First,? Digital operations and basic concepts

1. Operating knowledge system

The operation knowledge of numbers in primary school mathematics includes three aspects:

Basic operation method: addition, subtraction, multiplication and division

Basic principles of operation: basic principles of thinking and four algorithms.

Basic concepts of operation: sum, difference, division, average score, etc.

The operation process is the process of thinking, which is carried out in the brain. In this process, the relationship between numbers and operations is the relationship between thinking materials and thinking methods, both of which are indispensable.

The root of students' high calculation error rate lies in their poor understanding of related concepts and laws. Only when students have a thorough understanding of the basic concepts of logarithm (number of digits, counting unit and advancing rate) and operation (addition, subtraction, multiplication and division) can they demonstrate the steps and methods of operation. The idea of operation is arithmetic.

2. Reveal the operational significance with the concept of "harmony"

The concept of "harmony" essentially reflects the relationship between parts and the whole. Combining the two parts is the whole, and removing one part from the whole is the other part. In this way, with the concept of "harmony" as the core, through the relationship between part and whole, the connotation and internal relationship of the meaning of addition and subtraction operation are revealed. (You can review that the inverse operation of adding points is subtraction, and the equation solution is x+3= 10, 9-x=6). If the whole is formed by merging several parts, each part has the same number, then the relationship between parts and the whole is transformed into a "part" relationship. Then "share" becomes a special form between part and whole, and then the concept of "share" is used to reveal the connotation and internal relationship of multiplication and division. In the process of teaching, it can deepen students' understanding of operation by constantly clarifying such internal relations.

(1) initially established the concept of "harmony"

From the beginning of understanding, it is gradually established by studying the relationship between part and whole. For example, there are 1 apple on the left and 1 apple on the right. There are two apples in all. Q: Which two parts of these two apples are combined? Experience "harmony" and how to divide a * * * three flowers into several parts? Which two parts? Experience "points".

(2) Understand the operational significance of addition: the operation of combining two numbers into one number.

Understanding quantitative relations: What are the two parts of a unique number? I wonder how many kittens there are. what do you think? (Demonstrate merging with gestures)

With the help of the relationship between parts and the whole, students gradually establish a mathematical model of addition-combining the two parts into a whole and calculating by addition. If we master this model, the so-called problem that needs reverse thinking is no longer called a problem: the white rabbit took three carrots and there are five left in the bamboo basket. How many carrots were there in the original bamboo basket? According to the students' analysis, the white rabbit took three carrots, that is, the carrots in the original bamboo basket were divided into two parts-the three carrots taken by the white rabbit and the five carrots left in the bamboo basket. If you want to know how many there are in the bamboo basket, you should combine the three that took the white rabbit away with the original five in the bamboo basket.

(3) Understand the operational significance of subtraction.

Subtraction is still based on the concept of "sum", which is understood through the relationship between parts and the whole. There are five butterflies in the garden, and two have flown away. How much is left? Understand the quantitative relationship: flying away 2 just flies away from a few butterflies 2? This is how to divide five butterflies into several parts. Which two parts? I wonder how many butterflies are left. what do you think? Reveal the significance of subtraction: to remove a part from the whole and calculate it by subtraction.

Then change the situation: there are five butterflies in the garden, fly away a little, and there are three left. How many butterflies flew away?

Re-understand the quantitative relationship: "flying away" means dividing five butterflies into several parts. Which two parts?

(4) Reasoning training outside teaching.

First of all, the addition algorithm within 20 should be clear. Teachers should leave enough time for students to operate, so that they can understand the arithmetic of "ten-complement method" while operating. From 8+5 to 38+5 to 38+25 to 38+65. Students have the knowledge base and thinking conditions of "from ten to ten", and naturally infer from ten to one. What about a hundred to ten? Whoever arrives at ten o'clock will enter the next one.

