Teaching plan of the basic nature of mathematical proportion in the first volume of the sixth grade of primary school by People's Education Press
Teaching content: the content of pages 50 ~ 5 1 in the sixth grade mathematics textbook of primary school published by People's Education Press and related exercises.
Teaching objectives:
1. Understand and master the basic properties of the ratio, and apply the basic properties of the ratio to simplify the ratio, and initially master the method of simplifying the ratio.
2. In the process of independent exploration, communicate the relationship between ratio, division and score, and cultivate mathematical abilities such as observation, comparison, reasoning, generalization, cooperation and communication.
3. Infiltrate and transform the mathematical thought initially, so that students can know that there is an internal connection between knowledge.
Teaching emphasis: understanding the basic nature of ratio
Difficulties in teaching: Correctly using the basic properties of ratio to simplify ratio.
Teaching preparation: courseware, answer sheet, physical projection.
Teaching process:
First, review the introduction.
1. Teacher: Students, let's recall first. How much do you know by comparison?
Presupposition: the meaning of ratio, the name of ratio part, the relationship between ratio and fraction, division, etc.
2. Can you say 700? 25 business?
(1) What do you think?
(2) What is the basis?
Do you remember the basic nature of music score? Give examples.
An important factor that affects students' learning is what they already know, so this link is intended to let students communicate the relationship between ratio, division and score, reproduce the invariable nature of quotient and the basic nature of score, and lay the foundation for the basic nature of analogy and deduction ratio. At the same time, there is also a mechanism that permeates the transformed mathematical thought, which makes students feel that there is a close internal connection between knowledge.
Second, explore new knowledge.
(A) the basic nature of the conjecture ratio
1. Teacher: We know that ratio is closely related to division and fraction, while division is quotient invariant and fraction has the basic properties of fraction. Think about these two properties: what kind of laws or properties will proportion have?
Default: Basic attribute of ratio.
The students have guessed the basic nature of the ratio.
Default: The first and last items of the ratio are multiplied or divided by the same number at the same time (except 0), and the ratio remains unchanged.
3. According to the students' guess, the teacher wrote on the blackboard that the first and second items of the ratio are multiplied or divided by the same number at the same time (except 0), and the ratio remains unchanged.
Design intention is more suitable for cultivating students' analogical reasoning ability than basic nature learning. On the basis of mastering the basic properties of quotient invariance and score, students can naturally associate with the basic properties of ratio, which not only stimulates students' interest in learning, but also cultivates students' language expression ability.
(B) the basic nature of the verification ratio
Teacher: As everyone thinks, ratio, like division and fraction, has its own regularity, so do you want to guess with everyone? The first term and the last term of the ratio are multiplied or divided by the same number at the same time (except 0), and the ratio remains unchanged? What about the same? This needs us to prove through research. Next, please study in groups of four to verify whether the previous guess is correct.
1. The teacher explained the cooperation requirements.
(1) Complete independently: Write a ratio and verify it in your favorite way.
(2) Group discussion and study.
① Each student presents his own research results to the students in the group and communicates in turn (other students indicate whether they agree with the student's conclusion).
(2) If there are different opinions, give examples, and then the students in the group will discuss and study.
③ Choose a classmate to speak on behalf of the group.
2. Collective communication (ask the group spokesperson to explain with specific examples on the booth).
Preset: verify according to the relationship of ratio, division and score; Verify according to the proportion.
3. Category verification.
;
;
16:20=( 16○□):(20○□)。
4. Perfect induction and summarize the basic nature of ratio.
How to fill in the ○ in the above questions □ Can I fill in any number? Why?
(1) Students express their opinions and explain the reasons, and teachers improve the blackboard writing.
(2) The basic nature of students' reading ratio with books open, and teachers write on the blackboard. (Basic attribute of ratio)
5. Questioning discrimination and deepening understanding.
Make an accurate judgment by using the basic nature of the ratio;
( 1) ( )
(2) ( )
(3) ( )
(4) The former term of the ratio is multiplied by 3, and the latter term of the ratio should be divided by 3 to keep the ratio unchanged. ( )
Learning with design intent based on conjecture must be verified by students' independent inquiry. Cooperative inquiry is a good learning method, but cooperative learning cannot become a mere formality. Cooperative learning should first let students think independently, produce their own ideas, and then carry out cooperative exchanges, so that each student can experience the learning process of independent inquiry. In the process of communication, it not only cultivates students' reasoning and generalization ability, but also internalizes students' guesses. What is the basic nature of ratio? , thus greatly improving the effectiveness of cooperative learning.
