(A) Teaching objectives

1. Make students know the meaning and function of using letters to represent numbers, use letters to represent the learned operation rules and formulas, and use letters to represent common quantitative relations in specific situations. Initially learn to find the value of the formula containing letters according to the value of letters.

2. Make students understand the meaning of the equation, understand the basic properties of the equation, and use the properties of the equation to solve simple equations.

3. Make students feel the connection between mathematics and real life, and learn to make equations to solve some simple practical problems. Cultivate students' awareness and ability to choose algorithms flexibly according to specific situations.

(B) Teaching materials and teaching suggestions

Textbook description

1. Content structure and status of this unit.

The main content of this unit is to express numbers and solve simple equations with letters, and the application of simple equations in solving some practical problems.

These contents are learned on the basis that students have learned some arithmetic knowledge (such as four operations of integers and decimals and their applications) and contacted some algebraic knowledge (such as using letters to represent operation rules and using ○, △ or □ to represent numbers).

Generally speaking, teaching simple equations in primary schools has the following significance.

First, it helps to cultivate students' abstract generalization ability and develop students' flexibility of thinking. Because for primary school students, abstracting numbers from the number of concrete things is a leap in understanding, and now it is a leap in understanding to transition from concrete and certain numbers to abstract and variable numbers with letters. Moreover, on the basis of using letters to represent unknowns, the mathematical tool for students to solve practical problems has developed from listing formula solutions to listing equation solutions, which is another leap in understanding mathematical thinking methods, and it will improve students' ability to solve practical problems by using mathematical knowledge to a new level.

Second, it helps to consolidate and deepen the understanding of the arithmetic knowledge. Students' understanding of these knowledge can be deepened by using letters to express the learned quantitative relations, operation rules and formulas for calculating the perimeter and area of some graphics. At the same time, because using the alphabet is more concise and easy to remember than using words, it is convenient for students to consolidate their knowledge.

Third, it is conducive to strengthening the convergence of mathematics in primary and secondary schools. Giving students a little knowledge of algebra can help them get rid of some limitations in the arithmetic thinking method (reverse thinking, the unknown does not participate in the operation, which is equivalent to lacking a condition and increasing the thinking steps), so as to prepare and pave the way for further learning algebra knowledge.

The content of this unit is divided into two parts. The main content of the first section is to use letters to represent numbers, indicating operation rules, calculation formulas and quantitative relations. The main content of the second section is the significance of the equation, the basic properties of the equation and the solution of simple equations, as well as the use of column equations to solve some simple practical problems. The arrangement of these contents is as follows.

As can be seen from the above table, the four parts of the two textbooks have internal logical connections. Representing numbers with letters is the basis of learning equations, the meaning of equations is the basis of learning to solve equations, and slightly complicated equations are the development of solving equations.

2. The writing characteristics of this unit textbook.

Compared with the original textbook, the main improvements of this unit textbook are as follows.

(1) The arrangement of letters representing numbers in textbooks is closer to students' cognitive characteristics.

It is abstract for primary school students to represent numbers by letters. Especially, it is more difficult to express the quantitative relationship with formulas containing letters. For example, it is known that the father's age is 30 years older than the son's, and A indicates the son's age, so a+30 not only indicates the age relationship that the father's age is always 30 years older than the son's age, but also indicates the father's age. This is a difficult point for students when they are beginners. First of all, they should understand the relationship between father and son's age and turn this relationship described in language into a formula containing letters; Secondly, they are often not used to treating a+30 as a quantity, and students often think that it is a formula, not a result. The expression of a quantity by a formula is an indispensable basis for learning the sequence equation. Therefore, in order to ensure the foundation and break through the difficulties, the teaching material has arranged the teaching content of using letters to represent numbers, which is closer to students' cognitive characteristics. That is, first learn to express a specific number by letters (for example 1), then learn to express a general number by letters, that is, use letters to express operation rules and calculation formulas (for example 2 and 3), and then learn to express the relationship between quantities by letters (for example 4). This is convenient for students to gradually understand and adapt to the characteristics of alphabetic algebra.

(2) Solve the equation based on the basic properties of the equation, not on the inverse operation relationship.

