When studying this topic, students should master calculation techniques, gain strong computational skills, memorize the results of addition and subtraction within 10, as well as the composition of the numbers of the first 10, recognize and show the components and results of two arithmetic operations and understand their names in the teacher's speech.
As students master natural sequence numbers and the properties of this series, one must also be introduced to the techniques of addition and subtraction, based on this property of the natural series of numbers. Children learn these techniques to add and subtract one from a number, i.e. count and count down by 1.
When students have mastered counting techniques, the teacher introduces them to counting techniques.
If first grade students master counting techniques quite quickly, then counting techniques are much slower.
The difficulty is that the counting method is based on good knowledge counting backwards, and counting backwards is difficult for many first grade students. In addition, students have trouble remembering how much needs to be taken away, how much has already been taken away, how much more needs to be taken away.
When studying each number in the first ten, students also gain an idea of the composition of these numbers.
At the beginning, it is necessary to give exercises in which one of the terms is perceived by children visually, and they look for the second by representation.
When performing addition and subtraction operations within given number solutions to examples with a missing component are introduced. It is indicated by dots, frames, question marks, etc., for example:
I – 3, 4 +... = b, ? – 2 = 4. b - ? = 2.
Let's write 1-1=0 (the absence of objects is indicated by the numbers O). More examples are solved when the difference is zero.
You should introduce the number zero as a subtrahend and then as a addend in a large number of exercises. The meaning of operations with zero will be better understood by students if zero as a subtrahend and zero as a addend are not introduced simultaneously. Then exercises are carried out to differentiate examples in which zero will be added and subtracted.
The first grade teacher should draw students' attention to the fact that the sum is always greater than each of the terms, and the remainder is always less than the minuends.
The minuend is greater than or equal to the subtrahend, otherwise the subtraction cannot be performed.
Already from the first grade, students should be accustomed to checking the correctness of solutions to examples.
Analysis of Moreau's textbook
The student will know:
The specific meaning and name of the actions of addition and subtraction;
Know and use the names of components and the results of addition and subtraction when reading and writing numerical expressions;
Know the commutative property of addition;
Know the table of addition within 10 and the corresponding cases of subtraction;
Units of length: cm and dm, the ratio between them;
Unit of mass: kg.
Find the meaning of numerical expressions in 1–2 steps without parentheses;
Apply calculation techniques:
when adding - adding parts; rearrangement of numbers;
when subtracting - subtracting a number by parts and subtracting based on knowledge of the corresponding case of addition;
Perform addition and subtraction with the number 0;
Find a number that is several units greater or less than a given one;
Be able to solve one-step addition and subtraction problems.
Studying at joint activities with a teacher will have the opportunity to learn:
- group objects according to a given characteristic;
- solve puzzles, magic squares, circular examples, ingenuity tasks, puzzles, chains of examples, joke tasks, logic problems;
- build polygons and broken lines.
Cognitive UUD:
1. Find your bearings in textbooks (notation system, text structure, headings, vocabulary, content).
2. Search necessary information to execute educational assignments using reference materials textbook (under the guidance of a teacher).
3. Understand information presented in the form of text, pictures, diagrams.
4. Compare objects, objects: find commonalities and differences.
5. Group, classify objects based on essential features, according to specified criteria.
Regulatory UUD:
1. Organize your workplace under the guidance of a teacher.
2. Carry out control in the form of comparing your work with a given standard.
3. Make the necessary additions and corrections to your work if it diverges from the standard (sample).
4. In collaboration with the teacher, determine the sequence of studying the material, based on the illustrative series of the “route sheet”.
Communicative UUD:
1. Follow the simplest standards speech etiquette: say hello, say goodbye, thank you.
2. Engage in dialogue (answer questions, ask questions, clarify anything unclear).
3. Cooperate with comrades when performing tasks in pairs: establish and follow the order of actions, correctly report errors to a comrade.
4.Participate in a collective discussion of an educational problem.
Compare different methods of calculations, choose a convenient one.
Simulate situations illustrating an arithmetic operation and the progress of its execution.
Use mathematical terminology when writing and performing arithmetic operations (addition, subtraction).
Simulate studied arithmetic dependencies.
Forecast calculation result.
Monitor and carry out step-by-step control of the correctness and completeness of the execution of the arithmetic operation algorithm.
Use various techniques for checking the correctness of finding a numerical expression (based on algorithms for performing arithmetic operations, estimating the result).
Plan solution to the problem.
Explain choosing arithmetic operations for solutions.
Act according to a given plan for solving the problem.
Use geometric images to solve the problem.
Control: detect and eliminate errors of an arithmetic (calculation) nature.
Observe for changing the solution to a problem when its conditions change.
Fulfill short note in different ways, including using geometric images (segment, rectangle, etc.).
Research situations requiring comparison of quantities and their ordering.
Characterize phenomena and events using quantities.
11) Methodology for studying arithmetic operations. Addition and subtraction of numbers of the second ten (tasks of the topic, cases considered, addition and subtraction based on knowledge of numbering, cases of addition and subtraction without moving through the rank - include justification for the techniques!!!).
The study of numbering and actions within 20, i.e. the second and 1st center, takes place in the 2nd grade of a correctional school.
Objectives of the second concentration: to give the concept of ten as a new unit; teach counting to 20, counting and counting by one, ten and equal number groups (2, but 5, 4); introduce the decimal composition of numbers; develop an understanding of single and double digit numbers; teach to denote numbers from 1 to 20 with digits; introduce the principle local significance numbers; teach adding and subtracting in aisles 20; give the concept of new actions: multiplication and division; (introduce table multiplication and division within 20.
When selecting or producing special aids, you must remember that they must show the decimal composition of the numbers of the second ten, so the ten and ones must be clearly highlighted.
These benefits include: 20 sticks (10 sticks scattered and 10 tied in a bundle, i.e. 1 dozen); 20 cubes and 2 bars of 10 cubes; 20 squares and 2 stripes of 10 squares; ruler 20 cm long, all cardboard strips 10 cm long each, divided into 10 equal parts; coin box; class and individual abacus; digit table with units and tens digits; digital cash register; a table with numbers from 1 to 20 written in one and two rows; tables for counting in equal number groups of 2, 3, 4, 5; table with numbers from 1 to 20 showing even and odd numbers different colors; a set of tablets (10 pieces) with the number 10 for compiling and decomposing numbers (into tens and ones) from 11 to 20; signs with the number 20.
The basis for understanding the numbering of numbers of the second ten is the selection of ten and a clear idea that ten is ten units and at the same time it is new unit counting, which can be counted in the same way as units, adding one to the numbers, etc., the names of this counting unit, for example, one ten-ten.
The numbering of numbers within 20 consists of several stages: 1) obtaining one ten; 2) obtaining the second ten from 11 to 19 by counting several units to one; 3) obtaining the number 20 from two tens 1) written numbering of numbers from 11 to 20; 5) obtaining the second ten by counting one to the previous number and counting one bird from the subsequent number.
The score is within 20.
First, students need to repeat the numbering of the numbers of the first ten: obtaining numbers in a number series by adding to the previous number and subtracting 1 from the subsequent one, the relationship between neighboring numbers, the name of the numbers and their meaning in numbers. The teacher draws students' attention to the fact that each number from 0 to 10 is denoted by a new one, is not associated with another word, and to denote each of the numbers from O) 9 there is special sign, which is called a number. The number m is denoted by two digits 1 and 0. The teacher reports that there are only 10 digits. First, the counting in units within 10 is repeated and the receipt of a ten is shown. It is important to differentiate the concepts of “ten units” and “od > ten”. Ten is a whole, one.
The next stage in working on the numbers of the second ten is counting up to 20. Students must remember the names of numerals in the order of the number series, count objects, represent them with sounds, jumping, hitting the ball, clapping themselves given number several times, count a given number of objects in the aisles 20, counting is carried out by counting and reading one by one. When familiarizing yourself with numbering within 20 it is advisable. , introduce students to the unit of measurement dm.
Adding and subtracting numbers within 20 without jumping through place value
Repeat the decimal composition of numbers from 10 to 20, forward and backward counting from 1 to 20
Strengthen computational skills within 20 without going over the rank
The number series is from 10 to 20, but some numbers are missing digits.
each of you must take a number from my bag, with eyes closed guess it and put it in its place.
10,1., 1., 1., 14, 1., 1., 1., 1., 1., 2..
Repeating the decimal composition of a number
The teacher calls the decimal composition of the number, and the students show this number.
1dec.3 units, 1dec. 6 units, 1 des. 9 units, 2 des., 1 des. 2 units, 1 des. 8 units.
How many tens and ones are in the number 15? (In 15 there are 1 tens and 5 units.)
How can you get the number 15?
Mathematical dictation.
The teacher gives an example, and the students write down only the answer.
10 + 5 15 – 1 15 – 10 14 + 1 15 – 5
Answers: 15, 14, 5, 15, 10, 10.
Check: one student reads the answers, and everyone else checks.
Underline single digit numbers with one line.
Which numbers did you underline?
Solving a word problem.
Task: “The guys at the labor lesson were preparing decorations for the Christmas tree. On the first day they made 12 toys, and on the second day they made 2 fewer toys. How many toys did the guys make on the second day?
Work on the content of the task.
What does the problem say?
Who made the toys?
How many days did you make the toys?
Writing a short note.
How many toys did you make on the first day?
What does it say about the second day? (Said 2 less toys)
What does the problem ask? (The problem asks how many toys did the guys make on the second day?)
