “Find and correct errors: 5 cm + 2 cm - 4 cm = 1 cm, 7 kg + 1 kg - 5 kg = 4 kg.” In the Rainbow program, these are tasks of the same type, but by “naming” they mean any name with numerical values, and not just the names of measures of quantities, for example: 2 cows + 3 dogs + + 4 horses = 9 animals.

You can mathematically perform an operation with named numbers in the following way: perform actions with the numerical components of named numbers, and add a name when writing the answer. This method requires compliance with the rule of a single name in action components. This method is universal. In elementary school, this method is also used when performing actions with compound named numbers. For example, to add 2 m 30 cm + 4 m 5 cm, children replace the composite named numbers with numbers of the same name and perform the action: 230 cm + 405 cm = 635 cm = 6 m 35 cm or add the numerical components of the same names: 2 m + 4 m = 6 m, 30 cm + 5 cm = 35 cm, 6 m + 35 cm = 6 m 35 cm.

These methods are used when performing arithmetic operations with numbers of any kind.

Units of some quantities

Chapter 4

Studying quantities in primary school

Lecture 15,

Basic quantities studied

in primary school

1. The concept of magnitude

3. Mass and capacity.

4. Area.

6. Speed.

7. Actions with named numbers.

Concept of magnitude

In mathematics under size understand the properties of objects that can be quantitative assessment. The quantitative assessment of a quantity is called measurement. The measurement process involves comparing a given value with a certain measure, adopted for a unit when measuring quantities of this kind.

Quantities include length, mass, time, capacity (volume), area, etc.

All these quantities and their units of measurement are studied in elementary school. The result of the process of measuring a quantity is a certain numerical value, showing how many times the selected measure “fitted” into the measured value.

In elementary school, only those quantities are considered whose measurement result is expressed as a positive integer (natural number). In this regard, the process of introducing a child to quantities and their measures is considered in the methodology as a way to expand the child’s ideas about the role and possibilities of natural numbers. In the process of measuring various quantities, the child not only practices the actions of measurement, but also gains a new understanding of the previously unknown role of the natural number. Number- it is a measure of magnitude and the very idea of ​​numbers

was largely generated by the need to quantify
assessment of the process of measuring quantities. ,

When getting acquainted with quantities, some general stages can be identified, characterized by the commonality of the child’s objective actions aimed at mastering the concept of “quantity”.

At the 1st stage the properties and qualities of objects that can be compared are highlighted and recognized.

You can compare without measuring lengths (by eye, by application and overlay), mass (by estimate on the hand), capacity (by eye), area (by eye and by application), time (focusing on the subjective feeling of duration or some external signs of this process : seasons differ according to seasonal characteristics in nature, time of day - according to the movement of the sun, etc.).

At this stage, it is important to bring the child to understand that there are qualities of objects that are subjective (sour - sweet) or objective, but do not allow an accurate assessment (shades of color), and there are qualities that allow an accurate assessment of the difference (how much more - less ).

At the 2nd stage to compare values, an intermediate measure is used. This stage is very important for forming an idea of ​​the the idea of ​​measurement through intermediate measures. The measure can be arbitrarily chosen by the child from the surrounding reality for the container - a glass, for the length - a piece of lace, for the area - a notebook, etc. (The boa constrictor can be measured in both Monkeys and Parrots.)

Before the invention of the generally accepted system of measures, humanity actively used natural measures - step, palm, elbow, etc. From natural measures of measurement came the inch, foot, arshin, fathom, pound, etc. It is useful to encourage the child to go through this stage in the history of the development of measurements, using the natural measures of your body as intermediate.

Only after this can you move on to becoming familiar with generally accepted standard measures and measuring instruments (ruler, scales, palette, etc.). It will already be 3rd stage work on becoming familiar with quantities.

Familiarity with standard measures of quantities in school is associated with the stages of studying numbering, since most standard measures are focused on the decimal number system: 1 m = 100 cm, 1 kg = 1000 g, etc. Thus, the activity of measurement in school is very quickly replaced by the activity converting numerical values ​​of measurement results. The student is practically not directly involved in measurements and working with quantities; he performs arithmetic operations with the conditions of the assignment or problem given to him with numerical values.

quantities (adds, subtracts, multiplies, divides), and also deals with the so-called translation of the values ​​of quantities expressed in some names into others (converts meters into centimeters, tons into centners, etc.). Such activity actually formalizes the process of working with quantities at the level of numerical transformations. To be successful in this activity, you need to know by heart all the tables of ratios of quantities and have a good command of calculation techniques. For many schoolchildren, this topic is difficult only because of the need to know by heart large volumes of numerical relationships between measures of quantities.

The most difficult thing in this regard is working with the quantity “time”. This value is accompanied by the largest number of purely conventional standard measures, which not only need to be memorized (hour, minute, day, day, week, month, etc.), but also learn their relationships, which are not given in the usual decimal number system (day - 24 hours, hour - 60 minutes, week - 7 days, etc.).

As a result of studying quantities, students should master the following knowledge, skills and abilities:

1) get acquainted with the units of each quantity, get on
a clear idea of ​​each unit, as well as learn in relation to
relationships between all studied units of each quantity,
i.e. know the tables of units and be able to apply them when solving practical problems
technical and educational tasks;

2) know what tools and instruments are used to measure
each value, have a clear understanding of the measurement process
knowledge of length, mass, time, learn to measure and construct a segment
ki using a ruler.

