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Quantities and their measurements in elementary school. Using multi-level tasks when studying the topic “Quantities” in elementary school. Percentage

The concept of magnitude.

Weight and capacity.
Speed.
Actions with named numbers.

1. The concept of magnitude

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

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

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 primary 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 capabilities 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. The number is measure of magnitude , and the very idea of number was largely generated by the need to quantify 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.).

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 measuring through intermediate measures . A measure can be arbitrarily chosen by a child from the surrounding reality for a container - a glass, for length - a piece of lace, for an area - a notebook. (Boas 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. From natural measures of measurement came the inch, foot, arshin, fathom, pood. It is useful to encourage the child to go through this stage of the developmental history of measurements, using the natural measurements of his body as intermediate ones.

Only after this can you move on to getting acquainted with generally accepted standard measures and measuring instruments (ruler, scales, palette.). 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. Thus, the activity of measurement in school is very quickly replaced by the activity of converting the numerical values of the results measurements. The student is practically not directly involved in measurements and working with quantities; he performs arithmetic operations with the numerical values of quantities given to him by the conditions of the task or problem (adds, subtracts, multiplies, divides), and also engages in the so-called translation of the values of quantities expressed in some names into others (converts meters to centimeters, tons to centners.). 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.), but also to learn their relationships, which are not given in the usual decimal number system (day - 24 hours, hour - 60 minutes, week - 7 days.).

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

get acquainted with the units of each quantity, get a visual representation of each unit, and also learn the relationships between all the studied units of each quantity, i.e. know the tables of units and be able to apply them when solving practical and educational problems;

know what tools and instruments are used to measure each quantity, have a clear understanding of the process of measuring length, mass, time, learn to measure and construct segments using a ruler.

GENERAL QUESTIONS OF METHODOLOGY

1. Objectives of the study.

2. The meaning and place of the section “Quantities and their measurement” in the initial course of mathematics.

3. Stages of studying each of the basic quantities.

4. Features of lessons on familiarization with quantity and its measurement.

5. Methodology for developing the concept of “area” in younger schoolchildren, studying area measures and developing the corresponding skills.

Additional literature

Tikhonenko A.V. Didactic and methodological foundations formation of the concept of “area” // NSh, 1999, No. 12.

Tikhonenko A.V. Study of measures of time //NSh, 1998, No. 1, pp. 94-101.

Gryshkova I.M. Farmed egg bath at an hour // PSh, 2000, No. 7

Istomina N.B. IOM in primary school-M., 1999, chapter 2, paragraph 2.10

Medvedskaya V.N. Workshop - BrGU, 2000

1. STUDY OBJECTIVES

In the elementary grades, they consider the basic quantities (length, mass, capacity, time, area) and derivatives: speed, productivity, yield, etc.

In relation to the basic quantities of the program primary classes The following tasks are set:

1) formation correct ideas about these quantities;

2) practical familiarization with relevant measuring instruments;

3) formation of practical skills and skills for their measurement;

4) familiarization with the system of units of measurement and the table of measures of these quantities;

5) developing skills in converting the values of quantities and performing actions on them (on nominal numbers).

The solution of these problems contributes to the disclosure of the concepts of “length”, “mass”, ..., “magnitude” and their common basic properties (active-scalar quantities) (see: reference diagram No. 21 and tasks for it in the “Workshop” by V.N. Medvedskaya)

Acquaintance with derived quantities is carried out, as a rule, through solving word problems with proportionally dependent quantities (price, quantity, cost, speed; time, distance, etc.) The main attention is paid to both the specific meaning of the corresponding quantity and the relationship between quantities.

The study of quantities, as well as other objects of reality, in mathematics is associated with the problem of their mathematization, mathematical modeling, i.e. translation into the language of numbers and relationships between them. In a general way The solution to this problem is to introduce functions (more precisely, a functor), certain rules, allowing each object to be associated with a number, and the relations between real objects turn into certain relations between numbers.

Elementary examples functors are the operations of counting and measurement.

Quantity - general property(cardinality) of a class of finite sets of objects.

Mass, area etc. - also a general property of a class of objects.

Check is a function. What are the rules?

Measurement- function.

Introduction……………………………………………………………………………….
The concept of quantity and its measurement in the initial course of mathematics…….
Length of a segment and its measurement……………………………………………..
Area of the figure and its measurement…………………………………………….
Mass and its measurement………………………………………………………
Time and its measurement……………………………………………………..
Volume and its measurement…………………………….…………………….
Modern approaches to the study of quantities in the initial course of mathematics………………………………………………………………………………….
Conclusion………………………………………………………………..
Bibliography………………………………………………………
Lesson summary……………………………………………………………..

Introduction.

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

According to the traditional program, at the end of the third (fourth) grade, children must: - 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).

However, the result of the training shows that children do not sufficiently master the material related to quantities: they do not distinguish between a quantity and a unit of quantity, make mistakes when comparing quantities expressed in units of two names, and poorly master measurement skills. This is due to the organization of the study of this topic. In textbooks on the traditional curriculum, there are not enough tasks aimed at: clarifying and clarifying schoolchildren’s ideas about the quantity being studied, comparing homogeneous quantities, developing measurement skills, adding and subtracting quantities expressed in units of different names.

The concept of quantity and its measurement in the initial course of mathematics.

Length, area, mass, time, volume - quantities. Initial acquaintance with them occurs in elementary school, where quantity, along with number, is a leading concept.

QUANTITY is a special property of real objects or phenomena, and the peculiarity is that this property can be measured, that is, the number of quantities that express the same property of objects are called quantities Oof this kind or homogeneous quantities. For example, the length of a table and the length of a room are homogeneous quantities. Quantities - length, area, mass and others have a number of 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 “equal”, “less than”, “greater” take place and for any quantities, and 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 the given 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) Quantities of the same kind can be added; as a result of the addition, a quantity of the same kind is obtained. Those. for any two quantities a and b, the quantity a+b is uniquely determined, it is called Withatmmoy quantities a and b. For example, if a is the length of the segment AB, b is the length of the segment BC (Fig. 1), then the length of the segment AC is the sum of the lengths of the segments AB and BC;

3)Size atmultiplied by real number, resulting in a quantity of the same kind. Then for any value a and any non-negative number x there is a unique value b = x a, the value b is called work quantities a by number x. For example, if a is the length of segment AB multiplied by

x= 2, then we get the length of the new segment AC. (Fig. 2)

4) Values of this kind are subtracted, determining the difference in values through the sum:

the difference between the values a and b is a value c such that a=b+c. For example, if a is the length of segment AC, b is the length of segment AB, then the length of segment BC is the difference between the lengths of segments AC and AB.

5) Quantities of the same kind are divided, determining the quotient through the product of the quantity by the number; the quotient of a and b is a non-negative real number x such that a = x b. More often this number is called the ratio of the quantities a and b and is written in this form: a/b = X. For example, the ratio of the length of segment AC to the length of segment AB is 2. (Figure No. 2).

6) The relation “less” for homogeneous quantities is transitive: if A Quantities, as properties of objects, have one more feature - they can be assessed quantitatively. To do this, the value must be measured. Measurement consists of comparing a given quantity with a certain quantity of the same kind, taken as a unit.

scalar

Length of a segment and its measurement.

The length of a segment is a positive quantity defined for each segment so that:

1/ equal segments have different lengths;

2/ if a segment consists of a finite number of segments, then its length is equal to the sum of the lengths of these segments.

Let's consider the process of measuring the lengths of segments. From a set of segments, select some segment e and take it as a unit of length. On the segment a, segments equal to e are laid out successively from one of its ends, as long as this is possible. If segments equal to e were deposited n times and the end of the last coincided with the end of the segment e, then they say that the value of the length of the segment a is a natural number n, and write: a = ne. If segments equal to e have been deposited n times and there remains a remainder smaller than e, then segments equal to e = 1/10e are deposited on it. If they were deposited exactly n times, then a=n, n e and the value of the length of the segment a is a finite decimal fraction. If the segment e has been deposited n times and there is still a remainder smaller than e, then segments equal to e = 1/100e are deposited on it. If we imagine this process continuing indefinitely, we find that the value of the length of the segment a is an infinite decimal fraction.

So, with the chosen unit, the length of any segment is expressed as a real number. The opposite is also true; if a positive real number n, n, n, ... is given, then taking its approximation with a certain

accuracy and having carried out the constructions reflected in the recording of this number, we obtain a segment, the numerical value of the length of which is a fraction: n ,n ,n ...

Area of a figure and its measurement .

Any person has the concept of the area of a figure: we are talking about the area of a room, the area of a plot of land, the area of a surface that needs to be painted, and so on. At the same time, we understand that if the land plots are the same, then their areas are equal; that a larger plot has a larger area; that the area of an apartment is made up of the area of the rooms and the area of its other premises.