1 – 12 games.
2 – ? games., for 2 games. less.
Finding a solution to the problem.
So, how many toys were made on the first day? (12)
What is said about the second day?
What does “2 toys less” mean? (2 toys less - this is the same as on the first day, but without two).
How can we find out how many toys there are on the second day? (by subtraction)
How do we write the solution to the problem?
Did you answer the task question?
Recording the solution to a problem.
12 games. – 2 games. = 10 games.
Recording a response.
Answer: 10 toys.
sequence and techniques for learning addition and subtraction within 20.
I. Methods of addition and subtraction based on knowledge of the decimal composition of numbers (10+3, 13-3, 13-10) and numbering of numbers within 20 (16+1, 17-1).
When solving these examples, the relationship between addition and subtraction, the commutative property of addition, the names of components and results of actions are fixed. At the same time, students gradually stop using visual aids, but they are required to explain the actions.
II. Addition and subtraction without passing through ten.
The execution of actions is based on the decomposition of components into tens and ones: a two-digit number is added to a one-digit number. Subtract a one-digit number from a two-digit number. First we need to consider cases where the number of units in 1 slug. number is greater than in the second term (13+2, 1+3), and only then include cases of the form 11+6, 13+5, although their solutions are the same, --5
Explanation followed by use visual aids And detailed record solutions, for example: 13+2. The first term (13) consists of 1 ten and 3 units: 1 ten sticks and 1e 3 sticks. The second term is 2. Add 2 sticks. 3 sticks and 2 sticks - 5 sticks and 1 dozen sticks. Get 1 ten (sticks) and 5 units (sticks) - this is the number 15. Shechit, 13+2=15. The cases of you are explained in a similar way.
It is important to constantly emphasize that units are added and subtracted when solving such examples. When writing an example, students can underline the units: 14+2 = 16, 16-2 = 14. Sometimes it is advisable to write units and tens in different colors. You can circle them on the board.
When solving addition examples, students’ ability to use the commutative law of addition is strengthened: the solution to example 2 + 14 is carried out on the basis of the solution to example 14 + 2. It is useful to compare examples for addition and subtraction within 20 with examples for the same operations within 10:
7+ 2= 9 9-2= 7 5+ 3= 8- 3=
2+ 7= 9 9-7= 2 3+...= 8-...=
17+ 2=19 19-2 = 17 17+ 2= 19- 2=
2+17=19 19-7=12 2+...= 19-...=
b) obtaining the sum of 20 and subtracting a single-digit number from 20:
Solving examples of this type, especially subtraction, causes significant difficulties for many mentally retarded schoolchildren. Students are confused by the fact that when adding ones in the ones place, the result is zero. Dividing 20 into two tens and subtracting from one ten specified quantity units, children forget to add this result to ten and get the wrong answer: 20-3 = 7.
The use of visual aids, updating existing knowledge and relying on it helps to overcome these difficulties. It is necessary to repeat the table of addition and subtraction within 10. addition of a single-digit number to ten, subtraction from 10.
The explanation of addition does not represent anything new compared to the explanation of solving examples of the form 13 + 2, except for the formation of 1 ten: 5 + 5 = 10 (or 1 ten); 1 dec. + 1 dec.=2 dec.=20. ^"Consider an example of subtraction: 20-3. The number 20 has zero units, but you need to subtract 3 units. We take 1 ten, split it into 10 units and subtract 3 units, we get 7 units. In total, 1 ten and 7 units remain, or 17. Conducted reasoning
The movement is written like this: 20-3=17.
In case of difficulties in understanding and accepting calculations, an explanation can be carried out using sticks tied in bundles. For example, 20 is 2 tens (we take 2 bunches of sticks) and zero units. We take 1 ten and split it into 10 units (untie the bundle of sticks). 10 units minus 3 units equals 7 units. There are only 1 ten and 7 units left, or 17.
Examples of rearranging terms are solved, compiled according to the model, by analogy:
The operations of addition and subtraction are compared: 15+5=20; 20-5=15;
c) subtracting a two-digit number from a two-digit number: 15-12; 20-15. x The solution to examples of this type can be explained in different ways:
1. decompose the minuend and subtrahend into tens and ones and subtract tens from tens, ones from ones;
2. decompose the subtrahend into tens and ones. Subtract tens from the minuend, and units from the resulting number.
It is difficult for students to become familiar with two techniques at once, and even difficult to consistently become familiar with first one and then another technique. Mentally retarded schoolchildren They cannot independently choose when it is more appropriate to use one or another technique. Therefore, familiarity with two techniques only confuses them. It is better to work well on one method of calculation and teach students to use it independently.
Beginning of the form
End of form
12) Methodology for studying arithmetic operations. Addition and subtraction of numbers of the second ten (topic problems, cases considered, addition and subtraction with transition through place value; methods of familiarization with the combinatory property of addition, the rule for subtracting a number from a sum and a sum from a number).
Addition and subtraction within 20.
Mastering the computational techniques of addition and subtraction within 20 is based on a good knowledge of addition and subtraction within 10, knowledge of the numbering and composition of numbers within 20.
When studying the operations of addition and subtraction within 20, as well as when studying the corresponding operations within 10, great value has clarity and practical activities with benefits from the students themselves. Therefore, all types of visual aids used in the study of numbering will also find application in the study of arithmetic operations.
It is more appropriate to study the operations of addition and subtraction in parallel after becoming familiar with certain case addition study the corresponding case of subtraction versus addition.
In second grade, students should know the names of the components of addition and subtraction.
1. Techniques of addition and subtraction based on knowledge of the decimal composition of numbers.
2. Addition and subtraction without passing through ten:
a) a one-digit number is added to a two-digit number. A one-digit number is subtracted from a two-digit number;
b) obtaining the sum of 20 and subtracting a single-digit number from 20;
c) subtracting a two-digit number from a two-digit number: 15-12, 20-15.
Solving examples of this type can be explained in different ways:
1. Decompose the minuend and subtrahend into tens and ones and subtract tens from tens, ones from ones.
2. Decompose the subtrahend into tens and ones. Subtract tens from the minuend, and units from the resulting number.
3. Addition and subtraction with transition through series presents the greatest difficulties for students with psychophysical disorders. subtraction by passing through ten also requires a number of operations;
Divide the minuend into tens and ones
Decompose the subtrahend into two numbers, one of which is equal to the number of the minuend.
Subtract units
Subtract the remaining number of units from ten
Preparatory work should consist of repetition:
a) table of addition and subtraction within 10,
b) the composition of the numbers of the first ten (all possible options
of two numbers)
c) addition of numbers up to 10
d) decomposition of a two-digit number into tens and units
d) subtraction from ten single digit numbers
f) consideration of cases of type 17-8, 15-5.
Students work with the numbers 11, 12, 13, 14, 15, 16, 17, 18, 19).
student: “9+8=. We need to add 9 to 10, 8 is 1 and 7. 9 and 1 is 10. All that remains is to add 7, 10+7=17, which means 9+8=17. I'll do it in another way: 8+9=. 9 is 2 and 7, 8+2=10, 10 +7=17, which means 8+9=17. Rearranging the terms does not change the sum. So the calculation is done, right. Let's write the expression in your notebook 9+8=17.
placement of single-digit numbers with transition through ten
Let's do the addition by parts:
7 + 9 = (7 + 3) + 6 = 10 + 6 = 16 Answer: 7 + 9 = 16.
→ Arithmetic operations
Arithmetic operations
Finding one new number from several given numbers is called arithmetic operation. There are six operations involved in arithmetic: addition, subtraction, multiplication, division, exponentiation, root extraction.
1. Addition. This action consists of using several numbers, called addends, to find a number called their sum.
Example: 4+3=7, where 4 and 3 are terms, and 7 is their sum.
2. Subtraction- an action by which, from a given sum (minuend) and a given term (subtrahend), the desired term (difference) is found.
This is the reverse of addition.
Example: 7 – 3 = 4, where 7 is the minuend, 3 is the subtrahend, and 4 is the difference.
3. Multiplication. To multiply a certain number (multiplicand) by an integer (factor) means repeating the multiplicand as a summand as many times as there are units in the factor. The result of multiplication is called a product.
Example: 2 ∙ 3 = 6, where 2 is the multiplicand, 3 is the multiplier, and 6 is the product. (2 ∙ 3 = 2 + 2+ 2 = 6)
If the multiplier and the multiplicand change their roles, then the product remains the same. Therefore, the multiplier and the multiplicand are also called factors.
Example: 2 ∙ 3 = 3 ∙ 2, that is (2 + 2 + 2 = 3 + 3)
It is assumed that if the factor is 1, then a ∙ 1 = a.
For example: 2 ∙ 1 = 2, 44 ∙ 1 = 44, 13 ∙ 1 = 13.
4. Division. By dividing by this work(divisible) and given factor (divisor) find the required factor (quotient).
This is the inverse of multiplication.
Example: 8: 2 = 4, where 8 is the dividend, 2 is the divisor, and 4 is the quotient.
Checking division: the product of divisor 2 and quotient 4 gives dividend 8. 2 ∙ 4 = 8
Division with remainder
If, when dividing an integer by an integer, the quotient results in an integer, then such division of integers is called accurate, or that the first number completely divided(or simply - divided) by the second.
For example: 35 is divisible (by an integer) by 5, the quotient is the integer 7.
The second number is called a divisor of the first, and the first is a multiple of the second.
In many cases, you can find out without performing division Is it completely divisible? one integer divided by another (see signs of divisibility).
