METHODOLOGY FOR STUDYING VALUES IN THE PRIMARY SCHOOL

The study of values ​​​​and their measurements in the course of mathematics in elementary school is of great importance in terms of the development of younger students. This is due to the fact that through the concept of magnitude, the real properties of objects and phenomena are described, the knowledge of the surrounding reality takes place; acquaintance with the dependencies between quantities helps to create in children holistic ideas about the world around them; the study of the process of measuring quantities contributes to the acquisition of practical skills and abilities necessary for a person in his daily activities. In addition, the knowledge and skills associated with quantities and acquired in elementary school are the basis for further study of mathematics.

According to the traditional program, at the end of grade 4, children should:

Know the tables of units of quantities, the accepted designations of these units and be able to apply this knowledge in the practice of measurement and in solving problems,

Know the relationship between such quantities as price, quantity, cost of goods; speed, time, distance, be able to apply this knowledge to solving word problems,

Be able to calculate the perimeter and area of ​​a rectangle (square).

THE CONCEPT OF VALUE AND ITS MEASUREMENTS IN MATHEMATICS

One of the features of the reality around us is its diverse and continuous change. For example, the weather changes, the age of people, their living conditions. To give a scientific justification for these processes, you need to know their definition, properties, qualities, such. Like time, area, mass... These and other properties are called quantities.

In accordance with the definition of N.B. Istomina:

Firstly, magnitude is a property of objects.

Secondly, magnitude - this is a property of objects that allows them to be compared and set pairs of objects that have this property equally.

Thirdly, magnitude - this is a property that allows you to compare objects and establish which of them has this property to a greater extent.

Values ​​are homogeneous and heterogeneous. Quantities that express the same property of objects are called quantities of the same kind or homogeneous quantities . For example, the length of a table and the length of a room are homogeneous values. Heterogeneous quantities express different properties of objects (for example, length and area).

Homogeneous quantities have a number properties .

1) Any two quantities of the same kind are comparable: they are either equal, or one is less (greater) than the other. That is, for quantities of the same kind, the relations “is equal to”, “less than”, “greater than”, and for any quantities one and only one of the relations is true: For example, we say that the length of the hypotenuse of a right triangle is greater than any leg of this triangle ; the mass of a lemon is less than the mass of a watermelon; the lengths of opposite sides of the rectangle are equal.

2) Values ​​of the same kind can be added, as a result of addition, a value of the same kind will be obtained. Those. for any two quantities a and b the quantity a + b is uniquely determined, it is called the sum of the quantities a and b. For example, if a- length of segment AB, b- the length of the segment BC, then the length of the segment AC is the sum of the lengths of the segments AB and BC;

3) The value is multiplied by a real number, resulting in a value of the same kind. Then for any value a and any non-negative number x there is a single value b=x * a, the value b is called the product of the quantity a per number x. For example, if a is the length of the segment AB, multiply by x= 2, then we get the length of the new segment AC.

4) The values ​​​​of this kind are subtracted, determining the difference in values ​​through the sum: the difference in values a and b such a value is called With that a=b+c. For example, if a is the length of segment AC, b- the length of segment AB, then the length of segment BC is the difference between the lengths of segments AC and AB.

5) Values ​​of the same kind are divided, defining the quotient through the product of the value by the number; private values a and b such a non-negative real number is called X, what

a=x*b. This number is often referred to as the ratio a and b and write it like this:

6) The relation "less than" for homogeneous quantities is transitive: if A<В и В<С, то А<С. Так, если площадь треугольника F1 меньше площади треугольника F2, площадь треугольника F2 меньше площади треугольника F3, то площадь треугольника F1 меньше площади треугольника F3.

Quantities, as properties of objects, have one more feature - they can be quantified. To do this, the value must be measured. Measurement consists in comparing a given quantity with some quantity of the same kind, taken as a unit. As a result of the measurement, a number is obtained, which is called numerical value with the selected unit.

The comparison process depends on the kind of quantities under consideration: for lengths it is one, for areas - another, for masses - a third, and so on. But whatever this process may be, as a result of measurement, the quantity receives a certain numerical value with the chosen unit.

In general, if given a value a and the unit of measure is chosen e, then as a result of measuring the quantity a find such a real number x that a=x e. it the number x is called the numerical value of the quantity a when e is unity. This can be written like this: x=m (a).