This everyday idea of area is used when defining it in geometry, where they talk about the area of a figure. But geometric figures are arranged in different ways, and therefore when they talk about area, they distinguish a special class of figures. For example, they consider the area of polygons and other limited convex figures, or the area of a circle, or the surface area of bodies of revolution, and so on. In the initial mathematics course, only the areas of polygons and bounded convex plane figures are considered. Such a figure can be composed of others. For example, figure F (Fig. 4) is made up of figures F1, F2, F3. By saying that a figure is composed (consists) of figures F1, F2,..., Fn, they mean that it is their union and any two given figures do not have common interior points. FIG areaatry is a non-negative quantity defined for each figure so that:

I/ equal figures have equal areas;

2/ if a figure is made up of a finite number of figures, then its area is equal to the sum of their areas. If we compare this definition with the definition of the length of a segment, we will see that the area is characterized by the same properties as the length, but they are specified on different sets: the length is on the set of segments, and the area is on the set of plane figures. The area of figure F is denoted by S(F). To measure the area of a figure, you need to have a unit of area. As a rule, the unit of area is taken to be the area of a square with a side equal to the unit segment e, that is, the segment chosen as the unit of length. The area of a square with side e is denoted by e. For example, if the side length of a unit square is m, then its area is m.

Measuring area consists of comparing the area of a given figure with the area of a unit square e. The result of this comparison is a number x such that S(F)=x e. The number x is called the numerical value area for the selected unit of area.

Mass and its measurement .

Mass is one of the basic physical quantities. The concept of body mass is closely related to the concept of weight-force with which the body is attracted by the Earth. Therefore, body weight depends not only on the body itself. For example, it is different at different latitudes: at the pole the body weighs 0.5% more than at the equator. However, despite its variability, weight has a peculiarity: the ratio of the weights of two bodies remains unchanged under any conditions. When measuring the weight of a body by comparing it with the weight of another, a new property of bodies is revealed, which is called mass. Let's imagine that some body is placed on one of the cups of a lever scale, and a second body b is placed on the other cup. In this case, the following cases are possible:

1) The second pan of the scales fell, and the first one rose so that they ended up on the same level. In this case, the scales are said to be in equilibrium, and bodies a and b have equal masses.

2) The second pan of the scale remained higher than the first. In this case, we say that the mass of body a is greater than the mass of body b.

3) The second cup fell, and the first rose and stands higher than the second. In this case, we say that the mass of body a is less than body b.

From a mathematical point of view, mass is a positive quantity that has the following properties:

1) The mass is the same for bodies balancing each other on scales;

2) Mass adds up when bodies are connected together: the mass of several bodies taken together is equal to the sum of their masses. If we compare this definition with the definitions of length and area, we will see that mass is characterized by the same properties as length and area, but is defined on a set of physical bodies.

Mass is measured using scales. This happens as follows. Select a body e whose mass is taken as unity. It is assumed that it is possible to take fractions of this mass. For example, if a kilogram is taken as a unit of mass, then in the measurement process you can use its fraction as a gram: 1 g = 0.01 kg.

A body is placed on one pan of scales, the body mass of someone is measured, and on the other – bodies chosen as a unit of mass, that is, weights. There should be enough of these weights to balance the first pan of the scale. As a result of weighing, a numerical value of the mass of a given body is obtained for the selected unit of mass. This value is approximate. For example, if the body mass is 5 kg 350 g, then the number 5350 should be considered as the value of the mass of this body (with a mass unit of grams). For numerical values of mass, all statements formulated for length are valid, that is, comparison of masses, actions on them are reduced to comparison and actions on numerical values of mass (with the same unit of mass).

Basic unit of mass - kilogram. From this basic unit other units of mass are formed: gram, ton and others.

Time intervals and their measurement .

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. Thus, a lecture at an institute lasts the same amount of time as two lessons at school.

Time intervals are measured. But the process of measuring time is different from measuring length, area or mass. 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 the second. Along with the second, other units of time are also used: 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 month is not a very specific unit of time; it can consist of thirty-one days, thirty and twenty-eight, twenty-nine days leap years(days). But this unit of time has existed since ancient times and is associated with the movement of the Moon around the Earth. One turn around

The Moon makes it around the Earth in about 29.5 days, and in a year it makes about 12 revolutions. These data served as the basis for the creation of ancient calendars, and the result of their centuries-long improvement is the calendar that we use today.

Since the Moon makes 12 revolutions around the Earth, people began to count the full number of revolutions (that is, 22) per year, that is, a year is 12 months.

The modern division of the day into 24 hours also dates back to ancient times, it was introduced in Ancient Egypt. The minute and second appeared in Ancient Babylon, and the fact that there are 60 minutes in an hour and 60 seconds in a minute is influenced by the sexagesimal number system,

invented by Babylonian scientists.

Volume and its measurement.

The concept of volume is defined in the same way as the concept of area. But when considering the concept of area, we considered polygonal figures, and when considering the concept of volume, we will consider polygonal Figures.

The volume of a figure is a non-negative quantity defined for each Figure so that:

1/equal figures have the same volume;

2/if a figure is made up of a finite number of figures, then its volume is equal to the sum of their volumes.

Let us agree to denote the volume of the figure F as V(F).

To measure the volume of a figure, you need to have a unit of volume. As a rule, the unit of volume is taken to be the volume of a cube with a face equal to a unit segment e, that is, the segment chosen as the unit of length.

If measuring the area was reduced to comparing the area of a given figure with the area of a unit square e, then, similarly, measuring the volume of a given figure consists of comparing it with the volume of a unit cube e 3 (Fig.b). The result of this comparison is a number x such that V(F) = x e. The number x is called the numerical value of the volume for the selected unit of volume.

So. if the unit of volume is 1 cm, then the volume of the figure shown in Figure 7 is 4 cm.

Modern approaches to the study of quantities in the initial course of mathematics.

In elementary grades, quantities such as length, area, mass, volume, time and others are considered. Students must obtain specific ideas about these quantities, become familiar with their units of measurement, master the ability to measure quantities, learn to express measurement results in various units, and perform various operations on them.

Quantities are considered in close connection with the study of natural numbers and fractions; learning to measure is associated with learning to count; Measuring and graphical operations on quantities are visual tools and are used in solving problems. When forming ideas about each of these quantities, it is advisable to focus on certain stages, which are reflected: mathematical interpretation of the concept of quantity, relationship this concept with the study of other issues in the initial course of mathematics, as well as psychological characteristics younger schoolchildren.

N.B. Istomina, a mathematics teacher and author of one of the alternative programs, identified 8 stages in the study of quantities:

1st stage : clarification and clarification of schoolchildren’s ideas about a given quantity (referring to the child’s experience).

2nd stage : comparison of homogeneous quantities (visually, with the help of sensations, by imposition, by application, by using different measures).

3rd stage : familiarization with the unit of a given quantity and with the measuring device.

4 - 1st stage : formation of measurement skills.

5th stage : addition and subtraction of homogeneous quantities expressed in units of the same name.

6th stage : acquaintance with new units of quantities in close connection with the study of numbering and addition of numbers. Conversion of homogeneous quantities expressed in units of one denomination into quantities expressed in units of two denominations, and vice versa.

7th stage : addition and subtraction of quantities expressed in units of two names.

8th stage : multiplying and dividing quantities by a number.

Developmental education programs provide for consideration of basic quantities, their properties and relationships between them in order to show that numbers, their properties and actions performed on them act as special cases of already known general patterns quantities Structure this course mathematics is determined by considering the sequence of concepts: QUANTITY –> NUMBER

Let's take a closer look at the methodology for studying length, area, mass, time, and volume.

Methodology for studying length and its measurement.

In traditional elementary school, the study of quantities begins with the length of objects. Children have their first ideas about length as a property of objects long before school. From the first days of school, the task is to clarify spatial concepts children. An important step in the formation of this concept is familiarity with a straight line and a segment as a “carrier” of linear extension, essentially devoid of other properties.

First, students compare objects by length without measuring them. They do this by overlay (application) and visually (“by eye”). For example, students are asked to look at the drawings and answer the questions: “Which train is longer, with green cars or with red cars? Which train is shorter?” (M1M “1” p. 39, 1988)

Then you are asked to compare two objects different color and different in size (length) practically - overlapping. For example, students are asked to look at the pictures and answer the questions: “Which belt is shorter (longer), light or dark?” (M1M 1-4 p. 40, 1988). Through these two exercises, children are led to understand length as a property that manifests itself in comparison, that is: if two objects coincide when superimposed, then they have the same length; if any of the compared objects overlaps part of the other without covering it completely, then the length of the first object is less than the length of the second object. After considering the lengths of the objects, they move on to studying the length of the segment.

Here the length acts as a property of the segment.

At the next stage, we become familiar with the first unit of measurement for segments. From a set of segments, a segment is selected that is taken as a unit. This is centimeter. Children learn its name and begin to measure using this unit. In order for children to get a clear idea of the centimeter, they should perform a number of exercises. For example, it is useful for them to make a model of the centimeter themselves; Draw a line 1cm long in your notebook. They found that the width of the little finger is approximately 1 cm.

Next, students are introduced to the measuring device and measuring segments using the device. So that children clearly understand the process of measurement and what the numbers obtained during measurement show. It is advisable to gradually move from the simplest technique of laying out a centimeter model and counting them to a more difficult one - measuring. Only then do they begin to measure by applying a ruler or tape measure to the drawn segment.