Exact division is not always possible. In this case, perform the so-called division with remainder. In this case, find the largest number that, when multiplied by the divisor, will give a product that does not exceed the dividend. This number is called incomplete private. The difference between the dividend and the product of the divisor and the partial quotient is called remainder of the division.
The dividend is equal to the divisor multiplied by the partial quotient plus the remainder. The remainder is always less than divisor.
Example: The partial quotient of dividing the number 27 by 4 is 6, and the remainder is 3. Obviously, 27 = 4∙6 + 3 and 3˂4.
5. Exponentiation. Raising a certain number to an integer power (to the second, third, etc.) means taking this number as a factor two, three times, etc. In other words, exponentiation is accomplished by repeated multiplication.
The number that is taken as a factor is called degree basis; a number indicating how many times a base is repeated is called exponent; the result of raising a number to a power is called power of this number.
Example: 2∙2∙2 = 2³ = 8; where 2 is the base of the degree, 3 is the exponent, 8 is the degree.
The second power of a number is also called square, third degree – cube. The first power of a number is the number itself.
6. Root extraction is an action by which, according to a given degree ( radical number
) And this indicator degrees ( root exponent) find the desired base (root).
This is the opposite of raising to a power.
Example: ³√64 = 4; where 64 is the radical number, 3 is the root exponent, 4 is the root.
Root extraction check: 4³=64. Raising the number 4 to the 3rd power gives 64.
The root of the second degree is also called square; root of the third degree - cubic.
At the sign square root It is customary to omit the root exponent: √36 = 6 means ²√36 = 6.
Liter used:
Guide to elementary mathematics- Vygodsky M.Ya., “Science”, 1974
Handbook of Mathematics. Manual for students 9-11 grades. - Shakhno K.U., "Uchpedgiz", 1961
We will divide the questions of methodology for studying arithmetic operations into two parts. In this part, we will look at how to form students’ ideas about addition, subtraction, multiplication, division, the concept of an arithmetic operation, and their properties, and in the next part of the chapter, how to develop computational skills.
7.3.1. Goals and results of studying arithmetic operations. Arithmetic operations – key concepts number theory and the most important characteristics of number sets. Their study is an integral part of the formation of the concept of number and computational skills. In mathematics, the generalization of arithmetic operations led to the concept of an operation, and then to such concepts as mathematical structure, group, ring, field, which play a huge role in modern mathematics and in its application in various areas of life. Learning arithmetic operations allows children to intuitively come into contact with many mathematical ideas, in particular, with the ideas of functionality, mathematical structure, mathematical modeling, and the principle of duality. Arithmetic operations have rich potential for the development of thinking, speech, formation and development of universal educational actions.
Arithmetic operations in modern forms records are convenient for observing and discovering patterns and constructing numerical sequences. They allow the invention of methods for performing actions and corresponding algorithms, methods for converting numerical expressions, and therefore can serve as a means of developing independent thinking and creative abilities. The task of teaching calculations has not lost its importance, although the role of computing skills has now changed. The goals of studying arithmetic operations and the requirements for the results of their study have also changed.
Learning Objectives arithmetic operations younger schoolchildren – personal and intellectual development, development of ideas about number and arithmetic operations, formation of computational skills, propaedeutic acquaintance with key ideas mathematics, achieving planned results.
Personal and meta-subject results are ensured by a) the nature of students’ presentation of arithmetic operations, including consideration of not only narrowly substantive, but also interdisciplinary, humanitarian aspects of them; b) increased attention to the meanings of arithmetic operations, to logical connections and conclusions, to the use of arithmetic operations to describe the world around us; c) inclusion in the process of studying the existing and emerging subjective numerical experience of children, the experience of cognition.
Personal results studying arithmetic operations – a formed attitude towards the world, people, oneself, learning, numbers and arithmetic operations. Meta-subject results related to arithmetic operations is the ability to use them as models substantive actions and means of obtaining new information in different areas of knowledge and everyday life, this is the ability to use drawings, diagrams, tables as a means of understanding the meanings and properties of arithmetic operations; knowledge of general arithmetic methods for solving problems; modeling situations using arithmetic operations. The meta-subject results of studying arithmetic operations also include UUDs formed during the study of any educational material.
Subject results- this is what every student will know about arithmetic operations as mathematical objects, what they will learn and have the opportunity to learn and learn. The teacher’s responsibility is to ensure that all students, upon graduation from primary school, achieve the planned results of studying arithmetic operations in accordance with the requirements of the Federal State Educational Standard. A version of the planned subject results is presented below.
As a result of studying arithmetic operations, a primary school graduate will learn: use arithmetic operations to describe and explain surrounding objects, processes, phenomena, their quantitative and spatial relationships, to solve word problems(in 2 – 3 steps); perform oral addition, subtraction, multiplication and division of single-digit, double-digit and three-digit numbers in cases that can be reduced to actions within 100 (including with zero and the number 1); perform arithmetic operations with multi-digit numbers using written calculation algorithms (addition, subtraction, multiplication and division by single-digit, double digit numbers within 10,000), use a calculator to check the accuracy of oral and written calculations; isolate the unknown component of an arithmetic operation and find its value; calculate the value of a numerical expression containing 2-3 arithmetic operations, with and without parentheses.
Graduate will have the opportunity to learn: use the properties of arithmetic operations to simplify and rationalize calculations; perform actions with value values; check the correctness of calculations, including calculators (using reverse action, estimation and evaluation of the result of the action).
Having formulated the planned results, it is necessary to specify diagnostic tools and diagnostic materials that make it possible to identify the degree to which a primary school graduate has achieved the planned results. Below is one possible task option for final assessment subject and meta-subject results.
A. Basic level.
1. Part of the wall of the house model is made of 5 identical wooden blocks shaped like a parallelepiped. (The dimensions of the block are 10 cm × 2 cm × 2 cm. The bars are stacked on the desk.) By measuring the lengths of the sides and the operations of addition, subtraction, multiplication and division, characterize this part of the wall by answering the questions: 1.1. What is the length, thickness, height of this part of the wall? 1.2. What is the surface area of the inside of the wall? 1.3. Compare the lengths of the sides of the block using the questions “Are they equal or unequal?”, “How many centimeters more (smaller)?”, “How many times more (smaller)?”
2. 4560 kg of rice cereal in bags of 80 kg each and 64 bags of buckwheat were brought to the warehouse. How many bags of cereals were brought to the warehouse?
3. Find the meanings of the expressions: (360 – 24 ∙ 5) : 40; 450:50; 78:4; 73 + 89; 0 ∙ 256; (36: 9 – 3) ∙ 17; 32 ∙ (1462 + 748) : (7846 – 7781)
IN. Increased level.
1. Part of the wall of the house model is made of 5 identical wooden blocks shaped like a parallelepiped. (The dimensions of the bar are 10 cm × 2 cm × 2 cm. The bars are stacked on the desk.)
By measuring the lengths of the sides and the operations of addition, subtraction, multiplication and division, characterize this part of the wall by answering the questions: 1.1. What is the length, width and thickness of this part of the wall? 1.2. What is the surface area of the inside of the wall? 1.3. What is the volume of the block? wall volume? 1.4. Compare the lengths of the sides of the block using the questions “How many centimeters more (smaller)?”, “How many times more (smaller)?” 1.5. Compare the volume of part of the wall and the volume of the block.
2. In the warehouse there are 4560 kg of rice cereal in bags of 80 kg each and 3840 kg of buckwheat in 64 bags. Which bag of cereal is heavier and by how much? Which grain has more bags and by how many?
3. Find the values of numerical expressions using mental calculations and properties of arithmetic operations: (480 – 24 ∙ 6) : 16; 354 + 188; 162:4; 18∙4 – 1345∙0; 317: 50; 45:45; (27 - 108: 9) ∙ 17.
4. Find the values of numerical expressions using written calculation algorithms: 26 (1672 + 1448) : (4825 – 4773)
“The skill being tested: the ability to perform arithmetic operations using studied algorithms (addition, subtraction, multiplication and division by single-digit and double-digit numbers within 10,000). Setting the baseline. Calculate: 2072: 37. Advanced level task. Petya performed the multiplication and saw that the same number was repeated four times in the record. He covered this number with cards and invited Misha to guess this number. What is this number?
Mark the correct answer ✔. □ 0 □ 4 □ 5 □ 6.”
« Skill: understand the meaning of division with a remainder, highlight the incomplete quotient and remainder. Setting the baseline. We bought candy for gifts. There are 199 candies in total. You need to put 5 candies in each gift. How many candies will be left? We bought 18 tickets for one compartment carriage for the football team. Ticket numbers from 1 to 18. How many compartments will football players be accommodated in if each compartment can accommodate 4 people?”
“Ability: to estimate and check the result of an arithmetic operation. Task 31 basic level. What number is the result of the action 12064: 4? Circle the answer number. 1) two-digit; 2) three-digit; 3) four-digit; 4) five-digit.
Task 32 advanced level. Is 1,000 rubles enough to buy four books at a price of 199 rubles per book and a calendar for 250 rubles? Write down and explain your answer. Answer: …
Explanation. Answer: not enough. An example of an explanation: after purchasing four books, there will be a little more than two hundred rubles left. This money is not enough to buy a calendar for 250 rubles. ..." 18 A possible explanation: “It’s not enough. In 1000 rub. contains 5 times 200 rubles. They pay 4 times for 1 ruble. less than 200, i.e. for 4 r. less than 4 times for 200 rubles. After paying for four books, there will be only 4 rubles left. more than 200, which is less than 250.” If the explanation is given “It’s not enough, because: 199 ∙ 4 = 796 (r.); 1000 – 796 = 204 (r.); 204< 250», то оценить владение прикидкой и оценкой по этому ответу нельзя, так как в этом обосновании они не показаны.
