1, control mode
How to correctly understand and apply mathematical concepts? The common method of primary school mathematics is comparison. According to the meaning of mathematical problems, the method of solving problems through understanding, memorizing, identifying, reproducing and transferring mathematical knowledge is called contrast method.
The thinking significance of this method lies in training students to correctly understand, firmly remember and accurately identify mathematical knowledge.
Example 1: The sum of three consecutive natural numbers is 18, so what are the three natural numbers from small to large?
By comparing the concept of natural numbers with the properties of continuous natural numbers, we can know that the average sum of three continuous natural numbers is the middle number of these three continuous natural numbers.
Example 2: True or False: The number divisible by 2 must be even.
Let's compare the two mathematical concepts of "division" and "even number". Only by fully understanding these two concepts can we make a correct judgment.
2. Formula method
Methods to solve problems by using laws, formulas, rules and rules. It embodies the deductive thinking from general to special. Formula method is simple and effective, and it is also a method that primary school students must learn and master when learning mathematics. But students must have a correct and profound understanding of formulas, laws, rules and regulations, and can use them accurately.
Example 3: Calculate 59? 37+ 12? 59+59
59? 37+ 12? 59+59
=59? (37+ 12+ 1) ..................................................................................................................................................
=59? 50 ...............................................................................................................................................................................
=(60- 1)? 50 ...............................................................................................................................................................................
=60? 50- 1? 50 ...............................................................................................................................................................................
= 3000-50 ......................................................................................................................................................................
= 2,950 ..........................................................................................................................................................................
3. Comparative method
By comparing the similarities and differences between mathematical conditions and problems, we study the reasons for the similarities and differences, so as to find a solution to the problem, which is the comparative method.
Comparative law should pay attention to:
(1) Finding similarities means finding differences, and finding differences means finding similarities, and being indispensable means being complete.
(2) Find the connection and difference, which is the essence of comparison.
(3) Comparison must be conducted under the same relationship (same standard), which is the basic condition of "comparison".
(4) To compare the main contents, try to use the "exhaustion method" as little as possible, which will make the key points less prominent.
(5) Because of the rigor of mathematics, comparison must be meticulous, and often a word or a symbol determines the right or wrong conclusion of comparison.
Example 4: Fill in the blanks: the digit of 0.75 is (), and the digit of the decimal part of this number is (); Compared with the tenth digit 4, the tenth digit 4 has the same (), but the tenth digit 4 is different, the former is smaller than the latter ().
The purpose of this question is to distinguish between "the difference between the digits of a number and the digits of the decimal part" and "the difference between digits and values".
Example 5: The sixth grade students planted a batch of trees. If everyone plants 5 trees, there are 75 trees left. If each person plants 7 trees, there will be a shortage of 15 seedlings. How many students are there in the sixth grade?
This is a comparison between the two schemes. The similarities are: the number of sixth grade students remains unchanged; The difference is that the conditions in the two schemes are different.
Find a connection: the number of trees planted by each person has changed, and the total number of trees planted has also changed.
Solution (Method): Everyone is 7-5=2 (tree), so the whole class is 75+ 15=90 (tree), and the class size is 90? 2=45 (person).
Step 4 classify
According to the similarities and differences of things, things are divided into different categories, which is called classification. Classification is based on comparison. According to the * * * similarity between things, they are grouped into larger classes, and the larger classes are subdivided into smaller classes according to differences.
Classification is to pay attention to the different levels between categories and subcategories to ensure that subcategories in categories are not duplicated, omitted or crossed.
Example 6: Natural numbers can be divided into several categories according to the number of divisors.
A: It can be divided into three categories. (1) A number with only one divisor is a unit number with only one number 1; (2) There are two divisors, also called prime numbers, and there are countless; (3) There are three divisors, also called composite numbers, and there are countless 1.
5. Analytical method
A way of thinking that decomposes the whole into parts, decomposes complex things into various parts or elements, and studies and deduces these parts or elements is called analytical method.
Foundation: The whole is made up of parts.
Thinking: In order to better study and solve the whole, first separate all parts or elements of the whole, and then compare the requirements separately, so as to straighten out the problem-solving ideas.
That is to say, starting from the problem to be solved, the two required conditions are correctly selected and deduced in turn until the problem is solved. This problem-solving model is "tracing the cause from the result". Analytical method is also called inverse method. "Branch diagram" is often used to illustrate this idea.
Example 7: The toy factory plans to produce 200 toys every day, which has been produced for 6 days, * * * producing 1260 toys. How many pieces exceed the standard on average every day?
Thinking: How many pieces do you want to exceed the plan on average every day? You must know how many pieces you plan to produce every day and how many pieces you actually produce every day. How many pieces are planned to be produced every day is known, and how many pieces are actually produced every day is not mentioned in the question, so we have to know. How many toys are required to be actually produced every day must be known: how many days are actually produced and how many pieces are actually produced, both of which are known.
6. Integrated approach
A way of thinking that combines all parts or aspects or elements of an object into an organic whole to study, deduce and think is called comprehensive method.