In order for students to better understand the relationship between number and quantity, that is, to understand that as a result of measurement they get a number that can be added and subtracted, it is useful as visual aid Use the same ruler for addition and subtraction. For example, students are given a strip; You need to use a ruler to determine its length. The ruler is applied so that 0 coincides with the beginning of the strip, and its end coincides with the number 3 (if the length of the strip is 3 cm). Then the teacher asks questions: “And if you apply a ruler so that the beginning of the strip coincides with the number 2, what number on the ruler will the end of the strip coincide with? Why?". Some students immediately name the number 5, explaining that 2+3=5. Anyone who finds it difficult resorts to practical action, during which he strengthens his computational skills and acquires the ability to use a ruler for calculations. Similar exercises with a ruler and the reverse action - subtraction - are possible. To do this, students first determine the length of the proposed strip, for example, 4 cm, and then the teacher asks: “If the end of the strip coincides with the number 9 on the ruler, then what number will the beginning of the strip coincide with?” (5; 9-2 = 5). To develop measurement skills, a system of various exercises is included. This is the measurement and drawing of segments; comparison of segments to answer the question: how many centimeters is one segment longer (shorter) than another segment; increasing and decreasing segments by several centimeters. During these exercises, students develop the concept of length as the number of centimeters that fit in a given segment. Later, when studying the numbering of numbers within 100, new units of measurement are introduced - the decimeter, and then the meter. The work proceeds in the same way as when getting acquainted with a centimeter. Then relationships between units of measurement are established. From this time on, they begin to compare lengths based on comparison of the corresponding segments.

The introduction of the millimeter is justified by the need to measure segments smaller than 1 centimeter.

When becoming familiar with the kilometer, it is useful to carry out practical exercises on the ground in order to form an understanding of this unit of measurement.

In grades 3-4, students compile and memorize a table of all the studied units of length and their relationships.

Starting from grade 2 (1-3), children in the process of solving problems become familiar with finding length indirectly. For example, knowing the length of a given class and the number of classes on the second floor, calculates the length of the school; Knowing the height of the rooms and the number of floors in the house, you can approximately

calculate the height of the house and the like.

Work on this topic can be continued at extracurricular activities, for example, consider the ancient Russian measures: verst, fathom, vershok. Introduce students to some information from the history of the development of the system of measures.

Methodology for studying area and its measurement.

The method of working on the area of a figure has much in common with working on the length of a segment, that is, the work is carried out almost similarly.

Introducing students to the concept of “area of a figure” begins with clarifying the ideas that students have about this quantity. Based on their life experience, children easily perceive such a property of objects as size, expressing it in terms of “more”, “less”, “equal” between their sizes.

Using these ideas, you can introduce children to the concept of “area” by choosing for this purpose two figures such that when superimposed on each other, one fits entirely into the other.

“In this case,” says the teacher, “in mathematics it is customary to say that the area of one figure is greater (smaller) than the area of another figure.” When the figures coincide when superimposed, then they say that their areas are equal or coincide. Students can draw this conclusion on their own. But it is also possible that one of the figures does not fit completely into the other. For example, two rectangles, one of which is a square (Fig. 8). After unsuccessful attempts to fit one rectangle into another, the teacher turns the figures back, and the children see that one figure contains 10 identical squares, and the other 9 identical squares (Fig. 9).

The students, together with the teacher, conclude that to compare areas, as well as to compare lengths, you can use a measure.

The question arises: what figure can be used as a measure for comparing areas?

The teacher or the children themselves suggest using as measurements a triangle equal to half the area of the square M - M, or a rectangle equal to half the area of the square M - M or 1/4 the area of the square M . This can be a square M or a triangle M (Fig. 10).

Students place different measurements in rectangles and count the number of measurements in each.

So using the M1 measure, they get 20M1 and 10MG. Measuring with an M2 measure gives 40M2 and 36M2. Using the M3 measure - 20МЗ and 18МЗ. Measuring the rectangles with an M4 measure, we get 40M4 and 36M4.

In conclusion, the teacher may suggest measuring the area of one rectangle using the M1 measure, and the area of another rectangle (square) using the M2 measure.

As a result, it turns out that the area of the rectangle is 20, and the area of the square is 36.

“How is it,” says the teacher, “it turns out that there are fewer measurements in a rectangle than in a square? Maybe the conclusion we made earlier, that the area of a square is greater than the area of a rectangle, is incorrect?

The question posed helps to focus children's attention on the fact that to compare areas it is necessary to use a single yardstick. To understand this fact, the teacher can suggest laying out different figures from four squares on a flannelgraph or drawing them in a notebook, denoting the square with a cell (Fig. 11). After the task is completed, it is useful to find out;

How are the constructed figures similar? (they consist of four identical squares).

Can we say that the areas of all figures are the same? (children can check their answer by placing the squares of one figure on the squares of others).

Before introducing students to a unit of area, it is useful to conduct practical work associated with measuring the area of a given figure using various measures. For example, measuring the area of a rectangle with squares, we get the number 10; measuring with a rectangle consisting of two squares, we get the number 5. If the measure is 1/2 square, then we get 29, if 1/4 square, then we get 40. (Fig. 12)

Children notice that each subsequent measure consists of the two previous ones, that is, its area is 2 times larger than the area of the previous measure.

Hence the conclusion that by how many times the area of the measure has increased, the numerical value of the area of a given figure has increased by the same amount.

For this purpose, you can offer children such a situation. Three students measured the area of the same figure (the figure is first drawn in notebooks or on pieces of paper). As a result, each student received the first answer - 8, the second - 4, and the third - 2. The students guess that the result depends on the measure that the students used when measuring. Tasks of this type lead to the realization of the need to introduce a generally accepted unit of area -1 cm (a square with a side of 1 cm). The 1cm model is cut out of thick paper. Using this model, the areas of various figures are measured. In this case, students themselves will come to the conclusion that measuring the area of a figure means finding out how many square centimeters it contains.

By measuring the area of a figure using a model, schoolchildren are convinced that placing 1cm in a figure is inconvenient and time consuming. It is much more convenient to use a transparent plate on which a grid of square centimeters is applied. It's called a palette. The teacher introduces the rules for using the palette. It is superimposed on an arbitrary figure. The number of full square centimeters is calculated (let it be equal to a). Then the number of partial square centimeters is calculated (let it be equal to b) and divided by 2.(a+b):2. The area of the figure is approximately equal to (a + b): 2 cm. By placing the palette on a rectangle, children can easily find its area. To do this, count the number of square centimeters in one row, then count the number of rows and multiply the resulting numbers: a b (cm). Measuring the length and width of the rectangle with a ruler, students notice or the teacher draws their attention to the fact that the number of squares that fit along the length is the numerical value of the length of the rectangle, and the number of lines coincides with the numerical value of the width.

After students have verified this experimentally on several rectangles, the teacher can introduce them to the rule for calculating the area of a rectangle: to calculate the area of a rectangle, you need to know its length and width and multiply these numbers. Subsequently, the rule is formulated more briefly: the area of a rectangle is equal to its length multiplied by its width. In this case, the length and width must be expressed in units of the same name.

At the same time, students begin to compare the area and perimeter of polygons so that children do not confuse these concepts, and in the future clearly distinguish between methods for finding the area and perimeter of polygons. While doing practical exercises with geometric shapes, children count the number of square centimeters and immediately calculate the perimeter of the polygon in centimeters.

Along with solving problems of finding the area of a rectangle given the length and width, they solve inverse problems of finding one of the sides, given the area and the other side.

Area is the product of numbers obtained by measuring the length and width of a rectangle, which means that finding one of the sides of the rectangle comes down to finding the unknown factor from the known product and factor. For example, area garden plot 100m, section length 25m. What is its width? (100:25=4)

Except simple tasks, composite problems are also solved, in which, along with the area, the perimeter is also included. For example: “The vegetable garden has the shape of a square, the perimeter of which is 320 m. What is the area of the vegetable garden?

1) 320:4=80(m) - length of the garden; 2) 80*80=1600 (m) - area of the garden. Volume of a figure and its measurement.

The mathematics program provides, along with the quantities discussed, an introduction to volume and its measurement using a liter. The volume of spatial geometric figures is also considered and such units of volume measurement as cubic centimeter and cubic decimeter, as well as their ratios, are studied. Methodology for studying time and its measurement. Time is the most difficult quantity to study. Temporal concepts in children develop slowly in the process of long-term observations, accumulation of life experience, and study of other quantities.

Temporal ideas in first-graders are formed primarily in the process of their practical (educational) activities: daily routine, keeping a nature calendar, perception of the sequence of events when reading fairy tales, short stories, when watching movies, daily entry work dates in notebooks - all this helps the child to see and understand changes in time, to feel the passage of time.

Starting from the first grade, it is necessary to begin comparing familiar time periods that are often encountered in children’s experience. For example, what lasts longer: a lesson or a break, a school term or winter break; What is shorter than a student’s school day at school or a parent’s working day? Such tasks help develop a sense of time. In the process of solving problems related to the concept of difference, children begin to compare the ages of people and gradually master important concepts: older - younger - same in age. For example, “My sister is 7 years old, and my brother is 2 years old.” older than sister. How old is your brother?" “Misha is 10 years old, and his sister is 3 years younger than him. How old is your sister?" (M1M “1-3”, p. 68, M2, 13-respectively, 1994) “Sveta is 7 years old, and her brother is 9 years old. How old will each of them be in 3 years?”