Length

Length is a characteristic of the linear dimensions of an object (length).

Children become familiar with length and its units of measurement throughout all years of primary school.

Children receive their first ideas about length in preschool age; they identify the linear extent of an object: length, width, distance between objects. By the beginning of school, children should correctly establish the relationships “wider - narrower”, “further - closer”, “longer - shorter”.

In 1st grade, from the first mathematics lessons, children complete tasks to clarify spatial concepts: which is thinner, a book or a notebook; which pencil is longer? who is higher, who is lower. In 1st grade, children are introduced to the first unit of length - the centimeter.

Centimeter - metric measure of length. A centimeter is equal to one hundredth of a meter, a tenth of a decimeter. It is written like this: 1 cm (without a dot).

In 1st grade, children receive a visual understanding of the centimeter. They perform the following tasks:

1) measure the length of the strips using a centimeter model;

2) measure the length of the strips using a ruler.

To measure the length of a strip, you need to attach a ruler to it so that the beginning of the strip corresponds to the number 0 on the ruler. The number corresponding to the end of the strip is its length.

Children perform the following types of tasks:

1) comparison of the lengths of strips using measurements of arbitrary length:

Compare the lengths of the segments:

When completing a task, the child refers to the count of measurements: more measurements fit along the length of the segment, which means the segment is longer.

2) finding equal and unequal segments; definition, on
how much one segment is greater or less than another;

3) measuring segments and comparing them using a ruler (from
measure the length of a segment; compare the lengths of the segments, draw a segment
wire of a given length).

In grade 2, children are introduced to such units of length as decimeter and meter.

decimeter - metric measure of length. A decimeter is equal to one tenth of a meter. It is written like this: 1 dm (without a dot).

Children get a visual idea of ​​a decimeter as a segment equal to 10 cm and complete tasks of the following nature:

1) measuring objects using a decimeter model (album,
book, desk);

2) drawing a segment 1 dm long in a notebook;

3) comparison of the studied values:

1 dm * 1 cm 14 cm * 4 dm

4) transformation of quantities:

Fill in the blanks:

2 dm = ...cm

The basis for completing tasks for comparing and converting quantities is knowledge of the ratio: 1 dm = 10 cm

Meter - basic measure of length. The meter came into use at the end of the 18th century. in France.

In grade 2, children receive a visual understanding of the meter and become familiar with basic metric relationships:

10 dm - 1m; 100cm=1m

Children learn to designate a new unit of length: m (without a dot), measure objects using a new unit of length (cord, board, classroom). The tool used is a meter ruler or tailor's tape.

Students complete the following tasks:

1) comparison:

Put a comparison sign 1 m * 99 cm 1 m * 9 dm

2) transformation of quantities:

Express the units of quantities of one name in terms of others:

3 m 2 dm = ... dm

When performing transformations, children use tables of ratios of units of length: 1 m = 10 dm, 3 m is 3 times more, which means 3 m = 30 dm, and even 2 dm - a total of 32 dm.

Fill in the blanks: 56 dm = ... m... dm

Reasoning: 56 dm - as many meters as there are 56 tens in the number.

In previous textbooks of the 1-4 system, children were introduced to the kilometer in the 3rd grade; in the new edition of this textbook (2001), the kilometer is studied in the 4th grade.

Kilometer - This is a metric measure of length. A kilometer is equal to 1000 m. It is written as 1 km (without a dot). Children can be introduced to the fact that “kilo” translated into Russian means “thousand”, “kilo-meter” - a thousand meters. It is quite difficult to give a visual representation of a kilometer, since it is a fairly large measure of length. Teachers often offer this image: unwind a spool of thread, and then imagine that 10 spools of thread are unwound and stretched out in length - this is a kilometer (a standard spool contains 100 m). It is useful to do this experiment with at least one spool, since it is difficult for a child to even imagine the length of a spool of thread, let alone a kilometer:

Compare: Fill in the blanks:

1 km * 1000 m 1,000 cm = ... m

2 m 50 cm * 2 m 5 cm 5,000 m =... km

In grade 4, a new unit is introduced into tasks for converting and comparing quantities:

Millimeter - metric measure of length. A millimeter is equal to one thousandth of a meter, i.e. a tenth of a centimeter. It is written like this: 1 mm (without a dot).

1 cm - 10 mm

Students perform tasks like:

1) measuring objects (nail, screw), expressing the result
tov in millimeters;

2) drawing segments of different lengths: (9 mm, 6 mm, 2 cm 3 mm);

3) transformation of quantities:

Fill in the blanks: 620 mm = ... cm

Reasoning: There are as many centimeters in 620 mm as in the number 620 tens.

Fill in the blanks: 72 km 276 m = ... m

Reasoning: first we convert kilometers to meters: 1 km = 1000 m, 72 km = 72,000 m and also 276 m - 72,276 m

4) comparison:

Compare: 1 km * 100 m 7200 mm * 72 km

In grade 4, a summary table is compiled:
1 km = 1000 m 1 m = 100 cm 1 cm = 10 mm

1 m = 10 dm 1 dm = 10 cm

After compiling this table, children are offered tasks to select suitable units of measurement:

Fill in the blanks: 1... = 10 ... 1... = 100 ... 1... = 1000 ...