Seven-grade mathematics knowledge points
Relationship between variables
Theoretical understanding
1, if y changes with x, then x is an independent variable and y is a dependent variable.
The independent variable is a quantity that changes actively, the dependent variable is a quantity that changes with the change of the independent variable, and the quantity whose value remains unchanged is called a constant.
3. If the top angle of an isosceles triangle is y and the bottom angle is x, then the relationship between y and x is y= 180-2x.
2. The relationship between variables can be determined: related formula ① Distance = speed× time ② Rectangular circumference =2× (length+width) ③ Trapezoidal area = (upper bottom+lower bottom )× height ÷ ④ Sum of principal and interest = principal+interest rate× principal× time. ⑤ Total price = unit price × total amount. 6. Average speed = total distance/total time
Table method: The combination of numerical tables is adopted, and tables can be used to express the relationship between two variables. When listing, select some data that can represent independent variables, list them in order from small to large, and then calculate the corresponding values of dependent variables respectively. The characteristic of tabular method is intuitive, and the corresponding values of independent variables and dependent variables can be found directly from the table, but the disadvantage is that it has limitations and can only represent a part of dependent variables.
3. Relational method: Relational formula is an equation that uses mathematical formula to express the relationship between variables. With the relation, we can get the value of the corresponding dependent variable according to the value of any independent variable, and we can also get the value of the corresponding independent variable by knowing the value of the dependent variable.
Fourth, image attention: a. Seriously understand the meaning of the image and pay attention to choosing the image that can reflect the meaning of the problem; B. Understand the meaning (coordinates) of special points on the image from the practical meaning of horizontal axis and vertical axis, especially the starting point, inflection point and intersection point of the image.
8. Description of the changing trend of things: There are generally two descriptions of the changing trend of things:
1. With the gradual increase (large) of independent variable X, dependent variable Y also gradually increases (large) (or it can be described in functional language: dependent variable Y increases (large) with the increase (large) of independent variable X;
2. When the independent variable X increases (increases), the dependent variable Y decreases (or it can be described in functional language: when the independent variable X increases (increases), the dependent variable Y decreases).
Note: If the changing trend of things in the whole process is different, it can be described in sections. For example, in what range, as the independent variable X gradually increases (large), the dependent variable Y gradually increases (large), and so on.
Nine, estimate (or estimate) things estimate (or estimate) there are three kinds:
1. estimate (or estimate) with the changing law of things. For example, every time the independent variable x increases by a certain amount, the dependent variable y changes; Every average change (year) (every average change = (mantissa-prefix)/time or annual difference) and so on;
2. Using the image: firstly, make the corresponding image according to several corresponding group values, and then find the value of the dependent variable y corresponding to the corresponding point on the image;
3. Use relational expressions: first find relational expressions, and then substitute them directly for evaluation.
Junior one mathematics knowledge points
Solving a linear equation with one variable:
1. General steps for solving linear equations with one variable
Removing the denominator, removing brackets, moving terms, merging similar terms, and converting the coefficient into 1 are just the general steps to solve the linear equation with one variable. According to the characteristics of the equation, all the steps are to gradually transform the equation into the form of x = a.
2. When solving a linear equation with one variable, first observe the form and characteristics of the equation. If there is a denominator, generally go to the denominator first; If there are both denominators and brackets, and the denominator can be eliminated after the items outside the brackets are multiplied by the items inside the brackets, the brackets should be removed first.
3. When solving an equation similar to "ax+bx=c", merge the left side of the equation into one term according to the method of merging similar terms, that is, (A+B) x = C.
The equation is gradually transformed into the simplest form of ax=b, which embodies the idea of reduction.
When the coefficient of ax=b is changed to 1, the calculation should be accurate. Once it is clear whether the two sides of the equation are divided by a or b, especially when a is a fraction; Second, we must accurately judge symbols. The same sign X of A and B is positive, and the different sign X of A and B is negative.