To understand the passage of time (M1M “1-3”. p. 84, No. 2, 1994). Familiarity with units of time helps to clarify children's time concepts. Knowledge of the quantitative relationships of time units helps to compare and evaluate the duration of periods of time expressed in certain units.

Using a calendar, students solve problems to find the duration of an event. For example, how many days is spring break? How many months last summer holidays? The teacher calls the beginning and end of the holidays, and the students count the number of days and months on the calendar. We need to show how to quickly calculate the number of days, knowing that there are 7 days in a week. Inverse problems are solved similarly.

Units of time that children become familiar with in elementary school: week, month, year, century, day, hour, minute, second.

A table of measures, which should be hung in the classroom for a while, helps to master the relationships between units of time, as well as systematic exercises in converting quantities expressed in units of time, comparing them, finding different fractions of any unit of time, and solving problems on calculating time.

In grade 3 (1-3), the simplest cases of addition and subtraction of quantities expressed in units of time are considered. The necessary conversions of time units are performed here along the way, without first replacing the given values. To prevent errors in calculations that are much more complex than calculations with quantities expressed in units of length and mass, it is recommended to give calculations in comparison:

30min 45sec - 20min58sec;

30m 45cm - 20m 58cm;

30c 45kg - 20c 58kg;

To develop time concepts, we use the solution of problems to calculate the duration of events, their beginning and end.

The simplest problems of calculating time within a year (month) are solved using a calendar, and within a day - using a clock model.

Methodology for studying mass and its measurement.

Children receive their first ideas that objects have mass in life before school. Conceptual ideas about mass come down to the properties of objects “to be lighter” and “to be heavier.”

In elementary school, students are introduced to units of mass: kilogram, gram, centner, ton. With a device with which the mass of objects is measured - scales. With the ratio of mass units.

At the stage of comparing homogeneous quantities, weighing exercises are performed: 1,2,3 kilograms of salt, cereals, etc. are weighed out. In the process of completing such tasks, children should actively participate in working with scales. Along the way, you get acquainted with the recording of the results obtained. Next, children get acquainted with a set of weights: 1kg, 2kg, 5kg and then begin to weigh several specially selected objects, the mass of which is expressed in whole kilograms. When studying the gram, quintal and ton, their relationships with the kilogram are established, and a table of mass units is compiled and memorized. Then they begin to transform quantities expressed in units of mass, replacing small units with large ones and vice versa. For example, the mass of an elephant is 5 tons. How many centners is this? kilograms? (M4M.1 -4, :, Education, 1989) Express in kilograms: 12t 96kg, 9385g, 68t, 52t 5 kg; in grams: 13kg 125g, 45kg 13g, 6ts, 18kg? (MZM 1 - Z.M:, Linka press, 1995)

They also compare the masses and perform arithmetic operations on them. For example, insert numbers into the “boxes” to get the correct equalities:

7t 2ts+4ts=_ts; 9t 8ts-6ts=_ts.

During these exercises, knowledge of the table of mass units is consolidated. In the process of solving simple and then composite problems, students establish and use the relationship between quantities: the mass of one object - the number of objects - the total mass of these objects; they learn to calculate each of the quantities if the numerical values of the other two are known.

Conclusion.

Quantities, as properties of objects, have one more feature - they can be assessed quantitatively. To do this, the value must be measured. Measurement consists of comparing a given quantity with a certain quantity of the same kind, taken as a unit.

Quantities that are completely determined by one numerical value are called scalar quantities. These, for example, are length, area, volume, mass and others. In addition to scalar quantities, vector quantities are also considered in mathematics. To determine a vector quantity, it is necessary to indicate 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.

Measuring quantities allows us to reduce their comparison to comparing numbers

Bibliography

Anipchenko Z.A.

Problems related to quantities and their application in mathematics courses in primary school. M.: 1997 pp.2-5

Alexandrov A.D.

Foundations of geometry. Ed. "SCIENCE" Novosibirsk, 1987

Vapnyar N.F., Pyshkalo A.M., Yankovskaya N.A.

Notebook on mathematics for 1st grade 1-3, 7th ed.-M.: PROSVSHCHENIE, 1983. p.17

Volkova S.I.

“Cards with mathematical tasks and games” for 2nd grade 1-4: A manual for teachers - M.: ENLIGHTENMENT, 1990. pp. 32-36

Lesson summary

Volume and its measurement

The volume of a figure is a non-negative quantity defined for each figure so that:

1) equal figures have the same volume;
2) if a figure is made up of a finite number of figures, then its volume is equal to the sum of their volumes.

Let us agree to denote the volume of the figure F as V(F).

If the measurement of the area was reduced to comparing the area of a given figure with the area of a unit square e, then, similarly, the measurement of the volume of a given figure consists of comparing it with the volume of a unit cube e 3. The result of this comparison is a number x such that V(F)=xe. The number x is called the numerical value of the volume for the selected unit of volume.

Modern approaches to the study of quantities in the initial course of mathematics

Quantities are considered in close connection with the study of natural numbers and fractions; learning to measure is associated with learning to count; Measuring and graphical operations on quantities are visual tools and are used in solving problems. When forming ideas about each of these quantities, it is advisable to focus on certain stages, which are reflected: the mathematical interpretation of the concept of quantity, the relationship of this concept with the study of other issues in the initial course of mathematics, as well as the psychological characteristics of younger schoolchildren.

N.B. Istomina, a mathematics teacher and author of one of the alternative programs, identified 8 stages in the study of quantities:

Stage 1: clarification and clarification of schoolchildren’s ideas about this quantity (referring to the child’s experience).
Stage 2: comparison of homogeneous quantities (visually, with the help of sensations, by imposition, by application, by using various measures).
Stage 3: familiarization with the unit of a given quantity and with the measuring device.
Stage 4: formation of measurement skills.
Stage 5: addition and subtraction of homogeneous quantities expressed in units of the same name.
Stage 6: familiarization with new units of quantities in close connection with the study of numbering and addition of numbers. Conversion of homogeneous quantities expressed in units of one denomination into quantities expressed in units of two denominations, and vice versa.
Stage 7: addition and subtraction of quantities expressed in units of two names.
Stage 8: multiplying and dividing quantities by number.

Developmental education programs provide for the consideration of basic quantities, their properties and relationships between them in order to show that numbers, their properties and the actions performed on them act as special cases of already known general patterns of quantities. The structure of this mathematics course is determined by considering the sequence of concepts: quantity > number.

Let's take a closer look at the methodology for studying length, area, mass, time, and volume.

In traditional elementary school, the study of quantities begins with the length of objects. Children have their first ideas about length as a property of objects long before school. From the first days of school, the task is to clarify children's spatial concepts. An important step in the formation of this concept is familiarity with a straight line and a segment as a “carrier” of linear extension, essentially devoid of other properties.

Then it is proposed to compare two objects of different colors and different in size (length) practically - by superposition. For example, students are asked to look at the pictures and answer the questions: “Which belt is shorter (longer), light or dark?” . Through these two exercises, children are led to understand length as a property that manifests itself in comparison, that is: if two objects coincide when superimposed, then they have the same length; if any of the compared objects overlaps part of the other without covering it completely, then the length of the first object is less than the length of the second object. After considering the lengths of the objects, they move on to studying the length of the segment.

Here the length acts as a property of the segment.

At the next stage, we become familiar with the first unit of measurement for segments. From a set of segments, a segment is selected that is taken as a unit. This is what a centimeter is. Children learn its name and begin to measure using this unit. In order for children to get a clear idea of the centimeter, they should perform a number of exercises. For example, it is useful for them to make a model of the centimeter themselves; Draw a line 1cm long in your notebook. They found that the width of the little finger is approximately 1 cm.

In order for students to better understand the relationship between number and quantity, that is, to understand that as a result of measurement they get a number that can be added and subtracted, it is useful to use the same ruler as a visual aid for addition and subtraction. For example, students are given a strip; You need to use a ruler to determine its length. The ruler is applied so that 0 coincides with the beginning of the strip, and its end coincides with the number 3 (if the length of the strip is 3 cm). Then the teacher asks questions: “And if you apply a ruler so that the beginning of the strip coincides with the number 2, what number on the ruler will the end of the strip coincide with? Why?". Some students immediately name the number 5, explaining that 2+3=5. Anyone who finds it difficult resorts to practical action, during which he strengthens his computational skills and acquires the ability to use a ruler for calculations. Similar exercises are possible with a ruler and on reverse action- subtraction. To do this, students first determine the length of the proposed strip, for example, 4 cm, and then the teacher asks: “If the end of the strip coincides with the number 9 on the ruler, then what number will the beginning of the strip coincide with?” (5; 9-4 = 5). To develop measurement skills, a system of various exercises is included. This is the measurement and drawing of segments; comparison of segments to answer the question: how many centimeters is one segment longer (shorter) than another segment; increasing and decreasing segments by several centimeters. During these exercises, students develop the concept of length as the number of centimeters that fit in a given segment. Later, when studying the numbering of numbers within 100, new units of measurement are introduced - the decimeter, and then the meter. The work proceeds in the same way as when getting acquainted with a centimeter. Then relationships between units of measurement are established. From this time on, they begin to compare lengths based on comparison of the corresponding segments.

