Textbook training speech

Unit 4 Mathematics in Beer Production-Proportion

I. Status of teaching materials

This unit is taught on the basis of students' comparative knowledge and is an important basis for further study of scales and other disciplines. Through the study of proportional knowledge, we can deepen our understanding of the relationship between quantity and quantity, so that students can initially understand how one quantity changes with the change of another quantity, obtain the initial concept of function, and use this knowledge to solve some simple practical problems.

Two. Teaching objectives of this unit

1. Understand the meaning and basic nature of proportion in specific situations; Solution ratio.

2. Understand the significance of positive-negative ratio in specific situations, get a preliminary understanding of positive-negative ratio images, correctly judge the size of positive-negative ratio, and use the knowledge of ratio to solve simple practical problems.

3. Further develop reasonable reasoning ability in the process of exploring the basic nature of proportion.

4. In the process of solving practical problems, further experience the connection between mathematics and life and feel the value of mathematics.

Three. Unit teaching content

Information window topic knowledge points

Information window-the significance of transportation barley malt proportion, the basic nature of proportion, and the proportion of solution

The significance of producing and recording positive scale and positive scale images in information window II

Information Window 3 Significance of Inverse Ratio of Beer Production Plan

Information window 4: Solve practical problems with positive and negative proportions when filling beer.

Four. Outstanding characteristics of module preparation

1. Start learning new knowledge on the basis of students' existing knowledge and experience.

Students have been exposed to many quantitative relationships in their previous studies. The textbook of this unit is compiled on the basis of students' existing knowledge and experience, so that students can re-understand the relationship between quantity and quantity from the perspective of proportion. For example, the meaning of proportion is learned on the basis of the meaning of proportion with the help of the relationship between transportation volume and transportation times; On the basis of the meaning of ratio, with the help of the relationship between working hours and total workload, the meaning of positive ratio is studied. The significance of inverse proportion is learned through the relationship between the tonnage produced every day and the number of days required for production.

2. Material selection is close to life.

This unit selects life materials that students are interested in and introduces them into the study of mathematics knowledge, which can not only closely link the learning content with the reality of life, but also stimulate students' interest in learning and desire for exploration.

Verb (abbreviation of verb) the overall planning of unit class hours

Information Window One Information Window Two Information Window Three Information Window Four

The meaning of proportion, exercise: 1 class hour, positive proportion meaning, positive proportion image, basic exercise: 1 class hour, negative proportion meaning, basic exercise: 1 class hour, positive and negative proportion knowledge solving, basic exercise: 1 class hour.

Basic Properties of Proportion, Solution Ratio and Exercise: 1 Class Consolidation Exercise: 1 Class Comprehensive Exercise of Positive and Negative Proportions: 1 Class Consolidation Exercise: 1 Class Consolidation Exercise.

Review and practice: 2 class hours

Suggestions on teaching intransitive verbs

Information window 1:

1, teaching content: the meaning of proportion, the basic nature of proportion, and solving proportion.

2. Information window introduction:

The information window presents a close-up of the transportation of barley malt, and shows the relevant data of the transportation of barley malt in tables, so that students can ask mathematical questions according to these data. By solving the question "What is the ratio of transportation volume to transportation times?" What must they do? "These two questions, the significance of learning proportion. This unit has three red dots.

The first red dot: the meaning of proportion.

The second red dot: the basic properties of proportion.

The third red dot: solution ratio.

3, information window teaching suggestions:

First, combine the situation diagram to put forward mathematical problems.

Solving practical problems in life is an important concept of the new curriculum. In teaching, it is necessary to talk about beer with students in combination with the information window. Beer can be seen everywhere in our lives and is closely related to our lives. It can be derived from the topic of the main raw materials for producing beer. Some students may know that it is grain and barley malt. If they don't know, they can tell them, so beer is also called "liquid bread". Starting from this lesson, we will understand and solve the mathematical problems in beer production together. I would like to remind all teachers that in teaching, we should focus on guiding students to pay attention to the quantitative relationship contained in the information window materials and not discuss the beer production process too much.

Second, on the basis of students' existing knowledge and experience, they begin to learn new knowledge.