Third, the application of the basic nature of the ratio
Teacher: Students, do you still remember the basic purpose of our study scores? What is the simplest score?
The basic properties of the ratio we found today also have a very important purpose-we can simplify the ratio and then get the simplest integer ratio.
(1) Understand the meaning of the simplest integer ratio.
1. Guide students to learn the simplest integer ratio by themselves.
Default: The prime integer ratio of the former and the latter is called the simplest integer ratio.
2. Find the simplest integer ratio from the following ratios and briefly explain the reasons.
3:4; 18: 12; 19: 10; ; 0.75:2。
(2) Preliminary application.
1. Simplify the ratio of front and back terms to integers. (Courseware shows page 50 of the textbook, for example 1)
Students try independently and communicate after simplification.
( 1) 15: 10=( 15? 5):( 10? 5)=3:2;
(2) 180: 120=( 180? □):( 120? □)=( ):( )。
Default: Divide by the greatest common factor, divide by the common factor step by step, but emphasize the method of dividing by the greatest common factor.
2. Simplify the proportion of fractions and decimals in the preceding and following items. (Courseware demonstration)
Teacher: for the ratio of the front and back terms to integers, we just need to divide by their greatest common factor, but like: and 0.75:2,
These two ratios are not the simplest integer ratios. Can you find a way to simplify them yourself? Discuss and study in groups of four, and try to simplify.
Students study and write down the specific process, summarize the methods, and select representatives to show and report. Teachers compare different methods and guide students to master general methods.
Default: Ratios containing fractions and decimals should be converted to integer ratios before simplification. Least common multiple of denominator with fraction; If there are decimals, they should be converted into integers before simplification.
3. Summary: Through their own efforts to explore, the students summed up the methods of transforming various proportions into the simplest integer proportions. When simplifying, if the first term and the last term of the ratio are integers, they can be divided by their greatest common factor at the same time; When encountering decimals, first convert them into integers, and then simplify them; When you encounter a score, you can multiply it by the least common multiple of the denominator at the same time.
4. Method supplement, distinguishing between simplified ratio and calculated ratio.
What other methods can simplify the scale? (find the ratio)
What's the difference between simplifying ratio and seeking ratio?
Default: the final result of simplifying the ratio is a ratio, and the final result of finding the ratio is a number.
5. Try to practice.
Turn the following ratios into the simplest integer ratios (show the textbook page 565438 +0? Do it. )。
32: 16; 48:40; 0. 15:0.3;
; ; .
Should the new curriculum standard fully reflect the design intent? Student-oriented development? Teaching philosophy, give full play to the main role of students, so that students can become the masters of learning. Therefore, in the teaching process of simplifying ratio by using the basic nature of ratio, students are encouraged to explore independently and find ways to simplify ratio through self-study, independent exploration and group cooperation.
Fourth, consolidate practice.
Basic exercises
1. textbook page 53, question 4.
Convert the following ratio to 100.
(1) The ratio of the number of surviving plants to the number of school plants is 49:50.
(2) Prepare the medicament, and the ratio of the mass of the medicament to the total mass of the medicament is 0. 12: 1.
(3) The ratio of the actual output value to the planned output value of the enterprise in the previous year was 2.75 million: 2.5 million.
2. Question 6 on page 53 of the textbook.
(2) Expanding exercises (PPT courseware demonstration)
The students finished their oral answers.
In the ratio of 1.2:3, the former item should be increased by 12, and the latter item should be increased to keep the ratio unchanged.
The number of boys in Class 2.6 (1) is 1.2 times that of girls. The ratio of boys to girls is (), the ratio of boys to class is (), and the ratio of girls to class is ().
The design of design intention exercises should closely focus on the key points and difficulties of teaching, and the arrangement of exercises should reflect the hierarchy from easy to difficult. 1 is a basic exercise for the basic properties of ratio, and also lays the foundation for the following percentage learning. The second problem is to simplify the ratio of two different quantities in training units and cultivate students' ability to examine questions. Expanding exercises not only develop the flexibility of students' thinking and cultivate their creativity, but also consolidate the knowledge of this lesson. At the same time, this kind of problems is also the basic training of fractional application problems and proportional application problems, and also lays a solid foundation for the study of fractional application problems and proportional application problems in the future.