The introduction of the millimeter is justified by the need to measure segments smaller than 1 centimeter.

When becoming familiar with the kilometer, it is useful to carry out practical exercises on the ground in order to form an understanding of this unit of measurement.

In grades 3-4, students compile and memorize a table of all the studied units of length and their relationships.

Work on this topic can be continued in extracurricular activities, for example, consider ancient Russian measures: verst, fathom, vershok. Introduce students to some information from the history of the development of the system of measures.

The method of working on the area of a figure has much in common with working on the length of a segment, that is, the work is carried out almost similarly.

Using these ideas, you can introduce children to the concept of “area” by choosing for this purpose two figures such that when superimposed on each other, one fits entirely into the other.

“In this case,” says the teacher, “in mathematics it is customary to say that the area of one figure is greater (smaller) than the area of another figure.” When the figures coincide when superimposed, then they say that their areas are equal or coincide. Students can draw this conclusion on their own. But it is also possible that one of the figures does not fit completely into the other. For example, two rectangles, one of which is a square. After unsuccessful attempts to fit one rectangle into another, the teacher rotates the shapes reverse side, and the children see that one figure contains 10 identical squares, and another 9 identical squares.

The students, together with the teacher, conclude that to compare areas, as well as to compare lengths, you can use a measure.

The question arises: what figure can be used as a measure for comparing areas?

Students place different measurements in rectangles and count the number of measurements in each.

So using the M1 measure, they get 20M1 and 10M1. Measuring with an M2 measure gives 40M2 and 36M2. Using the M3 measure - 20МЗ and 18МЗ. Measuring the rectangles with an M4 measure, we get 40M4 and 36M4.

In conclusion, the teacher may suggest measuring the area of one rectangle using the M1 measure, and the area of another rectangle (square) using the M2 measure.

As a result, it turns out that the area of the rectangle is 20, and the area of the square is 36.

The question posed helps to focus children's attention on the fact that to compare areas it is necessary to use a single yardstick. To understand this fact, the teacher can suggest laying out different figures from four squares on a flannelgraph or drawing them in a notebook, denoting the square with a cell. After the task is completed, it is useful to find out:

* how are the constructed figures similar? (they consist of four identical squares).
* can we say that the areas of all figures are the same? (children can check their answer by placing the squares of one figure on the squares of others).

Before introducing the unit of area to schoolchildren, it is useful to carry out practical work related to measuring the area of a given figure using various measures. For example, measuring the area of a rectangle with squares, we get the number 10; measuring with a rectangle consisting of two squares, we get the number 5. If the measure is equal to 1/2 of a square, then we get 29, if it’s 1/4 of a square, then we get 40.

Children notice that each subsequent measure consists of the two previous ones, that is, its area is 2 times larger than the area of the previous measure.

Hence the conclusion that by how many times the area of the measure has increased, the numerical value of the area of a given figure has increased by the same amount.

For this purpose, you can offer children such a situation. Three students measured the area of the same figure (the figure is first drawn in notebooks or on pieces of paper). As a result, each student received the first answer - 8, the second - 4, and the third - 2. Students guess that the result depends on the measure that the students used when measuring. Tasks of this type lead to the realization of the need to introduce a generally accepted unit of area - 1 cm (a square with a side of 1 cm). The 1cm model is cut out of thick paper. Using this model, the areas of various figures are measured. In this case, students themselves will come to the conclusion that measuring the area of a figure means finding out how many square centimeters it contains.

By measuring the area of a figure using a model, schoolchildren are convinced that placing 1cm in a figure is inconvenient and time consuming. It is much more convenient to use a transparent plate on which a grid of square centimeters is applied. It's called a palette. The teacher introduces the rules for using the palette. It is superimposed on an arbitrary figure. The number of full square centimeters is calculated (let it be equal to a). Then the number of incomplete square centimeters is calculated (let it be equal to b) and divided by 2. The area of the figure is approximately equal to (a + b) : 2 cm. By placing the palette on a rectangle, children can easily find its area. To do this, count the number of square centimeters in one row, then count the number of rows and multiply the resulting numbers: аЧb (cm). Measuring the length and width of the rectangle with a ruler, students notice or the teacher draws their attention to the fact that the number of squares that fit along the length is the numerical value of the length of the rectangle, and the number of lines coincides with the numerical value of the width.

Along with solving problems of finding the area of a rectangle given the length and width, they solve inverse problems of finding one of the sides, given the area and the other side.

Area is the product of numbers obtained by measuring the length and width of a rectangle, which means that finding one of the sides of the rectangle comes down to finding the unknown factor by famous work and a multiplier. For example, the area of a garden plot is 100 m, the length of the plot is 25 m. What is its width? (100:25=4)

In addition to simple problems, composite problems are also solved, in which, along with the area, the perimeter is also included. For example: “The vegetable garden has the shape of a square, the perimeter of which is 320 m. What is the area of the vegetable garden?

1) 320:4=80(m) - length of the vegetable garden; 2) 80*80=1600(m) - area of the vegetable garden. The volume of a figure and its measurement.

The mathematics program provides, along with the quantities discussed, an introduction to volume and its measurement using a liter. The volume of spatial geometric shapes and such units of volume measurement as cubic centimeter and cubic decimeter are studied, as well as their relationships. Methodology for studying time and its measurement. Time is the most difficult quantity to study. Temporal concepts in children develop slowly in the process of long-term observations, accumulation of life experience, and study of other quantities.

Temporal ideas in first-graders are formed primarily in the process of their practical (educational) activities: daily routine, keeping a nature calendar, perception of the sequence of events when reading fairy tales, stories, when watching movies, daily recording of work dates in notebooks - all this helps the child to see and understand changes in time, feel the passage of time.

Starting from the first grade, it is necessary to begin comparing familiar time periods that are often encountered in children’s experience. For example, what lasts longer: a lesson or a break, a school term or the winter vacation; What is shorter than a student’s school day at school or a parent’s working day? Such tasks help develop a sense of time. In the process of solving problems related to the concept of difference, children begin to compare the ages of people and gradually master important concepts: older - younger - same in age. For example, “My sister is 7 years old, and my brother is 2 years older than my sister. How old is your brother?" “Misha is 10 years old, and his sister is 3 years younger than him. How old is your sister?" “Sveta is 7 years old, and her brother is 9 years old. How old will each of them be in 3 years?” - awareness of the passage of time. Familiarity with units of time helps to clarify children's time concepts. Knowledge of the quantitative relationships of time units helps to compare and evaluate the duration of periods of time expressed in certain units.

Using a calendar, students solve problems to find the duration of an event. For example, how many days is spring break? How many months do summer holidays last? The teacher calls the beginning and end of the holidays, and the students count the number of days and months on the calendar. We need to show how to quickly calculate the number of days, knowing that there are 7 days in a week. Inverse problems are solved similarly.

30min 45sec - 20min58sec;
30m 45cm - 20m 58cm;
30c 45kg - 20c 58kg;

To develop time concepts, we use the solution of problems to calculate the duration of events, their beginning and end.

The simplest problems of calculating time within a year (month) are solved using a calendar, and within a day - using a clock model.

Children receive their first ideas that objects have mass in life before school. Conceptual ideas about mass come down to the properties of objects “to be lighter” and “to be heavier.”

In elementary school, students are introduced to units of mass: kilogram, gram, centner, ton. With a device with which the mass of objects is measured - scales. With the ratio of mass units.

At the stage of comparing homogeneous quantities, weighing exercises are performed: 1, 2, 3 kilograms of salt, cereals, etc. are weighed out. In the process of completing such tasks, children should actively participate in working with scales. Along the way, you get acquainted with the recording of the results obtained. Next, children get acquainted with a set of weights: 1kg, 2kg, 5kg and then begin to weigh several specially selected objects, the mass of which is expressed in whole kilograms. When studying the gram, quintal and ton, their relationships with the kilogram are established, and a table of mass units is compiled and memorized. Then they begin to transform quantities expressed in units of mass, replacing small units with large ones and vice versa. For example, the mass of an elephant is 5 tons. How many centners is this? kilograms? Express in kilograms: 12t 96kg, 9385g, 68t, 52t 5 kg; in grams: 13kg 125g, 45kg 13g, 6ts, 18kg?

They also compare the masses and perform arithmetic operations on them. For example, insert the numbers into the “boxes” to get the correct equalities:

7t 2ts+4ts=_ts; 9t 8ts-6ts=_ts.

On this topic:

“Organization of design and research activities in 1st grade when studying the topic “Quantities and their measures”

Completed by: primary school teacher

MKOU Anoshkinskaya secondary school

Liskinsky district,

Voronezh region

Smorodinova Larisa Vasilievna,

Introduction

Relevance .