In the previous study, students' understanding of comparison has already had a certain foundation. In teaching, students can read the information in the information window first, and directly ask students to ask mathematical questions about comparison. First, let the students find out the ratio of the traffic volume to the number of times of transportation on the first day and the second day respectively. On this basis, let students observe the relationship between the two ratios, find that the ratio of the two ratios is equal, and then list the equations. The teacher further explained: two expressions with equal proportions are called proportions, and the meaning of proportions is actually a rule. Students only need to find out what it is, and don't need to study why. Here, the teacher should also ask students to compare "ratio" and "proportion" in a timely manner, and then introduce the names of each part of the proportion after clarifying the difference between them.

In order to let students further understand the meaning of proportion, you can show them some proportions and let them find out which ones can constitute proportions. You can also practice questions 3, 4 and 5 by yourself; You can also show four numbers that can form a proportion, such as 2, 3, 4 and 6, so that students can form different proportions. Through these forms of practice, we can deepen our understanding of the meaning of proportion.

Third, let students explore independently and further develop their rational reasoning ability.

When teaching the second question marked with a red dot, teachers should give students more thinking space according to the intention of compiling the textbook, so as to achieve "what is the relationship between the two external items and the two internal items in proportion?" Guided by this question, let students guess first, and then verify by calculation, so that students can experience the process of exploration independently. Then on the basis of group communication, the basic nature of proportion is summarized: in proportion, the product of two external terms is equal to the product of two internal terms. Here, teachers should pay attention to provide students with a lot of materials and give them enough time to explore, because a law can only be drawn with a lot of examples. Instead of "only letting students see what is the relationship between the product of external terms and internal terms", it reminds students of the direction of thinking, sets up thinking channels, narrows the exploration space, and makes students lose an excellent opportunity to exercise their thinking.

4. Independent practice analysis

The problem 1 of "Independent Exercise" is an exercise to consolidate the meaning of proportion. When practicing, students can think independently and finish independently. The focus of communication is how to judge whether two proportions can form a proportion according to the meaning of proportion.

Questions 3 and 4 are the significance and basic nature of consolidating proportion. When practicing, let the students finish it independently, and then organize communication. When communicating, talk about your own ideas. It can be judged according to the meaning of proportion and its basic nature. As long as what the students say is reasonable, they should be sure.

Question 5 provides a form of practice for group activities. In practice, teachers can first show a set of ratios, and students can name another set of ratios that can be proportional to them and explain their thinking methods. Then let every student participate in the exercise to consolidate the significance and basic nature of proportion.

Question 8 is the meaning and basic nature of flexible use of proportion. In practice, students can think independently, then communicate fully, and sum up the solution to the problem: first, find out two ratios with equal ratios, and then write the ratios according to their meanings; You can also find two groups of numbers with equal products first, and then write the proportion according to the basic properties of the proportion.

When practicing the ninth question, the teacher should help students understand the meaning of the question so that students are not affected by the interference factors (volumes).

The question * 12 is an open question. In practice, students can be guided to think according to the basic nature of proportion: if the two numbers on one side of the equation are the internal terms of proportion, and the two numbers on the other side are the external terms of proportion, then write the proportion. You can also ask students to give a few more examples to complete.

Information window 2:

1, teaching content: proportional meaning, proportional image.

2. Information window introduction:

The situation diagram presents a corner of the beer production workshop, shows some data of the total amount of work and working hours in beer production in the form of tables, guides students to ask questions, and introduces the learning of proportional quantity and proportional relationship. This window has two red dots.

The first red dot: the meaning of direct proportion

The second red dot: a proportional image.

3, information window teaching suggestions:

First of all, by observing a large number of real data, analyzing their quantitative relationship, mathematical knowledge is abstracted.

In teaching, teachers can introduce the topic of beer production, show the situation map, guide students to observe the record table of beer production, ask questions according to information, and sort out the questions raised by students, thus introducing proportional learning. The teaching content of positive and negative proportion reflects the relationship between quantity and quantity, and a large number of related quantities need to be analyzed, summarized and abstracted, which puts forward higher requirements for students' ability of observation, analysis, reasoning and abstract generalization, and is also a good teaching carrier for developing students' logical thinking ability. The method of "enumeration-observation-discussion-induction" can be used to study the meaning of positive proportion.

Second, give students sufficient space for thinking and communication, and guide students to carry out independent mathematics activities.

When teaching the first question marked with a red dot, teachers should create an open question situation and a relaxed learning atmosphere, so that students can experience the process of "doing mathematics" and independently construct the meaning of positive proportion.