Verb (abbreviation of verb) course summary
What did you learn from this course? Is there a problem?
Reflection after class:
Teaching design of solving problems by using proportional distribution
Teaching content: Example 2 and related exercises on page 54 of the first volume of the sixth grade mathematics textbook published by People's Education Press.
Teaching objectives:
1. Can understand the practical significance of proportional distribution in the analysis of examples.
2. Initially master the problem-solving method of proportional distribution, and use the knowledge learned to solve the practical problems of proportional distribution.
3. Through learning examples close to students' life, let students feel the fun of mathematics learning and activities in observation, discussion and communication.
Teaching emphasis: understand the significance of proportional distribution, and use the significance of ratio to solve the practical problems of proportional distribution.
Teaching difficulties: independently explore the strategies to solve the practical problems of proportional distribution, and can use different methods and angles to solve the practical problems of proportional distribution.
Teaching preparation: courseware.
Teaching process:
First, situational introduction
The courseware shows that the ratio of girls to boys is 5:7.
Teacher:? The ratio of girls to boys is 5:7? What information did you get from this sentence?
Designing a simple real life message can not only make students realize the connection between mathematics and life, but also stimulate their interest in learning and cultivate their ability to analyze and solve problems.
Second, the case study
(A) independent exploration
1. shows that there are 48 students in Class 6 (2), and the ratio of male to female students is 5:7.
Teacher: What can you get from these two messages? How many boys and girls are there? Can you calculate?
2. Students try independently.
3. Communicate at the same table.
Teacher: Talk to your deskmate about your ideas and practices. You can write it down in different ways. (Teachers' patrol guidance)
4. Report:
Please practice different students to perform on stage and exchange reports.
Default (1): 48? (5+7)=4 (person);
Girl: 4? 5=20 (person);
Boy: 4? 7=28 (person).
Teacher: Tell me what you think. What is the first step? What do the second and third steps mean respectively? What does this method seek first? What is it?
Teacher: Are there any different solutions?
Preset (2): girl: (person);
Boy: (person).
Teacher: So, what does it mean? And then what?
5. Summary: Just now, the students solved the same problem in different ways. Let's take a look again (with courseware demonstration).
The first method is to see how many copies a * * * is divided into according to the significance of comparison, first find out the number of one copy, and then calculate the number of several copies; The second method is to look at the scores of boys and girls in the total according to the relationship between the ratio and the score, and then solve it with the knowledge of the score. Both methods are good, which one do you prefer? Why?
The design intention is to guide students to explore, not directly using examples from books, but using the actual situation of the ratio of male to female students in the class. Because it is an example that students are very familiar with, students are willing to explore, communicate and practice. This design not only reduces the difficulty of learning, but also stimulates students' interest in learning.
(2) Reveal the topic
Teacher: Like the above question, the method of distributing the quantity according to a certain proportion is called proportional distribution. Today, let's learn about proportional distribution. (blackboard title: proportional distribution)
(3) Practical attempt
Example 2: This is a dilution bottle of detergent concentrate, and the ratio marked on the bottle indicates the volume ratio of concentrate to water. According to these ratios, diluents with different concentrations can be prepared.
1.
What do you mean by concentrated solution and diluted solution? (The concentrated solution is pure detergent, and the diluted solution is detergent after adding water. )
Teacher: Can you solve the problem just now? Students solve problems independently and exchange reports. )
2. Analyze the answer.
Default (1): 500 per copy? 5= 100(mL), and the concentration is 100? 1= 100 (ml), water is 100? 4=400 ml.
Teacher: What does 5 mean here? (Divide the total volume into 5 parts on average. )
Preset (2): concentrated solution has (mL) and water has (mL).
Teacher: What does this mean? (Concentrated solution accounts for the total volume; )
And then what? (Water accounts for the total volume. )
3. Review and reflection.
Teacher: What methods can be used to verify the results?
Default: See if the ratio of concentrated solution to water is equal to 1:4.
Summary: In the process of solving problems, it is necessary to see clearly which two quantities are the ratio of 1:4.