Project research activities of students are prescribed in the education standard. Therefore, every student should be trained in this activity.

It is becoming increasingly relevant in modern pedagogy. And this is no coincidence. After all, it is in the process of proper independent work on creating a project that the culture of mental work of students is best formed.

A child is born a researcher. The thirst for new experiences, curiosity, the desire to observe and experiment, to independently seek new information about the world is the normal, natural state of a child. It is this internal desire for knowledge through research that gives rise to research behavior and creates the conditions for research learning. The basis for the formation of the basicsResearch culture is precisely the primary school.

Currently, high demands are placed on the level of knowledge of students, which are necessary for successful adaptation in society. To do this, it is necessary to move away from the classical formation of knowledge, skills and abilities and give primacy to creative teaching methods, where research activities occupy a special place. It is in primary school that the foundation of knowledge, skills and abilities of active, creative, independent activity of students, methods of analysis, synthesis and evaluation of the results of their activities and research work should be laid - one of the most important ways in solving this problem. Purpose of use research work is to stimulate the development of the intellectual and creative potential of primary schoolchildren through the development and improvement of research abilities and research behavior skills. Accordingly, the main tasks can be identified: training in conducting educational research for junior schoolchildren already in the first grade;

development of creative research activity of children;

stimulating children's interest in fundamental and applied sciences - familiarization with scientific picture peace.

The problem of choosing the necessary method of work has always arisen for teachers. But in new conditions, new methods are needed that allow us to organize the learning process and the relationship between teacher and student in a new way. How to organize learning through desire? How to activate a student, stimulate his natural curiosity, motivate his interest in independently acquiring new knowledge? We need activity-based, group, game, role-playing, practice-oriented, problem-based reflective and other forms and methods of teaching. The project method is not fundamentally new in world pedagogy. It was proposed and developed in the 1920s by the American philosopher and educator J. Dewey on the basis of humanistic ideas and developed by his student J. Dewey proposed teaching on an active basis, using the purposeful activities of students, taking into account their personal interest in knowledge and ultimately obtaining real results .

In Russia, the ideas of project-based learning arose almost at the same time. Already in 1905 Russian S.T. Shchatsky and a small group of colleagues (A.G. Avtukhov, P.P. Blonsky, B.V. Vsesvyatsky, Sh.I. Ganelin, V.F. Natali) tried to actively use project methods in teaching practice. After October revolution their ideas and work experience began to be widely introduced into school practice, but not sufficiently thought out and consistently, and in 1931, by a resolution of the Central Committee of the All-Union Communist Party of Bolsheviks, the project method was condemned, and its use in the work of a teacher was prohibited. At the same time, in foreign practice it developed very successfully and gained popularity. At present, when your country also needs qualitatively new characteristics educational systems, emphasis is placed on students’ mastering the values and methods of human activity in the sociocultural environment; project methods are again in demand and popular.

The purpose of this project– create conditions for individualized and productive, creative self-learning of students based on the use of modern technologies training.

Object of study– research activities of students when studying the topic “Quantities and their measures”

Subject of study– organization of design and research activities in 1st grade with the aim of mastering the material in the process of discovering “new things”, as a way of personal achievement of goals and opportunities.

Research hypothesisincludes the assumption that the use of modern technologies of design and research teaching activities contributes to the independent acquisition of new information, organic skills of analytical and creative attitude.

Chapter 1. The concept of quantity and its measurement in the initial course of mathematics.

Length, area, mass, time, volume - quantities. Initial acquaintance with them occurs in elementary school, where quantity, along with number, is a leading concept.

QUANTITY is a special property of real objects or phenomena, and the peculiarity is that this property can be measured, that is, the number of quantities that express the same property of objects are called quantities of the same kind or homogeneous quantities. For example, the length of the table and the length of the rooms are homogeneous

quantities.

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

Length of a segment and its measurement.

The length of a segment is a positive quantity defined for each segment so that:

1/ equal segments have different lengths;

2/ if a segment consists of a finite number of segments, then its length is equal to the sum of the lengths of these segments.

Area of a figure and its measurement.

Any person has the concept of the area of a figure: we are talking about the area of a room, the area of a plot of land, the area of a surface that needs to be painted, and so on. At the same time, we understand that if land are the same, then their areas are equal; that a larger plot has a larger area; that the area of an apartment is made up of the area of the rooms and the area of its other premises.

The area of a figure is a non-negative quantity

defined for each figure so that:

I/ equal figures have equal areas;

2/ if a figure is made up of a finite number of figures, then its area is equal to the sum of their areas.

Mass and its measurement.

Let's imagine that some body is placed on one of the cups of a lever scale, and a second body b is placed on the other cup. In this case, the following cases are possible:

1) The second pan of the scales fell, and the first one rose so that they ended up on the same level. In this case, the scales are said to be in

equilibrium, and bodies a and b have equal masses.

2) The second pan of the scale remained higher than the first. In this case, we say that the mass of body a is greater than the mass of body b.

3) The second cup fell, and the first rose and stands higher than the second. In that

In this case, we say that the mass of body a is less than body b.

From a mathematical point of view, mass is a positive quantity

which has the following properties:

1) The mass is the same for bodies balancing each other on scales;

2) Mass adds up when bodies are connected together: the mass of several bodies taken together is equal to the sum of their masses. If we compare this definition with

definitions of length and area, we see that mass is characterized by the same properties as length and area, but is defined on a set of physical bodies.

Mass is measured using scales. This happens as follows. Select a body e whose mass is taken as unity.

It is assumed that it is possible to take fractions of this mass. For example, if a kilogram is taken as a unit of mass, then in the measurement process you can use its fraction as a gram: 1 g = 0.01 kg.

The basic unit of mass is the kilogram. From this basic unit other units of mass are formed: gram, ton and others.

Time intervals and their measurement.

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. Thus, a lecture at an institute lasts the same amount of time as two lessons at school.

In Ancient Rus', the week was called a week, and Sunday was a weekday (when there is no work) or simply a week, i.e. a day of rest. The names of the next five days of the week indicate how many days have passed since Sunday. Monday - immediately after the week, Tuesday - the second day, Wednesday - the middle, the fourth and fifth days, respectively, Thursday and Friday, Saturday - the end of things. A month is not a very specific unit of time; it can consist of thirty-one days, thirty and twenty-eight, twenty-nine in leap years (days). But this unit of time has existed since ancient times and is associated with the movement of the Moon around the Earth. The Moon makes one revolution around the Earth in about 29.5 days, and in a year it makes about 12 revolutions. These data served as the basis for the creation of ancient calendars, and the result of their centuries-long improvement is the calendar that we use today. Since the Moon makes 12 revolutions around the Earth, people began to count the full number of revolutions (that is, 22) per year, that is, a year is 12 months.

invented by Babylonian scientists.

Volume and its measurement.

Chapter 2. Methodology of design and research activities

What is a learning project? In modern pedagogy, the project method is used as a component of the education system. The project methods are based on:

development of students' cognitive skills,
the ability to independently construct one’s knowledge,
ability to navigate the information space,
analyze the information received,
independently put forward hypotheses,
decision making skills,
development of critical thinking,
skills of research and creative activity.

This approach fits seamlessly with a group approach to learning.

From the point of view of students, an educational project is an opportunity to do something interesting on their own, in a group or on their own, making the most of their knowledge and capabilities; This is an activity that allows you to demonstrate, try your strength, apply your knowledge, bring benefit and show publicly achieved result; This is an activity aimed at solving an interesting problem, when the found method of solving the problem is practical in nature and has applied significance.

An educational project from a teacher’s point of view is didactic tool, allowing you to teach design, that is, purposeful activity to find a way to solve a problem.

All activities in an educational project are subject to a certain logic, which is implemented in the sequence of its stages. Following the presentation of the project (title, topic and problem), students must independently formulate goals and objectives, organize groups, distribute roles in groups, then select methods, plan work and implement it. Implementation ends educational project presentation of the results obtained. Since students’ activities in the project are mostly independent, it is during the presentation that students imagine what was done during independent project work. Thus, the educational project method is one of the student-oriented technologies, a way of organizing students’ independent activities aimed at solving the problem of an educational project.

This method integrates:

problematic approach
group methods,
research methods,
search techniques.

The project method is a wonderful didactic tool for teaching design - the ability to find solutions to various problems that constantly arise in the life of a person who takes an active position in life.It allows you to cultivate an independent and responsible personality, develops creative beginnings And mental capacity– necessary qualities of developed intelligence.

When implementing a project, I distinguish the following stages:

immersion in the project;
organization of activities;
carrying out activities;
presentation of results.

The project-based teaching method involves the development process of creating a project (prototype, prototype, supposed or possible object or state). The research method of teaching involves organizing the process of developing new knowledge. Fundamental difference research from design is that the research does not involve the creation of any pre-planned object, even its model or prototype. Research, in essence, is the process of searching for the unknown, new knowledge, one of the types cognitive activity. Thus, as noted by A.I. Savenkov, “design and research are initially fundamentally different types of activity in focus, meaning and content. Research is a disinterested search for truth, and design is the solution to a specific, clearly understood problem.”