The transition from integer addition and subtraction to decimal addition and subtraction is difficult in counterpoint. Because when students learn integer addition and subtraction, the same numbers are aligned and more intuitive. It is easy for students to transfer this cognition to decimal addition and subtraction. In teaching, we should firmly grasp the concepts of "number", "counting unit" and "forward speed", and emphasize that the same counting unit can be added or subtracted. For example, 35.6+7.98, let students know that these two addends are composed of dozens of tenths. When the unit of calculation is clear, the alignment of numbers will be clear, and then the calculation will be clear.

(5) Thinking training

1. basis. ?

2. expand. Expand the two parts into three parts.

Example 1: There are 6 black goldfish, 7 red goldfish and 4 Huang Jinyu goldfish. How many goldfish are there in a * *?

Example 2: There are 12 goldfish. First, three goldfish swam, and then seven goldfish swam. How many goldfish are left?

Second,? Explain the significance of multiplication and division with the concept of "share" as the core

(A) understand the significance of multiplication.

Key points: Mastering "several numbers" makes students feel the close relationship between multiplication and addition, and finds that multiplication is a simple operation of addition.

Thinking training:

(2)? Understand the operational significance of division

[ 1.？ Understand the concept of average score

Give two students 10 loaves, and each student will get two loaves. In this way, each share gets the same amount, which is called average score.

(1) How to divide 12 bottles of mineral water equally among three students? 12-3-3-3 = 0. There are four 3s in the name12.

(2) 12 bottles of mineral water, 4 bottles for each student, how many students can you share? 12-4-4 = 0, that is to say, there are three 4s in12.

2. Grasp the "average score" to understand the meaning of division.

Thinking training:

According to the observation chart, there are 3 flowers in each bundle, and there are 4 such bundles, and one * * * has 12 flowers. According to the operational significance of division and multiplication, three formulas can be listed: 3× 4 = 12, 12 ÷ 3 = 4, 12 ÷ 4 = 3. In this way, the bridge between multiplication and division can be built through the internal relationship between the three quantities.

(3) Reasoning training in multiplication and division teaching.

The focus of reasoning training should be the arithmetic of multiplying multiple digits by two digits and multiplying two digits by two digits.

1. Multiple digits times one digit.

First of all, understand the oral calculation of multiplying multiple digits by one digit 4× 2, 40× 2 and 400× 2; Then, the pen algorithm of multiplying multiple digits by one digit is studied. 24×3; Display the multiplication algorithm with the aid of addition algorithm;

2. Multiply two digits by two digits

First, calculate the breakthrough point, from 30×2 to 30 × 20. "30" means three tens and "20" means two tens. Ten tens are 1 100, two or three get six, and six is 600.

The key to writing arithmetic is to understand where the number on the tenth digit is written in the last digit of the product.

3. Fractional multiplication.

0.72×5, 0.72 means 72 percentage points, and the result of multiplication is 360 percentage points, so the decimal point should be after 3.

(3) Reasoning training in division operation teaching.

Key point: understand that divisor is the arithmetic of division of one digit.

1. The divisor is the division of a single digit.

The first is to understand that the divisor is a single digit, 6 ÷ 3, 60 ÷ 3, 600 ÷ 3; Then study the pen division with divisor of one digit, 52÷2, and divide 52 branches into 2 parts of 5 bundles 10. If the highest digit of the dividend is not enough, such as 237÷6, let's look at the highest digit of the dividend first, and divide the two hundred into six parts, each of which won't get hundreds. This requires the combination of two hundred decimals and three tens, and the 23 decimals are divided into six parts, and each part gets three tens, so the ten quotient is 3. Then divide these 57 decimals into six parts.

A divider is the division of two digits.

First of all, the breakthrough point is 80÷20 (there is a situation), eight tens divided by two tens, 270÷90. In addition, the problem of simple operation by using quotient invariance should also be solved according to arithmetic. 3800 ÷ 500 = 7 ... 300,3800 divided by 500 leaves 300.

3. Fractional division.

For fractional division, arithmetic is exactly the same.