(2) The algebraic expressions on both sides of the listed equations must have the same meaning, unified units and equal quantitative relations.
(3) We should develop the good habit of "testing", that is, the results we seek should make practical problems meaningful.
(4) Don't leave out the "answer". Don't lose the company name in both "set" and "answer".
(5) The analysis process can only be written on draft paper, but it must be serious.
learning target
1. Understand the concept of linear equation with one variable, and flexibly use the basic properties of the equation and the rule of shifting terms to solve the linear equation with one variable, which will test the solution of the equation;
2. Cultivate students' computing ability by flexibly applying the steps of solving linear equations with one variable;
3. Through the teaching of solving equations, we can understand the idea that "unknown" can be transformed into "known".
Knowledge interpretation
I. Analysis of key points and difficulties
This section focuses on the concept and solution of the law of shifting term and the linear equation of one yuan. The difficulty lies in the flexible use of the steps of solving the linear equation of one yuan. Mastering the methods of shifting terms, changing signs and removing denominator and brackets is the key to correctly solve the linear equation with one variable. Attention should be paid to the following points in learning:
1. During the shift change.
Any term in the equation can be moved from one side of the equation to the other after sign change, that is, the term on the right side of the equation can be moved to the left side of the equation after sign change, or the term on the left side of the equation can be moved to the right side of the equation after sign change. A common mistake in moving terms is forgetting to change the symbol. Note that there are essential differences between moving terms and two terms on one side of the exchange equation. If the position of items on the same side of the equal sign changes, these items will remain unchanged. Because changing the order of a term in a polynomial is a variation based on additive commutative law and involution law, if some terms are moved from one side of the equal sign to the other, they will all change their signs.
2. About denominator
The denominator is to multiply each term on both sides of the equation by the least common multiple of the denominator according to the property 2 of the equation. A common mistake is to omit the multiplication term without denominator. For example, to convert this term into the least common multiple of denominator, in order to avoid such mistakes, you can write one more step when solving the problem, and then expand it with the distribution law. Another mistake that is easy to make is the incomplete understanding of the fractional line. The fractional line has two meanings. On the one hand, it is a divisor, on the other hand, it represents brackets.
3. About deleting brackets.
The mistake that is easy to make when removing brackets is that there is a negative sign in front of brackets, and if you remove brackets, you will forget the symbol; Multiply a number by a polynomial, and omit the subsequent items of the polynomial when deleting the brackets. For example, and are both wrong.
4. Ideas for solving equations:
Solving a linear equation with one variable is actually to use the properties of equality to transform an equation into a new form and then solve it.
Second, the knowledge structure
Three. Suggestions on teaching methods
1. The purpose of the first two examples in this section is to introduce the shift rules. The displacement rule is not only suitable for solving equations, but also
Suitable for solving inequalities; It is not only suitable for moving algebraic expression items, but also suitable for moving meaningful non-algebraic expression items. Therefore, it is unreasonable to say that the transfer rule is the inference of equality property 1, but it is easy for junior one students to introduce the transfer rule of equality property 1.
The first example is solving equations. After students see this equation, if they first think of using the bad luck they learned in primary school.
Then the teacher should tell the students that we should learn a new solution now, which can be used to solve more complicated problems.
Miscellaneous equation, please recall the solution of the first chapter of this textbook, and then inspire students to solve this according to the equation property 1
Equation.
In the process of analyzing the solution of the equation, the textbook puts forward the rule of shifting terms, that is, after changing the sign, the term on the left side of the equation can be moved to the right side of the equation; In the process of analyzing the solution of the equation, the textbook also puts forward that the term on the right side of the equation can be changed and moved to the left side of the equation. Through these two examples, students should be guided to sum up the law of moving terms-any term in the equation, which can be moved from the opposite side to the other side after changing its sign. In teaching, two pictures in the textbook can be used to explain the law of moving items and help students understand.
2.① To judge whether an equation is a linear equation, firstly, the equation is deformed by removing the denominator, brackets, moving terms and merging similar terms. If it can be reduced to the simplest form or standard form, then it is a linear equation; Otherwise, it is not a linear equation.
