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The learning method of mathematical algebraic expressions in the first grade of junior high school: mathematical concept learning method. There are many concepts in mathematics. How to make students master the concept correctly should explain what kind of process is needed and to what extent. Mathematical concept is a form of thinking that reflects the essential attributes of mathematical objects. Its definition is descriptive, indicating the extension of alien species, and there is a way to add concepts to classes. A mathematical concept needs to remember the name, describe the essential attributes, realize the scope involved, and use the concept to make accurate judgments. These questions are not required by teachers. Without learning methods, it is difficult for students to study regularly. Let's summarize the learning methods of mathematical concepts: reading concepts and remembering names or symbols. Recite the definition and master the characteristics. Give two positive and negative examples to understand the scope of conceptual reflection. Practice and judge accurately.
The learning method of mathematical formula is abstract, and the letters in the formula represent infinite numbers in a certain range. Some students can master the formula in a short time, while others have to experience it repeatedly to jump out of the ever-changing digital relationship. Teachers should clearly tell students the steps needed in the process of learning formulas, so that students can master formulas quickly and smoothly. The learning method of the mathematical formula we introduced is: write the formula and remember the relationship between the letters in the formula. Understand the ins and outs of the formula and master the derivation process. Check the formula with numbers and experience the law embodied in the formula in the process of concretization. Make various transformations on the formula to understand its different forms of change. Imagine the letters in the formula as an abstract framework, so that the formula can be used freely.
Learning methods of mathematical theorems. A definite reason consists of two parts: conditions and conclusions. This theorem must be proved. Proving process is a bridge connecting conditions and conclusions, and learning theorem is to better apply it to solve various problems. Let's summarize the learning method of mathematical theorems: reciting theorems. Conditions and conclusions of discriminant theorem. The proof process of understanding theorem. Applying theorems to prove related problems. Understand the internal relations between theorems and related theorems and concepts. Some theorems contain formulas, such as Vieta Theorem, Pythagorean Theorem and Sine Theorem, and their learning should be combined with the learning method of the formula with the same sign.
Learning methods of geometric proof for beginners. In the second semester of senior one, students have just started to learn solid geometry, and students always find it difficult to get started. Many old teachers agree with the following methods, which can be carried out in class or taught by themselves. Look at the questions and draw pictures. (Look, write) Examine the questions and find ideas (Listen to the teacher's explanation) Read the proof process in the book. Recall and write the proof process.
A reduction method to improve the ability of geometric proof. After mastering the basic knowledge and methods of geometric proof, how to improve the ability of geometric proof on the basis of fluent and accurate expression of the proof process? It is necessary to accumulate the proof ideas of various geometric problems and know some proof skills. In this way, we can achieve the above goal by focusing on the teacher's explanation or reading some geometric proof questions. Reduction is a method to classify the unknown into the known. When we encounter a new geometric proof problem, we need to pay attention to its question type and find the key steps, and it will be over when it falls into the known question type. At this time, the most important thing is to remember the steps of transformation and the idea of proving the problem, and no longer pay attention to the detailed expression process.
The habit of extracurricular learning is to carry out extracurricular activities in mathematics to broaden students' horizons. Under the guidance of teachers, students who have the spare capacity for learning carry out a variety of extracurricular activities, such as answering interesting math questions, reading extracurricular books about math, writing monographs on learning math, telling stories about math and mathematicians, summing up mathematical thinking methods, and solving practical problems within their power. I also give myself a space to develop my mathematical ability through activities such as math lectures or mathematicians' reports, math lectures and math competitions.
Knowledge points of algebraic expression in senior one mathematics: algebraic formula and rational formula
1, the formula that connects numbers or letters representing numbers with operation symbols is called algebraic expression. A single number or letter is also algebraic.
2. Algebraic expressions and fractions are collectively called rational expressions.
3. Algebraic expressions including addition, subtraction, multiplication, division and multiplication are called rational expressions.