The design intention is to take Example 2 in the book as the test question, let the students try and communicate independently, and finally make a summary. This not only cultivates students' ability of independent inspection and analysis, but also further deepens students' understanding of the two methods, so that students can have a first taste of success.
Third, practical application.
(1) Basic exercises
1. Teacher: Open page 55 of the textbook and look at the first question.
(1) Teacher: Use your favorite method to calculate independently and see who can calculate quickly and accurately.
(2) Communication: Tell me about your method.
2. Show me: Uncle Li's vegetable field is 800 square meters, and he is going to grow cucumbers and eggplants.
Teacher: Please design it. How to distribute it?
Default 1: 1: 1.
Teacher: If we allocate by 1: 1, what are the planting areas of cucumber and eggplant respectively? (Students calculate independently)
Teacher: Through calculation, it is found that the distribution according to 1: 1 is actually what we learned before? Average score? . Yes, the average score is distributed according to 1: 1, which is a special case of proportional distribution.
For other distribution methods, let the students do a quick calculation before communication.
(2) development and improvement
1. Teacher: Can you make it more difficult? Let me change the question.
Show me page 56 of the textbook, question 7: Uncle Li's vegetable field is 800 square meters. He is going to plant tomatoes, and the rest will plant cucumbers and eggplants according to the area ratio of 2: 1. How many square meters are these three vegetables?
(1) Comparison: What's the difference between this question and the previous one?
(2) Analysis: The question is which quantity to allocate and what proportion to allocate? Did this quantity tell us directly? So what should we calculate first? Can you calculate?
(3) Students try.
(4) AC algorithm.
Teacher: How do you calculate it? (Show students' homework) Do students have any other methods? Introduce your method.
Teacher: What are the similarities between these students' methods? What is the difference?
2. Presentation: According to the number of three classes in Grade 6, the school assigned 70 trees to each class. There are 46 students in Class One, 44 students in Class Two and 50 students in Class Three. How many trees should be planted in each of the three classes?
(1) Comparative analysis:
Teacher: What's the difference in this question? Not directly? Than? I can't match directly according to the score. What should I do?
Teacher: We can work out the proportion first, and then distribute it in proportion.
(2) Students try independently and exchange algorithms.
(3) Summary
Teacher: By answering the above two questions, what do you think should be paid attention to when answering the question of proportional distribution?
Teacher: That's right. When answering such questions, we should carefully examine the questions and see clearly which quantity is distributed according to what proportion. If the topic does not directly give a ratio, we must first find the ratio according to the topic information, and then allocate it according to the ratio.
The design intention is to create problem situations, from basic exercises to comprehensive questions, and then to problems without direct comparison, so that students can feel the fun and value of learning in the process of solving practical problems, which not only cultivates students' ability to solve problems independently, but also allows students to verify and taste their learning results in practical exploration and feel the fun brought by success again.
Fourth, class summary.
1. Teacher: After learning this, who can tell us what we learned in this class today? Tell me about your gains and feelings. (Answer by name)
2. Extracurricular expansion.
Teacher: Ratio is widely used in life. Please collect life examples after class, make up a topic of proportional distribution, and exchange learning in the next class.
Is it designed for students to master by themselves? Harvest? 、? Feeling? Summarizing the class can help students to sort out what they have learned again, cultivate the ability of evaluation and reflection, and let students feel the charm of mathematics more deeply.
Primary school mathematics knowledge points jingle.
First, carry addition within 20.
Look at a large number, divide it into decimals, round it to ten, and add a fraction.
(master? Make up ten Advocate? Recursive method. )
Second, the subtraction of abdication within 20 years
Less than 20 abdicates, and the oral calculation method is simple.
Ten people return one, one person makes up, and the writing is accurate and fast.
Third, the meaning of addition, vertical calculation
When two numbers are added, the result of the addition is called sum.
Don't forget the numbers on the right, one for every ten.
Fourthly, the significance of vertical calculation subtraction.
Subtraction is used from big to small, and the result of subtraction is called difference.
Digital alignment starts from the right, and when it is not enough, take the front position.
Five, two-digit multiplication
Multiplication of two digits is not difficult. There are three points in the calculation process:
The multiplier must be calculated first and then multiplied by ten digits.
The last digit of the product is the key, which is opposite to the ten digits;
After the two products are added up, calculate the number of layers in your mind.
