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Knowledge points of the first volume of mathematics in the first day of Beijing Normal University Edition
Learning methods of the first volume of seventh grade mathematics
Summary of mathematical knowledge points in the first volume of junior one
Knowledge points of the first volume of mathematics in the first day of Beijing Normal University Edition
First, the basic knowledge of algebra.
1. Algebraic formula: The formula that uses the operation symbol "+-××………" to express the relationship between letters and numbers is called algebraic formula (the number obtained by letters should ensure that the formula in which it is located is meaningful, and the number obtained by letters should also make real life or production meaningful; A single number or letter is also algebraic)
2. Some points for attention in column algebra:
Multiplying (1) numbers by letters or letters by letters usually uses "?" Multiply, or omit not to write;
(2) When multiplying numbers, we should still use "×" instead of "?" Multiplication, multiplication sign cannot be omitted;
(3) When a number is multiplied by a letter, the number is usually written in front of the letter in the result. For example, a×5 should be written as 5a;
(4) When the band fraction is multiplied by letters, the band fraction should be changed to a false fraction, for example, a× should be written as a;
(5) When there is a division operation in the algebraic expression, the division method and the division method are generally connected by a fractional line, such as the form written in 3 A;
(6) The difference between A and B should be written in alphabetical order; If we only talk about the difference between two numbers, when we set two numbers as A and B respectively, we should classify them and write them as a-b and B-A. 。
Secondly, several important algebraic expressions (m and n represent integers).
(1) The square difference between A and B is: A2-B2; The square of the difference between a and b is: (a-b) 2;
(2) If a, b and c are positive integers, the two-digit integer is 10a+b and the three-digit integer is10a+10b+c;
(3) If both m and n are integers, the quotient m is divided by 5, and the remainder n is 5m+n; Even number is 2n, and odd number is 2n+1; Three consecutive integers are: n- 1, n, n+1;
(4) If b>0, positive number is: a2+b, negative number is: -a2-b, non-negative number is: a2, and non-positive number is: -a2.
Third, rational number.
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; π is not rational number;
(2) Classification of rational numbers: ① ②
(3) Note: among rational numbers, 1, 0 and-1 are three special numbers with their own characteristics; These three numbers divide the numbers on the number axis into four areas, and the numbers in these four areas also have their own characteristics;
2. 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) Note: The inverse of a-b+c is-A+B-C; The inverse of a-b is b-a; The inverse of a+b is-a -a-b；;
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: the absolute value of knowledge points in the first volume of Senior One, which is often discussed in categories;
(3) |a| is an important non-negative number, that is | a | ≥ 0; Note: |a|? |b|=|a？ b|，
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.
Fourth, rational number rules and algorithms.
(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.
2. 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).
3. Rational number subtraction rule: subtracting a number is equal to adding the reciprocal of this number; That is, a-b=a+(-b).
4. The 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.
5. Arithmetic of rational number multiplication:
(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.
6. Rational number division rule: dividing by a number is equal to multiplying the reciprocal of this number; Note: Zero cannot be divisible.
7. The power law of rational numbers:
(1) Any power of a positive number is a positive number;
Definition of verb (abbreviation of verb) 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;
(3) According to the law, the decimal point of the cardinal number moves by one place and the decimal point of the square number moves by two places.
Six: Addition and subtraction of algebraic expressions.
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 degree term is called the degree of polynomial; Note: (If A, B, C, P and Q are constants) are two common quadratic trinomials.
Algebraic expression: monomials and polynomials are collectively called algebraic expressions.
Seven. Algebraic expressions are classified as.
1. Similar items: monomials with the same letters and the same index are similar items.
2. Rules for merging similar items: When the coefficients are added, the letter index remains unchanged.
3. Rules for deleting (adding) brackets: When deleting (adding) brackets, if there is a "+"before the brackets, all items in the brackets remain unchanged; If there is a "-"before the brackets, all items in the brackets should be changed.
4. Algebraic addition and subtraction: Algebraic addition and subtraction is actually to combine similar terms of polynomials on the basis of removing brackets.
5. Power-on and power-off arrangement of polynomials: arranging the items of a polynomial from small to large (or from large to small) according to the exponent of a letter is called power-on arrangement (or power-off arrangement) of this letter. Note: The final result of polynomial calculation should generally be power-on (or power-off arrangement).
Eight, a linear equation
1. Equation and Equivalence: An equation connected by "=" is called an equation. Note: "Equivalent value can be substituted"!
2. The nature of the equation:
Properties of the equation 1: Add (or subtract) the same number or the same algebraic expression on both sides of the equation, and the result is still an equation;
Property 2 of the equation: both sides of the equation are multiplied (or divided) by the same non-zero number, and the result is still an equation.
3. Equation: An equation with an unknown number is called an equation.
4. Solution of the equation: the value of the unknown quantity that makes the left and right sides of the equation equal is called the solution of the equation; Note: "The solution of the equation can be substituted"!