For a long time, primary schools have always taught simple equations, and the basis of equation deformation is always the relationship between addition, subtraction or multiplication and division. This is actually using arithmetic to find the unknown. In middle school, we should start a new stove, introduce the basic properties of the equation or the principle of the same solution of the equation, and then re-learn and understand the equation according to the basic properties of the equation or the original solution of the equation. Moreover, the more firmly the primary school's ideas and algorithms are mastered, the more obvious the negative transfer to the initial teaching of algebra in middle school. Now, according to the requirements of the standard, the basic properties of the equation have been introduced since primary school, and the solution method of the equation has been deduced on this basis. In this way, the phenomenon of two ideas and two mathematical explanations of the same content is completely avoided, which is conducive to strengthening the connection of mathematics teaching in primary and secondary schools.

From the previous experiments in some parts of China, the mathematical facts reflected by the basic properties of the equation are relatively simple, and it is not difficult for primary school students to find its changing law with their own knowledge and experience. As long as it is handled properly, it is feasible to use it as the basis for solving simple equations.

(3) Adjust the content of simple equations and highlight the advantages of solving equations by using their basic properties.

After introducing the basic properties of equations as the cognitive basis for solving simple equations, a corresponding measure is to adjust the basic content of simple equations so that simple equations such as a-x=b and a \x = b do not appear for the time being. This is because primary school students have not learned the four operations of positive and negative numbers. Using the basic properties of the equation to solve a-x=b, the process of equation deformation and its mathematical explanation are more troublesome. As for the equation with the shape a÷x=b, it is essentially a fractional equation. According to the basic properties of the equation, it is necessary to remove the denominator first, which is also not suitable for primary school learning. In fact, avoiding these two kinds of simple equations does not affect students to solve practical problems through column equations. Because when it is necessary to list the equation in the form of a-x=b or a÷x=b, the equation in the form of x+b=a or bx=a can always be listed according to the quantitative relationship of actual problems. This also reflects the advantages of using equations to solve problems, which can often turn reverse thinking into positive thinking.

After the content is adjusted, it is easier to reflect the advantages of solving the equation by using the basic properties of the equation. For example, equations with solutions, such as x+a=b and x-a=b, can all be summed up as subtracting (adding) A from both sides of the equation to get x=b-a, and equations with solutions, such as ax=b and x÷a=b, can all be summed up as: dividing (multiplying) A by both sides of the equation to get X = B+A,. Obviously, it is more unified than the original solution of the equation based on the inverse operation relationship.

(4) The teaching of solving equations is organically integrated with solving practical problems.

In the past, the teaching of solving equations and the teaching of solving application problems with equations were carried out separately. The former belongs to calculation and the latter belongs to application. Now the natural connection between calculation and application has been restored, which is embodied in this unit. When learning "slightly complicated equation", the equation is introduced from practical problems, and the equation is solved and tested in the realistic background. This treatment helps students to understand the process of solving equations, strengthen the connection between mathematical knowledge and the real world, and cultivate students' awareness of mathematical application.

Teaching suggestion

1. Pay attention to the abstract generalization process from concrete to general.

The knowledge in this unit is mostly abstract. In teaching, we should make full use of students' original cognitive basis and pay attention to the abstract generalization process from concrete examples to general meanings. Whether learning to express the quantitative relationship with letters or to learn the concept or properties of equations, we should not only play the supporting role of concrete examples in abstract generalization, but also guide students to get rid of the concreteness of examples in time and realize the necessary abstract generalization.

2. Make good use of teaching materials and appropriately expand the scope of combining with practice.

In this unit, letters are used to express the quantitative relationship, and equations are used to solve practical problems, all of which are easy to combine theory (mathematical knowledge) with practice (real life). The textbook carefully selects and designs many vivid and meaningful realistic themes from the perspective of the sex of senior primary school students, such as the relationship between the weight lifting quality of people on the earth and the moon in section/kloc-0, the relationship between standard weight and height, etc. Another example is the relationship between Fahrenheit and Celsius in the second quarter, the composition of the earth's surface, the ocean area and the land area, and so on. In teaching, we should make full use of the resources provided by textbooks, and then supplement some subjects around students from the local and school characteristics to further stimulate students' enthusiasm for learning and cultivate students' awareness of mathematics application.

