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Application problems are the key and difficult points in primary school mathematics teaching, especially some complex application problems. Because of the hidden quantitative relationship, it is difficult for students to find the correct problem-solving ideas when solving problems, and there will be such problems. Therefore, in the teaching of practical problems, teachers should teach students to use existing mathematical knowledge, boldly imagine, try to explore from different angles through different methods, and cultivate divergent thinking ability. Therefore, we should pay attention to the training of various problem-solving ideas.

First, the corresponding thinking training

Example 1: A farmer keeps 240 chickens, and an average of 5 chickens have to be fed 4.5 kilograms of feed for 6 days. According to this calculation, how many kilograms of feed should these chickens feed 15 days?

Write conditional questions in the questions:

5 chickens 6 days 4.5 kg

240 chickens 15 days? kilogram

From the above correspondence, two methods can be analyzed:

① The feed of 1 chicken 1 day was obtained by normalization method, and then the feed of 240 chickens 15 days was obtained. that is

4.5 ÷ 5 ÷ 6× 240× 15 = 540 (kg)

Answer: 240 chickens 15 days need 540 kilograms of feed.

② The average daily feed used by each chicken is certain. According to the multiple relationship, this problem can be solved as long as 240 is only several times that of 5 chickens and 15 days is several times that of 6 days.

4.5× (240 ÷ 5 )× (15 ÷ 6) = 540 (kg) (omitted)

Second, the combination of numbers and shapes to look at the picture analysis training

Example 2: The road team built a section of highway in three days, the first day was 40%, the second day was 1/2, and the third day was 2.5km. How many kilometers is this road?

Firstly, segment drawing:

Attached drawings (drawings)

Re-analysis and solution: If the whole road section is regarded as "1", then the 2.5km built on the third day is exactly the whole road section (1-40%- 1/2), corresponding to 2.5, so the length of the whole road section is:

2.5÷ (1-40%-1/2) = 25 (km) (omitted)

Example 3: Two-fifths of a barrel of oil was taken out for the first time, and 20 kilograms was taken out for the second time, leaving 28 kilograms of oil in the barrel. How much does the whole barrel of oil weigh?

Firstly, segment drawing:

Attached drawings (drawings)

Taking the whole barrel of oil as the unit of "1", it can be clearly seen from the figure that the sum of the oil taken out in the last two times is exactly the remaining part after the first oil extraction, that is, (1-2/5), which corresponds to (20+28).

Column calculation: (20+28) ÷ (1-2/5) = 80 (kg) (omitted)

Third, the training of multiple solutions to one question.

In order to cultivate students' thinking ability and guide them to explore ways to solve problems, we can analyze and compare the quantitative relationship of a problem and communicate the internal relations of knowledge from multiple angles and levels.

Example 4: Students take part in camping activities, and one of them asks the teacher in charge of logistics for a bowl. The teacher asked him how much he got in the exam, and he said he got 55; He asked "how many people eat", and he said "one person has a bowl of rice, two people have a bowl of vegetables, and three people have a bowl of soup". Calculate, how many bowls did this classmate take to camping activities?

Solution 1: General solution

Take the number of rice bowls as "1", the number of vegetable bowls is 1/2, the number of soup bowls is 1/3, and the total number of bowls is 55 (1+ 1/2+ 1/3).

55÷ (1+1/2+1/3) = 30 (piece)

According to the question, the number of people is the same as the rice bowl. (short answer)

Solution 2: Equation solution

There are x people taking part in camping activities. According to the meaning of the question, the number of rice bowls is X, the number of vegetable bowls is X/2 and the number of soup bowls is X/3. The equation is X+X/2+X/3 = 55 and the solution is X = 30. (short answer)

Solution 3: Proportional allocation solution

If the number of rice bowls is "1", then

Number of rice bowls: number of dishes: number of soup bowls

= 1∶ 1/2∶ 1/3=6∶3∶2

The number of rice bowls is 55× 6/6+3+2 = 30.

There are as many people as bowls. (short answer)

The solution to this problem is not limited to the above three, there are other solutions, so I won't go into details here.

Fourthly, group training of transformational problems.

There are many application problems with different topics, but the quantitative relationship is the same and the solutions are exactly the same. Putting this kind of application problems together will help students grasp the essence of the problems and find out the laws to solve these problems.

Here is a set of questions:

(1) It takes 12 days for Team A to build a project, and 20 days for Team B. How many days does it take for two teams to build together?

(2) It takes 2 hours for Party A to walk from Dongzhuang to Xizhuang and 3 hours for Party B to walk from Xizhuang to Dongzhuang. If Party A and Party B leave from Dongzhuang at the same time, how many hours will it take to meet?

(3) A and B children's wear factories jointly produce a batch of children's wear for export. A factory works alone for 20 days, and B factory works alone for 30 days. How many days can the cooperation between the two factories be completed?

(4) A pool is equipped with two water inlet pipes, A and B. It takes 6 minutes to fill a single open pipe, 4 minutes to fill a single open pipe, and how many minutes to fill a two-way pipe?

Analysis: (1) Let the total project amount be "1".

Party A completes the project every day112, Party B completes the project every day 1/20, and Party A and Party B jointly complete the project112+120, and the number of days required to complete the whole project is1.

(2) Let the distance from Dongzhuang to Xizhuang be "1".

The speeds of Party A and Party B are 1/2 and 1/3 respectively. Party A and Party B complete the race every hour (1/2+ 1/3), and the time required for them to meet is 1 ÷ (1/2).

(3) Let the total amount of these children's clothes be "1".

The daily workload of A factory is 1/20, and that of B factory is 1/30. The total amount completed by two factories in one day (1/20+ 1/30), and the number of days required after the work is completed is 1 ÷ (65438+).

(4) Set the volume of the pool as "1". According to the meaning of the question, a pipe can be filled with water 1/6, a pipe can be filled with water 1/4, a pipe and a pipe can be filled with water (1/6+ 1/4) every minute, and the time required for filling is/kloc.

Through the above analogy training, students can understand engineering problems, problems encountered, work problems and water pipe problems. Although the themes are different, their quantitative relationship is the same. This makes the connection between knowledge form in students' minds.

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