Students can observe the record form first and discuss and communicate in groups: focus on the following aspects: ① How many quantities are there? 2 how to change? ③ What is the law of change? (4) What is the quantitative relationship? On the basis of students' group inquiry and class communication, the preliminary impression is that there are two quantities in the table, namely, the total amount of homework and the time of homework; The total amount of work changes with the change of working hours. The longer the working hours, the greater the total amount of work, and the shorter the working hours, the smaller the total amount of work. According to the corresponding data of each group, the work efficiency can be calculated, and then students can be guided to find that the ratio of the total amount of work to the working time is the work efficiency, and the equal ratio means that the work efficiency is certain, and then it is concluded that the total amount of work and the working time = the work efficiency (certain). Finally, the teacher introduced to the students: the total amount of work changes with the change of working hours; The work efficiency is constant, that is, the ratio of the total amount of work to the working hours is certain, so we say that the total amount of work is directly proportional to the working hours, and their relationship is called direct ratio.

Third, encourage students to discover the rules through multiple examples, and enhance students' credibility of the learned rules.

After learning the concept of direct ratio, teachers can give several examples of direct ratio in life, and then let students find out which two quantities are also direct ratio in life. Here, students must be guided to grasp the key of direct ratio: (the ratio is certain). Through a large number of examples, on the one hand, it can deepen students' understanding of the meaning of proportionality, enhance students' credibility of the laws they have learned, on the other hand, it can also make students feel the close connection between mathematics and life.

Fourthly, with the help of the study of the image of direct proportion, the understanding of the meaning of direct proportion is further strengthened, and the function thought is moderately infiltrated.

The second red dot is mainly about the learning of scale images, which is arranged according to the requirements of the standard: "Draw a picture on the grid paper with the coordinate system according to the given data with proportional relationship, and estimate the value of another quantity according to the value of one quantity", which lays the foundation for learning the knowledge of scale line segments and functions in the future. The three aspects of design reflect the three steps of proportional image teaching. The first step is to draw an image. According to the left part of the textbook, that is to say, draw points first and know the specific meaning of each point. Understand that each point represents the total production in a certain period of time, and these points are drawn on grid paper according to the corresponding data of working hours and total work. Then connect the points according to the tips of the children on the right. The second step is to know the shape of the image. In the first question below, I found that the image in direct proportion is a straight line. I understand that the image with direct scale is a straight line, which can play two roles in future drawing: one is to draw an image with direct scale (such as question 9 on page 75), and many points of the image can be traced according to each set of data provided, and then connected into a straight line in turn; Second, if the points drawn in direct proportion are not in a straight line, it means that there is something wrong with the points drawn, and it should be corrected in time. The third step is to correctly analyze the image, which is the second and third questions prompted below. Estimate the number of beer produced in 4.5 hours and the time required to produce 80 tons of beer. Students should be instructed to use the skills of drawing vertical lines or parallel lines to make the numbers as accurate as possible. If the tonnage generated in 4.5 hours is estimated, it is necessary to find the point representing 4.5 hours on the horizontal axis, draw the vertical line of the horizontal axis through this point, and get the intersection point between the vertical line and the image, and then take the intersection point as the vertical line of the vertical axis, and the tonnage generated can be estimated according to the position of the vertical foot on the vertical axis.

Pay attention to this question:

(1) Do you need to explain the reasons in detail when judging the positive and negative ratio?

Compared with traditional textbooks, mechanical terms such as correlation quantity are eliminated. When judging whether two quantities are directly proportional or inversely proportional, it is unnecessary to say that "time and distance are two related quantities." When time changes, speed changes, and the product of speed and time is a certain distance, then time and distance are inversely proportional, and the relationship between them is inversely proportional. " This fixed format. As long as students can correctly judge the relationship, explain the reasons in their own words. What needs attention here is to give students the opportunity to express their reasons as much as possible. As long as the expression is sufficient, we can clear our minds and fully reflect the order of thinking. In practice, we should pay special attention to let students state their reasons. For example, the second question is a consolidation exercise with a positive proportional meaning. Through this question, let students further clarify the essential characteristics of positive proportion, that is, one quantity changes with another quantity, and the ratio of the two quantities is determined. In question (1), the ratio of broadcast time to broadcast words is certain, so the broadcast time is directly proportional to broadcast words; Question (2) Although the number of words broadcast and the number of words not broadcast are also two related quantities, the ratio of the number of words broadcast and the number of words not broadcast is not necessarily, so it is not directly proportional.