7.3.2. The sequence of studying arithmetic operations in elementary school. Traditionally, arithmetic operations are studied in the sequence: addition and subtraction, multiplication, division (whole) and division with a remainder. This order can be seen in many elementary school mathematics textbooks. However, there are other approaches to sequencing action learning.
In the history of Russian primary education, the operations of addition and subtraction were introduced and studied sequentially for a long time, with a significant gap in time. Then the opinion became recognized that long-term work with one arithmetic operation makes it difficult to master both operations, since students manage to develop a certain stereotype, which then needs to be destroyed. The simultaneous or sequential introduction of addition and subtraction in successive lessons creates conditions for comparing actions, which contributes to a better assimilation of meanings. Therefore, since the middle of the last century, the operations of addition and subtraction in our school were recommended to be studied simultaneously and introduced in one or sequential lessons.
There is no disagreement regarding the sequence of introducing multiplication and division. Multiplication is usually introduced slightly before division. Division begins to be studied after students have mastered the meaning of multiplication. Sometimes, after introducing multiplication, they study table multiplication, and only then division. But more often, table division is considered simultaneously with table multiplication in the same or consecutive lessons after the introduction of division.
There are different points of view regarding learning sequences full divisions And division with remainder. According to one of them, whole division, its meanings, and tabular cases of division are first introduced. After their assimilation, division with a remainder is introduced as a special action, with its own meanings, properties, and algorithms based on table division as a whole. Then the basic non-tabular methods of division by whole and division with a remainder are considered, and written division as division with a remainder, a special case of which is division by a whole - with remainder 0.
According to another point of view, division in whole and division with a remainder can be introduced as a designation for dividing a group of objects into parts equal to a given base (in accordance with the set-theoretic and magnitude meanings of the action of division) simultaneously or in a series of sequential lessons. The result of such an introduction will be the ability of students to designate the subject actions of dividing according to content and into equal parts by records of the form 12: 3, 13: 3, 12: 3 = 4, 13: 3 = 4 (rest. 1), and vice versa, perform objective actions or make drawings as written.
After mastering the subject meanings of division, which are the same for division by whole and division with a remainder, they move on to discussing the question of how to find the results of division without subject actions. The answer is sought by establishing the connection between division and multiplication first for integer division and focusing on tabular cases, properties of integer division, and properties of multiplication/division tables. Cases of division with a remainder are addressed incidentally during this period, consolidating its understanding, providing students with the opportunity to find the quotient and remainder based on an intuitive understanding of the connection between division by whole and division with remainder. After mastering table multiplication and division, the features, properties, methods and algorithms of division with a remainder are considered.
The justification for the latter point of view is that the presence or absence of a remainder does not change the course of practical division. For example, let's divide 12 and 13 cubes into equal parts of 3 cubes each. We proceed in the same way in both cases: take 3 cubes and put them aside. We repeat this action until we can take 3 cubes. Designated: 12: 3 and 13: 3. As soon as there are no cubes left or less than three left, we count the resulting parts. Their number will be private. In both cases, 4 equal parts of 3 cubes each were formed - the quotient will be the number 4. In the case of 12 cubes, there will be no “undivided” cubes left, and when dividing 13 cubes by 3, 1 cube will remain undivided. We get: 12: 3 = 4, 13: 3 = 4 (remaining 1).
We will divide 12 and 13 cubes into 3 equal parts. We take as many cubes as equal parts are required and arrange them one at a time. Then again we take as many objects as there are parts and arrange them one by one to those already laid out. We continue this way until there are no cubes left or there are fewer pieces left than the required number of pieces. In both cases, the quotient is 4 (each of the three equal parts has 4 cubes). When dividing 12: 3 there is no remainder, when dividing 13: 3 the remainder is 1. Entry: 12: 3 = 4 and 13: 3 = 4 (remaining 1).
In objective activities, when starting the division process, most often they do not know whether there will be a remainder. IN childhood experience There are many situations of practical division. Children share toys, candy, are divided into teams in games and much more. Complete division does not always work out. By introducing only complete division, it is necessary to protect children from situations where complete division is impossible. And if the period of meetings only with division is completely long, then children develop a stereotype: when dividing numbers, they always get one number - the quotient. This makes division with a remainder difficult to understand. This is partly why division with a remainder is considered a difficult operation, and word problems in which it can be used are either not considered (with the exception of simple tasks when introducing division with a remainder), or they are classified as problems of increased difficulty.
Based on the above reasoning, the sequence learning multiplication and division may look like this: introducing multiplication, mastering its meanings; introduction of division as a whole and with a remainder, mastering the meaning of division; table multiplication and division (integers); oral computational algorithms for division with remainder based on table division; algorithms for off-tabulation (oral) multiplication and division, including division with a remainder; written multiplication algorithms; algorithms written division as algorithms for division with a remainder, a special case of which is division with a zero remainder - division by an integer; multiplication and division using a calculator.
The study of each arithmetic operation can be presented in stages: preparation for the introduction of an arithmetic operation or actions; introduction of an action (actions), motivation to study, planning work on studying an arithmetic action (or actions), forming the meaning of the action being studied; studying the properties of arithmetic operations; studying algorithms for performing actions and developing computational skills.
Preparing to introduce an arithmetic operation or operations consists in creating a subject-activity basis for arithmetic operations, which is implemented in actions with groups of objects (set-theoretic approach) and with objects according to a given value (magnitude approach), in “walking” through a series of numbers, including the number 0 and the natural series (ordinal approach). Here it is necessary to clarify, deepen ideas about number, update methods of objective actions, and use them to solve text problems corresponding to arithmetic operations.
The main objectives of the lessons introducing an arithmetic action (or actions) and forming the meaning of the action being studied are: creating positive motivation for learning an action, isolating, performing and designating with a new action the objective actions underlying the introduced arithmetic operation; students’ mastery of terms and methods of symbolic designation and verbal description of actions; inclusion of a new arithmetic operation in the system of existing numerical representations.
Positive motives for learning action can be formed through children’s emotional experience of arithmetic action as a short and quick way to preserve and transmit information about action with objects, as a means of enrichment writing, as expanding communication opportunities, as a means of modeling task situations, as a means of obtaining new information. The subject of interest for children can and should be the properties of actions, the peculiarities of the behavior of individual numbers in relation to arithmetic operations, unusual methods of calculation, numerical sequences built on patterns expressed in the language of arithmetic operations. This is possible through the disclosure of the meanings of arithmetic operations, through the possibility of generating one’s own, personal meanings.
Let us remind you: arithmetic operations are mathematical operations on a number set (in elementary school on the set of non-negative integers). Operation – correspondence between a set of pairs of numbers from number set and elements of the same set. The match can be specified by an enumeration and a characteristic property. Such properties are included in the definition of an action. In the recording this is indicated by an action sign. In the entries 3 + 4, 17 – 9, 25 ∙ 7, 12: 6, 17: 5, operations are specified since specific pairs of numbers are indicated, and the sign indicates the method of obtaining the corresponding number. In the equalities 3 + 4 = 7, 17 – 9 = 8, 25 ∙ 7 = 175, 12: 6 = 2, 17: 5 = 3 (remaining 2), the corresponding number or numbers are specified not only by the characteristic property, but also by the enumeration .
Note that on initial stage mastering an arithmetic operation, as well as when studying properties, when generalizing some characteristics of an action, it is useful to use symbols for numbers invented by children, for example: ⌂ + ○; ⌂ – ○ = ◊ , ⌂ + ○ = ○ + ⌂ or ☼ +☺; ☼ +☺=☻. Such records allow us to consider an action and its properties when children cannot yet write down the necessary numbers, as well as when a specific numerical characteristic of groups of objects or an object cannot be accurately determined when it is necessary to show general view expressions and equalities. Moreover, such conventional signs carry the emotional component of their authors or “choices”.
Properties of arithmetic operations can be discovered by students in the process of educational and research activities organized by the teacher. It is important that each property is a solution to the problem accepted by the students, an answer to the question that arose in their minds. This can happen when, from the first days of education, we teach children to notice and identify similarities and differences between any objects, including between actions with objects, between their notes.
The main questions that lead to the discovery of the properties of arithmetic operations are questions about the possibility of replacing some expressions, and therefore a sequence of arithmetic operations, with others containing the same numbers and having the same numerical value as the original expression, but different actions or a different sequence actions.
