Summary of knowledge points of mathematics in the first semester
Knowledge point classification-real number
1. Classification by definition: 2. Classification by natural symbols:
Note: 0 is neither positive nor negative.
Knowledge point 2 Related concepts of real numbers
1. Inverse
Algebraic meaning of (1): 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.
(2) Geometric meaning: On both sides of the origin on the number axis, two points with the same distance from the origin represent two opposite numbers, or on the number axis, the points corresponding to two opposite numbers are symmetrical about the origin.
(3) The sum of two opposites is equal to 0.a and B are opposites a+b=0.
2. Absolute value |a|≥0.
3. The reciprocal (1)0 has no reciprocal. (2) Two numbers whose product is 1 are reciprocal. A and b are reciprocal.
4. Square root
(1) If the square of a number is equal to a, it is called the square root of a, a positive number has two square roots, and the two square roots are in opposite directions. 0 has a square root, and the square root itself is 0; Negative numbers have no square root. The square root of a(a≥0) is written as.
(2) The positive square root of a positive number is called the arithmetic square root of a, and the arithmetic square root of a(a≥0) is recorded as.
5. Cubic root
If x3=a, then x is called the cube root of a, and positive numbers have positive cube roots; Negative numbers have negative cubic roots; The cube root of zero is zero.
Knowledge point 3 Real number and axis
Definition of number axis: the straight line defining the origin, positive direction and unit length is called number axis, and the three elements of number axis are indispensable.
Comparison of real numbers of knowledge point four
1. For any two points on the number axis, the point on the right represents a larger number.
2. Positive numbers are all greater than 0, negative numbers are all less than 0, and two positive numbers, the greater the absolute value, the greater the positive number; Two negative numbers; The absolute value is large but small.
3. The relative size of irrational numbers:
The second volume of the first day of junior high school mathematics review materials
1. Binary linear equation: There are two unknowns whose exponents are both 1. Equations like this are called binary linear equations, and the general form is ax+by=c(a≠0, b≠0).
If an equation contains two unknowns, and all the unknowns are 1 power, then the whole equation is called a binary linear equation with infinite solutions, and if conditions are added, there are finite solutions. Binary linear equations generally have one solution, sometimes no solution, and sometimes countless solutions.
2. Binary linear equations: two binary linear equations are combined into one binary linear equation.
3. Solution of binary linear equations: Generally, the value of unknown quantity that makes the values on both sides of binary linear equations equal is called the solution of binary linear equations.
4. Solution of binary linear equations: Generally speaking, the common * * * solution of two equations of binary linear equations is called binary linear equations.
5. Elimination method: the idea of changing the number of unknowns from changeable to small and solving them one by one is called elimination thought.
Induction: the basic idea: "eliminate yuan"-change "binary" into "unitary".
6. Substitution elimination method: an unknown number is represented by a formula containing another unknown number, and then it is substituted into another equation to realize elimination, and then the solution of this binary linear equation group is obtained. This method is called substitution elimination method, or substitution method for short.
7. Addition, subtraction and elimination method: When the coefficients of the same unknown in two equations are opposite or equal, the two sides of the two equations can be added or subtracted respectively to eliminate the element. This method is called addition, subtraction and elimination, or addition and subtraction for short.
Teaching plan of square root of mathematical arithmetic in seventh grade of junior middle school
First, the teaching objectives
1. Understand the meaning of the square root of a number and an arithmetic square root;
2. To understand the meaning of the root sign, the root sign will be used to represent the square root and arithmetic square root of a number;
3. Improve students' logical thinking ability through the training in this section;
4. By learning that the operation of power sum root is reciprocal operation, we can understand the dialectical relationship of the unity of opposites between things and stimulate students' interest in exploring the mysteries of mathematics.
Second, the teaching focus and difficulties
Teaching emphasis: the concept and solution of square root and arithmetic square root.
Teaching difficulties: the connection and difference between square root and arithmetic square root.
Third, teaching methods.
Emphasize the combination with practice.
Fourth, teaching methods.
mixed-media
Teaching process of verbs (abbreviation of verb)
(1) ask questions
1. Given that the area of a square is 50 square meters, what should its side length be?
2. Given that the square of a number is equal to 1000, what is this number?
3. What is the side length of a cubic container with a volume of 0.1.25m3?
The common characteristics of these problems are: knowing the result of power, finding the value of base, how to solve these problems? This is what we will learn in this section. Let's do a little exercise: fill in the blanks.
1.( )2=9; 2.( )2 =0.25;
5.( )2=0.008 1.
When students finish this exercise, the most common mistake is to lose the negative solution, which should be corrected in teaching.
Introduce the concept of square root through practice.
(B) the concept of square root
If the square of a number is equal to a, then this number is called the square root (quadratic root) of a.
Expressed in mathematical language: If x2=a, then x is called the square root of a. 。
From practice, we know that 3 is the square root of 9;
0.5 is the square root of 0.25;
The square root of 0 is 0;
0.09 is the square root of 0.008 1.
From this we can see that 3 and -3 are the square roots of 9, and the square root of 0 is 0. Let's look at such a question and fill in the blanks:
( )2=-4
After thinking, the students come to the conclusion that there is no answer to this question. Why? Because the squares of positive numbers, 0 and negative numbers are all nonnegative, it can be concluded that negative numbers have no square root. Let's sum up the nature of the square root (students can sum up and teachers can sort it out).
(3) the nature of the square root
1. Positive numbers have two square roots in opposite directions.
2.0 has a square root, which is 0 itself.
Negative numbers have no square root.
(4) Square root
The operation of finding the square root of a number is called square root operation.
It can be seen from practice that the square of 3 and -3 is 9, and the square root of 9 is 3 and -3. It can be seen that the square operation and the square root operation are reciprocal operations. According to this relationship, we can find the square root of a number by square operation. Different from other algorithms, it can only operate non-negative numbers, and the result of positive numbers is two.
(5) Representation method of square root
The positive square root of a positive number is represented by the symbol "",A is called the root number, 2 is called the root number, and the negative square root of a positive number is represented by the symbol "-".The square root of a is collectively called the quadratic root number and A under the quadratic root number. When the root sign is 2, this 2 is usually omitted, so the square root of a positive number can also be recorded as.
Exercise: 1 Use the correct symbol to represent the square root of the following numbers:
①26②247③0.2④3⑤
Solution: ① The square root of 26 is
② The square root of 247 is
③ The square root of 0.2 is
The square root of ④3 is
The square root of ⑤ is
Sort out the related articles about the knowledge points of the first volume of mathematics in senior one;
★ Induction of knowledge points in the first volume of junior high school mathematics.
★ Summarize the knowledge points in the first volume of junior high school mathematics.
★ Summary of knowledge points in the first volume of junior high school mathematics
★ Summary of mathematical knowledge points in the first volume of the first day of junior high school.
★ Knowledge points in the first volume of first grade mathematics
★ arrangement of key knowledge in the first volume of junior one mathematics
★ Summary of the knowledge points in the first volume of mathematics in the first grade education edition.
★ Complete collection of knowledge points in the first volume of junior high school mathematics.
★ Mind map of knowledge points in the first volume of junior high school mathematics
★ Summarize the knowledge of the first volume of seventh grade mathematics.