Zhejiang education printing plate seventh grade mathematics first volume teaching material teaching plan
Chapter 1 Rational Numbers
1. 1 positive and negative numbers
Positive and negative numbers in 1
Teaching objectives:
1. Understanding positive and negative numbers is the need of real life.
2. Will judge whether a number is positive or negative.
3. Positive numbers and negative numbers will be used to represent quantities with opposite meanings.
Teaching emphasis: judge positive and negative numbers, use positive and negative numbers to represent quantities with opposite meanings, and understand the meaning of quantities with opposite meanings.
Teaching difficulty: the introduction of negative numbers.
Interactive design of teaching and learning;
(A) the creation of situations, the introduction of new courses
The courseware shows Mount Everest and Turpan Basin, so that students can feel the different situations above and below the water level.
(2) Cooperation and exchange, interpretation and exploration
For example, the temperature is 7℃ above zero and 5℃ below zero, 90 desks have been bought and 80 desks have been sold, and the car travels 50 meters east and west 120 meters.
Think about it. These are quantities with opposite meanings. Can you use the numbers in elementary school arithmetic to represent each pair of quantities? Can you name some quantities with opposite meanings in daily life? How to express them?
In order to use numbers to represent quantities with opposite meanings, we stipulate that quantities with only one meaning, such as temperature above zero, advance, income, rise and rise, are positive, while quantities with opposite meanings, such as temperature below zero, retreat, expenditure, decline and fall, are negative. Positive quantities are expressed by arithmetically learned numbers, and negative quantities are preceded by "-"(read
Each group of students cooperates and communicates with each other. One student said two quantities with opposite meanings, and the other students expressed them with positive numbers and negative numbers.
Discuss what kind of numbers are negative. What kind of numbers are positive numbers? Is 0 a positive or negative number? List positive and negative numbers yourself.
A plus number is a number greater than 0, and a minus number is a number preceded by a "-". 0 is neither a positive number nor a negative number, it is the dividing point between positive and negative numbers.
(3) Application of migration, integration and improvement
Example 1 gives several pairs of quantities with opposite meanings, which are represented by positive numbers and negative numbers respectively.
The quantities with opposite meanings are rising and falling, before and after, above and below, gain and loss, income and expenditure.
In a table tennis test, what does it mean that a table tennis ball exceeds the standard mass by +0.02g, which means +0.02g and -0.03g?
Example 3 A scientific research takes 45 minutes as 1 time unit, and every morning 10 is recorded as 0, which is negative before 10 and positive after 10. For example, 9: 15 is recorded as-1, 65438+.
3C。 -2.5d-7.45
It is the key to know the meaning of the problem by dialing. The difference between 7:45 and 10:00 135 minutes.
(4) Summing up reflection and expanding sublimation.
Negative numbers are introduced to represent quantities with opposite meanings in real life. Positive numbers are the numbers we have learned in the past (except zero). Adding "-"before a positive number is a negative number. It cannot be said that "the number with a positive sign is positive and the number with a negative sign is negative". In addition, 0 is neither positive nor negative.
1. The following table shows the funds in and out of Xiao Zhang's piggy bank within one week (deposits are indicated by "+"):
Sunday one two three four five six
(Yuan)+16+5.0-1.2-2.1-0.9+10-2.6
(1) How much did Xiao Zhang spend this week? How much money have you saved?
(2) Is there more or less money in the piggy bank than before?
(3) How to keep accounts without using positive or negative numbers? Compare the advantages and disadvantages of various bookkeeping methods.
2. Math game: 4 students stand in a row or squat in a row. From left to right, the number of each student is: 1, 2, 3, 4. Use "+"to mean "standing" and "-"(minus sign) to mean "squatting".
(1) If a classmate shouts:+1, -2, -3, +4, 1 and the fourth classmate is standing, the second and third classmates are squatting, and then shout:-1, -2,+.
(2) Increase the difficulty of the game and adjust the order of the four students, but each student records his original number and repeats the game in (1).
