Organization: GBOU Secondary School No. 74
Location: Moscow
Tasks: Improve the ability to measure quantities using a conventional measure. Create conditions for the development of logical thinking, intelligence, and attention. Develop coherent speech and the ability to give reasons for your statements.
Activities: gaming, communicative, educational and research.
Materials: doll, pieces of fabric: pink 4*24cm, yellow 9*18cm; a piece of chalk; measurements 4*6 cm and 3*18cm, numbers from 1 to 20, a piece of chalk, chips.
Progress of the lesson:
Organization of children
One two three four five
Let's play fun
Compare and measure
Answer questions!
Repeating a number series.
Guys, look at the board. What do you see? (Numbers). Let's count together from 1 to 20. And now back from 20 to 1.
Explanation of the topic of the lesson.
Guys, we need to sew a dress for the Dasha doll. To sew a dress you need material. We will use the material different color: pink and yellow. In order to sew a beautiful dress, you need to measure the fabric.
Today we will measure fabric using a conventional measure. Look, each of you has pieces of fabric on the table. I have exactly the same ones in my hands. I suggest you compare your pieces. Are they the same for everyone?
Guys, pay attention to the board. Today I will show you a way to measure a piece of fabric with a yardstick. Apply the measure so that the edges of the measure and the piece of fabric coincide, use chalk to draw a line along the edge of the measure and set aside one chip. Next, we apply the measurement to the line marked on the fabric. Now I suggest you measure your pieces of fabric yourself, remember that the measurement starts from the edge of the piece of fabric.
But now it's time to rest.
Physical education minute.
It's easy fun -
Turns left - right.
We all know for a long time -
There is a wall, and there is a window. Turns the body to the right and left.
We squat quickly and deftly.
You can already see skill here.
To develop muscles,
You have to do a lot of squats. Squats.
And now walking in place,
This is also interesting. Walking in place.
Now let's count how many chips you got. How many times has your measurement been completely striped? (6 times).
Now we measured the pink piece of fabric along its length. Now let's measure the width of the yellow piece of fabric using another measure using the same measurement method.
Independent work children.
How many times does your measurement fit completely into a piece of fabric? (3 times).
Well done! So the length of the pink fabric is 6 measures, and the width of the yellow fabric is 3 measures. Now we can sew a beautiful dress for the doll, and we will definitely have enough fabric.
Alla Cheremisova
Summary of GCD on FEMP in older children preschool age“Measuring a segment using a conventional measure”
Municipal Autonomous preschool educational institution
municipal formation of the city of Nyagan "Kindergarten
general developmental type with priority implementation of activities
in cognitive and speech direction of development children
№ 6 "POCK"
Summary of GCD on FEMP in children of senior preschool age
Subject: «»
Developer:
Cheremisova Alla Ivanovna, teacher
first qualification category
MADO DOU, Nyagan D/s No. 6 "Pock"
Abstract math classes in preparatory group.
Subject: « Measuring a segment using a conventional measure»
Goals: Learn measurement using a conventional measure, form children arbitrariness and ability to perform actions simultaneously, develop imagination children, learn children navigate on a sheet of paper, consolidate in memory children names of the days of the week in their sequence, learn to associate the events of your life with certain days of the week.
Equipment:
Demonstrative material
Calendar, strip of paper 3 cm wide, 24 cm long (the length of the strip is a multiple of 4 and 6 cm, two cardboard measurements: one 4cm long, the other 6cm, circle.
Handout
A strip of paper 3 cm wide, 24 cm long (the length of the strip is a multiple of 4 and 6 cm, two cardboard measurements: one 4cm long, the other 6cm, pencil.
GCD move
1. Organization children
One two three four five
Let's play fun
Compare and to measure,
Answer questions!
Guys, we sit down at the tables as quietly as leaves fall to the ground.
2. Repeating the number series.
Guys, let's look at the board. -What do you see? (numbers). Let's count together from 0 to 10. Now let's count back from 10 to 0.
Well done, now let’s count to ourselves, with our eyes, to 10 and back.
3. Explanation of the topic of the lesson
Guys, today we will be with you measure a segment using a conventional measure. Look, each of you has a strip of paper on the table. I have exactly the same one in my hands.
(The children are given measurements. One group children's measure 4 cm long, another 6cm. Do not warn about the difference in advance.)
I suggest you compare your strips. Are they the same for everyone?
