In this lesson, we will look at units of length, area, and a table of area units. Consider the various units of measurement for length and area, find out in which cases they are used. We systematize our knowledge using a table. Let's solve a number of examples for converting one unit of measure into another.

You are familiar with the various units of length. What units of length are convenient to use when measuring the thickness of a match or the length of a ladybug's body? I think you said millimeters.

What unit of length is convenient to use when measuring the length of a pencil? Of course, in centimeters (see Fig. 1).

Rice. 1. Length measurement

What units of length are convenient to use when measuring the width or length of a window? It is convenient to measure in decimeters.

And the length of the corridor or the length of the fence? Let's use meters (see Fig. 2).

Rice. 2. Length measurement

To measure larger distances, for example, distances between cities, a larger unit of length than a meter is used - a kilometer (see Fig. 3).

Rice. 3. Length measurement

There are 1000 meters in 1 kilometer.

Express the distance in kilometers.

1 kilometer is a thousand meters, so the number of thousands will mean kilometers.

8000 m = 8 km

385007 m = 385 km 7 m

34125 m = 34 km 125 m

In the number of hundreds, tens and units indicate meters.

You can argue differently: 1 km is a thousand times more than 1 meter, which means that the number of kilometers should be 1000 times less than the number of meters. Therefore, 8000: 1000 = 8, the number 8 means the number of kilometers.

385007: 1000 = 385 (rest 7). The number 385 denotes kilometers, the remainder is the number of meters.

34125: 1000 = 34 (rest. 125), that is, 34 kilometers 125 meters.

Read the table of units of length (see Fig. 4). Try to remember it.

Rice. 4. Table of units of length

Different measurements are used to measure areas. A square centimeter is a square with a side of 1 cm (see fig. 5), a square decimeter is a square with a side of 1 dm (see fig. 6), a square meter is a square with a side of 1 m (see fig. .7).

Fig.5. square centimeter

Rice. 6. Square decimeter

Rice. 7. Square meter

To measure large areas, a square kilometer is used - this is a square whose side is 1 km (see Fig. 8).

Rice. 8. Square kilometer

The words "square kilometer" are abbreviated with the number as follows - 1 km 2, 3 km 2, 12 km 2. In square kilometers, for example, the areas of cities are measured, the area of ​​\u200b\u200bMoscow S \u003d 1091 km 2.

Calculate how many square meters are in one square kilometer. To find the area of ​​a square, multiply the length by the width. We are given a square with a side of 1 km. We know that 1 km \u003d 1000 m, so to find the area of ​​​​such a square, we multiply 1000 m by 1000 m, we get 1,000,000 m 2 \u003d 1 km 2.

Express in square meters 2 km 2. We will argue as follows: since 1 km 2 is 1,000,000 m 2, that is, the number of square meters is a million times greater than the number of square kilometers, so we multiply 2 by 1,000,000, we get 2,000,000 m 2.

56 km 2: multiply 56 by 1,000,000, we get 56,000,000 m 2.

202 km 2 15 m 2: 202 ∙ 1,000,000 + 15 = 202,000,000 m 2 + 15 m 2 = 202,000,015 m 2.

To measure small areas, a square millimeter (mm 2) is used. This is a square whose side is 1 mm. The words "square millimeter" with a number are written as follows: 1 mm 2, 7 mm 2, 31 mm 2.

Calculate how many square millimeters are in one square centimeter. To find the area of ​​a square, multiply the length by the width. We are given a square with a side of 1 cm. We know that 1 cm = 10 mm. So, to find the area of ​​such a square, we multiply 10 mm by 10 mm, we get 100 mm 2.

Express in square millimeters 4 cm 2. We will argue as follows: since 1 cm 2 is 100 mm 2, that is, the number mm 2 is 100 times greater than the number cm 2, so we multiply 4 by 100, we get 400 mm 2.

16 cm 2: multiply 16 by 100 \u003d 1600 mm 2.

31 cm 2 7 mm 2: this is 31 ∙ 100 + 7 = 3100 + 7 = 3107 mm 2.

In life, such units of area as ar and hectare are often used. Ap is a square with a side of 10 m (see Fig. 9). With numbers ap, they write shorter: 1 a, 5 a, 12 a.

Rice. 9. 1 ar

1 a \u003d 100 m 2, therefore it is often called a hundred.

A hectare is a square with a side of 100 m (see Fig. 10). The word "hectare" with numbers is abbreviated as follows: 1 ha, 6 ha, 23 ha. 1 ha \u003d 10000 m 2.

Rice. 10. 1 hectare

Calculate how many ares are in 1 hectare.

1 ha \u003d 10000 m 2

1 a \u003d 100 m 2, then 10000: 100 \u003d 100 a

Now carefully consider the table of area units (see Fig. 11), try to remember it.