According to the definition, any quantity can be represented as a product of a certain number and a unit of this quantity. For example, 7 kg \u003d 7 * 1 kg, 12 cm \u003d 12 * 1 cm, 15 h \u003d 15 * 1 h. Using this, as well as the definition of multiplying a quantity by a number, you can justify the process of transition from one unit of quantity to another. Let, for example, you want to express 5/12h in minutes. Since 5/12 hours = 5/12*60 minutes = (5/12*60) minutes = 25 minutes.

Quantities that are completely determined by one numerical value are called scalars . Such, for example, are length, area, volume, mass and others. In addition to scalar quantities, in mathematics they also consider vector quantities . To determine a vector quantity, it is necessary to specify not only its numerical value, but also its direction. Vector quantities are force, acceleration, electric field strength and others.

In elementary school, only scalar quantities are considered, and those whose numerical values ​​are positive, that is, positive scalar quantities.

The size of a physical quantity- quantitative certainty of a physical quantity inherent in a specific material object, system, phenomenon or process.

The broad use of the word "size" is sometimes objected to, arguing that it only refers to length. However, we note that each body has a certain mass, as a result of which bodies can be distinguished by their mass, i.e. by the size of the physical quantity (mass) of interest to us. Looking at things BUT and AT, one can, for example, argue that they differ from each other in length or size of length (for example, A > B). A more accurate estimate can only be obtained after measuring the length of these objects.

Often in the phrase “size of a quantity”, the word “size” is omitted or replaced by the phrase “value of a quantity”.

In mechanical engineering, the term "size" is widely used, meaning by it the value of a physical quantity - the length inherent in any part. This means that two terms (“size” and “value”) are used to express one concept “the value of a physical quantity”, which cannot contribute to the ordering of terminology. Strictly speaking, it is necessary to clarify the concept of "size" in mechanical engineering so that it does not contradict the concept of "size of a physical quantity" adopted in metrology. GOST 16263-70 gives a clear explanation on this issue.

A quantitative assessment of a specific physical quantity, expressed as a certain number of units of a given quantity, is called "the value of a physical quantity".

An abstract number included in the "value" of a quantity is called a numerical value.

There is a fundamental difference between size and value. The size of a quantity really exists, whether we know it or not. You can express the size of a quantity using any of the units of a given quantity, in other words, using a numerical value.

For a numerical value, it is characteristic that when a different unit is used, it changes, while the physical size of the quantity remains unchanged.

If we designate the measured value through x, the unit of magnitude - through x 1 , and their ratio through q 1, then x = q 1 x 1  .

The size of x does not depend on the choice of unit, which cannot be said about the numerical value of q, which is entirely determined by the choice of unit. If to express the size of the quantity x instead of the unit x 1 , use the unit x 2  , then the unchanged size x will be expressed by a different value:

x = q 2 x 2  , where n 2 n 1 .

If q = 1 is used in the above expressions, then the sizes of the units

x 1 = 1x 1  and x 2 = 1x 2 .

The sizes of different units of the same value are different. Thus, the size of a kilogram is different from the size of a pound; the size of a meter is from the size of a foot, etc.

1.6. Dimension of physical quantities

Dimension of physical quantities - this is the ratio between the units of quantities included in the equation, connecting the given quantity with other quantities through which it is expressed.

The dimension of a physical quantity is denoted dim A(from lat. dimension - dimension). Let us assume that the physical quantity BUT associated with x, Equation A = F(X, Y). Then the quantities X, Y, A can be represented as

X = x [X]; Y=y [Y];A = a [A],

where A, X, Y - symbols denoting a physical quantity; a, x, y - numerical values ​​of quantities (dimensionless); [A];[X]; [Y]- corresponding units of data of physical quantities.

The dimensions of the values ​​of physical quantities and their units are the same. For example:

A=X/Y; dim(a) = dim(X/Y) = [X]/[Y].

Dimension - a qualitative characteristic of a physical quantity, giving an idea of ​​the type, nature of the quantity, its relationship with other quantities, the units of which are taken as the main ones.

Value is something that can be measured. Concepts such as length, area, volume, mass, time, speed, etc. are called quantities. The value is measurement result, it is determined by a number expressed in certain units. The units in which a quantity is measured are called units of measurement.

To designate a quantity, a number is written, and next to it is the name of the unit in which it was measured. For example, 5 cm, 10 kg, 12 km, 5 min. Each value has an infinite number of values, for example, the length can be equal to: 1 cm, 2 cm, 3 cm, etc.