(2) Equation or, only if, is a linear equation; On the contrary, if the equation is explicitly pointed out or is a linear equation, the known condition is implicit.
3.① What moves is the term in the equation, from one side of the equation to the other, instead of exchanging the positions of two terms on one side of the equation;
(2) when moving items, should change the logo, not change the logo can't move items.
4. After defining the one-dimensional linear equation, the textbook summarizes the general steps to understand this kind of equation. At this time, it should be emphasized that these five steps are not necessarily used in solving the equation because of the different forms of the equation, and they are not necessarily in this order. For example, example 1 and example 2 in this section of the textbook do not have the problem of removing brackets, and example 3 and example 4 do not have the problem of removing denominators; For example, when solving an equation, it is easier to move the term first than to remove the brackets first. Therefore, the general steps of solving a linear equation with one variable should be used flexibly according to the specific situation, and it is pointed out that the fourth step of the above general steps, "merging similar terms" and "transforming equations into forms", is an essential step and should be emphasized in teaching.
5. Examples 7 and 8 are two examples of the last small stage in this section. Example 7 is a slightly more complicated topic, and the denominator of the equation contains decimals. You can explain it to the students, usually by converting the decimal in the denominator into an integer, and then removing the denominator.
Steps to solve it. In addition, when the equation is complex, because there are many steps to solve the problem, it is easy to make mistakes. Students are required to check the roots and check whether the answer is correct, but testing is not a necessary step.
Example 8 can be regarded as an application of solving a linear equation with one variable: in a formula, one letter represents an unknown number, and when other letters represent known numbers, find the value of this unknown number. This kind of problem is used in practice. After students study physics,
Chemistry and other courses are often encountered, so we should pay enough attention to them in teaching.
Typical example
Example 1 Judge whether the following transposition is correct, and if not, how to correct it?
(1) obtained from;
(2) from;
(3) from;
(4) from;
Analysis: The key to judging whether the shifted item is correct depends on whether the sign has changed after the shifted item, and we must keep in mind the "sign change of the shifted item". Note: If there are no moving items, the symbol should not be changed; In addition, the items on the same side of the equal sign exchange positions with each other, and the signs of these items remain unchanged.
Solution: (1) is wrong. If the 7 on the left of the equal sign is moved to the right of the equal sign, the sign will change.
(2) Right.
(3) No, the -2 at the left end of the equal sign changed its sign when it moved to the right of the equal sign, but it did not change the equal sign when it moved to the left of the equal sign.
(4) No, if you move the right side of the equal sign to the left side of the equal sign, it will become correct, but the -2 on the right side of the equal sign is still on the right side of the equal sign, so there is no shift, so it should be correct:
Example 2 Solving Equation:
( 1) ; (2)
(3) ; (4)
Analysis: This problem is a simple equation. As long as the unknown coefficient on the left side of the equal sign is changed to 1 according to the properties of the equation, the solution of the equation can be obtained.
Solution: (1) Change the coefficient to 1. According to the properties of the equation, 2. While divide by 3 on both sides of that equation.
Check left and right
Left = right.
So it is the solution of the original equation.
(2) Change the coefficient of to 1, and divide it by 4 on both sides of the equation according to the property of the equation.
Test: left and right =2,
Left = right
So it is the solution of the original equation.
(3) Convert the coefficient of to 1. According to property 2 of the equation, multiply on both sides of the equation at the same time.
Check, left
Right/right hand side
Left =-right,
So it is the solution of the original equation;
(4) Convert the coefficient of to 1, and multiply it by -2 on both sides of the equation according to the property 2 of the equation to obtain:
Check: left, right,
Left = right.
So it is the solution of the original equation.
Description: ① When the coefficient of the unknown quantity is changed to 1 by applying the property 2 of the equation, it depends on the coefficient of the unknown quantity. Generally speaking, when the unknown coefficient is an integer, division is more appropriate. When the unknown coefficient is a fraction (or decimal), multiplication should be used. (Multiply by the reciprocal of the unknown coefficient).
I wish you progress in your study.