Look, my stripe is the same size too (compare with stripes of some children)
Guys, pay attention to the board. Now I'll show you the way measuring the strip with a yardstick. Let's apply measure like this to the edges measurements and stripes matched, With using a pencil, draw a line along the edge measurements. Next we apply measure already to the line marked on the strip.
And now, I suggest you, on your own measure your pieces, not forgetting that measurement starts from the edge of the strip. The edges of the strip and measurements must match, at the next measuring stick applied to the line marked with a pencil.
Let's start the task. Until then I'll finish measuring your strip.
4. Independent work children
Guys, next to measured put a number in a strip indicating how many times your measure completely laid out on a strip of paper.
My measure fit completely 4 times.
(Ask a question to a child who has measured 6 times)
How many times is yours measure completely fit into the strip?
(Answer: "Six times")
Six? And I only have four.
Were the stripes the same? (Address the children with whom the strip was previously compared)
Does anyone else have measured six times? Raise the number "6".
Does anyone have measure did you fit in completely four times? Show the number "4".
The stripes were the same. I saw that everything measured correctly. Why did some get 6, others 4?
(Statements children)
Right, the measurements were different.
(Invite to the board one person from those who got 4 and one who got 6)
Guys, let's make sure that the measurements are not the same. I will ask two people to come to me, one who has measure fit 4 times and one of whom measured 6 times.
The method of application, we are convinced that different measurements.
How many times did your measure? (address the child with less yardstick)
(Answer :6 times)
And you?
(4 times)
Look how interesting it is! Small measured six times, large - four times.
Physical education minute:
Get up quickly.
Quickly stand up, smile,
Pull yourself higher, higher.
Come on, straighten your shoulders,
Raise, lower,
Turned left, right,
Hands touched knees.
Sat down, stood up, sat down, stood up
And they ran on the spot.
5. Game "Name the neighbors"
Now guys, put the stripes aside and measurements and lay out a series of numbers from 0 to 10 in front of you.
Let's play a game "Name the neighbors".
I will name a number, and you must put forward two numbers - "neighbors" and explain why these two numbers are neighbors. For example, the number 7 is "neighbours" 6 and 8, because 6 is one less than 7, and 8 is one more than 7.
(the game is repeated 3-4 times)
6. Working with the template
Guys, pay attention to the board. You see sample drawings (fish, ladybug, mouse, eye, fungus, piglet, etc.). Look at them carefully.
What did you notice? (the teacher traces the shape of the leaf in each drawing with his finger)
(answers children)
Right! All drawings have a similar shape "Listika"
I would like to invite you to create your own drawings. In front of you on the table lies a landscape sheet, a template in the shape of a leaf, with its with help you can create your own drawings using all the free space on the album sheet.
(children draw)
I see many have finished. Who wants to come to the board and talk about their drawing?
(those who wish tell and show about their drawing, which is shown in the right top corner, bottom left, middle, etc.)
7. Game - relay race "Name yourself"
Well done! And now you and I will play a game "Name yourself"
- The rules of the game are as follows:
everyone sits on chairs, then I show the direction from whom we start and everyone stands up one by one, saying their first and last name, the next one gets up as soon as the previous one sits down. He gets up no earlier and no later than this time. When all the children have named themselves, the game is repeated, starting from the last, which now becomes the first.
(play 2-3 times)
8. Working with the calendar.
Guys, what month is it now?
What number?
What day of the week? Right. Let us mark this day on the calendar together.
Look, on the board, I put up a sign with the name of today's day of the week. What is the name of the previous day of the week?
(answers children)
What day will it be tomorrow?
(a card with the name is displayed)
Let's remember what day the week begins. What days of the week are missing? Let's arrange the cards with names in order and name the days of the week in unison.
Guys, let's together remember the most memorable events that happen regularly these days and designate them with a symbol.
(draw on the board)
9. Summary of GCD
Let's summarize our lesson. Today we learned a lot of interesting things.
What did we talk about today?
What did you do?
Did you like it?
(answers children)
Well done boys! You were all active today! Each of you receives a cheerful emoticon.
Valentina Timofeeva
Lesson summary “Measuring length. Measure"
Progress of the lesson.