At the same time, both the project method and the research method are based on:

Development of cognitive skills of students;

Ability to navigate the information space;

The ability to independently construct your knowledge;

Ability to integrate knowledge from various areas sciences;

Ability to think critically.

Both methods are always focused on independent activity students (individual, pair, group), which they perform in the time allotted for this work (from a few minutes of a lesson to several weeks, and sometimes months). This is the task of personal orientation of pedagogy. Project technology and technology of research activities assume:

The presence of a problem that requires integrated knowledge and a research search for its solution;

Practical, theoretical, cognitive significance of the expected results;

Independent activity of the student;

Structuring the content of the project indicating phased results;

Using research methods, that is, defining the problem and the resulting research objectives;

Discussion of research methods, collection of information, presentation of final results; presentation of the resulting product, discussion and conclusions.

The use of these methods involves a departure from the authoritarian teaching style, but at the same time provides for a well-thought-out, reasonable combination of methods, formats and teaching aids.

And for this the teacher needs:

Possess an arsenal of research and search methods, be able to organize research independent work students;

Be able to organize and conduct discussions without imposing your point of view, without suppressing students with your authority;

Establish and support in groups working on a business project, emotional mood, directing students to find a solution to the problem;

Be able to integrate the content of various subjects to solve problems of selected projects.

Research is a selfless search for truth, always creativity. Research activities should initially be free, practically not regulated by any external settings. In the practice of working with primary schoolchildren, individual and collective games are often used. Each game study consists of two stages: training sessions and independent research. Work on projects and children's research is quite complex, so it is necessary to prepare elementary school students gradually.

Tasks of design and research activities.

Educational:activation and updating of knowledge acquired by schoolchildren while studying a certain topic. Systematization of knowledge.

Acquaintance with a complex of materials that are obviously beyond the scope of the school curriculum.

Developmental: developing the ability to think in the context of the topic being studied, analyze, compare, and draw your own conclusions; select and systematize material, abstract it; use ICT when reporting the results of the research.

Educational : creating a product that is in demand by others.

In elementary school, those skills are laid that will allow students to become the subject of their own activities; they develop the ability to independently obtain information from various sources, organize their activities, and successfully communicate with peers and adults. The teacher's task is to organize educational process so that general educational skills become the basis for acquiring knowledge. To solve this problem, it is necessary to use the project-research method in the classroom and organize research activities outside of class time.

Of course, primary school age imposes natural restrictions on the organization of project activities, but it is imperative to begin to involve primary school students in project activities. The fact is that it is precisely at primary school age that a number of value systems are laid down, personal qualities and relationships. If this circumstance is not taken into account, if this age is considered to be of little significance for the project method, then the continuity between the stages of development of educational and cognitive activity of students is disrupted and a significant part of schoolchildren subsequently fail to achieve the desired results in project activities.

Inclusion of project activities in the work of younger schoolchildren.

It is necessary to include schoolchildren in project activities gradually, starting from the first grade. In the beginning, these are affordable creative tasks, performed during lessons in the form of collective creative activities carried out outside of class time. And already in grades 3-4, students carry out quite complex projects with great interest; under the guidance of a teacher, they conduct collective Scientific research, which can include the results of each student’s design and research work.

Children's themes design work better to choose from content educational subjects or from areas close to them. The fact is that the project requires a personally significant problem that is familiar to primary schoolchildren and meaningful to them.

The problem of the project, which provides motivation for schoolchildren to engage in independent work, should be in the area of students’ cognitive interests and be in their zone of proximal development.

It is advisable to limit the duration of the project in the mode of classroom and extracurricular tasks to one lesson (in 1st grade), one or two weeks (in 2nd grade) and gradually move on to long-term projects designed for a month, quarter, half a year.

When involving parents in this work, it is important that they do not take on part of the children’s work on projects, otherwise the very idea of the project method will be ruined. But help with advice, information, expression of interest on the part of parents - important factor supporting motivation and ensuring the independence of schoolchildren when they carry out project activities. Thus, project activities primary school students is necessary and possible. Method creative projects along with others active methods education is the basis for organizing research activities of junior schoolchildren.

Algorithm of project activities.

stage. Providing a project theme. Topics for children's projects are selected based on the content of academic subjects or areas related to them.
stage. Selecting a problem.

At this stage, children answer the question: “What do we want to know?”

When discussing a problem round table children offer their own solutions.

stage. Formulation of subtopics.

At this stage, children identify all the subtopics that will be included in the problem-solving plan. Individual consultations are provided.

stage. Work planning. The ways to find the necessary information are determined.
stage. Implementation of the project.

At this stage, you can ask students questions: “Do you know everything to complete this project. What information do you need to obtain? What sources should you turn to?” The teacher must show tact and delicacy so as not to impose information on students, but to direct their independent search. Children turn to additional literature(dictionaries, encyclopedias, reference books, etc.), ask your parents for help. I would like to note that such work brings closer not only the teacher and the child, but also the child and the parents. And when a child sees that his parents share his interests with him, he is doubly happy and his desire to create a more interesting project increases.

stage. Project presentation.

This stage requires special attention. The results of research activities are demonstrated.

To successfully defend a project, it is imperative to help students self-assess the project. To do this, you can suggest answering next questions: does the idea you choose meet the initially put forward requirements; How did outsiders evaluate your work?

stage. Project evaluation.

A very important issue is the evaluation of completed projects. The teacher’s task at this stage is to prevent the presentation of projects from being awarded places in the competition.

3. Implementation of the project method, research activities.

In the 1st grade mathematics course, children are introduced to various quantities: length, mass, volume. When forming ideas about each of these quantities, it is advisable to focus on certain stages, which are reflected: the mathematical interpretation of this concept, its relationship with the study of other issues in the initial course of mathematics.

Stages of research activity:

1st stage. Clarification and clarification of schoolchildren’s ideas about this quantity (referring to the child’s experience).

Stage 2. Comparison of homogeneous quantities (visually, with the help of sensations, by imposition, by application, by using different measures).

3rd stage. Familiarity with the unit of a given quantity and the measuring device.

4th stage. Formation of measurement skills.

Stage 5. Addition and subtraction of homogeneous quantities expressed in units of the same name.

6th stage. Acquaintance with new units of quantities in close connection with the study of numbering and addition of numbers. Conversion of homogeneous quantities expressed in units of one denomination into quantities expressed in units of two denominations, and vice versa.

7th stage. Addition and subtraction of quantities expressed in units of two names.

So, let’s consider how students’ design and research activities are included at certain stages of concept formation.

Length and units of length.

Stage 1. Available to the child life experience allows him to realize the practical significance of the concept being studied, connect it with real objects and phenomena, and translate existing everyday concepts into the language of mathematics. Children are still in preschool age meet with the need in certain situations to compare real objects with each other, according to specific characteristics. Arriving at school, they already have the idea that two different subjects can be in some ways the same, interchangeable, and in some ways different.

Among all the characteristics of real objects that have certain properties, those are distinguished in relation to which (in the case when the objects are not the same) it is possible to introduce the relations “more”, “less”. If two strips are not the same length, then one is longer than the other.

Number the trees by height, starting with the tallest tree.
Color the most tall tree V green color, the lowest is brown, and the rest are yellow.
Color the largest flower red, the smallest flower blue, and the remaining flowers yellow.

Stage 2. The basis of the student’s activity at the stage of comparing quantities is the practical actions he performs in various game situations.

The following tasks can be offered:

Compare:

capital height and lowercase letters in your math textbook;

The length and width of the notebook and textbook;

The length of the school board and pointer;

According to the height of children from the class;

Length of pen and pencil.

Stage 3. Big role In children’s awareness of the measurement process, various situations of a problematic nature play a role, which activate the cognitive activity of students. The explanation should take place in an atmosphere of live search, trials, suggestions.

At this stage, the following problem situations can be proposed.

For example, two strips (90 cm and 60 cm) are attached to the board. The teacher asks the students: “Which strip do you think is longer?” " Students can make a correct guess, but it must be justified. At first they suggest a method of action known to them, but the teacher sets a condition: the strips cannot be removed. When looking for a new way of doing things, students can suggest using pencils, pens, strings, etc. for this purpose. The teacher invites them to use planks of various colors and sizes to justify their answer: red - 30 cm; blue - 15 cm. By laying the red bar along the length of the first strip, students, without yet realizing it, take a measurement. As a result of measuring the first strip, they get the number 3, and the second - 2, and independently come to the conclusion that the length of the first strip is greater than the second. “And now I’ll try myself, using strips (measurements), which strip is longer,” says the teacher. The students carefully monitor his actions (the teacher does not accompany them with any explanations). He takes the red plank (30 cm) and lays it along the length of the strip 90 cm (receives the number 3), then takes the blue plank (15 cm) and lays it along the length of the strip 60 cm (receives the number 4).

“It turned out that 3

Practical work.

A strip and two measurements are placed on each desk: one red, the other blue. One student measures the strip with a red measure, the other with a blue one. Different numerical values are obtained. This allows the teacher to ask problematic issue: “Can it be like this: the same strip was measured, but the numbers turned out different. What's the matter?"

