Teaching structure

1. Positive and negative numbers

As we know, the numbers that have been known in mathematics are all abstracted from social practice. Positive integer, positive fraction and zero in primary school all represent the quantity of a certain quantity. Positive numbers and negative numbers are introduced because there are a lot of quantities with opposite meanings in real life, and it is impossible to clearly express the opposite situation with the numbers learned in primary school. For example, at a certain time on a certain day, city A is above zero 10℃, and city B is below zero 10℃. The temperature difference between the two places cannot be described by "10" only. For example, A goes 5 kilometers north and B goes 5 kilometers south. This distance "5" can't describe the directions of A and B. We call these two opposite quantities "X above zero and X below zero" and "5 kilometers north and 5 kilometers south" as "5". If a quantity with one meaning is defined as a positive quantity, another quantity with the opposite meaning is defined as a negative quantity. If "above zero 10℃ is defined as positive 10℃, then negative 10℃ is negative 10℃. Remove the units of positive and negative quantities, and you get the concept of positive and negative numbers. Numbers greater than 0 such as 5, 1.5, 10 and 9840 are called positive numbers. Numbers preceded by "-"(pronounced negative) symbols, such as -5,-1.5,-10, -9840, are called negative numbers. Among them, the "+"sign before the positive number can be ignored.

Is there a situation of "not going up or down" or "not going north or south" in the problem of quantity with opposite meanings? The answer is yes. The dividing point between "positive quantity" and "negative quantity" is neither correct nor irresponsible, and should be expressed by "zero" learned in primary school. So zero is neither positive nor negative. It is the boundary between positive and negative numbers, and it is the only real neutral number. In the past, zero meant "nothing". After learning the quantity with opposite meaning, we know that it still has rich practical significance. For example, 0℃ does not mean no temperature, but a fixed temperature such as freezing point.

Although there are many quantities with opposite meanings in life, not all quantities can find quantities with opposite meanings. For example, "the road is 2 meters wide" has no opposite meaning.

It should be noted that the "+"and "-"symbols in primary schools are only symbols of addition and subtraction. Like positive numbers and negative numbers, "+"and "-"symbols are natural symbols of numbers.

We call positive integers and positive fractions in primary school as positive rational numbers. Add a negative sign before a positive integer to get a negative integer, and add a negative sign before a positive score to get a negative score. Negative integers and negative fractions are collectively called negative rational numbers. Positive rational numbers and zero-sum negative rational numbers are collectively called rational numbers. Among them, positive numbers and 0 are also called non-negative numbers.

Positive integer (natural number)

Positive rational number positive fraction

Rational number zero

Negative rational number negative integer

Negative score

Rational numbers can also be classified as follows:

Positive integer (natural number)

Integer zero

Rational number negative integer

Fraction positive fraction

Negative score

That is, "integers and fractions are collectively called rational numbers". It should be noted that sometimes, for the need of research, integers can also be regarded as fractions with the denominator of 1, in which case the fractions contain integers. Fractions in this chapter refer to fractions other than integers.

Also pay attention to the relationship between decimals and fractions: fractions can be converted into decimals (finite decimals or infinite circulating decimals); Finite decimals and infinite cyclic decimals in decimals can be divided into components, both of which are rational numbers. Infinitely cyclic decimal is not a fraction, not a rational number, such as π.

2. Counting axis

In our daily life, we often encounter digital measuring equipment, such as scales, thermometers, scales and so on. Marking numbers on such a straight object will bring great convenience to our research.

In order to mark a rational number on a straight line, we first determine the position of zero, that is, the dividing point between positive and negative numbers, which is the so-called origin. Then specify the positive direction and units. In this way, a straight line can be marked with rational numbers.

The straight line that defines the origin, positive direction and unit length is called the number axis. such as

-2- 1 0 1 2(A) 1 0- 1(B)

1

0 degrees celsius

- 1

All axes. But it is customary to draw a figure (a) and a straight line horizontally, and the direction from left to right is defined as the positive direction. (positive direction from the origin to the right and negative direction from the origin to the left), that is, the number on the right of the origin means positive, the number on the left of the origin means negative, and the origin means zero. It must be remembered that the origin, positive direction and unit length are the three elements of the number axis, and all three are indispensable.

The introduction of the number axis connects numbers with points on the graph, and all rational numbers can be represented by points on the number axis, which is a combination of numbers and shapes. The combination of numbers and shapes is an important method to learn mathematics.

3. Inverse number

Images 2 and -2 have the same distance from the origin of the number axis. Only the signs are different. We call these two numbers reciprocal.

There are only two numbers with different signs, and we say that one of them is opposite to the other. The antonym of 0 is 0.

By observing the position of antipodal on the number axis, we find that each group of antipodal is on both sides of the origin, and the distance to the origin is equal, but the signs are different. In this way, the geometric meaning of reciprocal is obtained:

On both sides of the origin on the number axis, two numbers represented by two points with the same distance from the origin are called reciprocal. The reciprocal of 0 is zero.

Generally speaking, the reciprocal of the number A is -a, where A represents any number, which can be positive, negative or 0. For example, when a=+7 and -a =-7, because the inverse of 7 is -7. When a =-5, -A =-(-5) a=-5, because the inverse of -5 is 5. When a=0, -a =-0 = 0, because the reciprocal of 0 is 0.

