Teaching objectives
I. Knowledge and skills
(1) Through examples, I feel the necessity and rationality of introducing negative numbers, which can be used to represent quantities with opposite meanings in life.
(2) Understand the meaning of rational numbers and the universality of rational numbers.
Second, the process and methods
Through the introduction of examples, it is recognized that negative numbers come from production and life, positive and negative numbers will be used to represent quantities with opposite meanings, and rational numbers can be classified according to needs.
Third, emotional attitudes and values
Feel the close connection between mathematics and real life, enhance students' awareness of mathematics application, and form a good habit of learning to analyze and solve problems.
Emphasis and difficulty in teaching
Teaching focus
Positive and negative numbers are meaningful, and the meaning of rational numbers can correctly classify rational numbers.
Teaching difficulties
Understand negative numbers and correctly classify rational numbers.
teaching tool
PPT multimedia courseware
teaching process
First, the introduction of new courses.
As we all know, mathematics is inseparable from numbers. Now let's recall what types of numbers we learned in primary school.
After the students answered, the teacher pointed out that the numbers learned in primary schools can be divided into three categories: natural numbers (positive integers), fractions and zeros (fractions contain decimals), all of which are due to practical needs.
To represent a person, two hands, …, we use the integer 1, 2, …
It means "no one", "no sheep", ... We need to use 0.
But in real life, there are still many quantities that cannot be expressed by the above natural numbers, zeros or fractions or decimals.
Second, the new lesson learning
1. On a certain day, the highest temperature in a city is 5℃ above zero and the lowest temperature is 5℃ below zero. To express these two temperatures, it is impossible to distinguish them clearly if only the number learned in primary school is recorded as 5℃. They are two quantities with opposite meanings.
In real life, there are many such opposite meanings ... for example, Mount Everest is 8848 meters above sea level, and Turpan Basin is 155 meters above sea level. The meanings of "above" and "below" are opposite. "In" and "out" have opposite meanings.
On the passbook, how does the bank distinguish deposits and withdrawals?
Can students give examples?
After the students answered, the teacher asked: How to distinguish the quantities with opposite meanings?
After thinking, please answer, comment and supplement.
The teacher summed up: the students became inventors. A student said, use different colors to distinguish. For example, red 5℃ means MINUS 5℃, and black 5℃ means above zero 5℃; Student b said to put different symbols in front of the numbers to distinguish them. For example, △5℃ means 5℃ above zero and ×5℃ means 5℃ below zero ... In fact, ancient mathematicians in China used different colors to distinguish them. It was called "positive black, negative red" in ancient times. This method is still used in bookkeeping. This is the origin of the so-called "deficit".
Now, it is distinguished by symbols in mathematics. It is stipulated that 5℃ above zero is marked as +5℃ (pronounced as positive 5℃) or 5℃, and negative 5℃ is marked as -5℃ (pronounced as negative 5℃). In this way, as long as "+"or "-"is added before the number learned in primary school, two quantities with opposite meanings are concisely expressed.
Ask the students to use the same method to express the opposite quantity in the above example:
8848 meters above sea level, recorded as +8848 meters; Altitude/kloc-below 0/55m, marked as-155m;
The teacher explained: a pair of quantities with opposite meanings, one represented by a positive number and the other by a negative number.
It should be emphasized that the number 0 is neither positive nor negative. It is the boundary between positive and negative numbers, indicating the "benchmark" number, and zero does not mean "nothing", but an actual quantity. It is also pointed out that the "+"and "-"symbols of positive and negative numbers are quantities representing opposite properties, and the symbols are written in front of the numbers, which are called property symbols.
Call positive numbers and zeros non-negative.
Story: Subzero Vacation
There are a lot of quantities with opposite meanings in daily life and production, so it is absolutely necessary to introduce negative numbers.
Historically, negative numbers have been criticized. Until16th century, most mathematicians in Europe did not recognize negative numbers. They think that "0 means nothing". What could be smaller than "nothing"? German mathematician Stephen said: "Negative numbers are false negatives", which is just a sign. Pascal, a French mathematician, thinks that subtracting 4 from 0 is nonsense.
We in China were the first to find negative numbers. In the book Mencius, there is a saying that "the people of neighboring countries will not increase or decrease, and our people will not increase or decrease", in which "increase or decrease" means decrease, that is, add negative numbers. It is clearly pointed out in the equation of Nine Chapters Arithmetic, an ancient calculation classic in Qin and Han Dynasties that selling is positive and buying is negative; The remaining money is positive and the loss is negative. During the Three Kingdoms period, wei ren Liu Hui further summarized the meanings of positive and negative numbers in the annotation of Nine Chapters Arithmetic. He clearly pointed out that two kinds of numbers with opposite gains and losses are called positive numbers and negative numbers respectively. The emergence of the concept of negative number is a great discovery in the history of world science and a great contribution of our people to the development of mathematics. We should be proud of this! In addition, Indian mathematicians put forward the concept of negative number in 625 AD (hundreds of years later than our country). He used "property" to represent positive numbers and "debt" to represent negative numbers, and used them to explain the addition and subtraction of positive and negative numbers.
Does 0 only mean no?
1. The number of gold coins in an empty can;
2. The temperature is 0℃;
3. The height of sea level;
4. Standard water level;
5. Benchmark of height comparison;
6. The dividing point between positive and negative numbers;
..... 0 is just a benchmark, which has rich meanings, rather than simply meaning nothing.
2. Give new concepts of integer and fraction.
After the introduction of negative numbers, the range of numbers has been expanded. Positive integers, negative integers and zero are collectively called integers, and positive and negative fractions are collectively called fractions.
3. Give the concept of rational number.
Integers and fractions are collectively called rational numbers.
4. Classification of rational numbers
In order to facilitate the study of some problems, it is often necessary to classify rational numbers, and the classification methods are often different. According to the definition of rational number, rational number can be divided into integer and fraction. Are there any other classification methods for rational numbers?
After thinking, please answer, comment and supplement.
Summary after class
Teacher's summary: According to the sign of rational numbers, they are divided into positive rational numbers, negative rational numbers and zero. Within the scope of rational numbers, positive numbers and zeros are collectively referred to as non-negative numbers. Emphasize to students that classification can use different classification standards according to different needs, but the objects to be discussed must be classified without any omission.