(E) classroom tracking feedback
Solid foundation
1. Fill in the blanks:
(1) If saving 30 tons of water is marked as +30 tons, then wasting 20 tons is marked as tons.
(2) If 4 years later is recorded as +4 years, then 8 years ago is recorded as one year.
(3) If the 7-ton shipment is recorded as -7 tons,+100 tons means.
(4) Xiao Liang gained 3kg in one year, which was recorded as+3 kg; Xiaoyang lost 2 Jin, and then Xiaoyang gained.
2. At noon 12, the water level was -0.5m lower than the standard water level and recorded as-0.5m. In the afternoon 1, the water level rose 1m, and at 5 pm, the water level rose by 0.5m. 。
(1) Record the water levels of 1 pm and 5 pm with positive or negative numbers;
(2) How much higher is the water level at 5pm than that at noon 12?
Improve ability
3. The standard weight of each bag of grain is 50kg, and the measured weights of the three bags of grain A, B and C are 52kg, 49kg and 49.8kg respectively. If the overweight part is represented by a positive number, please record the overweight and shortage of the three bags of grain A, B and C with positive and negative numbers.
(6) class summary
1. Compared with before, what is the meaning of 0?
2. How to use positive numbers and negative numbers to represent quantities with opposite meanings? (a positive number is used to represent a quantity of meaning, and the other is represented by a negative number)
The Application of Positive and Negative Numbers in the Second Classroom
Teaching objectives:
1. Through the discussion of the meaning of "zero", we can further understand the concepts of positive and negative numbers, and we can correctly use positive and negative numbers to represent quantities with opposite meanings (the quantity that changes in the specified direction is specified);
2. Further experience the wide application of positive and negative numbers in production and life, and improve the ability to solve practical problems.
Teaching emphasis: deepen the understanding of the concepts of positive and negative numbers.
Teaching difficulty: correctly understand and express the variation in the specified direction.
Interactive design of teaching and learning;
Knowledge review and understanding
Through the study of last class, we know that there are two quantities with different meanings in actual production and life. In order to distinguish them, we use positive numbers and negative numbers respectively.
[Question 1]: Why is "zero" neither positive nor negative?
Students think, discuss and illustrate with examples.
Reference example: use positive numbers, negative numbers and zero to indicate the temperature above zero, below zero and above zero.
What is the significance of thinking about "0" in practical problems?
Inducing "0" not only means "nothing" in practical problems, but also has certain practical significance.
For example, the change of water level when the water level does not rise or fall is recorded as: 0m.
[Question 2]: After introducing negative numbers, numbers are divided into "two quantities with opposite meanings". How many categories can they be divided into? What is the difference?
(B) to deepen understanding and solve problems
[Question 3]: (Example of textbook P3)
Example 1( 1) within one month, Xiaoming gained 2kg, Xiaohua lost 1kg, and Xiao Qiang's weight remained unchanged. Write down their weight gain this month.
Example 2(2) The changes in the total import and export volume of countries below a certain year compared with the previous year are as follows:
The United States decreased by 6.4% and Germany increased by 1.3%.
France fell by 2.4%, Britain by 3.5%,
Italian growth of 0.2%, China growth of 7.5%.
Write down the growth rate of total import and export of these countries this year.
Note: In the same question, positive numbers and negative numbers have opposite meanings. Write down the weight gain value and the growth rate of import and export, suggesting that the growth amount is expressed as a positive number. Similarly, there are rising water levels and rising incomes. When solving problems, we should pay attention to these quantities indicating the direction and correctly express them with positive and negative numbers.
Consolidation exercise
1. remind students to pay attention to the requirements when reviewing the questions through the example (2). What is sought in the question is the growth rate, not the growth value.
2. Ask students to name some commonly used quantities with opposite meanings.
3. 1990~ 1995, the annual average forest area (unit: km2) in the following countries changes as follows:
China decreased by 866, India increased by 72,
South Korea decreased by 130, New Zealand increased by 434,
Thailand decreased by 3,247, and Bangladesh decreased by 88.