3. Pay attention to the cultivation of good study habits.

The characteristics of simple equation learning content determine that it is particularly necessary and appropriate to cultivate students' habit of standardized writing and conscious testing through this unit.

As far as writing habits are concerned, whether writing with letters or solving equations, it is necessary to strengthen the necessary writing norms from the beginning. In order to play the powerful role of pre-perception and pre-concept, and promote the formation of good writing habits.

Judging from the test of solving mathematical problems, the test of solving equations is easy to learn and operate, and it is the easiest to show the effect of the test, so it is an important opportunity to cultivate students' test habits. Teachers should pay attention to it and grasp it.

(3) Description of teaching materials in each section and teaching suggestions 1. Use letters to represent numbers.

(Page 44-52) Textbook Description

The teaching in this section uses letters to represent numbers. This is the beginning of learning the basic knowledge of algebra. In arithmetic, people only study some specific and individual quantitative relations. After introducing letters to represent numbers, we can express and study the quantitative relationship with more universal significance. It can be said that learning algebra begins with learning to represent numbers with letters.

Four examples are arranged in this textbook. The four examples are not only progressive, but also have their own key points and are handled quite delicately. For example, the writing rules of some formulas containing letters are scattered in Example 2, Example 3 and Exercise 10, so that students can master them and reduce the memory burden.

Example 1 focuses on the transition from numbers represented by symbols to numbers represented by letters.

Example 2 introduces the writing method of omitting the multiplication sign in the formula with letters while teaching the arithmetic with letters.

Example 3 introduces the writing method of "square" and the writing habit of multiplying numbers with letters while teaching the calculation formula with letters, and then teaches substitution evaluation.

Example 4, focus on teaching the formula with letters to express the relationship between quantity and quantity, and continue to learn substitution evaluation.

In "Doing" and "Exercise 10", some corresponding exercises are arranged. There are consolidation exercises with examples and special exercises to pave the way for follow-up teaching. For example, the practice of expressing quantities with formulas containing letters can prepare for learning equations and solving practical problems in the future. Letters are used to express common quantitative relations, such as "distance = speed × time" and "S=vt". Examples are arranged in the original textbook. Now, considering that students can use letters to express calculation formulas, they can use analogy, so they are also used as exercises, interspersed in exercise 10. The whole exercise 10/3 exercise mainly focuses on algebraic writing and substitution evaluation, forming a series from basic exercise to variant exercise and comprehensive exercise.

Teaching suggestion

1. Let students feel the superiority of using letters to represent numbers.

In teaching, we should pay attention to let students feel the advantages of letter algebra through a series of teaching activities. For example, using letters to express arithmetic rules, especially multiplication and distribution rules, makes students feel that the symbolic language of mathematics is more concise than the written language. By abstracting the quantitative relationship expressed by letters from specific formulas, students can understand the cognitive needs from individual to general and feel the role of abstraction initially. Accumulating such experience and knowledge is helpful to improve learning interest and understand what you have learned.

2. Strengthen the training of expressing quantities with formulas containing letters.

The training of expressing quantities with formulas containing letters is also the training of writing algebraic expressions. This is the basis of the column equation. To strengthen this kind of training, we can use written homework or more oral answers, such as group oral answers, individual oral answers, group speeches, deskmate speeches, etc., to improve the efficiency of practice.

3. Pay attention to the idea of osmotic function.

It is mainly reflected in that when letters are used to express the inductive quantitative relationship, the corresponding relationship and dependence relationship between variables can be properly infiltrated. If the standard weight changes with the height, there is a one-to-one relationship between the two quantities. It is also reflected in the idea that the function definition domain can be properly penetrated when explaining the letter range. For example, in view of the textbook question, "Think about it, what numbers can the letters in the formula represent?" When teachers guide or evaluate students' answers, they can make students realize that there is often a certain range of numbers represented by letters in formulas, and this range needs specific analysis and cannot be generalized.