Second, algebraic expressions and fractions
1. Rational expressions without division or division but without letters are called algebraic expressions.
2. The rational number formula with division operation and the letters in division are called fractions.
3. Monomial and Polynomial
1, algebraic expressions without addition and subtraction are called monomials. (product of numbers and letters-including single numbers or letters)
2. The sum of several monomials is called polynomial. Each monomial is called a polynomial term, and the term without letters is called a constant term.
Note: ① According to whether there are letters in the division formula, algebraic expressions and fractions are distinguished; According to whether there are addition and subtraction operations in algebraic expressions, monomial and polynomial can be distinguished. ② When classifying algebraic expressions, the given algebraic expressions are taken as the object, not the deformed algebraic expressions. When we divide the category of algebra, we start from the representation.
monomial
1, an algebraic expression of the product of numbers and letters, is called a monomial.
2. The single numerical factor is called the single coefficient.
3. The index of all the letters in the monomial and the number of times called the monomial.
4. A single number or letter is also a monomial.
5. The coefficient of monomial with only letter factor is 1 or-1.
6. A number is a monomial, and its coefficient is itself.
7. The degree of a single nonzero constant is 0.
8. A single item can only contain multiplication or power operation, and cannot contain other operations such as addition and subtraction.
9. The coefficient of the monomial includes the symbol before it.
10, when the coefficient of the monomial is a fraction, it should be turned into a false fraction.
1 1, when the coefficient of a single item is 1 or-1, the number is usually omitted. 1? .
12, the number of monomials is only related to letters, and has nothing to do with the coefficient of monomials.
multinomial
The sum of 1 and several monomials is called a polynomial.
2. Each monomial in a polynomial is called a polynomial term.
3. The term without letters in polynomial is called constant term.
4. A polynomial has several terms, which are called polynomials.
5. Every term of polynomial includes the symbol before the term.
6. Polynomials have no concept of coefficient, but have the concept of degree.
7. The degree of the term with the highest degree in a polynomial is called the degree of the polynomial.
Integral expression
1, monomials and polynomials are collectively called algebraic expressions.
2. Both monomials and polynomials are algebraic expressions.
3. Algebraic expressions are not necessarily monomials.
4. Algebraic expressions are not necessarily polynomials.
5. Algebraic expressions with letters in denominator are not algebraic expressions; It is a fraction to learn in the future.
Fourth, the addition and subtraction of algebraic expressions.
The theoretical basis of 1. Algebraic addition and subtraction is: the rule of removing brackets, the rule of merging similar items, and the multiplication distribution rate.
Rules for deleting brackets: If there is a? Ten? Number, put the brackets together with the one in front? +? Delete the number, and all items in brackets keep the same symbol; What if it comes before the brackets? One? Number, put the brackets together with the one in front? One? Remove the symbols, everything in brackets has changed symbols.
2. Similar items: items with the same letters and the same letter index are called similar items.
Merge similar projects:
1). The concept of merging similar projects:
Merging similar terms in polynomials into one term is called merging similar terms.
2). Rules for merging similar projects:
The coefficients of similar items are added together, and the results are taken as coefficients, and the indexes of letters and letters remain unchanged.
3). To merge similar projects:
A. find similar projects accurately.
B. Reverse the distribution law, and add the coefficients of similar items together (with brackets) to keep the letters and the indexes of letters unchanged.
C. write the results after the merger.
4). Please note:
A. If the coefficients of two similar items are opposite, the result after merging similar items is 0.
B. Don't leave out items that can't be merged.
C. As long as there are no more similar terms, it is the result (which may be a single term or a polynomial).
Note: The key to merging similar items is to correctly judge similar items.
3, several general steps of algebraic expression addition and subtraction:
1) List algebraic expressions: enclose each algebraic expression in parentheses and then connect it with a plus sign or a minus sign.
2) Remove brackets according to the rules for removing brackets.
3) Merge similar items.