5. Moving term: after changing the sign, moving the term of the equation from one side to the other is called moving term. The shift term is based on the equality attribute 1.
6. One-dimensional linear equation: An integral equation with only one unknown number, degree 1 and non-zero coefficient is a one-dimensional linear equation.
7. The standard form of one-dimensional linear equation: ax+b=0(x is unknown, a and b are known numbers, a≠0).
8. The simplest form of linear equation with one variable: ax=b(x is unknown, a and b are known numbers, a≠0).
9. General steps for solving a linear equation with one variable: sorting out the equation ... removing the denominator ... dismantling the bracket ... changing the terms ... merging similar terms ... and converting the coefficient into 1 ... (testing the solution of the equation).
Do a linear equation with one variable to solve application problems.
(1) reading analysis method: reading analysis method
Read the stem carefully, find out the key words that express the equal relationship, such as "big, small, many, few, yes, * * *, combination, For, completion, increase, decrease, match-",list the literal equations with these key words, and set the unknown number according to the meaning of the question. Finally, using the relationship between quantity and quantity in the question, fill in the algebraic expression and get the equations.
(2) Drawing analysis method
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.
X. common formulas for solving application problems with column equations.
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Learning methods of the first volume of seventh grade mathematics
First, reading habits
This is the basic skill of self-study ability. According to the survey of dozens of famous universities in the United States and the former Soviet Union, 20%~25% of the knowledge of those outstanding scientists comes from the school, and 75%~80% of the knowledge is obtained through work, self-study and scientific research after they leave the school. According to the laws of psychology, junior high school students are able to read, but due to the influence of intuitive imitation habits in primary school, many students mistake math textbooks for problem sets. Therefore, from the first day of junior high school, we should pay attention to correcting our wrong study habits, establish the correct concept that math textbooks also need reading, and pay attention to summing up how to read math textbooks.
1. Before each class, you must form the habit of previewing, and try to find out the questions you don't understand in the preview, so that you can attend the class with questions.
Pay attention to how teachers read the text in class, and cultivate themselves to master how to analyze the key words, words and sentences in definitions and theorems and their connections with old knowledge.
2. Always sum up what you have learned and cultivate review habits.
At first, you can summarize the content of a class or a unit with the teacher. After a stage, you can learn the text with your own questions according to the review outline put forward by the teacher, and finally transition to your own induction, prompting you to read the text repeatedly, review in time and learn new things.
Second, the habit of taking notes.
A good memory is better than a bad writing. Middle school mathematics is rich in content, and the classroom capacity is generally large. If you want to learn math well systematically, you should pay attention to cultivating the habit of taking notes in class from junior high school, which can also restrain your mind wandering and improve the efficiency of listening to classes. Generally speaking, in addition to recording the lecture outline, class notes mainly record the key points, ideas, methods and content summary explained by the teacher in the lecture. Pay special attention to write down some experiences and problems in class at any time. Listening is the basis of two aspects: listening and recording. Don't just focus on memory, which will affect your listening.
In order to improve the quality of class notes step by step, students should communicate appropriately and learn from each other.
Third, the habit of hands-on practice, cooperation and communication.
"Practice makes true knowledge". Hands-on practice can concentrate attention, improve learning interest and deepen the impression and understanding of the learning object. In hands-on practice, we can link the knowledge in books with practical things to form a correct and profound concept. In hands-on practice, you can use your brain and practice to gradually form and develop your own cognitive structure, and you can form skills and develop abilities. Develop the habit of "guessing before doing-experiment-calculation result-induction and summary" in hands-on practice.
"three people walk together, there must be a teacher." Students exchange their learning results with each other, express their opinions and learn from each other's strengths. It can realize the functions of brain, mouth and hand, stimulate thinking, activate atmosphere and arouse enthusiasm.
Fourth, work habits.
Math homework is an important link to consolidate math knowledge, stimulate learning interest and cultivate math ability. Some students regard homework as a burden, and only answer in a hurry based on the impression in class after class, often with a single solution; Some handwriting is scrawled, careless, irregular and even plagiarized. This missed the training opportunity and seriously affected the learning effect. We should correctly understand the purpose of doing homework and cultivate good homework habits. Good work habits should include:
1. Get into the habit of reading before homework.
Before you do your homework, you should read and review the text carefully, and observe the problem-solving format, steps and methods of the examples. This is exactly "sharpening the knife and cutting the wood by mistake."
2. Develop the habit of examining questions.
After reading the topic, first find out what kind of topic it is, what conditions it has, what characteristics it has, and so on.
3. Develop the habit of working independently.
If there are special circumstances that can't be completed on schedule, you can explain the situation to the teacher: if you can't do it when you encounter problems, you can ask the teacher or classmates and finish it independently after understanding. Never copy to cope with the task.