Rice. 11. Table of units of area

In the lesson, we got acquainted with a new unit of length - km and units of area - m 2, km 2, a, ha.

1. Bashmakov M.I. Nefedova M.G. Mathematics. 4th grade. M.: Astrel, 2009.
2. M. I. Moro, M. A. Bantova, G. V. Beltyukova and others. Mathematics. 4th grade. Part 1 of 2, 2011.
3. Demidova T. E. Kozlova S. A. Tonkikh A. P. Mathematics. 4th grade 2nd ed., corrected. - M.: Balass, 2013.

1. School.xvatit.com().
2. Mer.kakras.ru ().
3. dpva.info().

Homework

1. Find the area of ​​a square with a side of 15 cm.
2. Express: in square meters: 5 ha; 3 ha 18 a; 247 acres; 16 a;
3. in hectares: 420,000 m 2; 45 km 2 19 ha;
4. in ares: 43 ha; 4 ha 5 a; 30 700 m 2; 5 km2 13 ha;
5. in hectares and ares: 930 a; 45 700 m2.

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

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

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

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

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

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

Measures

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

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

Units

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

Measures of weight/mass

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

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

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

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

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

Measures of time

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

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

• quarter - 3 months
• decade - 10 days

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

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

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

Measure abbreviations

Abbreviated names of measures are usually written without a dot:

Measuring instruments

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

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

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

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

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

In order to determine the division price, you must:

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

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

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

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

(30 °C - 20 °C) : 10 = 1 °C

Therefore, the thermometer shows 47 °C.

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

Today, each of us, when designating certain measures of measurement, uses only modern terms. And this is considered normal and natural. However, when studying history or reading literary works, we often come across such words as “spans”, “arshins”, “elbows”, etc.

And this use of terms is also normal, since it is nothing but ancient measures of measurement. What they mean, everyone should know. Why? First, it is the history of our ancestors. Secondly, such knowledge is an indicator of our intellectual level.

The history of the emergence of measures

The development of human society was impossible without comprehending the art of counting. But even this was not enough. To conduct many cases, certain units of length, mass and area were also needed. Their man invented in the most unexpected forms. For example, any distances were determined by transitions, or steps. The old measures of measurement relating to a person's height or to clarify the amount of tissue corresponded to the length of a finger or joint, arm span, etc., that is, everything that was a kind of measuring device that was always with you.

We learn about very interesting lengths of our ancestors from chronicles and ancient letters. This is “throwing a stone”, that is, throwing it, and “cannon shot”, and “shooting” (range of the arrow), and much more. Sometimes a unit of measurement indicated the distance at which the cry of one or another animal could still be heard. It was the “cock crow”, “bull roar”, etc. An interesting measure of length existed among the peoples of Siberia. She was called "beech", and she meant by herself the distance at which a person visually merged the horns of a bull into a single whole.

From the chronicles that have come down to us, we can conclude that the ancient measures of measurement in Rus' appeared in the 11-12 centuries. These were such units as a verst, sazhen, cubit and span. However, in those days, the methods of determining the length invented by man were still extremely unstable. They differed somewhat depending on the principality and constantly changed over time.

From the annals of the 13th-15th centuries, we learn that the ancient measures for measuring loose bodies (usually cereals) are kad, halves, quarters and octopuses. In the 16-17 centuries. these terms have disappeared from everyday life. Since the indicated period, the main measure of loose bodies has become a quarter, which approximately corresponded to six pounds.

In a number of documents of the era of Kievan Rus, the word "spool" is found. This weight unit had the same distribution as Berkovets and pood.

Length determination

Ancient measures of physical quantities were not very accurate. The same applies to determining the length in steps. Such a unit was used in ancient Rome, ancient Greece, Persia and Egypt. With a human step, the average length of which is 71 cm, distances were determined even between cities. A similar unit is used today. However, today special pedometer devices determine not the distance, but the number of steps taken by a person.

A measure of length, which was used in the Mediterranean countries, was such a unit as stages. The mention of it can be found in manuscripts dating back to the first millennium BC. e. A stadion was equal to the distance that a person at a calm pace could walk from dawn to the moment when the sun's disk appeared completely above the horizon.

As society developed, people began to need larger quantities. In this regard, the ancient Roman mile, equal to 1000 steps, appeared.

The ancient measures of measuring the length of different peoples differed from each other. So, Estonian sailors determined the distance with tubes. This was the path the ship took during the time it took to smoke a pipe full of tobacco. The Spaniards called the same measure of length a cigar. The Japanese determined the distance with "horse shoes". This was the path that the animal could go until the straw sole that replaced the horseshoe was completely worn out.