4. Absolute value

From the number axis (that is, the geometric meaning of absolute value), the absolute value of a number A is the distance between the point representing the number A on the number axis and the origin. The absolute value of the number a is recorded as |a|.

From the geometric meaning of the above absolute values, it is easy to know that |2|=2, |-2 | = 2, |0|=0. Described in written language is:

The absolute value of a positive number is itself; The absolute value of a negative number is its reciprocal; The absolute value of 0 is 0.

We use the formula to express the above relationship, namely

a(a & gt； 0)a(a≥0)a(a & gt； 0 )

|a|= 0 (a=0) or |a|= or |a|=

-a(a & lt； 0)-a(a & lt； 0 ) -a (a≤0)

Studying the absolute value from the above three different angles, we find that the absolute value of rational numbers can not be negative, but positive or 0, that is, the absolute value is non-negative.

5. Comparison of rational numbers

From the size of positive rational numbers, we can know that "the two numbers represented on the number axis, the number on the right is always greater than the number on the left", so it is stipulated that "the number represented by the right point on the number axis is greater than the number represented by the left point"

According to this law, we can know that all positive numbers are greater than 0; Negative numbers are all less than 0; Positive numbers are greater than all negative numbers.

We have known the size of two positive numbers since elementary school. Although we can determine the comparison size of two negative numbers according to their positions on the number axis, we hope to convert them into positive numbers for comparison, which will make the calculation easier. Such as |-3 | = 3, |-2 | = 2, because 3 >; 2, so |-3 | > |-2 | And we can know -3 from the number axis.

Key points of solving problems

Example (1) If the reservoir water level rises by 5cm, record it as +5cm, and then the water level drops by 3cm, why? What does it mean to rise by -2 cm?

Analysis: Because the rise and fall of the water level are quantities with opposite meanings, it is known that a rise of 5cm is recorded as +5cm, and a drop of 3cm in the water level should be recorded as -3 cm. A rise of -2 cm means that the water level drops by 2 cm.

Solution: the water level drops by 3cm, and it is recorded as -3 cm. A rise of -2 cm means that the water level drops by 2 cm.

Example (2) Judging whether it is true or false (the correct one is indicated by "√" and the incorrect one is indicated by "×").

1. Forward 10m and right 10m are opposite quantities.

The expenditure of 2.8 yuan is opposite to the income of 100 yuan.

3. Going east 15km and going west 1km are opposite quantities.

An increase in the price of an article in 25 yuan is just the opposite of a decrease in 20 yuan.

Analysis: In the question 1, going forward 10m and going right 10m are the same quantity, but going forward and right cannot be regarded as quantities with opposite meanings. In the second and third questions, expenditure and income are opposite quantities to the east and the west, and they are the same quantity. Although the increase and decrease in question 4 have opposite meanings, they are not the same quantity and cannot be regarded as quantities with opposite meanings.

Solution:1.×; 2.√; 3.√; 4.×

Question: What are the characteristics of quantities with opposite meanings?

The number of antonyms must have two characteristics: ① antonyms; 2 is the same amount.

Example (3) Use positive numbers and negative numbers to represent the following groups of quantities with opposite meanings, and point out their demarcation points.

1. 400 meters above sea level, 256 meters above sea level.

2. 44 degrees north latitude and 33 degrees south latitude.

Analysis: In general, we use a positive number to represent the altitude and a negative number to represent the altitude below the altitude. Although we can also express the height below sea level as a positive number.

Solution: 1 If the altitude is represented by a positive number, then 400 meters above sea level means +400 meters or 400 meters, while 256 meters above sea level means -256 meters. Sea level is their dividing point, which is expressed by 0 meters.

2. If a positive number is used to represent the degree of north latitude, then 44 degrees north latitude means +44 degrees or 44 degrees, and 33 degrees south latitude means -33 degrees. The equator is their dividing point, which is represented by the 0-degree latitude.

Example (4) Fill the following figures in the corresponding braces: +6, 0.003, 1, 43, 0, (2.3, -2, -5.0 1, -25, -0.2 1.

Positive integer set: …

Negative integer set: …

Positive fraction set: …

Negative diversity: …

A set of positive numbers: …

Integer set: …

Analysis: 0.003 and 12.3 are finite decimals, both of which can be divided into components, and both of them should be filled in the set of positive fractions. -0.2 1 is an infinite cyclic decimal number, which can also be converted into component values and should belong to a negative fraction set. 0 is an integer, because integers are a general term for positive integers, 0 and negative integers. Don't ignore "0" when considering integer sets. In addition, it needs to be clear that 0 is neither positive nor negative.

Solution: Positive integer set: +6, 1, 43, …

Negative integer set: -2, -25, …

Positive score set: 0.003, …

Negative score sets:-,-5.0 1, -0.2 1, …

A set of positive numbers: +6, 0.003, 1, 43, 12.3, …

Integer set: +6, 1, 43, 0, -2, -25, …

Q: Is -(-3) a negative number? Why?

-(-3) is positive. Because -3 represents a negative number, there is a "-"sign in front of it, which means a quantity with the opposite meaning to -3. Negative numbers and positive numbers have opposite meanings, so -(-3) represents a positive number 3.

Example (5) Please draw a number axis and use points A, B, C and D to represent 2,-1,-1 respectively.

Analysis: Painting axis must have three elements: origin, positive direction and unit length. The numbers represented on the number axis should be marked with real dots (black dots) and then added with letters.

Solution: d