Teaching Record and Commentary on "Preliminary Understanding of Negative Numbers"

Teaching content:

Jiangsu Education Edition Primary School Mathematics for Fifth Grade Mathematics Volume 1, Examples on Pages 1-2 1. Example 2, "Try it", questions 1-4 of "Exercise 1" on page 5.

Analysis of academic situation:

"Preliminary Understanding of Negative Numbers" (the textbook before the 2011 edition of the curriculum standard was called "Understanding Negative Numbers") only appears in the fifth grade of the Jiangsu Education Edition of primary school mathematics textbooks. On pages 1-5 of the first grade volume. Looking at the contents of the 12 volumes of the Jiangsu Education Edition of primary school mathematics textbooks, there is no knowledge of negative numbers before, and it has not been covered since. Only these 5 pages form a thin and independent unit. Before the fifth grade, the numbers students are exposed to are positive numbers and 0. Negative numbers are an unfamiliar concept, but the seeds of "negative numbers" have been buried in students' lives. For example, if there is -1 floor in the elevator, the temperature in winter will appear. How many degrees Celsius is negative, etc.

Teaching objectives:

1. Preliminarily perceive the background of negative numbers in the process of counting and measuring, and experience the significance of negative numbers in real life;?

2. Understand the meaning of positive numbers, negative numbers and "0", master the expression methods of positive and negative numbers, and be able to use positive and negative numbers to describe phenomena in real life. For example, "0" can be used for temperature, altitude, etc. Quantities with different meanings as intermediate quantities;?

3. Experience the close relationship between mathematics and life, mathematics and culture, and stimulate students' interest in mathematics learning.

Key and difficult points in teaching:

1. Understand the meaning of positive numbers, negative numbers and "0";

2. Use positive numbers, negative numbers and "0" Describe phenomena in life. ?

Teaching process:

1. Introduction: The seeds of negative numbers are bred in the process of “counting” and “measuring”

Teacher: Students, please look at the projection. (Show Hua Luogeng’s photo) Do you know him? (Some students said they knew him, some said he didn’t know him)

Teacher: He is a famous mathematician in my country. (Showing the three characters "Hua Luogeng") His name is... Tell me (student said), what is his last name? The character (华) is polyphonic, so when used as a surname, it should be pronounced huà. He once said a sentence. Because of the polyphonic characters, many people mispronounced it (showing "Number originates from number, and quantity originates from quantity"). Who will give it a try? (Look for students who raise their hands)

Teacher: (Turn the two "numbers" into red) Look at these two identical words. Do they have the same pronunciation? Which word should be read first? What about the second one? (Based on the students’ answers, show the pinyin of the two words accordingly)

Teacher: (Turn the two “quantities” into red) Are the pronunciation of these two words the same? (Based on the students’ answers, show the pinyin of the two words accordingly)

Teacher: Okay, let’s read this sentence together! (Students read together) What do you want to say after reading?

Student 1: Pay attention to polyphonic words when reading!

生2: Number (shù) comes from counting (shǔ), and quantity (liàng) comes from measuring (liáng)!

Teacher: It seems that learning mathematics is inseparable from "number" shǔ and "quantity"! (Shadows appear on "number" and "quantity")

[Commentary: Mathematician Hua Luogeng's surname is polyphonic, and one of his mathematical quotes also has polyphonic characters. It seems unintentional and coincidental, but it is actually mine. Designed with care. By guiding students to accurately read polyphonic words, students can feel that the concise language contains profound mathematical truths from Hua Luogeng's famous sayings. They can use this as a starting point to find the feeling of learning mathematics, and extend from mathematical famous quotes to the teaching and learning of new courses. ! In this process, students not only appreciate the charm of mathematical culture, but also taste the taste of mathematical culture. At the same time, they also grow and develop themselves, reflecting the taste of growth.

]

Teacher: Let us count them first, okay? (Okay)

Teacher: (shows a circle) Can you count? Count together (1) (show another disc) and then count (2) (show 10 discs in a row and let the students keep counting). If you continue counting like this, will you finish counting? (Infinite number) What should I do? (Students say the ellipsis and show the ellipsis)

Teacher: Now we start from 10 and count backwards (reduce the number of discs in turn and let the students count when one appears) Is there any number smaller than 1? (0, there is not a single disc in the courseware) What does 0 mean?

Teacher: 0 means there is no object at all. It seems that when expressing the number of objects, 0 is the smallest number!

Teacher: Okay, we just counted together, now let’s measure again!