Guys, of course, you know and love fairy tales. Today we will remember a fairy tale "Kolobok". But our fairy tale will be unusual, with mathematical tasks. And so... Once upon a time there lived an old man and an old woman. That's what the old man says old woman: “Bake a bun”. And the old woman answers: “I’ll bake it if you and the guys complete the tasks”:
1. If the ruler longer than the handle, then the pen? (shorter than a ruler).
2. If the rope is thicker than the thread, then is it the thread? (thinner than rope).
3. If the green stripe is wider than the yellow one, is it yellow? (already green).
4. If the table is higher than the chair, then the chair? (below the table).
Well done, you completed the tasks.
The old woman kneaded flour with sour cream, made a bun, fried it in oil and put it on the window to cool. The gingerbread man lay there, lay there, and then rolled off. I decided to visit the fox, congratulate her on her birthday and give her a ribbon.
Kolobok is rolling along the road, towards him Hare:
-Where are you going, Kolobok?
– Visit Lisa, it’s her birthday today.
- Take me with you.
- I'll take it if you help me. I decided to give Lisa a ribbon, not red, but green, but what’s her name? measure - I don't know. Help me, Hare.
Guys, let's help Kolobok together with the Hare measure the ribbon. What measure, called yardstick. What will we be measure the length of the ribbon? You have to choose measure. Merkoy maybe a piece of rope, a tube, a ruler, a cardboard strip, sticks. I suggest choosing a cardboard strip, as it will be more convenient to measure. Now we will see how many times the strip will fit along ribbon length. Let's remember the rules measurements linear quantities : you need to start exactly from the end, lay the strip - measure straight. We stack until there is no entire length measured. We make a mark where the end of the strip is, and again lay it exactly from the mark.
Children measure green ribbon using a conditional measurements. Length green ribbon – 3 measurements.
Now let's compare length red and blue stripes. Children determine length strips by applying and applying.
In the Kolobok store I bought the ribbon I needed length, The hare decided to buy a ball.
Wolf enters the store:
-Where are you going?
- To Lisa for her birthday.
- Take me with you.
- We'll take it if you play with us.
Physical exercise.
Hares are jumping:
Hop, hop, hop!
Yes, to a green meadow.
They sit down, listen,
Is there a wolf coming?
Once - bend over, straighten up.
Two – bend over, stretch.
Three - three claps of your hands,
Three nods of the head.
The wolf bought a mirror for the Fox as a gift.
-Where are you hurrying?
- To Lisa for her birthday.
- Take me with you.
– We’ll take it if you solve the problems.
Let's help Bear solve problems?
1. How many ears do two mice have? (4)
2. How many times did they say cat:
It's rude to eat without a spoon.
As soon as I run into the house,
Licking the porridge with his tongue.
It's even worse with a pig:
He was swimming in a puddle again.
And the naughty little goat
Ate four dirty pears.
How many were naughty? (3)
3. Four rabbits were walking from school.
And suddenly they were attacked by bees.
Two bunnies barely escaped.
How many didn’t make it? (2)
4. Five puppies played football
One was called home.
He looks out the window, thinks,
How many of them are playing now? (4)
Well done, you helped the bear. The animals ran further. The Bear decided to give the Fox a flag. It hung over his den.
They ran to the Fox, but she didn’t let anyone in house:
“I didn’t invite anyone, I won’t let you in until you complete the tasks.”
1. How many characters are there in the fairy tale? "Kolobok"?
2. How many animals came to visit me?
3. What kind of animals are we?
4. What other wild animals do you know?
5. What kind of domestic animals do you know?
Domestic and wild animals, birds, insects - all our little brothers. And, of course, they must be treated with care.
Lisa invited guests to her house.
Kolobok gave a ribbon.
The hare gave a ball.
The wolf gave him a mirror.
The bear gave a flag.
What shapes do the gifts look like?
What will we give to Lisa? I suggest making gifts from geometric shapes.
Bottom line: What did we learn today? to measure? How measured the ribbon? Let's remember the rules measuring length using a measuring stick.
Publications on the topic:
Intellectual and aesthetic development of children in the process of mastering basic quilling techniques Today we say with confidence that every normal child is born with congenital creative abilities. But creative people.
Using verbal and constructive methods in studying the topic “Quantities and their measurements” in primary school At any modern system general education mathematics occupies one of the central places. The need for research activities.
Measuring the length and height of objects using a standard (cuisenaire rods). Abstract on the development of mathematical concepts in senior group. Measuring the length and height of objects using a standard (cuisenaire rods).