The same value can be expressed in different units, for example, kilogram, gram and ton are units of weight. The same value in different units is expressed by different numbers. For example, 5 cm = 50 mm (length), 1 hour = 60 minutes (time), 2 kg = 2000 g (weight).

To measure a quantity means to find out how many times it contains another quantity of the same kind, taken as a unit of measurement.

For example, we want to know the exact length of a room. So we need to measure this length using another length that is well known to us, for example, using a meter. To do this, set aside a meter along the length of the room as many times as possible. If he fits exactly 7 times along the length of the room, then its length is 7 meters.

As a result of measuring the quantity, one obtains or named number, for example 12 meters, or several named numbers, for example 5 meters 7 centimeters, the totality of which is called composite named number.

Measures

In each state, the government has established certain units of measurement for various quantities. A precisely calculated unit of measurement, taken as a model, is called standard or exemplary unit. Model units of the meter, kilogram, centimeter, etc., were made, according to which units for everyday use are made. Units that have come into use and approved by the state are called measures.

The measures are called homogeneous if they serve to measure quantities of the same kind. So, grams and kilograms are homogeneous measures, since they serve to measure weight.

Units

The following are units of measurement for various quantities that are often found in math problems:

Measures of weight/mass

• 1 ton = 10 centners
• 1 centner = 100 kilograms
• 1 kilogram = 1000 grams
• 1 gram = 1000 milligrams

• 1 kilometer = 1000 meters
• 1 meter = 10 decimeters
• 1 decimeter = 10 centimeters
• 1 centimeter = 10 millimeters

• 1 sq. kilometer = 100 hectares
• 1 hectare = 10000 sq. meters
• 1 sq. meter = 10000 sq. centimeters
• 1 sq. centimeter = 100 sq. millimeters

• 1 cu. meter = 1000 cubic meters decimeters
• 1 cu. decimeter = 1000 cu. centimeters
• 1 cu. centimeter = 1000 cu. millimeters

Let's consider another value like liter. A liter is used to measure the capacity of vessels. A liter is a volume that is equal to one cubic decimeter (1 liter = 1 cubic decimeter).

Measures of time

• 1 century (century) = 100 years
• 1 year = 12 months
• 1 month = 30 days
• 1 week = 7 days
• 1 day = 24 hours
• 1 hour = 60 minutes
• 1 minute = 60 seconds
• 1 second = 1000 milliseconds

In addition, time units such as quarter and decade are used.

• quarter - 3 months
• decade - 10 days

The month is taken as 30 days, unless it is required to specify the day and name of the month. January, March, May, July, August, October and December - 31 days. February in a simple year has 28 days, February in a leap year has 29 days. April, June, September, November - 30 days.

A year is (approximately) the time it takes for the Earth to complete one revolution around the Sun. It is customary to count every three consecutive years for 365 days, and the fourth following them - for 366 days. A year with 366 days is called leap year, and years containing 365 days - simple. One extra day is added to the fourth year for the following reason. The time of revolution of the Earth around the Sun does not contain exactly 365 days, but 365 days and 6 hours (approximately). Thus, a simple year is shorter than a true year by 6 hours, and 4 simple years are shorter than 4 true years by 24 hours, that is, by one day. Therefore, one day (February 29) is added to every fourth year.

You will learn about other types of quantities as you further study various sciences.

Measure abbreviations

Abbreviated names of measures are usually written without a dot:

Kilometer - km Meter - m Decimeter - dm centimeter - cm Millimeter - mm	Measures of weight/mass ton - t centner - c kilogram - kg gram - g milligram - mg
Area measures (square measures) sq. kilometer - km 2 hectare - ha sq. meter - m 2 sq. centimeter - cm 2 sq. millimeter - mm 2	cube meter - m 3 cube decimeter - dm 3 cube centimeter - cm 3 cube millimeter - mm 3
Measures of time century - in year - y month - m or mo week - n or week day - from or d (day) hour - h minute - m second - s millisecond - ms	A measure of the capacity of vessels liter - l

Measuring instruments

To measure various quantities, special measuring instruments are used. Some of them are very simple and are designed for simple measurements. Such devices include a measuring ruler, tape measure, measuring cylinder, etc. Other measuring devices are more complex. Such devices include stopwatches, thermometers, electronic scales, etc.