The list of properties of arithmetic operations (on the set of natural numbers and zero) may be as follows:
Properties of the connection of relations "(directly) follow" and addition and subtraction: a + 1 = A′ And A′ – 1 = a(if you add 1 to a number, you get the next number; if you subtract 1, you get the previous number); commutative property of addition, multiplication 3 + 4 = 4 + 3, a + b = b + a, ab= ba; associative property addition ( a + b) + c = a + (b + c), multiplication ( ab)c = a(bc) or in the form of rules for adding a number to a sum and a sum to a number, multiplying a number by a product and product by a number; rules for subtracting a number from a sum and a sum from a number: (7 + 9) – 5 = (7 – 5) + 9 = 7 + (9 – 5), 9 – (4 + 3) = 9 – 4 – 3; rules for dividing a product by a number and numbers by a product: (12 8) : 4 = (12: 4) 8 = 12 (8 ; 4), 24: (3 4) = (24: 3) : 4; rule for dividing a sum by a number: if ac And bc (- is completely divisible), then ( a + b) : c = a:c + b:c, (60 + 12) : 6 = 60: 6 + 12: 6; distributive property of multiplication relative to addition (3 + 4) 5 = 3 5 + 4 5, 5 (3 + 4) = 5 3 + 5 4 or in the form of the rules for multiplying a sum by a number and numbers by a sum: ( 3 + 4) 5 = 3 5 + 4 5, 5 (3 + 4) = 5 3 + 5 4; rule for multiplying the difference by a number: (13 – 5) 2 = 13 2 – 5 2; properties reflecting the relationship between addition and subtraction, multiplication and division: a + b = c ↔ c – b = a And c – a = b; a : b = q ↔ a = bq And a : q = b, a : b = q(rest.r), r < b ↔ a = bq + r; dependencies between changes in components and the result of an action: a + b = c (a ± d) + b = c ± d (if one term is increased (decreased) by some number, then the sum will increase (decreased by the same number); a + b = c ↔ (a + d) + (b – d) = c (if one term is increased and the other is decreased by the same number, then the sum will not change); a – b = c ↔ (a ± d) – (b ± d) = c (if the minuend and the subtrahend are increased (decreased) by the same number, then the difference will not change); ab = c ↔ (a: d) b = c: d; ab = c ↔ (a: d)(bd) = (ad)(b: d) = c; a : b = q ↔ ad : b = CD; properties of division with a remainder: division with a remainder is feasible for any numbers (except division by zero); the remainder is less than the divisor; the dividend is equal to the sum of the product of the quotient and the divisor and the remainder .
If we take a closer look at the equalities expressing the properties of arithmetic operations, we will find that there is much in common in the properties of addition and multiplication, division and subtraction. This is where “ principle of duality 19, ..., which consists in the fact that each true statement of this section corresponds to a dual statement, which can be obtained from the first by replacing the concepts included in it with others, the so-called. concepts dual to them."
The principle of duality one of the important meaningful ideas of mathematics, which significantly expands the possibilities of knowledge. The idea of duality is discovered by children if the teacher organizes the study of a new action, the properties of this action on the basis of already learned actions, encouraging children to predict properties, check predictions, for example, using simple questions and tasks about similarities and differences: “How is subtraction similar to addition? ? How is it different?”, … “How is division similar to other arithmetic operations that you know? How is division similar to subtraction? How does division differ from subtraction?”, “You know that addition has commutative and combinative properties. Formulate the same properties for multiplication. Check their validity using several examples", "Formulate commutative and associative properties for division. Check their validity with several examples."
7.3.3. Learning addition and subtraction. The content of the study of actions significantly depends on the approach to the concept of number that the teacher adheres to, on the meanings that he puts into this concept. We will follow a universal approach, examining number with students in all its basic senses.
Set-theoretic meaning addition actions in a language accessible to students can be presented through tasks, describing the corresponding subject actions and drawings for them (Fig. 7.7). There are 4 apples on one plate and 3 on the other. How many apples are on the two plates? (Task to find the sum). There are 4 apples on one plate, and 3 more apples on the other. How many apples are on the other plate? There are 4 apples on one plate, which is 3 apples less than on the other. How many apples are on the other plate? (Problems with “more (less) by” relations in which the larger number is unknown.); There are 4 apples on one plate and 3 apples on the other. In how many ways can you choose one fruit? (Combinatorial problems specifying the sum rule for counting the number of combinations).
Tasks revealing set-theoretic the meaning of the subtraction action. a) There were 4 apples on the plate, 3 apples were eaten. How many apples are left? (Finding the remainder (difference)); b) There are 4 apples on one plate, and 3 less apples on the other. How many apples are on the other plate? There are 4 apples on one plate, which is 3 more apples than on the other. How many apples are on the other plate? There are 4 apples on one plate and 3 apples on the other. How many more apples are on the first plate than on the second? How many fewer apples are on the second plate than on the first? (Problems with relations “more (less) by”) with an unknown smaller number or how much one number is more or less than another (by difference comparison. (Fig. 7.8 a, b).
Meanings of addition and subtraction based on the concept of magnitude, express the operations of combining and removing objects with length, area, volume, mass and other quantities that can be shown practical action or drawing (Fig. 7.9)
Ordinal meanings of addition and subtraction manifests itself in a sequential transition from the first term to the number immediately following it, from it to the next as many times as the second term. Subtraction can be defined as a sequential transition from the minuend to the previous one as many times as the subtrahend. When introducing addition and subtraction, this meaning is represented by a rule that is formulated as a result of observing the position of a number to which a unit is added using actions with objects (from which a unit is subtracted) and the result of these actions: “If you add one to a number, you get the following number ; If you subtract one from a number, you get the previous number.”
Preparing to introduce addition and subtraction Exercises in actions with objects corresponding to the input actions, and the counting of objects and measures that accompanies these actions when measuring quantities in the simplest cases, are promoted. For example, counting steps when walking (measuring the length of a path), counting identical triangles, rectangles that make up a figure (measuring area), counting glasses of water poured into or poured out of a jar, movements of the second hand on a dial, etc. . Counting in twos, threes, fours, and fives is useful.
Possible types objective operations corresponding to addition and subtraction may be like this.
Place 3 cubes on the left. Place a card below the right number. Place 5 cubes on the right. Place a card with a number. Combine the cubes by moving them closer to each other. Find a strip of 3 units of length (3 measures consisting of three equal parts) and a strip of 5 of the same length units. Make one long strip from these two strips. What do the numbers 3 and 5 mean for dice? ... For stripes? ...What did you do with the cubes? ...What did you do with the stripes? ...
Count all the triangles. (8) Count all the red triangles. (3) Put them in an envelope. This jar contains 8 glasses of water. Pour out 3 glasses of water. Label with numbers.
Doing addition and subtraction. A feature of arithmetic operations, including addition and subtraction, which encourage children to study them, is the ability to reduce the recording of information many times over. To show this to students, as students complete the tasks above, the text appears on the board: Place 3 cubes on the left. Place 5 cubes on the right. Combined cubes. We took a strip 3 units long and a strip 5 units long. We made one long strip from two strips. (If subtraction is introduced simultaneously with addition, then the text will also contain sentences like: “There were 8 triangles. 3 triangles were removed”, “There were 8 glasses of water. 3 glasses were poured”). Below are the numbers written (or laid out on cards): 3 5 (8 3).
It is written on the board what you just did with cubes, with stripes, (with triangles, with water). Is it easy for you to read this text? (Not easy.) – But if you use the language of mathematics, you can write it down much more briefly. Maybe someone already knows how to denote our actions in mathematics? Together with the children, we construct a sample record (at first only the expression): 3 + 5 (8 – 5).
This entry replaces all of this text. How many digits are there in mathematical notation? (Total 3. With simultaneous introduction and subtraction - 6.) - How many characters are there in the text?
If the recording was made on interactive whiteboard, then by selecting the text it is easy to determine the number of characters: 163 (or subtracting 236!): 163! (or 236!) versus 3 (or 6!) the mathematical notation is more than 50 (almost 40 times) shorter! This discovery can be a point of surprise, which will give an emotional coloring to what is being studied and increase interest in it.
Perhaps some of you already know how to read this entry and what it means? (The children speak first, and then the teacher.) – The entry 3 + 5 is usually read “add five to three” (and “subtract five from eight”). Read it again with me. ... This entry means that there were 3 objects and 5 objects, and they were combined (There were 8 objects, 5 of them were taken and removed). Or that from two strips of length 3 and 5 units of length they made one strip of length 3 and 5 units of length. They also say that 3 + 5 is a notation for action addition(8 – 5 is an action record subtraction).
Next, three types of tasks are organized to develop the ability to move from subject actions to actions with numbers and from actions with numbers to subject actions: (1) subject actions are demonstrated (by the teacher, students, in pictures in a textbook or workbook, on an interactive board), and students mark them as corresponding numerical expressions, read expressions; (2) numerical expressions are named or shown (add two to four, subtract three from four, 4 + 2; 4 – 3), and students perform actions with objects, draw or select images of object actions that could be indicated by addition (subtraction ); (3) a correspondence is established between the image of objective actions and numerical expressions (drawings and expressions can be in manuals, on separate sheets, on a board, interactive or regular; these can be two sets of cards - with drawings of objective actions and with numerical expressions, or cards according to domino type).
Let's pay attention to several important points. Although the introduction to addition and subtraction comes from studying the numbers in the top ten, it is useful to consider the situations represented by addition and subtraction not only with the numbers in the top ten, but also with numbers in other number sets. For example, the teacher shows one box with 14 buttons, and another with 26 of the same buttons. On each box the corresponding number is written large. You need to put the same numbers on your desks with cards with numbers. Then he pours buttons from the second box into the first and asks students to put a card with the corresponding sign between the numbers. The resulting entry is: 14 + 26. With the help of the teacher, the children read the entry and say what it means.
At the beginning of the introduction of an arithmetic operation, we denote objective actions by a numerical expression or a numerical expression and equality. Equality requires naming and writing a specific number, the result of an action, while children do not yet know how to find it, other than objective actions and counting. A numerical expression does not name the number, the result of the action, but specifies the method of obtaining it with the sign of the action. In this case, we get the opportunity to consider action for any numbers and actions with any subject models of action. This is important for forming the meaning of action. Students also get the opportunity to determine the limit of applicability of calculations using objects, which motivates them to invent methods and algorithms without interacting with objects.