How to introduce Russian nesting dolls to preschoolers IN kindergarten children receive information about various phenomena life, learn a lot of new and interesting things about the past and present of our country.
Summary of a lesson on familiarization with elementary mathematical concepts in the middle group Topic: Generalization and consolidation of the material covered. Goal: 1. Develop intelligence and logical thinking. 2. Exercise children in quantitative skills.
Summary of a lesson in a preparatory group on speech development. Program tasks: introduce children to the vowel sounds [a], [o] Introductory part: speech warm-up. What time of year is it now? How many seasons are there in total? (list them). How many days are there in a week? From whom.
Target: Formation of initial mathematical knowledge.
Tasks:
- Continue learning to compose and solve problems involving addition and subtraction within 10.
- Improve the ability to measure the length of objects using a conventional measure.
- Improve your ability to orientate yourself on a sheet of squared paper.
- Strengthen the ability to name sequentially the days of the week, times and months of the year.
- Strengthen the ability to use “greater than”, “less than”, “equal” signs;
- Strengthen children's knowledge about geometric shapes;
- Develop attention, memory, thinking;
- Cultivate an interest in mathematics, the ability to listen to instructions from adults.
Didactic visual material:
Demonstration material. A ball, a card with a picture of a square, an envelope, cards with arithmetic signs, pictures for solving problems.
Handout. Cards with diagrams of the route from home to school, strips of cardboard (conventional measures), pencils, checkered pieces of paper.
Progress of the lesson
Educator . - Guys, we have a guest today, he’s funny (I put a picture of Dunno on the board).
Malvina asked homework Dunno, but he doesn’t know how to deal with it. So he came to you for help. Will you help him? (Yes.)
Then we’ll start our lesson, and Dunno will watch and learn from you.
And for each completed task, Dunno will give you a letter so that we can put them together into a word.
1 task "Think and answer"
Children stand in a semicircle. The teacher throws the ball to the child and gives the task. The child answers and returns the ball.
What are the 4 seasons?
What time of year is it now?
How many spring months are there? Name them.
Name the winter months.
What month comes after January?
How many days are there in a week?
What day of the week is today?
What was it like yesterday?
What day of the week will be tomorrow?
Name the days off.
What number must be added to 8 to make 10?
What number is less than 5 by 1?
Name the neighbors of the number 8; 4; 6
What number comes after the number 5; 1; 7.
What number comes before 8? 6; 4.
Educator. Well done! We completed the first task. And for this, Dunno gives you the letter “C” (I put it on the board).
Sit down at the tables. Don't forget that the back at the table should be straight. So are you ready? (Yes.)
Task 2. "Solve the problems"
And now I suggest you teach Dunno how to solve problems. But before we solve the problem, let’s remember what parts the problem consists of? (Condition, Question, Solution, Answer.)
What is a condition? (this is what we know).
What is a question for a problem? (this is what we need to find out).
What is a solution to a problem? (this is something that can be added or subtracted).
What is the answer? (this is what happened and is known to us).
1 task.
How many ducklings swam in the creek?
How many ducklings came to land?
Let's make a problem about them.
What does it say about ducklings?
(ducklings swam in the creek)
How many were there at first?
How many ducklings came ashore?
Are there fewer ducklings or are there more swimming?
If less, should you add or subtract?
What is the question in the problem? (how many ducklings are left to swim?)
Now let's create a problem.
Child. 9 ducklings swam in the creek. 1 went ashore. Ask a question about the problem.
Child. How many ducklings are left to swim?
Let's denote the solution with numbers and signs. (the called child lays out an example from the numbers for problem 9-1=8).
Task 2. Let's solve another problem. Look, there are 2 aquariums on the board. How many fish are in the aquarium on the left? How many fish are in the aquarium on the right?
What question can you ask in this problem? (how many fish are in 2 aquariums?)
Who will try to create a problem?
Child. There are 4 fish swimming in the aquarium on the left, and 3 fish swimming in the aquarium on the right.
Who will ask a question about the problem?
Child. How many fish swim in 2 aquariums?
Now you need to lay out the solution with numbers and signs on the board. And all the guys will write down the solution on pieces of paper. Well done. Dunno gives you the letter “P”.
Physical education moment.
Quickly stand up, smile,
Pull yourself higher, higher.
Well, come on. straighten your shoulders,
Raise, lower.
Turned left, right,
Hands touched knees.
Sat down, stood up, sat down, stood up
And they ran on the spot.