Teacher: (showing a blue ribbon) There is a blue ribbon here. Can you measure its length? (Yes) I believe everyone can measure it! (Show a ruler to measure the length of the strip) Is this the measurement? Its length is... (4cm) What does the 0 here mean besides meaning no length? (Guide students to tell the starting point of measurement, and "starting point" appears)

Teacher: A blue ribbon of the same length (move the blue ribbon downward), a student measured it like this, do you think it is okay? ? (Let the students talk about it)

Teacher: The teacher also thinks that this measurement is OK, but the starting point of the measurement is changed from 0 to 1 (show the red line), then the number 5 on the ruler should become How much? (The student’s answer is 4) What numbers do 2, 3, and 4 on the ruler turn into? (Student answers 1, 2, 3) What about 6, 7, 8 and 9? (Students answered 5, 6, 7, 8)

Teacher: Okay, then what number should the 0 on the ruler become? Ask students to write your answers in their exercise books! (Students write by hand, and the teacher inspects and asks students with different writing methods to write on the blackboard)

Teacher: Have you finished writing? Ever wonder how mathematicians represent this number? (Think) This is how mathematicians express this number (appears -1). If you write it like a mathematician, please raise your hand! (Some students wrote -1) The teacher gives you a sentence: You already have the potential of mathematicians, how awesome! Please put it down!

Teacher: Mathematicians add a small dash in front of 1 to represent this number. This small dash is not a minus sign, but a negative sign. This kind of number is the negative number we want to know today ( Part of the topic posted on the board: negative numbers)

Teacher: How to read this number? (Show "Pronounced as: minus one") Read it together! (Students read, teacher writes on the blackboard - 1) Read again! (Students read again)

Teacher: Now, the teacher will move the ruler 1cm to the left. What number should be used to represent 0 at this time? Please write it down in your exercise book (-2 appears). Students who wrote the same thing please raise their hands! (All students in the class write -2) Read it together. If we move another 1cm to the left, what number should be used to represent 0 at this time? (-3) What if we pan to the left again?

Teacher: (A straight line appears in the courseware) This is a straight line. If the point on the straight line is represented by 0, and these points are represented by 1, 2, 3, 4, and 5 respectively, then these points What numbers should be used to represent it? (Student answers: -1, -2, -3, -4)

Teacher summary: OK, students, during the counting process just now, we found that when expressing the number of objects, 0 is The smallest number; in the process of measuring, we found that 0 can also represent the starting point of measurement. At the same time, we got to know a new number called (pointing to the "negative number" on the blackboard)... (negative number) Now let's think about it together. Think about it, have you ever seen negative numbers in your life? (Students said that the teacher immediately asked you what the negative numbers you saw meant?)

[Commentary: The understanding of "negative numbers" is not simply to understand that numbers smaller than 0 are called negative numbers. The design of this link allows students to feel that when expressing the number of objects, 0 is the smallest number in the process of "counting"; in the process of "measuring", 0 not only means nothing, but also Represents "starting point". I changed the starting point from 0 to 1 in real time, and the other numbers on the ruler changed in turn, allowing students to think: How should 0 be expressed at this time? The students tried to express this number. Although only some students expressed it with -1, but the whole class later used -2 ??to express it, they established the concept of negative numbers.

Students have an intuitive perception of negative numbers in the unconscious process. This design is in line with children's age characteristics and cognitive rules, and has a strong flavor of children and the growth of knowledge, while also having a light cultural flavor. ]

2. Focus: Feel the existence of negative numbers in the process of understanding "temperature" and "altitude"

Teacher: (shows a thermometer) What is this? (Thermometer) Do you know it? Who will introduce it?

Student 1: There is a red liquid in the middle of the thermometer. When the temperature is high, it will rise, and when the temperature is low, it will fall!

Student 2: There is a 0 scale on the thermometer, which means 0 degrees Celsius.

Student 3 added: If it is higher than 0 degrees Celsius, it is called how many degrees Celsius above zero; if it is lower than 0 degrees Celsius, it is called how many degrees Celsius below zero.

Teacher: A thermometer is an instrument used to measure temperature! There is a small circle with an F in the upper right corner of the thermometer to represent the Fahrenheit temperature. This is used in other countries, but generally not in China. A small circle plus a C in the upper left corner represents the temperature in Celsius, which is commonly used by us Chinese to measure temperature. Just now a classmate said 0 degrees Celsius. Does 0 degrees Celsius mean there is no temperature? (No) What is the temperature represented by 0 degrees Celsius? The temperature of the ice-water mixture is set to 0 degrees Celsius. (Show a picture of the ice-water mixture) Can you feel 0 degrees Celsius?