Six, two-digit division
Divide two by two, and two is not enough to divide three.
Except quotient, the remainder is less than divisor.
Then, except for the next item, the test method should be flexible.
Master? Rounding? Method, what else? The same business comparison method? ,
Do you understand? Semi-fixed commercial law? , less than divisor quotient nine, eight. (where: same head, less high position 1)
Seven, mixed operation
Look at the formula carefully, calculate multiplication and division and add alkali.
When you encounter brackets, you should count them first, and the application rules should be changed.
There are some data to memorize and skills to master.
Eight, fast addition and subtraction
Add and subtract quickly, don't worry, you can see clearly when you get the formula.
Rounded integer close to the whole hundred, the following treatment is not fallacious.
If you don't add enough, you will reduce the complement and add the extra score to the end.
If the subtraction is insufficient, add the complement, and then subtract the excess score.
Nine, multi-digit reading method
The reading method is very simple. First, the fourth grade.
We should start from the highest place, thousands, hundreds, and ten.
Grade units read hundreds of millions, and no zeros are read at the end.
(Don't read the 0 at the end of the stage, and don't read the 0 at the end of the integer)
Read one with a zero in the middle, and the expression of Chinese characters is irrelevant.
Note that the reading is zero:
1, ten thousand levels, the first level has zero.
2, the whole class is zero.
There are zeros at the end of the superior and at the top of the subordinate.
4. There is a 0 in the middle of each level.
X. decimal addition and subtraction
Decimal addition and subtraction calculation problems, with point alignment.
An algorithm is like calculating an integer. After the calculation, move the point down.
XI。 Decimal multiplication
Decimals are multiplied by decimals in the same way as integers.
Fixed product decimal places, factors * * * together.
Twelve, divisor is the division of decimals
Divide the dash of the decimal point, (remove the decimal point)
Move the decimal point of dividend, move it to the right,
The number of decimal places of a divisor determines it.
Thirteen, prime songs
A prime number 2, 3, 5 and 7,
Add 1 before 1, 3, 7 and 9.
After 4, before 3 and 7, after 9 and 7, 1,
Add 7, 1 after 3, 4 and 6,
Add 9, 3 after 2, 5, 7 and 8,
Remember all 25 prime numbers.
Fourteen, fractional multiplication and division method
Fractional multiplication is easy to learn and understand, and numerator and denominator are multiplied separately. The meaning of the formula should be clear, and it is easier to make an appointment. Fractional division is wonderful, changing the original division symbol into multiplication symbol. The divisor is reversed, so calculation is essential.
Fifteen, about the point
Tangent point, tangent point, take the net, save time and effort. From top to bottom, from left to right, make the data clear and don't miss a number. Decimals are encountered, decimal points are integers, and there are not enough digits. Zero? To make up.
Practical application of elementary school mathematics knowledge jingle
? Find a number greater than the number. Application problems of
In the fourth volume of the six-year mathematics textbook? Find a number greater than the number. With what? Find a number less than the number. Comparing two types of application problems with two numbers, we can get a difference. Knowing the difference between two numbers and one of them, finding another number means finding a number that is more or less than one. So what? Bido? With what? Less than? Both types of application problems are inverse problems to find the difference between two numbers, and the structure of the problems is the same. In view of this situation. How much more? With what? How much is missing? The application problem is just two sides of the same problem. What are the most common mistakes students make in solving such problems? how much is it? Just add it, see? Less? Just use subtraction and judge the algorithm with a single word.
The teaching philosophy is:
1. Analyze the quantitative relationship and teach students the way of thinking.
2. Give full play to the role of line drawing and make application questions? Things? Become? Why? , and by what? Why? Become? Type? Express it intuitively, and then find the law.
Example: P 17 Example 5 bright primary school planted 300 willows. There are 70 more poplars than willows. How many poplars have been planted?
1. Question: What kinds of trees are there? (willow, poplar)
Who is better than who? (Yang Liubi)
Who has more? Who is less (more poplars)? (Little willow)
2. Calculation formula: number of willows+number of poplars more than willows = number of poplars.
3. The formula is: 300+70=370 (tree)
4. If the first condition is changed to a question and the question is changed to a condition, how should it be calculated?
5. Then get the key sentences: there are more known conditions than before (the number of requirements is higher than before) and the number of requirements is lower than after.
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