(1) positive and negative numbers indicate that the average forest area of these six countries has increased from 1990 to1995;
(2) How to express the reduction of forest area, and what is the relationship between the result and the increase?
(3) Which country has suffered the greatest loss of forest area?
(4) Based on the analysis of these data, what's your opinion?
Reading and thinking
(Textbook P6) Positive numbers and negative numbers are used to indicate allowable machining errors.
Question: 1. Are parts with diameters of 30.032mm and 29.97mm qualified?
2. Do you know any other events that can express the allowable error with positive and negative numbers? Please give an example.
(3) Application of migration, integration and improvement
1. The temperature of cold storage A is-12℃, and the temperature of cold storage B is 5℃ lower than that of cold storage A, so the temperature of cold storage B is.
2. The inner diameter of a part on the drawing is 9 0.05 (unit: mm), indicating that the standard size of this part is 9mm, and how much does the machining requirement not exceed the standard size? What is the minimum size not less than the standard size?
The motorcycle factory plans to produce 250 motorcycles every day this week. Because workers take turns off, the number of people who go to work every day is not necessarily equal. The actual daily output (compared with the planned quantity) is increased or decreased in the following table:
Monday 1234
Increase or decrease -5+7-3+4
According to the above records, Q: In which days did you produce more motorcycles than planned? On which day of the week do you produce the most motorcycles? How many motorcycles are there? What day of the week produces the least number of motorcycles? How many?
Analogy examples require students to pay attention to the writing format and experience the application of positive and negative numbers.
(4) class summary (completed by teachers and students)
1.2 rational number
Rational Numbers of Class 1
Teaching objectives:
1. Understand the meaning of rational numbers.
2. Be able to classify the given rational number as needed.
3. Understand the role of 0 in rational number classification.
Teaching emphasis: the given number will be filled in the set diagram where it is located.
Difficulties in teaching: mastering two classifications of rational numbers.
Interactive design of teaching and learning;
(A) the creation of situations, the introduction of new courses
Now, as you all know, there is another form of number besides what we learned in primary school, and that is negative number. Let's discuss what types of numbers you know so far.
(2) Cooperation and exchange, interpretation and exploration
3,5.7,-7,-9,- 10,0,,,-3,-7.4,5.2…
Tell me about it. Can you tell me something about the characteristics of these figures?
Students answer and complement each other: there are positive integers, zeros and fractions learned in primary school, as well as negative integers and fractions.
It means that we call these figures reasonable.
Just try it. Can the above figures be classified into a table?
rational number
If you do more than one thing, you can divide it by integers and fractions. Can you divide by nature (positive and negative)? Just try it.
rational number
A set of numbers
The set of all positive numbers is called the set of positive numbers.
Try to sum up what is a negative set, what is an integer set, what is a fractional set and what is a rational number set.
(3) Application of migration, integration and improvement
Example 1 Fill in the following figures in the corresponding set:
,3. 14 16,0,2004,-,-0.23456, 10%, 10. 1,0.67,-89
The following is the classification method of two students. Do you think their classification results are correct? Why?
rational number
(4) Summing up reflection and expanding sublimation.
Question: What knowledge have you gained today?
Summarize by the students themselves, and then the teacher summarizes: Today we learned the definition of rational numbers and two classification methods. We should be able to correctly judge which category a number belongs to, and pay special attention to the correct statement of "0".
The two circles below represent the set of negative numbers and the set of fractions respectively. Can you tell which group of numbers the overlapping parts of these two figures represent?
(E) classroom tracking feedback
Solid foundation
1. Fill the following figures in the corresponding brackets:
-7,0. 125,,-3,3,0,50%,-0.3
(1) integer set {};
(2) Score set {0};
(3) Negative score set {0};
(4) Non-negative set {};
(5){ rational number set}.
2. The following statement is true ()
A. Integers are natural numbers
B.0 is not a natural number.
C. positive numbers and negative numbers are collectively called rational numbers.
D.0 is an integer, not a positive number.
Improve ability
The letter a can represent a number. Within the scope of what we are learning now, can you try to explain what number A stands for?