Teacher: This is the lowest temperature in Nanjing, Sanya and Harbin on a certain day. (Example 1 picture)

Teacher: Let’s first look at what is the lowest temperature in Nanjing? (Enlarge the temperature in Nanjing) Who will read it? (Show 0℃), (Zoom in to the temperature in Sanya) What is the lowest temperature in Sanya on this day? (20℃ above zero) 20℃ above zero, we can also express it as +20℃ (show +20℃). This number is read as positive twenty degrees Celsius. Read it together (students read, writing on the blackboard: +20), (enlarge The temperature in Harbin) (show minus 20℃ and -20℃ based on the students’ answers, and write -20 on the blackboard).

Teacher: Here are the two expression methods of 20℃ above zero and +20℃, and 20℃ below zero and -20℃. Which one do you think is better? Tell us your reasons! (Student said) That’s true. (20℃ above zero and 20℃ below zero disappear)

Teacher: What is the difference in meaning between "+20℃" and "-20℃" here? (Guide students to say that numbers larger than 0 are represented by positive numbers, and numbers smaller than 0 are represented by negative numbers)

Teacher: It seems that the students have already understood the temperature represented by the thermometer. Do you want to try it? Give it a try?

Teacher: (Show "Give it a try") This is the highest peak in the world - Mount Everest! Its peak is located at the border between my country and Nepal. (Show the temperature map) This is the lowest temperature on Mount Everest on a certain day. Who will read it? (Students will show -17℃ after answering and write -17 on the blackboard)

Teacher: This is the Turpan Basin, the basin with the lowest altitude in the world. This is the highest temperature in the Turpan Basin on a certain day. Who will read it? (Based on the students’ answers, show +35℃ and write +35 on the blackboard). This is the lowest temperature in the Turpan Basin on this day. Who will read it? (Based on the students’ answers, show -5℃ and write -5 on the blackboard)

Teacher: Please think about it carefully. What is the temperature difference in the Turpan Basin on this day? Do you know what "temperature difference" is? (Student said) What is the temperature difference? (40℃) Is the temperature difference big? Why is the temperature difference so big? In fact, the reason for the large temperature difference in the Turpan Basin is related to its unique geographical location!

Teacher: So what is the unique geographical location of the Turpan Basin? Want to know more? (Think)

Teacher: If we imagine that the highest peak in the world - Mount Everest and the Turpan Basin, the lowest basin in the world - are moved together. (Example 2 picture)

Teacher: The word "elevation" appears here. (Move the two "elevations" together) Can anyone tell me what altitude means? (Student says first) Altitude is the vertical height above sea level! Usually, we stipulate that the average altitude of sea level is 0 meters.

How should we express Mount Everest, which is higher than sea level, and Turpan Basin, which is lower than sea level? Invite two people at the same table to discuss!

Teacher: Mount Everest is 8844.4 meters above sea level, we call it 8844.4 meters above sea level, (appears as 8844.4 meters above sea level), and can be recorded as "+8844.4 meters" (appears as +8844.4 meters, writing on the blackboard: +8844.4) How should the height of Turpan Basin be expressed? (Students answered that the altitude appeared to be minus 155 meters, -155 meters, and written on the blackboard -155)

Teacher: Do you want to know the average altitude of our Huai'an? (Thinking) (Showing the altitude of Huai'an) Do you know what the altitude of our Huai'an is?

Teacher: Okay, students, look at the blackboard together. Are all the numbers on the blackboard negative? (No) So what are negative numbers? (Student) Numbers like -1, -17.5, -20, -155 are called negative numbers (the board posted "Negative Numbers"). Can you name another negative number? Are you finished? what to do? (ellipsis)

Teacher: Numbers like +20, +35, +8844.4, +14.5 are called... (positive numbers, "positive numbers" are posted on the board). Can you name another positive number? Are you finished?

[Commentary: The introduction of the thermometer as an example 1 has laid a good foundation for the teaching. Only when students have an understanding of the components and functions of a thermometer can they consciously read the information expressed in the thermometer. temperature. The trial practice after Example 1 not only tested whether the students correctly read the temperature indicated by the thermometer, but also laid a natural foundation for the teaching of Example 2 to realize the natural link between "temperature" and "altitude". From students talking about negative numbers in life to finally revealing the concept of negative numbers, the process presents the flavor of life. ]

3. Influence: Experience the simplicity of mathematical symbols in the evolution of the history of negative numbers

Teacher: In fact, the generation and development of negative numbers have a long history. Let’s find out together.