When solving mathematical problems by comprehensive method, each problem is usually regarded as a part (or element). After analyzing the internal relationship between each part (or element) layer by layer, the problem requirements are gradually deduced. Therefore, the problem-solving mode of comprehensive method is based on cause and cause, also called forward deduction method. This method is suitable for mathematical problems with few known conditions and simple quantitative relationship.
Example 8: The difference between two prime numbers is a composite number less than 30, and their sum is a multiple of 1 1, which is an even number less than 50. Write down the number of groups suitable for the above conditions.
Idea: Even numbers with multiples of 1 1 less than 50 are 22 and 44.
Both numbers are prime numbers, and the sum is even. Obviously, there is no 2 in these two prime numbers.
The two prime numbers whose sum is 22 are 3 and19,5 and 17. Are their differences all a composite of less than 30?
The two prime numbers with the sum of 44 are 3 and 4 1, 7 and 37, 13 and 3 1 respectively. Is their difference a composite of less than 30?
This is the idea of comprehensive method.
7. Equation method
Unknown numbers are represented by letters, and expressions (equations) containing letters are listed according to the equivalence relation. Column equation is an abstract and generalized process, and solving equation is a deductive process. The characteristic of equation method is to treat the unknown as a known number, participate in formulation and operation, and overcome the deficiency that arithmetic method must avoid the formulation of knowledge. It is beneficial to the transformation from known to unknown, thus improving the efficiency and accuracy of solving problems.
Example 9: A number is increased by 3 times 100, and then reduced by 2 times/36 to get 50. Find this number.
Example 10: 40% of a barrel of oil was used for the first time, which was 10 kg more than the first time, leaving 6 kg. How much does this barrel of oil weigh?
It is easier to solve these two problems with equations.
8, parameter method
A method of expressing related quantities by letters or numbers that only participate in formulas and operations without solving them, and listing formulas according to the meaning of the questions is called parameter method. Parameters are also called auxiliary unknowns and intermediate variables. Parametric method is the product of the extension and expansion of equation method.
Example: 1 1: the average speed of a car when climbing a mountain15km, and the average speed when going downhill10km. What is the average speed of cars?
The average speed of going up and down the mountain cannot be divided by 2 by the sum of the speeds of going up and down the mountain. Instead, you should take advantage of the journey down the mountain? 2。
Example 12: it takes 4 days for Party A to do a job alone, and 5 days for Party B to do a job alone. How many days does it take two people to work together?
In fact, the total workload is regarded as "1", and this "1" is the parameter. If the total workload is regarded as "2, 3, 4 ...", this is only the most convenient operation.
9. Exclusion method
The result of eliminating opposition is called exclusion.
The logical principle of exclusion is that everything has its opposite. In all kinds of right and wrong results, excluding all wrong results, the rest can only be correct results. This method is also called exclusion, screening or disproof. This is an indispensable method of formal thinking.
Example 13: Why are all prime numbers odd except 2?
This requires the reduction to absurdity: all natural numbers greater than 2 are either prime numbers or composite numbers. Suppose a prime number greater than 2 has an even number, then this even number must be divisible by 2, that is, it must have a divisor of 2. A number has other divisors (divisor 2) besides 1 and itself. This number must be a composite number, not a prime number. This is contrary to the assumption that it is a prime number at first. Therefore, the original assumption is wrong.
Example 14: True or False: (1) Two straight lines on the same plane will intersect if they are not parallel. (error)
(2) The numerator and denominator of a fraction are multiplied or divided by the same number, and the size of the fraction remains unchanged. (error)
10, special case method
For topics involving general conclusions, the method of solving problems by taking special values, drawing special pictures or setting special positions is called special case method. The logical principle of special case method is that the generality of things exists in particularity.
Example 15: The radius of a big circle is twice that of a small circle, its circumference is () times that of a small circle, and its area is () times that of a small circle.
You can take the radius of the small circle as 1, so the radius of the big circle is 2. Calculate and you will get the correct result.
Example 16: Is the area of a square proportional to the length of its sides?
If the side length of a square is A and the area is S, then s:a=a = a (the ratio is uncertain).
Therefore, the area of a square is not proportional to its side length.
1 1, reduction method
Through some transformation process, the method to solve the problem by simplifying it into a typical problem is called simplification. Transformation is an important way of knowledge transfer and the first step to expand and deepen cognition. The logical principle of reduction is that things are generally related. Transformation is a common dialectical thinking method.
Example 17: a pharmaceutical factory produced a batch of anti-SARS drugs, which was originally planned to be completed in 25 people 14 days. Due to the urgent need, it needs to be completed four days in advance. How many more people do you need?
This requires that "total working days" be classified as "total workload" when considering problems.
Example 18: potatoes, tomatoes and cowpeas were delivered from the supermarket, of which potatoes accounted for 25%, and the weight ratio of tomatoes to cowpeas was 4: 5. It is understood that cowpea is 36 kilograms heavier than potato. How many kilograms of tomatoes were shipped in the supermarket?
It is necessary to classify "the weight ratio of tomatoes and cowpeas is 4: 5" as "how many percent of the total weight each", that is, to classify the application of proportion as the application of scores.