In addition, for classes that don't offer English classes or haven't learned English letters, the letters appearing in textbooks, such as A, B, C, H, S, T, V and X, can be recognized by students before Protestant school or in new teaching, so as to distinguish them from the pronunciation of Chinese Pinyin, thus removing obstacles for mathematics learning.

4. This part can be completed in 3 class hours.

1. Specific content description and teaching suggestions

1. Example 1.

Writing intention

Example 1 consists of three questions. The problem (1) is to find out the rule of the number of groups in each line, and determine the number expressed by numbers and letters according to the rule.

Problem (2) According to the known condition (an equation), it is equivalent to solving the equation to find the number represented by numbers and letters.

Problem (3) is to find out the law according to the given sequence, and then determine the number represented by the letters in the sequence.

As the beginning of formal learning to express numbers by letters, the three questions inherit the existing foundation of students and enrich their perceptual knowledge through various forms from symbols to letters. * * * is similar in that the symbols or letters here all represent a specific and concrete number, for example, the m in question (3) stands for 8.

Teaching suggestion

In teaching, students can think independently with three questions at the same time, try to find out the law, write unknown values, and then communicate. Students can also independently examine the questions and use their own words to describe the laws of each small question or the meaning of known conditions, such as:

(1) The sum of the left and right numbers is equal to the middle number; Or the number in the middle minus the number on the left is the number on the right.

(2) The sum of the three low points is12; Or 3 times of ● is 12.

Then work out the numbers represented by numbers or letters, and then talk about how you worked them out or what you think.

When summing up, you can ask: What are these three questions expressed in numbers or letters? Then it is pointed out that in mathematics, we often use letters to represent numbers. Then, ask the students to consider the question raised in the textbook: What other examples have you seen of numbers represented by symbols or letters? This leads to Example 2.

2. Example 2.

Writing intention

(1) Example 2 requires students to express the operation rules they have learned in letters. The textbook takes multiplication and commutative law as examples to illustrate the advantages of letter representation and introduce the usual writing methods of letter multiplication. Then it is required to use three numbers, A, B and C, and write other algorithms.

(2) "Do you know?" The list introduces the commonly used units of length, area and mass expressed by letters, so that students can further understand the various uses of letters and expand their knowledge.

Teaching suggestion

(1) When teaching Example 2, students can read the textbook first and then write other operation rules according to the requirements of the textbook. You can also ask students to say which algorithms they have learned first, then describe them in words and then in letters, and complete the table below.

Then read a book to learn how to omit the multiplication sign and change the places where the multiplication sign can be omitted in the table.

In teaching, special attention should be paid to guiding the understanding of the same operation law. It is troublesome to describe it in written language, and sometimes it is not easy to make it clear. If expressed in letters, it is clear at a glance, concise and easy to remember, and easy to apply. To this end, you can write on the blackboard appropriately. Take multiplication and division, for example. Expressed in language: When the sum of two numbers is multiplied by a number, you can multiply the two addends by this number respectively, and then add the products. Use letters: (a+b)c=ac+bc, which forms a sharp contrast and makes students feel the advantages of using letters.

Questions should also be asked: What numbers can A, B and C stand for here? Let the students understand that these three letters can represent any number we have learned.

For the rules of writing, we can only introduce here: the multiplication sign in the middle of the letter can be omitted or recorded as "",and it is emphasized that other operational symbols in the middle of the letter cannot be omitted. As for other writing rules, we will introduce them later.

(2) Reading materials can be read by students themselves. Students can also communicate the rules they find. For example, meters are represented by m, grams by g, kilometers and kilograms by k, decimeters, centimeters and millimeters by d, c and m, respectively. As for the representation of "square", wait until you learn Example 3. Teachers can point out that the letter representation of these units of measurement in the table is international.

3. Example 3 and "Do it".

Writing intention

(1) Example 3 takes the area and perimeter of a square as an example to teach how to express the calculation formula with letters and how to substitute the known data into the formula for evaluation.