At the first stage of action learning, it is necessary to focus children’s attention on the questions “ What What is “addition”?”, “What is “subtraction?” Here it is preferable to write the action as a numerical expression. When the answers to the questions “What ...?” will be understood and appropriated, we can move on to the question “ How find the result of the action (the value of the sum, difference)? Now addition and subtraction can be written and spoken as equalities.
Before moving on to equalities and finding results and writing equalities, we summarize subtotal, giving students the opportunity to show their understanding of addition (and subtraction if the operations are introduced in the same lesson).
So, you now know how to denote actions with objects for adding numbers. Show how you can do it. Read mathematical notations and say what each one could mean: 3 + 2, 1 + 3, 5 + 8, 10 + 4, 1000 + 5000, Ω + ☼. (On the board there are corresponding drawings, for example, for the entry 1000 + 5000 there is a drawing of two banknotes, for the entry in “magic” numbers - two containers with cargo on a railway platform, indicating the mass in tons Ω and ☼.).
You said correctly: that addition denotes situations when something has been added to something, combined. How can we indicate what results from such actions? - Observe Dima’s movement, measure with him the length of each part of the path, counting the steps. (Dima takes 4 steps from the desk to the board, stops, then takes another 3 steps to the window). - Record the action. (4 + 3). – Dima, go through it again, counting all the steps. How many steps are there in total? (7) – How to write this down? Complete the record of what you did with the result of the action. (After the children’s suggestions, we write: 4 + 3 = 7. – Read this equality. (With the help of the teacher, read: “We added three to four and got seven.”)
Next, children complete tasks of the above types (1), (2) and (3). In the case when the number of objects in a combination or the number of measures when measuring a quantity can be counted, students write down equalities, in other cases they write down only expressions.
During the same period, the terms were introduced term, term, sum; minuend, subtrahend, difference. It is useful to preface the introduction of terms with a conversation about names. Each of us has many names and titles. One group of names are proper names: Tanya, Lena, Valentina Sergeevna. Names are also given according to what we do - cyclist, pedestrian, passenger, passer-by, reader; by occupation and profession - teacher, student, tailor, turner, pilot and many other reasons - person, employee, friend, sister, daughter, grandson.
If this approach is applied to numbers, then proper names are “one”, “two”, “three hundred and seventy”, etc. Participation of numbers in arithmetic operations and their execution certain functions or roles allows you to name them according to these functions. First, let the children propose their names and justify them. You can even announce a competition! Only in the context of their own word creation will generally accepted terms be “alive”, memorable, and emotionally charged for children.
When students move freely from subject situations to notation by addition and subtraction and vice versa, the question “How to find the result of addition, subtraction without drawings, counting on fingers, measuring?” will become relevant.
During this same period, it is already necessary to begin including children in planning your academic work, encourage reflection on the teaching and its results, i.e. to form educational activities, gradually, as they master the appropriate learning activities, transfer them from externally controlled educational activities to independent ones.
For example, after introducing addition and subtraction we ask:
Do you now know what addition is and what subtraction is? (Yes.) - Everyone, you know everything about addition? About subtraction? (No, not all.) – What else do you think we should know about these actions? What to be able to do? ... - What questions about addition and subtraction would you like answers to? What to learn? ...
Based on this dialogue, during which the teacher writes down the children’s questions and suggestions on the board, organizes an exchange of opinions, students, with the participation of the teacher as the organizer and bearer of knowledge about existing agreements, build a sequence for learning addition and subtraction.
The next pedagogical task is developing table calculation skills, A learning task students - learn to find the results of addition and subtraction, sum and difference (the value of the sum and the value of the difference), explain calculations, test yourself, plan further actions.
Studying the properties of addition and subtraction. The peculiarity of studying the properties of addition and subtraction is that these are the first arithmetic operations that children become familiar with. The properties of actions are considered during the period of mastering the objective meaning of actions and are justified by these objective, intuitive properties of actions. All properties can be discovered by children in a process organized by the teacher educational activities. It is important that property statements and notations are not cumbersome.
Many calculations in first grade, especially in the first half of the year, are performed in ways in which known properties appear on an intuitive level. These properties are presented with the participation of children in a form accessible to them. For example, methods for adding and subtracting one, by one, by parts: 3 + 4 = 3 + 1 + 1 + 1 + 1 = 3 + 2 + 2; 7 – 4 = 7 – 2 – 2 = 7 – 1 – 3.
The first properties available to students may be properties that connect the concepts of “next”, “previous” (“immediately following”) with the operations of addition and subtraction. This properties of the natural series, which manifest the ordinal meaning of a number in arithmetic operations, which we formulated above. This was preceded by the invention of methods for quickly counting objects in the combination of two groups of objects, for example, counting one group of objects by another to a known number of objects: ⌂⌂⌂⌂⌂⌂ - 6 ⌂ 7⌂ 8 ⌂ 9. 9 items.
The consequence of this method is to find the results of addition and subtraction by “stepping” along the natural series, first in single steps, and then in steps of a different length (addition, subtraction in groups).
Discover commutative property of addition or rearrangement of terms students can in several situations.
1. Using objective actions, calculate the values of pairs of the form 4 + 3 and 3 + 4. Establish similarities and differences. Make assumptions about the value of other similar sums, check the assumption by calculating the values using available methods.
2. In the process of performing objective actions of combining two groups of objects, two objects, substances, it is established that when the location of the parts or the order in which the combination occurs changes, the quantitative characteristics of the result of the combination do not change. By denoting objective actions with numerical expressions, we obtain two expressions with different orders of terms and identical values.
3. Two students, located on opposite sides of the table, indicated by addition (the sum of two terms) the number of objects on the table (Chekin A.L. Mathematics, 1st grade 2011) and received two different expressions: 3 + 4 and 4 + 3. By putting themselves in the position of each, children make sure that both entries correctly indicate the same situation, the number of the same objects. On this basis, 3 + 4 = 4 + 3. Since any other number of objects can be placed on the table, for example, Ω and ☼, then Ω + ☼.= ☼ + Ω, where Ω and ☼ are arbitrary numbers.
An important characteristic of addition and subtraction is that these actions express relationships « more (less) by" Any of the equalities of the form a + b = c And m – n = k defines relationships in which three numbers are involved: the greater, the lesser, and a number that answers the question how much one number is greater (less) than the other. If an equality is given, for example, 5 + 3 = 8, then the numbers related by the relation “more (less) by” can be the numbers 5 and 8, and the number 3 will show how much 5 is less than 8, and 8 is more than 5. tee, or 3 and 8, then 5 will show how much 3 is less than 8, and 8 is more than 3.
Other properties of addition and subtraction operations can also be discovered by students with appropriate organization. To discover properties, the focus of tasks on comparison, classification, and observation of changes is of great importance. With the introduction of the operations of multiplication and division, rules for the order of operations, the distributive property of multiplication relative to addition, the rule for dividing a sum, differences by a number, products by a number, numbers by a product, and other properties related to one or more properties are studied.
Further expansion and deepening of knowledge about addition and subtraction is associated with the expansion of numerical sets and the transfer of previously studied techniques, algorithms, terms, properties to them, with the study of properties and mastery of computational skills, with the enrichment of terminology with the names of properties (combinative property, distributive property), names ranks and classes, names of multi-digit numbers, characteristics of numbers.
7.3.4. Learning multiplication and division. First, let us recall the main meanings of multiplication and division.
Set-theoretic meanings of multiplication operations And divisions Let's present them with text problems and pictures for them. a) “There are 4 apples on one plate. How many apples are on 3 such plates? (Fig. 7.10 a); b) 3 teams participated in the chess tournament, each of which included 4 chess players - a candidate master of sports and chess players of 1st, 2nd and 3rd categories. How many chess players took part in the tournament?"; c) “There are 4 apples on one plate, and 3 times more on the other. How many apples are on the other plate?”, “There are 4 apples on one plate, this is 3 times less than on the other plate. How many apples are on the other plate? (tasks with relations “more (less) by ... times”, in which the larger number is unknown) (Fig. 7.10, c); d) In how many ways can the pair “envelope, stamp” be made if there are 3 types of envelopes and 4 types of stamps? (tasks on counting the number of combinations, product rule) (Fig. 7.10, d).
Dividing numbers in the set-theoretic sense arose as a designation two types of practical division of a group of objects into parts equal in number of items, which in mathematics teaching methods are called division by content And division into equal parts. Division by content: a group of objects is divided into parts according to a given equal number of objects in each part and it is required to find out how many such parts are formed. Division into equal parts: a group of objects is divided into a given number of equal (by the number of objects) parts and you need to find out how many objects there will be in each part.
Subject action division by content- this is the sequential putting aside of a given number of items until all the items are laid out or until there are fewer items left than there should be in one part. The procedure of postponement corresponds to the objective meaning of subtraction and can be designated by subtraction. Division acts as a shorter notation
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15 Sanitary and epidemiological rules and regulations SanPiN 2.4.2.2821-10. http://www.rg.ru/2011/03/16/sanpin-dok.html Date of access: December 4, 2011.
16 See Ondar Ch. Ethnocultural aspects in the formation of numerical representations // Primary school, 2010. - No. 11. – P. 104 – 107; Tsareva S.E. Poems, riddles, proverbs, sayings, fairy tales in primary education mathematics Novosibirsk, 1998.