Measuring instruments, as a rule, have a measuring scale (or short scale). This means that dash divisions are marked on the device, and the corresponding value of the quantity is written next to each dash division. The distance between two strokes, next to which the value of the value is written, can be further divided into several smaller divisions, these divisions are most often not indicated by numbers.

It is not difficult to determine which value of the value corresponds to each smallest division. So, for example, the figure below shows a measuring ruler:

The numbers 1, 2, 3, 4, etc. indicate the distances between the strokes, which are divided into 10 equal divisions. Therefore, each division (the distance between the nearest strokes) corresponds to 1 mm. This value is called scale division measuring instrument.

Before you start measuring a quantity, you should determine the value of the division of the scale of the instrument used.

In order to determine the division price, you must:

1. Find the two nearest strokes of the scale, next to which the magnitude values ​​are written.
2. Subtract the smaller value from the larger value and divide the resulting number by the number of divisions in between.

As an example, let's determine the scale division value of the thermometer shown in the figure on the left.

Let's take two strokes, near which the numerical values ​​of the measured quantity (temperature) are plotted.

For example, strokes with symbols 20 °С and 30 °С. The distance between these strokes is divided into 10 divisions. Thus, the price of each division will be equal to:

(30 °C - 20 °C) : 10 = 1 °C

Therefore, the thermometer shows 47 °C.

Each of us constantly has to measure various quantities in everyday life. For example, to come to school or work on time, you have to measure the time that will be spent on the road. Meteorologists measure temperature, atmospheric pressure, wind speed, etc. to predict the weather.

Physical quantity called the physical property of a material object, process, physical phenomenon, characterized quantitatively.

The value of a physical quantity expressed by one or more numbers characterizing this physical quantity, indicating the unit of measurement.

The size of a physical quantity are the values ​​of the numbers appearing in the meaning of the physical quantity.

Units of measurement of physical quantities.

The unit of measurement of a physical quantity is a fixed size value that is assigned a numeric value equal to one. It is used for the quantitative expression of physical quantities homogeneous with it. A system of units of physical quantities is a set of basic and derived units based on a certain system of quantities.

Only a few systems of units have become widespread. In most cases, many countries use the metric system.

Basic units.

Measure physical quantity - means to compare it with another similar physical quantity, taken as a unit.

The length of an object is compared with a unit of length, body weight - with a unit of weight, etc. But if one researcher measures the length in sazhens, and another in feet, it will be difficult for them to compare these two values. Therefore, all physical quantities around the world are usually measured in the same units. In 1963, the International System of Units SI (System international - SI) was adopted.

For each physical quantity in the system of units, an appropriate unit of measurement must be provided. Standard units is its physical realization.

The length standard is meter- the distance between two strokes applied on a specially shaped rod made of an alloy of platinum and iridium.

Standard time is the duration of any correctly repeating process, which is chosen as the movement of the Earth around the Sun: the Earth makes one revolution per year. But the unit of time is not a year, but give me a sec.

For a unit speed take the speed of such uniform rectilinear motion, at which the body makes a movement of 1 m in 1 s.

A separate unit of measurement is used for area, volume, length, etc. Each unit is determined when choosing one or another standard. But the system of units is much more convenient if only a few units are chosen as the main ones, and the rest are determined through the main ones. For example, if the unit of length is a meter, then the unit of area is a square meter, volume is a cubic meter, speed is a meter per second, and so on.

Basic units The physical quantities in the International System of Units (SI) are: meter (m), kilogram (kg), second (s), ampere (A), kelvin (K), candela (cd) and mole (mol).

Basic SI units
Value	Unit	Designation
	Name	Russian	international
The strength of the electric current
Thermodynamic temperature
The power of light
Amount of substance

There are also derived SI units, which have their own names:

SI derived units with their own names
	Unit		Derived unit expression
Value	Name	Designation	Via other SI units	Through basic and additional SI units
Pressure				m -1 ChkgChs -2
Energy, work, amount of heat				m 2 ChkgChs -2
Power, energy flow				m 2 ChkgChs -3
Quantity of electricity, electric charge
Electrical voltage, electrical potential				m 2 ChkgChs -3 CHA -1
Electrical capacitance				m -2 Chkg -1 Hs 4 CHA 2
Electrical resistance				m 2 ChkgChs -3 CHA -2
electrical conductivity				m -2 Chkg -1 Hs 3 CHA 2
Flux of magnetic induction				m 2 ChkgChs -2 CHA -1
Magnetic induction				kghs -2 CHA -1
Inductance				m 2 ChkgChs -2 CHA -2
Light flow
illumination				m 2 ChkdChsr
Radioactive source activity	becquerel
Absorbed radiation dose

Andmeasurements. To obtain an accurate, objective and easily reproducible description of a physical quantity, measurements are used. Without measurements, a physical quantity cannot be quantified. Definitions such as "low" or "high" pressure, "low" or "high" temperature reflect only subjective opinions and do not contain comparison with reference values. When measuring a physical quantity, it is assigned a certain numerical value.