As far as the thinking process is concerned, the transition from the formula composed of specific numbers to the formula containing letters is an abstract process from individual to general, while substituting specific numbers into the formula containing letters to find its value is contrary to the above process, and it is a concrete process from general to individual. Therefore, using the letter formula to evaluate can help students better understand the meaning of using letters to represent numbers. In addition, the skills of substitution evaluation can be used not only to substitute various calculation formulas, but also to solve equations and check computations. You need to practice when you start to contact the letter formula, so it is one of the important learning contents in the textbook "Representing Numbers with Letters". Limited by students' knowledge level and acceptance ability, there are no terms such as algebraic expression and algebraic value in the textbook.

When substituting data into the formula for evaluation, students should be reminded to restore the omitted multiplication symbols. For example, when a=6, 4a=4×6.

(2) "Do one thing" arranges two questions, which match the two small questions in Example 3. 1 Practice using letters to express the formula for calculating the area and perimeter of a rectangle, and the second practice is to substitute the formula to find the area and perimeter of a rectangle.

Teaching suggestion

(1) When teaching Example 3 (1), students can first describe the calculation method of the area and perimeter of rectangles and squares in language. Then introduce letters, that is, the area is usually represented by S, the perimeter is represented by C, the side length of a square is represented by A, and the width of a rectangle is represented by B. Let the students try to calculate the area and perimeter of a square by themselves with letters, and then turn to the books to see how the textbooks are expressed. Of course, teachers can also explain writing habits.

(2) Regarding the representation of "square", teachers should emphasize the significance of a2 and its difference from 2a. that is

A2 represents the product of two A's, that is, A× A..

2a stands for the sum of two A's, which is A+A..

You can also add some oral arithmetic exercises, such as 32, 52 and 62, to help students understand. But this unit only requires students to use the formula for calculating the square area, which can be written as 6×6 in the textbook when it is substituted for evaluation.

(3) When teaching Example 3 (2), you can show the questions first, and let the students try to write the letter formula orally and then enter the evaluation score calculation process, then read a book and fill in the blanks in the example. You can also demonstrate the substitution calculation process of square area by the teacher: write the formula first, then substitute the calculation and write the answer. What needs to be pointed out here is that the unit name of the calculated number only needs to be written in the answer. Then let the students complete the substitution calculation of the square perimeter by themselves.

(4) "Doing one thing" can be done by students independently, but teachers need to remind students of the writing format.

4. Example 4 and "Do it".

Writing intention

(1) Example 4 Teaching uses a formula containing letters to express the quantitative relationship and a quantity, including two examples. The former is an example of addition and subtraction, and the latter is an example of multiplication and division. Both examples are based on inductive thinking, that is, the formulas represented by specific numbers are listed first, so that students can see these formulas, and each formula can only represent individual phenomena, which leads to cognitive conflicts. How to express the general situation with a formula? This leads to formulas that contain letters.

In the previous example, students are guided to complete the induction from individual to general, and the conclusion that A+30 represents the age of dad in any year is drawn. Then, students are asked to substitute in the evaluation, from general to individual, to further understand that when A is a specific age, A+30 is also a specific age. Therefore, through the positive and negative thinking process, students can really understand that A+30 can really represent their father's age. The latter example has a similar treatment.

(2) "Do one thing" gives the relationship between the standard weight expressed in words and the height, so that students can express it in letters and use it to calculate the standard weight of their father. This is not only the supporting exercise of Example 4, but also allows students to see the application of mathematics in physical health, which is helpful to broaden their knowledge.

Teaching suggestion

(1) When teaching example 4 (1) is a minor problem, conditions can be given for students to indicate their father's age when Xiaohong 1 year old, 2 years old and 3 years old. The teacher pointed out: if you keep writing, each item can only indicate the age of your father in a certain year. Then ask: how to express my father's age in any year concisely with a formula? Group discussion can be organized to let students express their opinions. With the learning foundation of the first three examples, most students will think of "please write for help". Students can choose a letter to indicate Xiaohong's age and write a formula to indicate his father's age. When communicating, you can compare other expressions that students think of on the blackboard, such as writing, so that students can see that expressions with letters are simpler and clearer.