17 Lysenkova S.N. When it's easy to learn. – M.: 1985.
18 Assessing the achievement of planned results in primary school. Task system. At 2 p.m. Part 1/ [M. Yu. Demidova, S. V. Ivanov, etc.]; edited by G. S. Kovaleva, O. B. Loginova - M. 2011. P. 58
19 http://slovari.yandex.ru/~books/TSB/Duality principle/.
Arithmetic operations
Objectives of studying the topic:
2) To acquaint students with the rules for the order of performing operations on numbers and, in accordance with them, develop the ability to find numeric values expressions.
3) Introduce students to identical transformations of expressions based on the properties of arithmetic operations.
There are 2 main stages in working on numerical expressions:
1) Studying the simplest expressions of the form: sum (2 + 3); difference(5 -1); product (3 4); private (12:4).
2) Studying complex expressions containing two or more actions, with and without parentheses.
1) When working with the simplest expressions in accordance with the requirements of the program, the teacher is faced with the task of developing in children the ability to read and write such expressions.
Students' first encounter with expressions occurs in the first grade in the topic "Numbers from 1 to 10", where children first become acquainted with the action signs "+" and "-". At this stage, children write down expressions and read them, focusing on the meaning of action signs, which they recognize as short designation the words "add" and "drop". This is reflected in the reading of expressions: 3 + 2 (3 yes 2); 3 - 1 (3 minus one).
Gradually, children’s ideas about these actions expand. Students will learn that adding a few units to a number increases it by the same number of units, and subtracting it decreases it. This is reflected when reading the expressions: 4 + 2 (4 increased by two units); 7 - 1 (7 decrease by one unit).
Then children learn the names of the plus and minus action signs. (When studying addition and subtraction of the first ten numbers). These expressions are read differently: 4 + 2 (4 "plus" 2); 7 - 1 (7 minus 1).
And only when familiarizing yourself with the names of the components and results of the action of addition, strict mathematical terminology is introduced, the name of this mathematical expression is given - “sum”, and a little later the term “difference” is similarly introduced.
The names of the next two mathematical expressions“product” and “quotient” are introduced similarly when studying the operations of multiplication and division in the second grade. Here, in the second grade, the terms “expression”, “meaning of expression” are introduced, which, like other mathematical terms, should be acquired by children naturally, just as they acquire other words that are new to them, if they are often used by others and find application in practice.
2) Along with the simplest mathematical expressions, complex expressions containing two or more actions, with and without parentheses, are also studied. Such expressions appear depending on the consideration of relevant issues in the mathematics course. However, their consideration is mainly subordinated to one didactic purpose– to develop the ability to find the meaning of an expression, and this is directly related to the rules for the order of performing arithmetic operations.
a) The first consideration is the rule about the order of operations in expressions without parentheses, when with numbers there is either only addition and subtraction, or only multiplication and division. The first such expressions of the form 5 + 1 + 1, 7 - 1 - 1 are found at the very beginning of the study of addition and subtraction of numbers within 10. Already here the main attention is paid to clarifying the question of how to reason when calculating the meaning of expressions. IN I-II grade there are exercises: 70 – 26 + 10, 90 – 20 – 15, 42 + 18 – 19; in grade II there are exercises: 4 · 10: 5, 60: 10 · 3, 36: 9: 2. Upon further examination of similar expressions, the conclusion is drawn: in expressions without parentheses, the actions of addition and subtraction (multiplication and division) are performed in the order how they are written: from left to right.
b) Then expressions containing brackets appear and again the main attention is paid to the rule about the order of actions in expressions with brackets. This way we actually introduce children to the second rule about the order of actions in expressions containing parentheses. Exercises: 80 – (34+13), 85 – (46 – 14), 60: (30 – 20), 90: (2 ·5).
In the second grade, when studying the operations of multiplication and division, we encounter expressions containing the actions of addition, subtraction, multiplication and division. To clarify the question of the order of execution of actions in such expressions, it is advisable for the first consideration to take the expression 3 · 5 + 3. Using the meaning of the multiplication action, we come to the conclusion that the value of this expression is 18. This implies the order of execution of actions. As a result, we actually get the third rule about the order of operations in expressions without parentheses containing the operations of addition, subtraction, multiplication and division: in expressions without parentheses, the operations of multiplication or division are performed first, and then the operations of addition or subtraction in the order in which they are written . At the same time, a sample of reasoning is given, where attention is drawn to pronunciation intermediate result, which allows you to warn possible errors children. Exercises: 21 + 9: 3, 34 – 12 2, 90: 30 – 2, 25 4 + 100.
Rules about the order of performing arithmetic operations deserve special attention. This is one of the complex and abstract questions of the initial mathematics course. Working on it requires numerous time-distributed training exercises. The ability to apply these rules in the practice of calculations is included in the basic requirements of the program at the end of each year, starting from the second grade and at the end of training in the primary grades.
Exercises:
1. From given pairs examples, select only those where calculations are performed according to the rules of the order of actions: 20 + 30: 5 = 10, 20 + 30: 5 = 26, 42 – 12: 6 = 40,
42 – 12: 6 = 5, 6 5 + 40: 2 = 50, 6 5 + 40: 2 = 35.
After explaining the errors, give the task: change the order of action so that the expression has set value.
2. Place parentheses so that the expression has the specified value:
72 – 24: 6 + 2 = 66, 72 – 24: 6 + 2 = 6, 72 – 24: 6 + 2 = 10, 72 – 24: 6 + 2 = 69
On last year teaching in primary school, the rules discussed are supplemented by new rules for children about the order of performing actions in expressions containing two pairs of brackets or two actions inside brackets. For example: 90 8 – (240 + 170) + 190, 469 148 – 148 9 + (30 100 – 26 909), 65 6500: (50 + (654 – 54)).
Familiarization with identical transformations of expressions. An identical transformation of an expression is a replacement given expression another whose value is equal to the value of the given expression. They perform such transformations of expressions based on the properties of arithmetic operations and the consequences arising from them (how to add a sum to a number, how to subtract a number from a sum, how to multiply a number by a product, etc.) For example: Continue writing so that the “=” sign is preserved :
76 – (20 + 4) = 76 – 20…
(10 + 7) 5 = 10 5…
60: (2 10) = 60: 10…
Using knowledge of the properties of actions to justify calculation methods, students perform transformations of expressions of the form:
36 + 20 + (30 + 6) =+ 20 = (30 + 20) + 6 = 56
72: 3 = (60 + 12) : 3 = 60: 3 + 12: 3 = 24
18 30 = 18 (3 10) = (18 3) 10 = 540
It is necessary to understand that all these expressions are connected by the “=” sign because they have the same meaning.
Identical transformations of expressions are also performed on the basis of the specific meaning of actions. For example, the sum of identical terms is replaced by the product: 6 + 6 + 6 + 6 = 6 4, and vice versa, 6 4 = 6 + 6 + 6 + 6. Also based on the meaning of the multiplication action, they transform more complex expressions: 8 4 + 8 = 8 5, 7 6 – 7 = 7 5.
If in expressions with brackets the brackets do not affect the order of actions, then they can be omitted: (30 + 20) + 10 = 30 + 20 + 10, (10 6) : 4 = 10 6: 4, etc.
Subsequently, using the studied properties of actions and rules for the order of actions, students practice transforming expressions with brackets into identical expressions without brackets. For example: write expressions without parentheses so that their values do not change: (65 + 30) – 20, (20 + 4) 3, 96 – (46 + 30)
Let's consider what theoretical and practical issues are studied in the topic “Arithmetic operations”, what is the level of their disclosure and the order of introduction.
The specific meaning of arithmetic operations, i.e., connections between operations on sets and corresponding arithmetic operations (for example, the connection between the operation of combining disjoint sets and the action of addition). Knowledge of the specific meaning of arithmetic operations must be acquired at the level empirical generalization: students must learn to practically establish connections between operations on sets and arithmetic operations when finding the results of arithmetic operations in a number of cases, as well as choosing arithmetic operations when solving text problems arithmetic problems.
Properties of arithmetic operations. These are mathematical provisions about identical transformations of mathematical expressions; they reflect under which transformations of a given mathematical expression its value does not change. The initial mathematics course includes properties that are theoretical basis computational techniques.
IN initial course mathematicians are studied following properties arithmetic operations: commutative and associative properties of addition, property of subtracting a number from a sum, property of subtracting a sum from a number, property of subtracting a sum from a sum, commutative and associative properties of multiplication, distributive property of multiplication relative to addition, property of dividing a sum by a number, property of dividing a number by a product .
The properties of arithmetic operations provided for by the program must be mastered at the level of conceptual generalization: students must know their formulation and practically apply them when justifying computational techniques, when solving problems, equations, exercises on identity transformations etc.
Other properties of arithmetic operations (existence and uniqueness of the result, monotonicity of the sum and product, etc.) are revealed at the level of empirical generalization: students practically operate with them, the formulation of the properties is not given.
Connections between components and results of arithmetic operations. These are mathematical statements that reflect how each of the components of arithmetic operations is expressed through the result and its other component.
In the initial mathematics course, the connection between the components and the result of the action of addition is first studied, and then the connection between the components and the result of the actions of subtraction, multiplication and division is studied.
Knowledge of connections must be acquired at the level of conceptual generalization: students must know the appropriate formulation and practically use this knowledge when solving equations and justifying computational techniques.
Changing the results of arithmetic operations depending on a change in one of the components, i.e., mathematical provisions that characterize how the value of an expression changes depending on a change in one of its components.