The main quantities for determining the length in Rus'

Recall proverbs with ancient measures of measurement. One of them is well known to us since childhood: "From the pot two inches, and already the pointer." What is this unit of length? In Rus', it was equal to the width of the index and middle fingers. At the same time, one vershok corresponded to one sixteenth of an arshin. Today, this value is 4.44 cm. But the Russian old measure - a nail - was 11 mm. Taken four times, it was equal to one inch.

In Rus', some ancient measures of measurement came into use in connection with the development of trade relations with other countries. So there was a quantity called arshin. The name comes from the Persian word for elbow. In this language, it sounds like "arsh". Arshin, equal to 71.12 cm, came along with merchants from distant countries who brought Chinese silks, velvets and Indian brocades.

When measuring the fabric, the Eastern merchants pulled it over their arm up to the shoulder. In other words, they measured the goods in arshins. It was very convenient, because such a measuring device was always with him. However, cunning merchants were looking for clerks with shorter arms, so that there was less fabric per arshin. But this was soon put an end to. The authorities introduced the official arshin, which everyone without exception had to use. It turned out to be a wooden ruler, which was made in Moscow. Copies of such a device were sent throughout Rus'. And so that no one could cheat and shorten the arshin a little, the ends of the ruler were bound with iron, on which the state brand was affixed. Today, this unit of measurement is no longer used. However, the word denoting such a value is familiar to each of us. Proverbs with ancient measures of measurement also tell about it. So, they say about a shrewd person that he "sees three arshins underground."

How else was distance determined in Rus'?

There are other ancient measures of length measurement. These include a sazhen. The mention of this term is first found in the "Word about the beginning of the Kiev-Pechersk Monastery", dating from the 11th century. Moreover, there were two types of sazhens. One of them is flywheel, equal to the distance between the tips of the middle fingers of the hands, spread out in different directions. The value of ancient measures of this type was equal to 1 m 76 cm. The second type of fathom is oblique. It was the length from the heel of the shoe on the right foot to the tip of the middle finger of the left hand, stretched up. The size of an oblique sazhen was approximately 248 cm. Sometimes this term is mentioned when describing a person of a heroic physique. They say that he has a slanting fathom in his shoulders.

Ancient Russian measures for measuring large distances - a field or a verst. The first mention of these quantities are found in manuscripts of the 11th century. The length of a verst is 1060 m. Moreover, initially this term was used to measure arable land. He meant the distance between the turns of the plow.

The old measures of measuring quantities sometimes had a playful name. So, since the reign of Alexei Mikhailovich (1645-1676), a very tall person began to be called This humorous term is not forgotten today.

Until the 18th century in Rus', such a unit of measurement as a boundary verst was used. She measured the distance between the boundaries of settlements. The length of this verst was 1000 fathoms. Today it is 2.13 km.

Another ancient measure of length in Rus' was the span. Its size was about a quarter of an arshin and was approximately 18 cm. There were:

- "smaller span", equal to the distance between the tips of the extended index and thumb;
- "large span", equal to the length between the spaced thumb and middle fingers.

Many proverbs about ancient measures of measurement point us to this value. For example, "seven spans in the forehead." So they say about a very smart person.

The smallest ancient unit of length is the line. It is equal to the width of a wheat grain and is 2.54 mm. Until now, watch factories use this unit of measurement. Only Swiss size is accepted - 2.08 mm. For example, the size of the men's watch "Victory" is 12 lines, and the women's "Dawn" - 8.

European units of length

From the 18th century Russia has significantly expanded its trade relations with Western countries. That is why there was a need for new measures of measurement that could be compared with European ones. And then Peter I carried out a metrological reform. By his decree, some English values ​​\u200b\u200bfor measuring distances were introduced in the country. These were feet, inches and yards. These units are especially widespread in shipbuilding and the navy.

According to the existing legend, the yard was first identified back in 101. It was a value equal to the length from the nose of Henry I (King of England) to the tip of the middle finger of his hand, extended in a horizontal position. Today this distance is 0.91 m.

The foot and yard are ancient measures of measurement that are closely related. Derived from the English word "foot" - foot, this value is equal to one third of a yard. Today a foot is 30.48 centimeters.

From the Dutch word for thumb, the unit of measure, inch, got its name. How was this distance originally measured? It was equal to the length of three dried grains of barley or the phalanx of the thumb. Today, one inch is 2.54 cm and is used to determine the inner diameter of car tires, pipes, etc.

Ordering the system of measures

In order to ensure ease of transition from one unit of measurement to another, special tables were published in Rus'. On the one hand, ancient measures were brought into them. Units of measurement of foreign origin, which corresponded to Russian, were placed through the equal sign. The same tables also included those units that were to be used in the country.

However, the confusion with the system of measures in Rus' did not end there. Different cities used their own units. This was put to an end only in 1918, when Russia switched to the metric system of measures.

Volume measurement

A person needed to measure bulk physical quantities and liquids. To do this, he began to use everything that he had in everyday life (buckets, vessels and other containers).