Next, lead the students to think: What numbers can A stand for here? A can it be 200? By answering, make the students clear that in a practical problem, the range of letters is determined by the actual situation.

Then ask the students to think: How old was Xiaohong when she was as old as most of our classmates, 1 1 years old? Students can be asked to fill in the process of substitution calculation in the textbook.

(2) When teaching example 4 (2) is a minor problem, students should tell the meaning of the problem after giving the conditions and explain why people can lift objects 6 times heavier than the ground on the moon. Usually, some students in a class know that this is because the gravity of the moon is smaller than that of the earth. On the basis of students' understanding of the meaning of the question, they can carry out the teaching process more freely than the sub-question (1).

Write: Use a formula containing letters to indicate the mass that people can lift on the moon.

Think about it: What numbers can the letters in the formula represent?

Calculate: How much can a young friend lift on the moon in the textbook illustration?

(3) In order to finish "doing something" in class, teachers should arrange for students to go home to know their father's height and weight before class. In class, let students use a formula containing letters to represent the standard weight formula of adult men, then substitute the centimeters of father's height to calculate the kilograms of standard weight, and then compare it with his father's actual weight to see if his father's weight is appropriate, overweight or thin.

If students are interested, you can also introduce the calculation method of the standard weight of adult women (height in centimeters, weight in kilograms)

Standard weight = height-1 10

During practice, the teacher can also report his height and let the students choose the corresponding calculation method to calculate the standard weight. The teacher will report his weight again, so that students can compare and judge whether the teacher's weight is up to standard.

5. Explanation of some exercises in Exercise 10 and teaching suggestions.

Question 1, omit the writing practice of multiplication sign. Four small questions correspond to four writing habits. That is, a×x, as long as the multiplication sign is omitted; X×x, expressed in square; B×8, omitting the multiplication sign and writing 8 in front; B× 1, 1 can be omitted, and students should be reminded when commenting.

Question 2, the consolidation exercise of square meaning. The upper and lower rows of formulas correspond side by side, in which a2 and a×2, 62 and 6×2 cannot be connected. When explaining, students can write a formula, which is connected with a2 and AX2 respectively.

Question 3: Arithmetic and writing consolidation exercises. The third sub-topic has a footnote, which allows students to read and understand by themselves, fill in the blanks by themselves, and cultivate students' self-study ability.

At this point, writing habits with letters appeared one after another. Therefore, teachers can guide students to summarize. For example:

The fourth question, the practice of writing algebra according to pictures, requires that the specified number be expressed by a formula containing letters according to the meaning of pictures. Four pictures correspond to four operations of addition, subtraction, multiplication and division.

The fifth question, the practice of writing algebra according to the text, four small questions, also correspond to four kinds of operations respectively, but it is more abstract than the previous question, which is conducive to cultivating students' reading comprehension ability. When practicing, remind students to read the questions carefully, and then write to fill in the blanks after understanding the meaning of the questions.

Questions 6 and 7 are exercises to express common quantitative relations with letters and substitute them for evaluation. Question 6 is about the relationship between distance, speed and time. Fill in the blanks with illustrations can play a role in prompting and paving the way. Students should be reminded to fill in the blanks in the illustrations before summarizing the relationship. Question 7 is about the relationship between unit price, quantity and total price. It is required to write the formula for calculating the total price first, then deform the formula by using the multiplication and division relationship, write the formula for calculating the unit price and quantity, and then select a formula to substitute for evaluation.

The practice ideas of questions 8 and 9 are just the opposite of the previous questions 4 and 5. According to the meaning of the question, it is required to explain the algebraic formula given, that is, to tell the actual meaning of the letter formula. This is very helpful to further cultivate students' mathematical reading comprehension ability. When practicing, let the students think independently first, then communicate with each other at the same table or in groups, and then communicate with the whole class.

10 ~ 12 requires writing algebraic expressions according to the meaning of the questions and substituting them for evaluation. The comprehensive degree of the quantitative relationship in the questions is slightly improved, and teachers can give appropriate guidance when practicing.

The question 13* is for students who have spare capacity. In fact, it is a geometric model of multiplication and division, that is, the multiplication and division method is explained intuitively through area calculation.