("Did you know?" appears and the recording is played: China is the first country to recognize and use negative numbers. According to "Nine Chapters of Arithmetic", as early as more than 2,000 years ago, ancient Chinese people had "grain" The idea of ????"entering a position is positive and exiting a position is negative; the money earned is positive and the money spent is negative". More than 1,700 years ago, Chinese mathematician Liu Hui first proposed the concepts of positive and negative numbers. More than 400 years ago, a French mathematician Girard used "+" to represent positive numbers and "-" to represent negative numbers for the first time. This method is still used today)

Teacher: Students, what have you read in the history of negative numbers? Student 1: China is the first country to recognize and use negative numbers! (Real-time guidance: We should be proud as Chinese)

Student 2: More than 1,700 years ago, there were concepts of positive and negative numbers.

Sheng 3: More than 400 years ago, "+" was used to represent positive numbers, and "-" was used to represent negative numbers.

Teacher: In the process of the creation and development of negative numbers, has it changed? What has changed?

Student: From expression methods, to Chinese characters, to symbols, it has become more and more concise!

Teacher: What the students said is really good! Some people say: Mathematical language is the most concise language in the world! It seems that there is some truth to this statement.

[Commentary: Regarding the teaching of "Did you know?", the general design is to put this part of the content at the end of the class and read it so that students can understand it. And I put the "Did you know?" teaching in the middle link, before the exercises, to highlight the mathematical culture as an important part of teaching, and to explore it to truly highlight the flavor of mathematical culture. This period of cultural history not only allows students to feel the pride of being Chinese, but also allows students to experience the gradual simplicity of "negative numbers" in the process of their emergence and development, trying to present symbolic ideas. ]

4. Perspective: Use a mathematical perspective to enter the world of negative numbers and improve the understanding of negative numbers

Teacher: Let us use a mathematical perspective to enter the "world of negative numbers" "!

Teacher: (Show "One point") Fill in the following numbers into the appropriate circles (students answer together, when asked about 8) What number is 8? (Positive numbers) There is a "+" sign in front of a positive number, but there is no "+" in front of 8. Why is it also a positive number? (Students try to explain why) In order to further simplify positive numbers, the "+" in front of positive numbers can also be omitted; of course, adding "+" in front of positive numbers is to be consistent with negative numbers in form.

Can the "-" sign in front of a negative number be omitted?

Student: No, if the symbol in front of a negative number is omitted, it will be confused with a positive number!

Teacher: (write 8 on the blackboard within the range of positive numbers) +20, +35, +8844.4, +14.5, these positive numbers, the "+" in front of them can also be omitted! What about 0? Is it a positive number? Is it a negative number? Invite two classmates at the same table to discuss! (Post "0" on the board, which is neither a positive number nor a negative number. It is the dividing point between positive numbers and negative numbers.)

Teacher: (Show "Lian Yi Lian" and let students observe first) When water boils What is the temperature of (100℃) and which chain? (No. 2) What is the temperature of water when it freezes? (No. 3) The lowest temperature on the earth’s surface is in Antarctica, which can reach... ( -89.2℃)

Teacher: (Show "fill in") Use a positive or negative number to indicate the altitude below. Who will answer the first picture? (Look for students who raise their hands to answer) Figure.

Teacher: (Showing "practical activities") This is a food packaging bag. (Click to enlarge) There is a mark like this on the packaging bag (500±2g flashes, click to shrink the graphic) 500 here. What does ±2g mean? Discuss with the tablemates! (Students report after discussion) The 500 here refers to the standard weight. Food packed in bags that are 2g more than 500 or 2g less than 500 is qualified. Why? Because there are usually errors in packaging. An error of 2g more or less than 500 is a reasonable error. If it exceeds this range, it is not a reasonable error. Do you understand?

Teacher: Here is what the quality inspector takes. Take out 5 bags of food for inspection. The test results are as follows: (show the table) Do you think these bags of food are qualified? (Student said)

Teacher: OK, students, for negative numbers, Today we only have a preliminary understanding (the complete topic posted on the board: "Preliminary understanding of negative numbers"). In future studies, we will gradually study in depth!

Teacher: Tell me what you gained today! ?

[Commentary: The teaching in the textbook that the "﹢" in front of a positive number can be omitted is a direct notification. As for why it can be omitted, I passed this knowledge point. The "divide one point" exercise, when should the number 8 be placed in the circle of positive numbers or the circle of negative numbers? Let students understand the conflict of cognition and further feel the simplicity of mathematics.]