Measurements are made using measuring instruments. There is a fairly large number of measuring instruments and fixtures, from the simplest to the most complex. For example, length is measured with a ruler or tape measure, temperature with a thermometer, width with calipers.

Measuring instruments are classified: according to the method of presenting information (indicating or recording), according to the method of measurement (direct action and comparison), according to the form of presentation of indications (analog and digital), etc.

The measuring instruments are characterized by the following parameters:

Measuring range- the range of values ​​of the measured quantity, on which the device is designed during its normal operation (with a given measurement accuracy).

Sensitivity threshold- the minimum (threshold) value of the measured value, distinguished by the device.

Sensitivity- relates the value of the measured parameter and the corresponding change in instrument readings.

Accuracy- the ability of the device to indicate the true value of the measured indicator.

Stability- the ability of the device to maintain a given measurement accuracy for a certain time after calibration.

From the course of mathematics, we know the actions that can be performed on numbers. You can add, subtract, and compare any number in mathematics. Such operations on physical quantities can be performed only if they are homogeneous, i.e., they represent the same physical quantity.

For example:

4 m + 3 m = 7 m;
9 kg - 5 kg = 4 kg;
30 s > 10 s.

In all three cases, we performed operations on homogeneous physical quantities. The length was added to the length, the mass was subtracted from the mass, and the time interval was compared with the time interval. It would be ridiculous and absurd to add 4 m and 5 kg or subtract 30 s from 9 kg!

But you can multiply and divide not only homogeneous, but also different physical quantities. For example:

1. 10 kg ÷2 kg = 5. Not only numerical values ​​are divided here (10 ÷ 2 = 5), but also units of physical quantities (kg ÷ kg = 1). The result shows how many times one physical quantity (mass) is greater than another.
2. 2 m. 4 m = 8 m 2. Numerical values ​​​​are multiplied (2. 4 \u003d 8) and units of physical quantities (m. m \u003d m 2). As a result of multiplying two physical quantities - lengths l 1 \u003d 2 m and l 2 \u003d 4 m - a new physical quantity was obtained - area S \u003d 8 m 2.
3. 10 m ÷ 2 s = 5 m/s. As a result of dividing two different physical quantities - length l = 10 m by a time interval t = 2 s, a new physical quantity 5 m / s was obtained. Its numerical value is 5, and the unit of the new physical quantity is m/s. This physical quantity v = 5 m/s is the speed.
4. 10 m ÷ 2 s = 20 m ÷ 4 s. The equal sign applies not only to numerical values, but also to units. An equal sign cannot be put if we compare 10 m ÷ 2 s and 20 m ÷ 4 min. Here m/s ≠ m/min.

Think and answer

1. What should be taken into account when adding and subtracting physical quantities? What will be the result of their addition and subtraction?
2. What physical quantities can be compared with each other? Give examples.
3. Is it possible to divide and multiply different physical quantities? What will be the result?
4. Determine the value of which physical quantity will be the result:
   1. 40 s - 10 s;
   2. 40 s ÷ 10 s;
   3. 3 m. 4 m. 2 m;
   4. 120 km ÷ 2 h.

Interesting to know!

Large units of time - a year and a day - were given to us by nature itself. But the hour, minute and second appeared thanks to man.

The currently accepted division of the day goes back to ancient times. In Babylon, not a decimal, but a sexagesimal number system was used. Sixty is divisible without remainder by 12, hence the Babylonian division of the day into 12 equal parts. In ancient Egypt, the division of the day into 24 hours was introduced. Later minutes and seconds appeared. The fact that there are 60 minutes in 1 hour and 60 seconds in 1 minute is also a legacy of the sexagesimal system of Babylon.

The definition of time units is very important. The basic unit of time - the second - was first introduced as 1/86400 of a fraction of a day, and then, due to the volatility of the day, as a certain fraction of a year. At present, the standard second is associated with the frequency of radiation of cesium atoms.