We rested. Go to your places and we will continue to help Dunno.
3 task. Game exercise “Measuring the road to school.” The teacher clarifies: “Where will you go on the first day of autumn? What is the name of the first month of autumn?
Children have cards with maps of the route from home to school.
The teacher asks the children to find out the length of the road from home to school: “How can I find out the length of the road to school? (Measure.) How will we measure the road? (First from the house to the turn, then from the turn to the school.)
How can you measure the length of a road? (Children's answers.)
Today we will measure the road from home to school using a conventional measure. Now I will remind you of the method of measuring with a conventional yardstick. You need to apply the measure so that the edge of the measure and the beginning of the road coincide. Using a pencil, draw a line along the opposite edge of the measurement. Next, we apply the measurement to the line and mark it again with a pencil.
Now measure the length of the road yourself in your picture. First, measure the length of the road before the turn and record how many times the measurement fits completely onto the strip. And then measure the length of the road after the turn, and also write down in a square how many times the measurement was taken after the turn.
After completing the task, the teacher asks: “What is the length of the road from the house to the turn? (Children give their answer according to the indicated figure.) What is the length of the road from the turn to the school? What is the length of the road from home to school? How did you find out the length of the road? (We added up the number of measures and indicated the result with a number.)
Well done, you coped with this task too. You receive the letter “A” from Dunno.
4 task. Game exercise “Drawing figures.”
The teacher asks the children to guess what figure is drawn on a piece of paper lying in an envelope. To do this, children must complete the task correctly: From a point from left to right, draw a line three cells long, then from top to bottom, draw another line three cells long, then draw 3 cells from right to left, and finally, from bottom to top, 3 cells.”
What kind of figure did you get? (the teacher shows a card with a picture of a square). For this task you receive the letter “C” from Dunno.
Task 5. "Auditory dictation."
Draw in a rectangle:
In the upper right corner there is a square;
In the lower left corner there is a ball;
In the lower right corner there is a triangle;
In the upper left corner there is a circle;
In the middle there is an oval.
Where did you draw the ball? (in the lower left corner)
Where did you draw the square? (in the upper right corner)
Where did you draw the oval? (in the middle of the rectangle).
Did everyone cope with the dictation? Well done. Here is the letter “I” as a gift for you.
6 task. “Compare the numbers and put the signs”(work at a desk)
The teacher shows cards with the signs “>”, “<», «=»и уточняет, что они обозначают:
“The bird turned its beak
Where there is more delicious food,
And where there is less - she turned away,
I didn’t eat anything.”
The open beak points to a larger number, and the corner points to a smaller number.
Give the children cards with numbers: 3 and 4, 5 and 4, 7 and 7, 5 and 5, 7 and 8, 9 and 8.
And almost everyone completed this task. Here's the letter "B"
Task 7. "Connect the dots in the picture"(children take turns going to the board). And the last task was completed. You receive the letter “O” as a gift.
So we helped Dunno complete all the tasks. Let's read what kind of word we got? (Children read: “THANK YOU.”)
Educator . This is Dunno thanking us for our help with our homework. I also say thank you. You guys did a very good job today. And for this Dunno gives you these badges.
Draw a sun with a smile if you liked it in class today, and if you didn’t like it, then a sad sun - without a smile.
Used Books:
Pomoraeva I. A., Pozina V. A. “Formation of elementary mathematical concepts” (preparatory group).
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Introduction………………………………………………………………………………. |
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The concept of quantity and its measurement in the initial course of mathematics……. |
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Length of a segment and its measurement…………………………………………….. |
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Area of the figure and its measurement……………………………………………. |
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Mass and its measurement……………………………………………………… |
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Time and its measurement…………………………………………………….. |
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Volume and its measurement…………………………….……………………. |
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Modern approaches to the study of quantities in the initial course of mathematics…………………………………………………………………………………. |
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Conclusion……………………………………………………………….. |
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Bibliography……………………………………………………… |
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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 and 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 quantities such 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 defined on different sets: the length is on the set of segments, and the area is on the set of flat 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.
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. 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 polyhedral 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: 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:
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 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.
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 children's spatial concepts. An important step in the formation of this concept is familiarity with 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 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?” (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 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 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 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.
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 my 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 awareness 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. 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 the 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 unknown multiplier by known product and factor. For example, area garden plot 100m, section length 25m. 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 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, 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 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 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?" (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, tension electric field 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