(2) Learning the image in direct proportion should be regarded as a way to understand the meaning in direct proportion. By analyzing the image, we can better understand the changing law of two quantities in direct proportion and the thought of osmotic function. You can't simply stay in skills training such as drawing points and connecting lines.

4, independent practice analysis:

The topic 1 of "independent exercise" is a basic exercise with proportional meaning. In practice, students can be guided to think first and judge whether distance and time are in direct proportion, and it is important to judge whether the ratio of the two is equal. Then, by calculating the ratio of each set of corresponding data, we can find out what is a constant, and then combine the meaning of direct ratio to judge: because distance and time = speed (certain), distance and time are directly proportional.

The second problem is to consolidate the meaning of positive proportion. Through this question, let students further clarify the essential characteristics of positive proportion, that is, one quantity changes with another quantity, and the ratio of the two quantities is determined. In question (1), the ratio of broadcast time to broadcast words is certain, so the broadcast time is directly proportional to broadcast words; In question (2), the proportion between the number of words broadcast and the number of words not broadcast is not necessarily, so it is not proportional. At the same time, let the students give a few more examples to judge with real-life examples. (The relevant quantity appears in the teaching reference)

Question 4 is a set of right or wrong questions. When practicing, let the students think first: How to judge whether two quantities are in direct proportion? After thinking clearly, let the students solve the problems one by one through independent thinking. When communicating, pay attention to let students use the meaning of direct proportion to explain. About a person's age and weight, although the weight changes with age, this change is irregular, so it is out of proportion.

The sixth problem is the subject of consolidating and applying direct scale images. In practice, students can first observe the images, understand some of the data, and judge whether the number of running weeks is proportional to the time spent according to the ratio of the corresponding data; You can also judge directly from the image. Then guide students to estimate according to the image: first find 9 on the horizontal axis, then find the corresponding point on the vertical axis, and then estimate. It takes about 16 hours to run for 9 weeks.

Question 9 is the topic of consolidating the knowledge of scale images. When practicing the second small question, we should follow three steps: first, distinguish what the horizontal axis and the vertical axis represent respectively, and second, draw corresponding points according to the provided data. Third, connect the points in order.

10 is a comprehensive problem to consolidate the knowledge of proportional ratio. This problem involves four quantities: radius, diameter, perimeter and area. Some are directly proportional (such as radius and diameter, radius and perimeter, diameter and perimeter), and some are not (such as radius and area, perimeter and area, diameter and area), so there may be confusion among students here. Pay attention to let students talk about the reasons and further deepen their understanding of the meaning of positive proportion. (The relevant quantity appears in the teaching reference)

Information window 3:

1, teaching content: the meaning of inverse proportion

2. Information window introduction:

The situation diagram presents a corner of the beer production workshop, introduces the tonnage of beer produced every day and the number of days required for production in tabular form, guides students to ask questions, and introduces the learning of inverse ratio and inverse ratio relationship.

There is only one red dot: the meaning of inverse proportion.

3. Teaching suggestions of information window

First, ask challenging questions, so that students can explore the meaning of inverse proportion independently.

This lesson is taught on the basis of students' learning the meaning of positive proportion. However, if students still use the same teaching procedure to learn inverse proportion on the basis of learning positive proportion knowledge and research methods, it will inevitably lead to students' "copying models" and "applying conclusions", and their thinking level will not be further developed. In the process of learning, children focus on finding answers rather than developing their understanding of knowledge. On the premise of insufficient cognitive understanding, students rigidly apply the interpretation mode of positive proportional meaning to define the meaning of inverse proportion, and students lack in-depth understanding of the nature of knowledge points. In view of this, I think we can design the following teaching:

Teacher: In this class, we will learn the inverse proportional quantity. What do you think will happen if the quantity is inversely proportional? (ask challenging questions. )

Students may have the following views:

1 ",it may be that the changes of the two quantities are opposite. "

Health 2: In direct proportion, one quantity is expanded several times, and the other quantity is also expanded by the same multiple. Their changes are consistent. I think in inverse proportion, it may be that one quantity is expanded several times, while the other quantity is reduced by the same multiple, and their changes are opposite.

Health 3: The quotient of the corresponding number in the positive proportional quantity is certain, and the product of the corresponding number in the inverse proportional quantity may be certain.

Health 4: It may be the same. One quantity is increasing and the other is decreasing, and their changes are opposite.