What old ones took place in Rus'? Our ancestors measured loose bodies:

1. Octopus, or octopus. This is an old unit, equal to 104.956 liters. A similar term was applied to the area, which was 1365.675 square meters. For the first time, the octopus is mentioned in documents of the 15th century. It was widely used in Rus' because of its practicality, as it had a volume half that of a quarter. There was even a certain standard for such a measure. It was a container to which an iron rower was attached. The grain was poured into such a measured octopus with a top. And then, with the help of rowing, the contents of the form were trimmed to the edges. Samples of such containers were made of copper and sent throughout Rus'.

2. A shackle, or a kadyu. These measuring containers were common in the 16th and 17th centuries. In later periods, they were extremely rare. Okov was the main measure of loose bodies in Rus'. Moreover, the name of this unit comes from a special barrel (vat), which was adapted for measurements. The measuring container was covered with a metal hoop on top, which made it impossible for the cunning to cut its edges and sell a smaller amount of grain.

3. A quarter. This measure of volume was used to determine the amount of flour, cereals and grains. In everyday life, a quarter was more common than luggage, as it had more practical dimensions (1/4 of the fetter). Such a unit of measurement was used in Rus' from the 14th to the 19th centuries.

4. Kulem. This ancient Russian measure, used for bulk bodies, was equal to 5-9 pounds. Some researchers believe that the word "kul" once meant "fur". This term was used for a receptacle sewn from the skin of animals. Later, such containers began to be made from woven materials.

5. Buckets. By such a measure, our ancestors determined the amount of liquid. It was believed that 8 mugs were placed in a trading bucket, the volume of each of which was equal to 10 cups.

6. Barrels. A similar unit of measure was used by Russian merchants when selling wine to foreigners. It was believed that one barrel contained 10 buckets.

7. Korchagami. This large clay pot found its use in measuring the volume of grape wine. For different parts of Rus', the korchaga ranged from 12 to 15 liters.

Weight measurement

The old Russian system of measures also included units for measuring mass. Without them, trading activity was impossible. There are various ancient measures of mass measurement. Among them:

1. Spool. Initially, this word meant a small gold coin, which was the unit of measure. By comparing its weight with other precious items, the purity of the noble metal from which they were made was determined.

2. Pud. This unit of weight was equal to 3840 spools and corresponded to 16.3804964 kg. Ivan the Terrible also ordered that any goods be weighed only at pudovschiks. And since 1797, after the Law on Measures and Weights was issued, they began to make spherical weights corresponding to one and two pounds.

4. Share. This unit of measurement in Rus' was the smallest. Its weight was 14.435 mg, which could be compared to 1/96 of a spool. Most often, the share was used in the work of mints.

5. Pound. Initially, this was called "hryvnia". Its value corresponded to 96 spools. From 1747, the pound became which was used until 1918.

Area measurement

Some standards were invented by our ancestors to determine the size of land plots. These are ancient measures for measuring area, including:

1. Square verst. The mention of this unit, equal to 1.138 sq. kilometers, found in documents dated 11-17 centuries.

2. Tithing. This is an old Russian unit, the value of which corresponds to 2400 square meters. meters of arable land. Today, the tithe is 1.0925 hectares. This unit has been used since the 14th century. She was known as a rectangle, the sides of which were 80 by 30 or 60 by 40 fathoms. Such a tithe was considered government and was the main land measure.

3. Quarter. This measure of arable land was a unit representing half a tithe. A quarter has been known since the end of the 15th century, and its official use continued until 1766. This unit got its name from a measure of the area on which rye could be sown in the amount of ¼ of the qadi volume.

4. Sokha. This unit of area measurement was used in Rus' from the 13th to the 17th centuries. Used it for tax purposes. Moreover, several types of plow were distinguished, depending on the area of ​​​​the best lands. So, such a unit was:

Servant containing 800 quarters of good plowing;
- church (600 quarters);
- black (400 quarters).

In order to find out how many sokhs are available in the Russian state, censuses of taxable lands were carried out. And only in 1678-1679. this unit of area was replaced by a yard number.

Modern application of ancient measures

About some units for determining volume, area and distance, which were widely used by our ancestors, we know today. So, in some countries, length is still measured in miles, yards, feet and inches, and in cooking the pound and spool are used.

However, most often old units are found in literary works, historical stories and proverbs.

Linear measures of length, measures of area, measures of volume, measures of mass. Three versions of the multiplication table. Decimal number system

Multiplication table. Option 1

Multiplication table from 1 (one) to 10 (ten). Decimal system

Multiplication table. Option 2

Multiplication table abbreviated from 2 (two) to 9 (nine). Decimal system

Multiplication table. Option 3

Multiplication table from 1 (one) to 20 (twenty). Decimal system