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
2 x 1 = 2 |
3 x 1 = 3 |
4 x 1 = 4 |
5 x 1 = 5 |
6 x 1 = 6 |
7 x 1 = 7 |
8 x 1 = 8 |
9 x 1 = 9 |
Multiplication table. Option 3
Multiplication table from 1 (one) to 20 (twenty). Decimal system
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
For measuring large areas use a square kilometer - 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 km2. 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.
- Bashmakov M.I. Nefedova M.G. Mathematics. 4th grade. M.: Astrel, 2009.
- M. I. Moro, M. A. Bantova, G. V. Beltyukova and others. Mathematics. 4th grade. Part 1 of 2, 2011.
- Demidova T. E. Kozlova S. A. Tonkikh A. P. Mathematics. 4th grade 2nd ed., corrected. - M.: Balass, 2013.
- School.xvatit.com().
- Mer.kakras.ru ().
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Homework
- Find the area of a square with a side of 15 cm.
- Express: in square meters: 5 ha; 3 ha 18 a; 247 acres; 16 a;
- in hectares: 420,000 m 2; 45 km 2 19 ha;
- in ares: 43 ha; 4 ha 5 a; 30 700 m 2; 5 km2 13 ha;
- in hectares and ares: 930 a; 45 700 m2.
At first glance, in the system of measures of length, mass, etc. there is nothing complicated, but for many schoolchildren, the transfer from one measure to another is very difficult. In some children and after elementary school and it is not possible to correctly correlate, for example, a decimeter with a millimeter, or a hectometer with a cubic meter.
Nevertheless, one cannot live without a clear knowledge of the system of measures at the present time; people encounter measurements of one or another value daily and several times.
Length units in tables
How from one to another without errors? One of the most effective ways the study of measures of length or weight are tables, this is recognized by teachers, parents, and the students themselves.
Properly selected pictures of length measures clearly explain to the student the dependence of one unit on another. The most useful table is the one in which the measures of values from the smallest gradually increase, that is, the student sees that, for example, 1000mm \u003d 100cm \u003d 10 dm \u003d 1 m, especially if each value is displayed as a picture.
Looking at the table, most of the schoolchildren start by simply memorizing the dependencies of certain quantities, however, understanding comes pretty soon: the student realizes that a meter contains, for example, 100 centimeters, or 1000 millimeters, but decimeters - only 10. Good a large ruler will be of help at this moment, so that the learned numbers can be correlated with the real length, and it is best remembered.
Why are there different units of length?
Some parents wonder why it is so necessary to operate with different units of length? Children get confused in centimeters and decimeters, and sometimes adults themselves cannot explain to them which value is larger and by how many times.
It will not take long to find the answer to this question. In what units of length is it convenient to measure the thickness of a match or the body of a ladybug? Of course, in millimeters. In what units of length is it convenient to measure the length of a pen or pencil? In centimeters.
If you need to measure the width or length of the window, decimeters will be a convenient unit. For the length of the fence, the best option would be meters. For the distance between cities - kilometers, for the distance between the continents - also kilometers, since this is the largest among the lengths.
Very often in school problems the task is given - to express the length given in meters or decimeters, in millimeters or kilometers, or vice versa. This is not difficult to do if you know the length ratio by heart, or use the table helper. It is much more difficult to convert volume measures - liters to square decimeters or vice versa, but there are also tables for volume measures that successfully help to assimilate the relationship between quantities.
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:
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Measures of weight/mass
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Area measures (square measures)
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Measures of time
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measure of vessel capacity
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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 by 10 identical 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:
- Find the two nearest strokes of the scale, next to which the magnitude values are written.
- 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.
Measure various quantities in Everyday life each and every one of us has to do. 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 to predict the weather. Atmosphere pressure, wind speed, etc.