In relation to this material, an empirical level of generalization is provided: students, performing special exercises, observe corresponding changes in specific examples establish either the nature of the change in the results of arithmetic operations depending on the increase or decrease of one of the components, or establish quantitative changes– how the result will change if one of the components is increased or decreased by several units or several times. Such observations will serve to further basis to introduce the concept of function, at the same time they are great exercises developmental in nature.
Relationships between components and between components and the results of arithmetic operations. These are mathematical provisions that reflect the relationships “greater than,” “less than,” “equal to,” either between components (the minuend is greater than or equal to the subtrahend), or between the components and the results of arithmetic operations (the sum may be greater than each of the terms, or may be equal to one or each of the terms). This material is also absorbed at the level of empirical generalization: students establish appropriate relationships by performing special exercises. Knowledge of these relations is used to check calculations; they also serve the purposes of functional propaedeutics.
Rules. These are, first of all, provisions that are consequences of the definition of arithmetic operations and their specific meaning: the rules of addition and subtraction with the number 0, multiplication and division with the numbers 1 and 0, as well as historically established provisions - rules on the order of performing arithmetic operations in mathematical expressions. Students must understand the wording of the rules and be able to practically use them.
Terms and symbols. In connection with the study of these issues related to theoretical material, the corresponding terminology and symbolism are introduced: the name of arithmetic operations, the symbols denoting them and their name, the name of the components and results of arithmetic operations, the name of the corresponding mathematical expressions. The terms must be included in active dictionary students and be used by them in formulating mathematical statements, students must also learn to correctly use the appropriate symbols. Terms and symbols are entered in close connection with the study of relevant arithmetic operations.
Along with theoretical material and in organic connection he is being treated practical questions: computational techniques and solving arithmetic problems. Computational techniques are techniques for finding the results of arithmetic operations. Computational techniques are revealed based on the explicit use of appropriate theoretical provisions. For example, based on the commutative property of addition, the technique of rearranging terms is introduced. Each center studies computational techniques over integers. non-negative numbers the corresponding segment of the natural series (in the first concentration - within 10, in the second - within 100, etc.). In the "Ten" concentration, only addition and subtraction techniques are studied, and in the remaining concentrations, techniques of all four arithmetic operations are studied.
The order of introduction of all the above questions is subject to main goal studying arithmetic operations - the formation of conscious, strong, automatic computational skills.
3. General provisions methods of forming concepts and ideas about arithmetic operations in primary schoolchildren.
Students' assimilation of theoretical material comes down to their assimilation of the essential aspects of the mathematical principles being studied at the level of generalization provided for by the program. Consequently, all students’ activities in acquiring knowledge should be aimed at highlighting and understanding the essential aspects of the theoretical principles being studied. This is carried out mainly by students performing an appropriate system of exercises, which is subject to the goals of each stage of knowledge formation. In the methodology of knowledge formation there are the following steps: preparatory stage, familiarization with new material, consolidation of knowledge.
At the stage of preparation for familiarization with new theoretical material First of all, exercises are provided to reproduce previously acquired knowledge, which are means for assimilation of new knowledge. In most cases, during this period it is advisable to create in the minds of children “ subject models» generated knowledge by performing operations on sets. For example, before becoming familiar with specific meaning addition actions should be carried out sufficient quantity exercises to perform the operation of combining disjoint sets (add 3 balls to 4 balls and find out how many balls there are), which will later serve as the basis for becoming familiar with the meaning of the operation of addition.
At the stage of familiarization with new material the essential aspects of the mathematical propositions being studied are revealed with the help of a system of exercises performed by students. When becoming familiar with the properties of arithmetic operations, connections and dependencies between their components and results, it is more advisable to use heuristic conversation method, failing students inductively to the “discovery” of the corresponding pattern and convincing of its validity using visual means. When familiarizing yourself with the rules, when introducing terminology and symbolism, use method of explanation, i.e. The teacher presents the material, and the students perceive it.
Upon review inductively with the specific meaning of arithmetic operations, with their properties, connections and dependencies between components and results, students are offered exercises in which the corresponding patterns appear when performed. Analyzing them, students identify the essential features of the knowledge being formed and, depending on the level of its generalization, or formulate a number of particular conclusions (with empirical level), or from them pass to general conclusion(at the conceptual level). It is important to highlight not only essential features, but also a number of non-essential features. For example, consider how you can introduce the commutative property of multiplication. Students are asked to arrange 6 squares in each row into 4 rows and find out total quantity squares that were laid out. At the same time, students’ attention is drawn to the fact that counting total number squares can be carried out in two ways: 6* 4 = 24 and 4* 6 = 24. When comparing the received records, students establish similar features (products are given, the same factors are equal, the values of the products are equal) and distinctive features(multipliers are swapped). Next, similar exercises are performed, one or two of them being children. After completing enough exercises to compare pairs of products, students establish that all pairs of products have the same factors and the values of the products in each pair are equal, with the factors swapped. These observations allow students to come to a generalizing conclusion, which is a formulation of the commutative property of multiplication: “If the factors are swapped, the value of the product will not change.”
With this method of introducing new material, the exercise system must meet a number of requirements:
· The system of exercises should provide a visual basis for the knowledge being formed. Therefore, when performing exercises, it is important in many cases to use clarity: operations on sets (in the example considered, the union of equal disjoint sets of squares) and the corresponding mathematical notations (6* 4 = 24 and 4* 6 = 24). This creates the opportunity for the children themselves to “discover” the patterns they are studying.
· Exercises must be selected so that the essential aspects of the knowledge being formed remain unchanged, and the non-essential ones change. So, for the commutative property of multiplication essential features will be: the products have the same factors, the products differ in the order of the factors, the values of the products are equal; The unimportant features are the numbers themselves and their ratio. Therefore, when selecting pairs of works, you need to take them from different numbers, and the numbers are in different ratios (6* 4 and 4* 6; 2*5 and 5* 2; 7* 3 and 3* 7, etc.). This will allow students to highlight not only essential, but also non-essential features of new knowledge, which will contribute to correct generalization.
· Students should be encouraged to create exercises similar to those discussed. The ability to compose such exercises will indicate that students have identified the essential aspects of the knowledge being formed.
· When familiarizing yourself with new material, situations often arise when children’s previous experience has both positive and negative impact to master new material. This must be taken into account when introducing new material and provide special exercises for comparing and contrasting issues that have some similarities. For example, before learning the commutative property of multiplication, you need to repeat the commutative property of addition and use the same technique. In this case, an analogy will help when mastering a new property. Before studying distributive property When multiplying relative to addition, it is useful to repeat the associative property of addition in order to prevent confusion of these properties and the occurrence of errors when learning a new property.
So, as a result of performing special exercises, students are led either to a generalized formulation of the mathematical proposition being studied, or only to specific conclusions.
At the stage of consolidation of knowledge As a result of students completing a system of exercises to apply the studied material, their knowledge is enriched with new specific content and included in the system of existing knowledge. Consolidation of knowledge of each mathematical position is accomplished as a result of students completing special system exercises, subject to general requirements:
· Each exercise of the system must have the potential to apply the knowledge being generated. Then the student, performing them, will each time highlight the essential properties of the knowledge being formed and thereby better assimilate it. In this case, the first ones to include are exercises that can be performed both on the basis of the application of the knowledge being formed, and other previously acquired knowledge. Performing such exercises with the appropriate technique creates real opportunities to generalize the knowledge being formed by each student.
· Exercises to apply knowledge should be based on various specific contents (solving arithmetic problems, comparing mathematical expressions, etc.). This will ensure the formation of meaningful and flexible knowledge and prevent its formal assimilation.
· The system of exercises should ensure the establishment of intra-conceptual connections (connections between arithmetic operations, between their properties, etc.) and inter-conceptual connections (connections between the components and results of arithmetic operations with the solution of equations). This determines the inclusion of new knowledge into the system of existing knowledge.
· There should be a sufficient number of exercises to ensure the strength of the knowledge being formed.
· Exercises should be accessible to students and ranged from simple to complex.
· The system should provide special exercises that prepare students to master questions of a practical nature: performing calculations, solving arithmetic problems, solving equations, etc.
· At this stage, more than at the previous one, exercises should be provided for comparing and contrasting new material with previously learned material, which will prevent confusion of similar issues and help establish intra-conceptual and inter-conceptual connections.
· When organizing student activities at this stage, the method should be used more often independent work, to contribute in every possible way to the mental development of students.
· In addition, it must be taken into account that junior schoolchildren They learn the material better if it is included in lessons in small parts, but for a sufficiently long time.
Appendix No. 1
Arithmetic operations
| Action name | Signs | Name of the sign | Component name | Name of expressions | Reading examples |
| Addition | + | "Plus" | 3 – term 5 – term 8 – sum or value of the sum | 3 + 5 sum | Add Add Increase by... More by... Sum 1st term, 2nd term |
| Subtraction | - | "Minus" | 7 – minuend 4 – subtrahend 3 – difference or difference value | 7 – 4 difference | Subtract Reduce by... Less by... Difference Minuend, subtracted |
| Multiplication | *, X | Multiplication sign | 2 – multiplier 3 – multiplier 6 – product or value of the product | 2* 3 piece | Multiply Increase in... More in... Product 1st factor, 2nd factor |
| Division | : | Division sign | 8 – dividend 2 – divisor 4 – quotient or value of the quotient | 8: 2 quotient | Divide Reduce by... Less by... Quotient Dividend, divisor |
Appendix No. 2
Related information.