A stripe is drawn on checkered paper. The teacher offers a situation: “Three students measured this strip, one got the number 8, the other - 4, and the third -2. Which one is right?

As a result of practical activities, students themselves draw a conclusion about the need to introduce a unit of length.

The children are very interested in the situation from the cartoon, when they measured the length of a boa constrictor (with parrots, monkeys, elephants), but could not decide how long it was.

Familiarity with each new unit of length is also associated with practical actions by schoolchildren. For example, when introducing a new unit of measurement - the decimeter - the teacher organizes the study of the material so that the children first of all understand its necessity. For this purpose, we can again return to comparing the lengths of two strips, for example 30 and 40 cm; Having offered students strips of 1 cm and 1 dm (you don’t have to tell the length of these strips at first), ask the question: “Which measure is more convenient to use to measure these strips?” Students find out in practice that using a 1 cm measure is inconvenient: it takes a lot of time. Using the second measure allows you to complete the task much faster. The teacher reports that the length of the second measure is 10 cm and is called a decimeter. After which the students find 1 dm on the ruler.

You can ask questions

1) The beginning of the segment coincides with the number 3 on the ruler. What number will be on the ruler at the end of a segment 1 dm long? (13, because 1 dm = 10 cm, 3 + 10 = 13)

2) The end of the segment coincides with the number 17 on the ruler. What number on the ruler coincides with the beginning of this segment if its length is 1 dm? (With the number 7, because 17 – 10 = 7)

3) How long can the segments be added to obtain a segment equal to 1 dm?

Stage 4. This stage offers the measurement of segments, the lengths of which can be designated by a number expressed in units of two names.

According to the traditional program with the value “capacity” and its

Students are introduced to the liter measure in 1st grade. At the stage of comparing homogeneous quantities, it can be suggested to first compare the containers by eye. For example, compare a saucepan and a mug, which are significantly different from each other.

Where can more water fit? (In a saucepan.) Then you can measure how many cups of water will go into the saucepan.

The following situation.Two water vessels are provided. One is narrow, the other is wide. The water level in both vessels is the same. In addition, on the teacher’s table there are two glasses of different capacities (let’s call them No. 1 and No. 2).

Using measure No. 1, find out which container has more water.

How many measures can fit in a wide container? (7.)

How many measures fit in a narrow container? (5.)

What can be concluded? (7>5. This means that there is more water in a wide vessel than in a narrow vessel.)

Using measure No. 2, find out which container has more water.

How many measures can fit in a wide container? (4.)

How many measures fit in a narrow container? (2.)

What can be concluded? (4>2. This means there is more water in a wide vessel.)

Is this a property of blood vessels? (Yes.)

Who knows what this property is called? (Capacity.)

What topic will we work on today? (Capacity.)

Then the teacher suggests measuring the amount of water in a wide vessel with measure No. 2, and in a narrow vessel with measure No. 1.

You can have a conversation like this:

How many measures No. 2 fit in a wide vessel? (4 measurements.)

How many measures No. 1 fit in a narrow vessel? (5 measures.)

What can be concluded? (4

Has the capacity of our vessels changed? (No.)

But, we said that the capacity of a wide vessel is greater than that of a narrow one. Is there a mistake in our reasoning? (Yes. We measured with different containers.)

Children come to the conclusion that a general measure is needed.

At the stage of getting acquainted with the unit of measurement of capacity, you can use the following situation. There are two vessels on the teacher’s table: one wide, the other narrow. One and the other are filled with water. The water level in a narrow vessel is higher than in a wide vessel.

The teacher asks a question:

Which vessel contains more water?

Children's opinions differ.

Was there one question?

How many opinions?

So what don't we know yet, what is the question?

What do you need to do to complete this task?

After the first situation has been dealt with, the students themselves will suggest using the third vessel for this purpose; it will perform the functions of a measure.

What can be concluded? (To make sure which container is larger (where there is more water), you need to use a measure.)

There is a generally accepted standard for measuring capacitance. Any guesses as to what it's called?

At this stage, it is possible not to communicate knowledge to finished form, but rely on the child’s experience. For example, show a juice box and ask:

How much juice does a box hold?

Perhaps one of the children will answer this question.

Weight. Kilogram. (lesson - project)

Immersion in the project:

Two identical boxes (same shape, size, color) are offered for comparison, but one is empty, the other is filled with clothespins.

First it is demonstrated at a distance. Children do not notice any differences. Then the children pick it up and discover the differences: one box is lighter and the other is heavier.

Organization of activities.

Which box is lighter? Which one is heavier? Does it make a difference to be lighter or heavier than another object? (Yes, this is a sign).

Both boxes are placed on the pans of the scales, balanced before this and the pans are moved.

Why did one cup go down and the other go up? (Because one box is heavier and the other is lighter).

Carrying out activities:

Exercise 1.

And again our friends are in trouble. Mom gave Petya an apple, Vova a pear, Katya a lemon, and Lena a strawberry. They just can’t decide whose object is the heaviest. What do you need to do to find out whose item is the heaviest? (Children's answers: Weigh). That's right, they need to be weighed and then we will know which object is heavy. What do we need for this? (Scales).

Weigh the apple and lemon. Does Petya or Katya have the hardest subject? (They weigh the same). Please tell me, does Petya or Vova have the heaviest subject? (Vova has the heaviest item). Now we will weigh Petit’s apple and Lena’s strawberry. Are Petit's apple and Lena's strawberry the same in weight? Whose object is heavier? (They are different. Petit's apple is heavier than Lena's strawberry).

Lighter - heavier - is this a property (attribute) of objects? (Children's answers).

We have become acquainted with a new property called mass.

Task 2.

Petya put an apple on one scale and strawberries on the other scale. Help Petya express the mass of an apple in strawberries. (One apple is equal to 5 strawberries).

Mass can be measured and the measurement result can be written down using a number. Mass is a quantity.

Task 3.

Katya the sweet tooth came to see us. Katya decided to measure the mass of the apple in the chocolate bars. How many chocolates did she use so that the mass of objects was the same? (She used two chocolates). How do we write in our notebooks? (An apple is equal to two chocolates).

It turns out that 5 strawberries = 2 chocolates? But 5 > 2. Is there an error here? (No. Because strawberry is a measure, and chocolate is also a measure).

Task 4.

Vova asks you to compare the mass of a melon and the mass of a bag of rice. What measure is used to measure the mass of a melon? (Measure – apple). What is the mass of a melon? (5 apples).

What is the mass of a bag of rice? (Weight of a bag of rice is expressed in bananas). What is the mass of a bag of rice? (The mass of a bag of rice is equal to 5 bananas). Is it possible to complete Vova's task? Why? (We cannot complete Vova’s task, since different measurements are used). What needs to be done so that this task can be completed? (You need to measure the mass with one measure).

Right. To complete Vova’s task, we need a measure that is the same for everyone. And such a measure exists. It is called 1 kilogram - this is one of the measures of mass. The number we get when measuring mass is a measure of mass.

Even in ancient times, people came to the conclusion that in order to correctly compare various items, we need a single measure of weight. And then they came up with an idea. We still use it today. There were other measures, but they are not currently used.

One kilogram of sugar is equal to one kilogram of salt. Since one measurement is used. One kilogram of sugar and the same amount of salt.

What conclusion can we draw with you? (Measures of masses measured by the same standards can be compared, added and subtracted).

Presentation of results:

Drawings illustrating the weighing of objects with different or equal weights.

Conclusion.

Everyone knows the truth - children love to learn, but one word is often left out: children love to study well! One of the powerful levers of the desire and ability to learn is the creation of conditions that ensure the child’s success in educational work, a feeling of joy on the path of progress from ignorance to knowledge, from inability to skill, i.e. awareness of the meaning and results of one’s efforts. Finding ways to enhance the cognitive activity of students is a task that educators, psychologists, methodologists and teachers are called upon to solve.

Design and research activities can be successfully implemented by students starting from the 1st grade, provided that a task that is feasible and interesting for children is set, as well as competent organization their work. Research activities help diversify children's activities in the classroom, maintain interest in mathematics and, most importantly, help them master the ability to solve assigned problems.

By defending design and research work, students become familiar with the basics oratory, gain experience public speaking, listen to their peers - all this activates cognitive interest students, helps to increase their intellectual level and creative potential.

Thus, design and research activities make it possible to reveal individual abilities younger children school age and gives them the opportunity to apply their knowledge, bring value and publicly demonstrate the results achieved. Usage research method in the practice of teaching and organizing the learning process of a primary school student has great importance, as it allows for the search orientation of students aimed at creative development

Literature:

1. Anipchenko Z.A. Problems related to quantities and their application in mathematics courses in primary school. M.: 1997 pp.2-5

2. “Organization of design and research activities in elementary school.” Materials of the regional scientific and practical seminar. Voronezh, 2011 pp.67-68, 183

3. Krom V.I. Activation of cognitive activity in mathematics lessons // Primary school - 1999 - No. 8 p. 27.

Quantities and their measurements in elementary school. Using multi-level tasks when studying the topic “Quantities” in elementary school. Percentage

1. The concept of magnitude

Volume and its measurement

Modern approaches to the study of quantities in the initial course of mathematics

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