14, the application of one-dimensional linear equation
1. Types of applied problems for solving linear equations of one variable
(1) Explore the problem of regularity;
(2) Quantity;
(3) Sales problem (profit = selling price-purchase price, profit rate = profit purchase price ×100%);
(4) engineering problems (① workload = per capita efficiency × number of people × time; (2) If a job is completed in several stages, the sum of workload in each stage = total workload);
(5) Travel problem (distance = speed × time);
(6) the problem of equivalent transformation;
(7) Sum, difference, multiplication and division;
(8) Distribution problem;
(9) Competition score;
(10) Current navigation problem (downstream speed = still water speed+current speed; Water velocity = still water velocity-water velocity).
2. The basic idea of solving practical problems by using equations:
First, find out the unknown quantity and all known quantities in the problem through examination, set the required unknown quantity as X directly or indirectly, and then use the formula containing X to express the related quantity, find out the equation between them, and solve it to get the answer, that is, set, column, solution and answer.
List five steps of solving application problems by linear equations of one variable.
(1) Examination: Carefully examine the questions, determine the known quantity and the unknown quantity, and find out the equivalent relationship between them.
(2) Assumptions: Assumptions about the unknown (X). According to the actual situation, it can be directly unknown (ask whatever you want) or indirectly unknown.
(3) Column: list the equations according to the equivalence relation.
(4) Solution: Solve the equation to obtain the value of the unknown quantity.
(5) Answer: Check whether the unknown value is correct and write a complete answer.
Math methods and skills in junior one.
1. Please summarize the learning methods.
Yue: "Learning mathematics, like learning other things, requires research methods. The method I recommend to you is: study in advance, develop associations, sum up more and find out what is reasonable.
Please talk about the benefits of studying in advance.
First of all, studying in advance can tap one's own potential and cultivate self-study ability. After studying in advance, you will find that you can solve many problems independently, which is very helpful to improve your self-confidence and cultivate your interest in learning. "
Secondly, it is enough to eliminate the "hidden danger" of new knowledge. Studying in advance can find out what's wrong with your understanding of new knowledge on the existing basis. On the contrary, if you listen to others directly. It seems that I can reach this level of understanding from the beginning. Practice has proved that this is not the case.
Thirdly, some contents in advanced learning were not fully understood at that time, but after careful consideration, even if they were left behind, the brain would subconsciously "process". When the teacher's progress reaches this content, we will have a second understanding, which will be much deeper.
Finally, studying in advance can improve the quality of lectures. After studying in advance, we find that most of the new knowledge is completely understandable. Only a few places need help from others. In this way, you can concentrate on understanding "these places", that is, "good steel is used in the cutting edge". In fact, there is not much time to concentrate in a class.
3. Please talk about association and summary.
Yue: Association and summarization run through the whole process of learning. The understanding of every kind of knowledge must have a cognitive basis. The process of finding cognitive basis is association, and cognitive basis is a summary of previous knowledge. The more concise, clear and reasonable the previous summary, the easier it is to associate. In this way, new knowledge can be integrated into the original knowledge structure, laying the foundation for the next association. Association and summarization are particularly effective in solving problems. Maybe you didn't know this before, but your ability to solve problems is very strong, which shows that you are smart and you used this method unconsciously. If you can clearly understand this, your ability will be stronger.
4. So how do we preview?
Say: "Let's talk about the goal of learning first: (1) Understand the background of knowledge generation and find out the process of knowledge formation.
(2) Know the position and function of knowledge sooner or later: (3) Summarize the laws of cognitive problems (or tell which laws were used in previous cognitive problems).
Let's talk about the specific method first: (1) Understanding of the concept. Mathematics is highly abstract. It is usually understood by concrete things. Sometimes with literal meaning: sometimes with other disciplines. Sometimes it is tacit to understand the field of concepts with the help of graphics. You must try to understand the concept before you do the problem.
(2) Preview the formula theorem, which is a summary of the most used "laws". Such as: complete square formula, Pythagorean theorem, etc. The proof of formula derivation theorem often contains rich mathematical methods and quite useful law of solving problems. Such as the proof of the theorem of bisector of triangle interior angle. You should first deduce the formula or prove the theorem yourself. If you can't do it, you should refer to others' practice. Whether you do it yourself or watch others, you should talk about how you came up with it.
(3) For the treatment of examples and exercises, see (2) above and Article 5 below.
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