Number axis of the second kind
Teaching objectives:
1. Master the three elements of the number axis and draw the number axis correctly.
2. Can represent the known number on the number axis, and can tell the number represented by the known point on the number axis.
Teaching emphasis: the concept of number axis.
Teaching difficulties: from intuitive understanding to rational understanding, thus establishing the concept of number axis.
Interactive design of teaching and learning;
(A) the creation of situations, the introduction of new courses
Courseware shows the "problems" in teaching material P7 (students draw pictures)
(2) Cooperation and exchange, interpretation and exploration
Teacher: In order to express it more clearly, we use positive numbers and negative numbers on the left and right sides of 0, that is, we use points on a straight line to represent positive numbers, negative numbers and 0. This is what we are going to learn in this section-number axis.
Hug (1) guides students to learn how to draw several axes.
Step 1: Draw a straight line and determine the origin.
Step 2: Specify that the direction from the origin to the right is positive (negative to the left).
Step 3: Choose the appropriate length as the unit length (as the case may be).
Step 4: Take out the teaching thermometer and let the students observe whether the structure of the thermometer is similar to that of the number axis.
Compare and think about what the origin is equivalent to; What is the positive direction; What is the unit length?
(2) With the above foundation, we can try to define the number axis:
The straight line that defines the origin, positive direction and unit length is called the number axis.
Do it. Students practice drawing several axes by themselves.
Just try it. Can you represent the numbers 4, 1.5, -3, -2, 0 with the points on the number axis drawn by yourself?
If a is a positive number, where is the point representing the number A on the number axis at the origin? How many unit lengths are there from the origin? Where is the point representing -a on the origin? How many unit lengths are there from the origin?
Can summing integers find a point representation on the number axis? What about the score?
It can be seen that all can be represented by points on the number axis; All on the left of the origin, all on the right of the origin.
(3) Application of migration, integration and improvement
Example 1 Is the axis drawn below correct? If not, point out what is wrong.
Try Example 2: Use the points on the number axis you draw to represent 4, 1.5, -3, -0.
Example 3 The following statement:
① Points on the number axis can only represent integers; ② The number axis is a straight line; ③ A point on the number axis can only represent a number; ④ There is no point on the number axis that represents neither positive nor negative numbers; ⑤ The numbers represented by points on the number axis are rational numbers. The correct statement is ()
1。
Example 4 represents -2 and 1 on the number axis, and points out all integers greater than -2 and less than 1 according to the number axis.
Example 5 The points representing integers on the number axis are called whole points, and the unit length of the number axis is 1cm. If a line segment AB with a length of 2000cm is randomly drawn on this axis, the whole point covered by the line segment AB is ().
A. 1998 or 1999 B. 1999 or 2000.
C.2000 or 200 1 D.200 1 or 2002.
(4) Summing up reflection and expanding sublimation.
The number axis is a very important tool, which establishes a one-to-one correspondence between numbers and points on a straight line. It reveals the internal relationship between number and shape, and provides new methods and new ideas for our further research in the future. To master the three elements of the number axis and draw the number axis correctly. Remind everyone that all rational numbers can be represented by related points on the number axis, and vice versa, that is, all points on the number axis do not represent rational numbers.
(E) classroom tracking feedback
Solid foundation
1. Specifies that the straight lines of, and are called number axes, and all rational numbers can be represented by points.
2.p Start from the origin on the number axis, move 2 unit lengths to the right, and then move 5 unit lengths to the left. At this time, the number represented by point P is.
3. Move the point representing 2 on the number axis by 5 unit lengths, and the number represented by the corresponding point is ().
a . 7 B- 3
C.7 or -3D. uncertain
4. On the number axis, the origin and the point to the left of the origin represent the number ().
A. positive number B. negative number
C. is not a negative number. D. not a positive number
5. The distance between the points representing 5 and -5 on the number axis and the origin is, but they represent respectively.
Improve ability
6. There are two points 3.5 unit lengths away from the origin, namely and.
7. Draw a number axis and indicate the following numbers on the number axis:
+2,-3,0.5,0,-4.5,4,3.