4, the general steps of algebraic evaluation:
Algebraic simplification of (1)
(2) Substitution calculation
(3) For some special algebraic expressions,? Overall substitution? Do a calculation.
V. Multiplication with the same base number
1, multiplied by n identical factors (or factors) A, is recorded as an, and read as the n power (power) of A, where A is the base, n is the exponent, and the result of an is called the power.
2. Powers with the same base are called same base powers.
3. Same base multiplication algorithm: same base multiplication, constant base, exponential addition. Namely: am ﹒ an = am+n
This rule can also be reversed, that is, am+n = am-an.
5. Start the power of different cardinality. If it can be converted into a power with the same base, first turn it into a power with the same base, and then apply the rules.
Sixth, the power of power.
The sum of powers of 1 refers to the multiplication of several identical powers. (am)n represents the multiplication of n am.
2. Power algorithm: power, constant basis, exponential multiplication. (am)n=amn .
3. This rule can also be reversed, that is, AMN = (am) n = (an) m.
Seven, the product of power
1, the power of product is the power of cardinal number and product.
2. Multiplication algorithm of product: Multiplication of product is equal to multiplying each factor in the product separately, and then multiplying the obtained power. That is, (ab)n=anbn.
3. This rule can also be reversed, namely: AnBN = (ab) n.
Eight, the same base power division
1, same base powers's division rule: same base powers divides, the base number remains the same, and the exponent is subtracted, that is, am? an=am-n(a？ 0)。
2. this rule can also be reversed, that is, am-n=am? Ann (a? 0)。
Nine, zero exponential power
1, the meaning of zero exponential power: any number that is not equal to 0 is equal to 1, that is, a0= 1(a? 0)。
Ten, negative exponential power
1, the -p power of any number that is not equal to zero is equal to the reciprocal of the p power of this number. Note: In same base powers's division, the base of zero exponential power and negative exponential power is not 0.
XI。 Multiplication of algebraic expressions
(1) Multiplies the monomial by the monomial.
1, the rule of monomial multiplication: the monomial is multiplied by the monomial, and their coefficients are multiplied by the power of the same letter, respectively, and the remaining letters, together with their exponents, remain unchanged as the factors of the product.
2, coefficient multiplication, pay attention to symbols.
3. The powers of the same letters are multiplied, the base is unchanged, and the exponents are added.
4. For the letters only contained in the monomial, write them together with its index as the factor of the product.
5. The result of multiplying the monomial by the monomial is still the monomial.
6. The multiplication rule of monomials also applies to the multiplication of three or more monomials.
(2) Multiplication of monomial and polynomial
1. Multiplication rule of monomial and polynomial: Multiplying monomial and polynomial means multiplying each term in polynomial by monomial according to distribution rate, and then adding the products. Namely: m(a+b+c)=ma+mb+mc.
2. Please pay attention to the product logo when operating. Every term of a polynomial is preceded by a symbol.
3. The product is a polynomial with the same number of terms as the polynomial.
4. When mixing operations, pay attention to the operation sequence. If there are similar items in the results, they should be merged to get the simplest result.
(3) Multiplication of Polynomials and Polynomials
1, polynomial and polynomial multiplication rule: polynomial multiplication, first multiply each term of one polynomial with each term of another polynomial, and then add the products. That is: (m+n)(a+b)=ma+mb+na+nb.
2. Polynomial multiplication must not be repeated or missed. Multiplication should be carried out in a certain order, that is, every term of one polynomial should be multiplied by every term of another polynomial. Before merging similar terms, the number of terms of the product is equal to the product of two polynomial terms.
3. Every term of a polynomial is preceded by a symbol. What should be used to determine the symbol of each term in the product? Is the same number positive and the odd number negative? .
4. If there are similar items in the operation results, they should be merged.
5. For multiplying two linear binomials with the coefficient of the linear term 1 containing the same letter, the following formula can be used to simplify the operation: (x+a)(x+b)=x2+(a+b)x+ab.