Because students learn inverse proportion on the basis of positive proportion, their minds will not be blank. In the form of "guessing", students are given imagination (guessing) space, their positive thinking is stimulated, the original knowledge base is reproduced, and the transfer and interaction between old and new knowledge are promoted. The teacher then displays the form in the information window.

The daily tonnage is 100 200 300 400 500. ...

Days required for production 60 30 20 15 12 ...

Let the students discuss and communicate in groups. Finally, the teacher summarizes the meaning of inverse proportion.

Second, combine life examples to deepen the understanding of concepts.

Like positive proportion, after learning the concept of inverse proportion, students should first find out which two quantities in life are also inverse proportion, and explain them with specific data to deepen their understanding of the meaning of inverse proportion.

Pay attention to the question:

Why learn the positive-negative ratio? Proportional knowledge is widely used in industrial and agricultural production and daily life. For example, drawing a map requires knowledge of scale, and there is a direct or inverse relationship between the two quantities often used in production and life. The knowledge of proportion is also the basis for further learning middle school mathematics, physics, chemistry and other knowledge. Not to mention the knowledge needed by all walks of life, such as mathematics, geography and physics. Almost inseparable from the proportion. Such as the inverse relationship between air temperature and air pressure, the inverse relationship between air temperature and altitude, the inverse relationship between air temperature and latitude, the inverse relationship between the wavelength emitted by an object and its own temperature, and the direct relationship between wind speed and horizontal air pressure gradient force. )

4. Independent practice analysis

Question 3 is a set of true or false questions. When practicing, let the students think first: how to judge whether two quantities are inversely proportional? After thinking clearly, let the students think independently and solve them one by one. When communicating, pay attention to let the students explain it in inverse proportion. Regarding the number of trees planted and the number of trees not planted, although the number of trees not planted varies with the number of trees planted, and the sum of these two quantities is also certain, their products are not necessarily, so the number of trees planted and the number of trees not planted are not inversely proportional. Through the practice of this question, let the students know clearly how to judge whether two quantities are directly proportional or inversely proportional.

"Do you know?" The purpose of introducing inverse proportion images in the column is to let students know that inverse proportion relations can also be represented by images, and students are not required to draw images in teaching.

Information Window 4- Transport Beer

1, teaching content: solving practical problems with positive and negative proportions.

2. Information window introduction: The picture shows the situation of beer transportation by car, with close-ups. By introducing the relevant data in beer packaging, students are guided to ask questions and learn to solve practical problems with proportional knowledge. This window has two red dots.

The first red dot: solving practical problems with positive proportion knowledge.

The second red dot: solving practical problems with inverse proportion knowledge.

3, information window teaching suggestions:

First, it not only encourages students to diversify their problem-solving strategies, but also attaches importance to the teaching of problem-solving in proportion.

In teaching, we can introduce the topic of transporting beer, introduce relevant information, and then present a situational map to guide students to observe, understand the meaning of the map and ask questions.

Proportional quantity is widely used in real life. Students have also been exposed to such problems in their previous studies, such as an application problem, but it was only a topic at that time and did not rise to the general law. After showing examples, teachers should guide students to think independently, solve problems in their own way, and then organize students to communicate. In communication, students may use what they have learned before to answer questions. At this time, the teacher should give affirmation, and then guide the students to answer with proportional knowledge, which can inspire students to think: which quantity is certain? What is the ratio of the total number of bottles to the number of boxes of beer? Why? Then list the equations (equations) according to the meaning of positive proportion, let the students answer independently, and then communicate.

When teaching the problem marked with the second red dot, we can imitate the teaching idea of the first red dot.

Second, guide students to compare the positive and negative proportions in solving problems in time.

After solving the two red dot problems, we should guide students to strengthen the comparison and find out the similarities and differences in solving problems, so that students can master the ideas and methods of solving problems with positive and negative proportion knowledge.

4. Independent practice analysis

The fifth topic is the flexible use of inverse proportion knowledge to solve practical problems. When practicing, we should pay attention to organizing students to carefully examine the questions, so that students can make it clear that the area of the ground is certain, and the area of each square brick is inversely proportional to the number of blocks. Therefore, we should first calculate the area of square brick according to the side length, and then solve it according to the knowledge of inverse proportion. This problem is the most common problem for students, and some students directly multiply the side length by the number of blocks. Let the students analyze the quantitative relationship. And then solve it.