Open investigation
8. There are three points with a distance of-1 unit length on the number axis, namely: put a piece of wood with a length of 3 units on the number axis, which can cover an integer point at most.
9. Among the following four numbers, the number between -2 and 0 is ().
A.- 1B. 1C。 -3D.3
Reciprocal in the third category
Teaching objectives:
1. Understand the concept of opposites with the help of the number axis and know the positional relationship of opposites.
Given a number, we can find its reciprocal.
Teaching emphasis: understanding the meaning of the opposite number.
Teaching difficulties: understanding and mastering the law of double symbol simplification.
Interactive design of teaching and learning;
(A) the creation of situations, the introduction of new courses
Ask a student to go to the podium and tell everyone, step forward and step back.
If communication is positive, what are five steps forward and five steps back?
(2) Cooperation and exchange, interpretation and exploration
1. Observe the following numbers: 6 and -6, 2 and -2, 7 and -7, and mark them on the number axis.
What are the characteristics of the logarithm on (1)?
(2) What are the characteristics of the points representing these four logarithms on the number axis?
(3) Can you write n groups of numbers with the above characteristics?
Observing two numbers with different signs like this is called the opposite number.
The corresponding points (except 0) of two numbers with opposite numbers on the number axis are two points with the same distance from the origin on both sides. That is, we write the inverse of a as -a, and stipulate that the inverse of 0 is zero.
Add a "-"sign in front of a positive number, and you get the reciprocal of this positive number, which is a negative number; Remove the "-"sign before a negative number to get the reciprocal of this negative number, which is a positive number.
Add a "-"sign before any number, and the new number is the reciprocal of the original number. For example, -(+5)=-5, which means that the reciprocal of +5 is-5; -(-5)=5, which means that the reciprocal of -5 is 5; -0=0, which means that the reciprocal of 0 is 0.
(3) Application of migration, integration and improvement
Example 1 Fill in the blanks
(1)-5.8 is the inverse number, the inverse number of A is -(+3), and the inverse number of A is; The inverse of a-b is 0, and the inverse of 0 is 0.
(2) The inverse of a positive number is, the inverse of a negative number is, and the inverse of is itself.
Example 2 The following judgment is incorrect ()
(1) Two opposite numbers must not be equal; (2) the points on the number axis that are opposite to each other must be on both sides of the origin; ③ All rational numbers have antonyms; The opposite number is two points with opposite signs.
1。
Example 3 Simplify the following symbols:
( 1)-[-(-2)]; (2)+{-[-(+5)]};
(3)-{-...-(-6)} ...} (* * n minus signs).
The law of induction and simplification is: even if there is a negative sign, the result is positive; There are odd negative signs, and the result is negative.
Example 4 On the number axis, point A represents +4, while point B and point C represent opposite numbers, and the distance from C to A is 2. What numbers do point B and point C correspond to respectively?
(4) Summing up reflection and expanding sublimation.
The concept and representation of inductive (1) inverse number.
(2) Algebraic meaning and geometric meaning of reciprocal.
(3) Simplification of symbols.
(E) classroom tracking feedback
Solid foundation
1. True or false
(1)-3 is the opposite number. ()
(2)-7 and 7 are opposites. ()
(3) The reciprocal of-A is A, and they are reciprocal. ()
(4) The numbers with different symbols are opposite. ()
2. Write the opposite numbers of the following numbers respectively and express them on the number axis.
1,-2,0,4.5,-2.5,3
3. If the reciprocal of a number is not a positive number, it must be ().
A. positive number B. positive number or 0
C. negative number D. negative number or 0
4. A number is smaller than its reciprocal, and this number is ().
A. positive number B. negative number
C. non-negative number D. non-positive number
5. If the distance between two points representing opposite numbers on the number axis is 4, then these two numbers are.
Improve ability
6. If A and a-2 are reciprocal, then the reciprocal of A is.
7. Given the positions of rational numbers m, -3 and n on the number axis as shown in the figure, the reciprocal of m, -3 and n is expressed on the number axis, and these six numbers are expressed as ".