Twelve. Variance formula
1, (a+b)(a-b)=a2-b2, that is, the product of the sum of two numbers and the difference between these two numbers is equal to their square difference.
2. A and B in the square difference formula can be monomials or polynomials.
3. The square difference formula can be reversed, that is, a2-b2=(a+b)(a-b).
4. The square difference formula can also simplify the operation of the product of two numbers. To solve this kind of problem, we must first see whether two numbers can be converted into
(a+b)？ (a-b), and then see if a2 and b2 are easy to calculate.
Thirteen, the complete square formula
1 、( a？ b)=a？ 2ab+b means that the square of the sum (or difference) of two numbers is equal to the sum of their squares, plus (or minus) twice their product.
2. A and B in the formula can be monomials or polynomials.
Fourteen Division of algebraic expressions
(a) The law of dividing the monomial by the monomial
1, the law that the monomial is divided by the monomial: Generally speaking, when the monomial is divided, the coefficient and same base powers are separated as factors of quotient; For the letter only contained in the division formula, it is used as the factor of quotient together with its index.
2. According to the law, the calculation method of monomial division is similar to that of monomial multiplication, and it is also divided into three parts: coefficient, same letter and different letter.
(b) The rule of polynomial divided by monomial
1, the law of polynomial divided by monomial: polynomial divided by monomial, first divide each term of this polynomial by monomial, and then add the obtained quotients.
2. Divide the polynomial by the monomial, and note that each term of the polynomial includes the previous symbol.
Guidance on mathematics learning methods in senior one 1. Do a good job of preview: read roughly in the unit preview to understand the learning content in the recent stage, read carefully in the classroom preview, pay attention to the formation process of knowledge, and record the difficult concepts, formulas and rules so as to listen to the class with questions. Persisting in preview, finding doubts and changing passive learning into active learning can greatly improve learning efficiency. Oh, interest is the best teacher.
2. Listen carefully: Listening should include listening, thinking and remembering. Listen, listen to the ins and outs of the formation of knowledge, and listen to the key and difficult points (remember the doubts in the preview? Listen carefully), listen to the solutions and requirements of examples, listen to the mathematical ideas and methods involved, and listen to the class summary. Thinking, one is good at association, analogy and induction, and the other is daring to question, ask questions and make bold guesses. Remember, of course, it means class notes, either more or more effective, you know? If it affects the class, it is better not to remember, but to remember when, just to learn. Remember the methods, skills, doubts, requirements and main points, and remember to take notes after class.
3. Seriously solve the problem: classroom practice is the most timely and direct feedback, and you must not miss it. Don't rush to finish your homework. Look at your notebook first. It is very important to review the learning content, deepen understanding and strengthen memory.
4. Timely error correction: class exercises, homework, tests, review in time after feedback, analyze the reasons for the wrong questions, and review the questions. Is there a problem? Is the concept blurred? Time is tight, no time? Won't you do it? Don't let your carelessness succeed. If the thinking is correct and the calculation is wrong, correct it in time and strengthen the training of relevant calculation when necessary. Fuzzy concepts and mistakes in examining questions all indicate that you are prone to be ignorant in your study, which is a taboo in learning mathematics and should be resolutely overcome. As for what you can't do, of course, you should ask your classmates and teachers in time. Don't let the problem hang in the air. Get into the good habit of doing things today.
5. Learn to summarize: Adults often say that mathematics is the link, which means that knowledge is closely linked. Summarizing by stages can not only play a role in reviewing and consolidating, but also find the connection between knowledge. The purpose, necessity and knowledge of learning can be clearly understood and integrated, and the methods of solving problems can be done quickly. Even if the questions are not practiced in class, they can be handy, that is, they can be generalized.
6. Learn to manage: manage your notebook, exercise book, corrected book and all the exercise papers and papers you have done, which is the most useful information when reviewing for the big exam.