Knowledge points in the first volume of seventh grade mathematics: chapter 1 rational number 1. knowledge frame
Two. The concept of knowledge
1. rational number:
(1) Any number that can be written in form is a rational number. Positive integers, 0 and negative integers are collectively referred to as integers. Positive and negative scores are collectively called scores; Integers and fractions are collectively called rational numbers. Note: 0 is neither positive nor negative; -a is not necessarily negative, and +a is not necessarily positive; P is not a rational number;
(2) Classification of rational numbers: 122. Number axis: The number axis is a straight line that defines the origin, positive direction and unit length.
3. The opposite number:
(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 still 0;
(2) Is the sum of opposites 0? a+b=0？ A and b are opposites.
4. Absolute value:
(1) The absolute value of a positive number is itself, the absolute value of 0 is 0, and the absolute value of a negative number is its inverse; Note: the absolute value means the distance between the point representing a number on the number axis and the origin;
(2) The absolute value can be expressed as: or; The problem of absolute value is often discussed in categories;
5. Rational number ratio: (1) The greater the absolute value of a positive number, the greater the number; (2) Positive numbers are always greater than 0 and negative numbers are always less than 0; (3) Positive numbers are greater than all negative numbers; (4) The absolute values of two negative numbers are larger than the size, but smaller; (5) Of the two numbers on the number axis, the number on the right is always greater than the number on the left; (6) large number-decimal number >; 0, decimal-large number < 0.
6. Reciprocal: Two numbers whose product is 1 are reciprocal; Note: 0 has no reciprocal; If a? 0, the reciprocal is; If ab= 1? A and b are reciprocal; If ab=- 1? A and b are negative reciprocal.
7. The rational number addition rule:
(1) Add two numbers with the same symbol, take the same symbol, and add the absolute values;
(2) Add two numbers with different symbols, take the symbol with larger absolute value, and subtract the one with smaller absolute value from the one with larger absolute value;
(3) Adding a number to 0 still gets this number.
8. Arithmetic of rational number addition:
The commutative law of (1) addition: a+b = b+a; (2) The associative law of addition: (a+b)+c=a+(b+c).
9. Rational number subtraction rule: subtracting a number is equal to adding the reciprocal of this number; That is, a-b=a+(-b).
10 rational number multiplication rule:
(1) Multiply two numbers, the same sign is positive, the different sign is negative, and the absolute value is multiplied;
(2) Multiply any number by zero to get zero;
(3) When several numbers are multiplied, one factor is zero and the product is zero; Each factor is not zero, and the sign of the product is determined by the number of negative factors.
1 1 rational number multiplication algorithm;
(1) The commutative law of multiplication: ab = ba(2) The associative law of multiplication: (AB) C = A (BC);
(3) Distribution law of multiplication: a(b+c)=ab+ac.
12. rational number division rule: dividing by a number is equal to multiplying the reciprocal of this number; Note: Zero cannot be divisible.
13. Power Law of Rational Numbers:
(1) Any power of a positive number is a positive number;
(2) The odd power of a negative number is a negative number; Even the power of negative numbers is positive; Note: When n is positive odd number: (-a)n=-an or (a -b)n=-(b-a)n, when n is positive even number: (-a)n =an or (a-b) n = (b-a) n. 。
14. Definition of power:
(1) The operation of seeking common ground factor product is called power;
(2) In power, the same factor is called base, the number of the same factor is called exponent, and the result of power is called power;
15. scientific notation: write a number greater than 10 as a? 10n, where a is an integer with only one bit. This notation is called scientific notation.
16. Approximation precision: a divisor rounded to that bit, that is, the divisor is accurate to that bit.
17. Significant digits: All digits from the first non-zero digit on the left to the exact digit are called significant digits of this approximation.
18. Mixed algorithm: multiply first, multiply then divide, and finally add and subtract.
This chapter requires students to correctly understand the concept of rational numbers, and understand the meanings of positive and negative numbers, antonyms and absolute values on the basis of real life and learning the number axis. Focus on solving practical problems with the algorithm of rational numbers.
An important reason for the development of experiential mathematics is the actual needs of life. Stimulate students' interest in learning mathematics, teachers cultivate students' ability of observation, induction and generalization, and enable students to establish a correct sense of numbers and the ability to solve practical problems. When teaching this chapter, teachers should create more situations to fully reflect the main position of students' learning.
Knowledge points in the first volume of seventh grade mathematics: Chapter II Addition and subtraction of algebraic expressions 1. Knowledge framework II. Knowledge concept
1. monomial: in algebraic expressions, if only multiplication (including power) operations are involved. Or algebraic expressions that contain division but do not contain letters in division are called monomials.
2. The coefficient and times of single item: the non-zero numerical factor in single item is called the numerical coefficient of single item, which is simply referred to as the coefficient of single item; When the coefficient is not zero, the sum of all the letter indexes in a single item is called the degree of the item.