4. Develop the habit of reflecting on homework.
Many students do not pay attention to reviewing and reflecting on their homework, which leads to the formation of wrong practices in their minds. Some questions are wrong. The teacher has corrected them, but you are still wrong. That's the reason. In this way, there will be more mistakes in new knowledge and new homework. In order to consolidate the results of homework, students must give feedback on the previous day's homework before each new assignment. The feedback content includes: (1) topic type; (2) Ideas and methods to solve problems; (3) the cause of the problem; (4) error correction; (5) Collect mistakes (that is, collect your own mistakes in one place and mark the above four items so that you can correct them when reviewing in the future).
Five, thinking habits
Scientific thinking methods and good thinking habits are the key to developing intelligence and ability. Psychology tells us that the first stage is an important period for students to change from image thinking to abstract thinking. At this time, we must pay attention to the cultivation of good thinking habits. According to the characteristics of junior high school mathematics content, good thinking habits include logicality, thoroughness, divergence, convergence and reversion.
1. logic.
This requires students to "answer with evidence" and avoid taking it for granted. In the process of reasoning and calculus, you can understand the basis of each step, don't write anything you don't understand, and try to understand before continuing reasoning and calculus.
2. thoroughness.
This requires students to consider the problem comprehensively. For example, it is known that point C is on a straight line AB, with line segment AB=8cm and line segment BC=3cm. Find the length of line segment AC. Considering the problem comprehensively, it is necessary to discuss the point C on the line segment AB and the point C on the extension line of the line segment AB: when the point C is on the line segment AB, AC = AB-BC = 8-3 = 5 cm; When point C is on the extension of line AB, AC = AB+BC = 8+3 =11cm. To cultivate this habit, we should pay special attention to the situation and reasons of "easy to make mistakes or incomplete thinking" pointed out by the teacher in class.
3. divergence.
This requires students to use a variety of methods to solve a problem. To cultivate this habit, we should pay special attention to the teacher's thinking method when talking about multiple solutions to one problem and the analysis when extending the problem, and strive to develop the habit of multiple solutions to one problem and changing one problem in the process of mathematics learning.
4. Convergence.
This is a summary based on divergent thinking, in order to solve many problems. The comprehensive application of divergent thinking and convergent thinking can complement each other.
5. reversibility.
This requires students to consider some formulas, laws and theorems in reverse. Such as calculation:
(-0.38)×4.58-0.62×4.58, the multiplication and division method can be applied in reverse, and a simple calculation method can be obtained.
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Summary of mathematical knowledge points in the first volume of junior one
Rational number and its operation block;
1. Integers include positive and negative integers, and fractions include positive and negative fractions.
Positive integers and fractions are generally called positive numbers, while negative integers and fractions are generally called negative numbers.
2. Numbers such as positive integer, 0, negative integer, positive fraction and negative fraction are called rational numbers.
3. Absolute value: The distance between the point corresponding to a number on the number axis and the origin is called the absolute value of the number, which is represented by "||".
Algebraic expression board:
1, monomial: A formula consisting of the product of numbers and letters is called a monomial.
2. The number of times of the monomial: The sum of the indexes of all the letters in the monomial is called the number of times of the monomial.
3. Algebraic expressions: monomials and polynomials are collectively referred to as algebraic expressions.
4. Similar items: items with the same letters and the same letter index are called similar items.
One-dimensional linear equation.
1, an equation containing unknowns is called an equation, and the value of the unknowns that make the left and right sides of the equation equal is called the solution of the equation.
2. Shift term: shift the sign of an item on one side of the equation to the other side, which is called shift term.
The summary of mathematics knowledge points in the first volume of the seventh grade actually includes a lot, but I think everything will not change.
Everyone should pay attention to sorting out and accumulating. Cooperate with more practice. Some knowledge points should be recorded in the notebook in time, and some wrong questions should be sorted out and reviewed in time. Go through the knowledge points one by one. I believe that as long as you have the heart, you can get high marks in the math exam.
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Related articles on knowledge points in the first volume of senior one mathematics published by Beijing Normal University;
★ Summary of knowledge points in Chapter 4, Volume 1, Grade 7 Mathematics
★ Summary of Mathematics Knowledge Points of Grade One in Beijing Normal University
★ Summary of knowledge points at the end of the first year of mathematics published by Beijing Normal University
★ Summarize and sort out the knowledge points of Grade One mathematics.
★ Summary of Mathematics Knowledge in Junior Middle School of Beijing Normal University
★ Beijing Normal University Edition Junior High School Mathematics Teaching Plan
★ Summarize the knowledge points of seventh grade mathematics.
★ Beijing Normal University Edition Seventh Grade Volume 1 Mathematics Catalogue
★ Beijing Normal University Edition seventh grade mathematics first volume teaching plan
★ The first volume of seventh grade mathematics in Beijing Normal University
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