3. Polynomial: The sum of several monomials is called polynomial.
4. Number and degree of polynomials: the number of monomials contained in a polynomial is the number of polynomial terms, and each monomial is called a polynomial term; In polynomial, the degree of the term with the highest degree is called the degree of polynomial.
Through the study of this chapter, students should achieve the following learning objectives:
1. Understand and master the concepts of monomial, polynomial and algebraic expressions, and find out the differences and connections between them.
2. Understand the concept of similar items, master the method of merging similar items, master the changing law of symbols when removing brackets, and be able to merge and remove brackets correctly. On the basis of accurate judgment and correct combination of similar items, add and subtract algebraic expressions.
3. Understand that the letters in the algebraic expression represent numbers, and the addition and subtraction operations of the algebraic expression are based on numbers; Understanding the basis of merging similar items and removing brackets is the distribution law; Understanding the operation rules and properties of numbers is still effective in the addition and subtraction of algebraic expressions.
4. Be able to analyze the quantitative relationship in practical problems and express it with a formula with letters.
In the study of this chapter, teachers can experience the formation process of concepts through group discussion and cooperative learning, and initially cultivate students' thinking ability and application consciousness such as observation, analysis, abstraction and generalization.
Knowledge points in the first volume of seventh grade mathematics: Chapter III Linear Equation of One Variable This chapter is the core of algebra and the basis of all algebraic equations. Colorful problem situations and happiness in solving problems can easily arouse students' interest in mathematics, so we should pay attention to guiding students to carry out effective mathematical activities and cooperative exchanges from the research of problems around us, so that students can acquire knowledge, improve their ability and experience mathematical thinking methods in the process of active learning and inquiry learning.
I. Knowledge framework
Two. The concept of knowledge
1. One-dimensional linear equation: An integral equation with only one unknown number and a degree of 1 and a non-zero coefficient is a one-dimensional linear equation.
2. The standard form of linear equation with one variable: ax+b=0(x is unknown, A and B are known numbers, A? 0).
3. The general steps to solve a linear equation with one variable: sorting out the equation? Denominator? Not wearing a seat belt? Mobile project? Merge similar projects? The coefficient is 1? Test the solution of the equation.
4. Set up a linear equation of one variable to solve application problems:
(1) reading problem analysis method: how to use it? Sum, difference, multiplication and division?
Read the questions carefully and find out the key words that mean equality, such as? Big, small, more, less, right, * * *, together, right, complete, increase, decrease, support-? Use these keywords to list text equations and set unknowns according to the meaning of the topic. Finally, the algebraic expression is filled with the relationship between quantity and quantity in the topic, and the equations are obtained.
(2) Drawing analysis method: how to use it? Travel problems?
Analyzing mathematical problems with graphics is the embodiment of the combination of numbers and shapes in mathematics. Read the question carefully, and draw the relevant figures according to the meaning of the question, so that each part of the figure has a specific meaning. Finding the equation relationship through the graph is the key to solve the problem, so as to get the basis of concise equation. Finally, using the relationship between quantity and quantity (unknown quantity can be regarded as known quantity), filling in the relevant algebraic expression is the basis of getting the equation.
1 1. Common formulas for solving application problems with column equations:
(1) Travel Problem: Distance = Speed? Time;
(2) Engineering problem: workload = work efficiency? Working hours;
(3) Proportion: Part = All? Ratio;
(4) Downstream problem: Downstream velocity = still water velocity+water velocity, and countercurrent velocity = still water velocity-water velocity;
(5) Commodity price problem: selling price = pricing? Fold? Profit = price-cost;
(6) Perimeter, area and volume: C circle =2? R, s cycle =? R2, c rectangle =2(a+b), s rectangle =ab, c square =4a,
S square =a2, S ring =? (R2-r2), V cuboid =abc, V cube =a3, V cylinder =? R2h, V cone =? R2h。
Knowledge points in the first volume of seventh grade mathematics: Chapter IV Preliminary understanding of graphics I. Knowledge framework
The main content of this chapter is a preliminary understanding of graphics. Starting from the familiar objects in our life, the understanding of the shape of objects gradually rose from perceptual to abstract geometric figures. By viewing and unfolding the three-dimensional graphics from different directions, we can initially understand the relationship between the three-dimensional graphics and the plane graphics. On this basis, we can know some simple plane figures? Lines, rays, segments and angles.
Second, the mathematical ideas involved in this chapter:
1. Discuss ideas by classification. When drawing a straight line through several points on the plane, we should pay attention to the discussion of these points; When drawing graphics, we should pay attention to the possibility of graphics.
2. Equation idea. When dealing with the calculation of angle size and line segment size, it is often necessary to solve it by column equation.
3. Graphics conversion ideas. When learning the concept of angle, we should fully understand the understanding of light rotation. When dealing with graphics, we should pay attention to the application of transformation ideas, such as the mutual transformation between three-dimensional graphics and plane graphics.
4. Turn to thinking. When calculating straight lines, line segments, angles and related figures, it always belongs to the concrete application of formula n(n- 1)/2.
& gt& gt& gt More exciting next page? Knowledge points in the second volume of seventh grade mathematics?