The idea of the combination of numbers and shapes is that you can think of the corresponding reflection in the mathematical formula when you see some characteristics of the figure, but you can associate the corresponding geometric expression in the figure when you see the characteristics of the mathematical formula. For example, after the number axis was introduced into the textbook, it laid the foundation for the idea of combining numbers with shapes. For example, the comparison of rational numbers, the geometric meaning of opposites and absolute positions, and the drawing analysis of solving application problems with equations. This combination of abstraction and image can train students' thinking.
The combination of numbers and shapes is a common way of thinking to solve mathematical problems. The combination of numbers and shapes can make some abstract mathematical problems intuitive and vivid, change abstract thinking into image thinking, and help to grasp the essence of mathematical problems. In addition, due to the combination of numbers and shapes, many problems are easy to solve and the solutions are simple.
The so-called combination of numbers and shapes is the idea of solving mathematical problems through the mutual transformation of numbers and shapes according to the corresponding relationship between numbers and shapes, which is often related to the following contents: (1) the corresponding relationship between real numbers and points on the number axis; (2) correspondence between function and image; (3) The correspondence between curve and equation; (4) Concepts based on geometric elements and geometric conditions, such as complex numbers and trigonometric functions; (5) The structure of a given equation or algebraic expression has obvious geometric significance. Such as equation.
Looking at the exam questions for many years, skillfully using the thinking method of combining numbers and shapes to solve some abstract mathematical problems can get twice the result with half the effort. The key point of the combination of number and shape is to study "helping number with shape"
Example 1: As shown in the figure: Compare the sizes of A, -A, B and-B..
Analysis: Point out the points represented by -a and -b on the number axis, and the relationship between the four numbers will be clear.
However.
Example 2: There are intersections. A starts from the intersection and goes straight south, and b goes straight east from the intersection to the west1500 m. It is known that a and b start at the same time. 10 minutes later, they are at the same distance from the intersection for the first time, and after 40 minutes, they are at the same distance from the intersection again. Find the speed of a and b ..
Analysis: Draw a "cross" diagram, analyze and display the positions of 10 minutes and 40 minutes, and then list the equations analyzed from the diagram.
Second, the change of the overall concept.
The idea of integral transformation refers to transforming a part of a complex algebra or geometric figure as a whole, which simplifies the problem.
Example 3: it is known that y=ax7+bx5+cx3+dx- 1, and when x=2 and y=4, then when x=-2,
y= .
Analysis: the value of 27a+25b+23c+2d is obtained by known conditions. When x=-2 is substituted by the whole,
The value of y.
Example 4: There is a six-digit number, and its unit math is 6. If you move 6 to the first place,
The six figures obtained are four times the original figures. Found six figures.
Analysis: Let the first five digits of this six-digit number be X, then this six-digit number is: 10x+8, an integer.
Body processing, the problem is simplified.
Third, the idea of classified discussion.
When solving some math problems, sometimes there will be many situations. Classify various situations, solve them one by one, and then solve them comprehensively. This is the classified discussion method. Classification discussion is not only a logical method, but also a mathematical thought. Mathematical problems related to classified discussion ideas are obviously logical, comprehensive and exploratory, which can train people's thinking order and generality, so they occupy an important position in the examination questions.
The general steps of classified comments are: defining the object of discussion, determining the overall object → determining the classification standard, correctly classifying → discussing step by step, achieving phased results → comprehensively summarizing and drawing conclusions.
The principles to be followed in classified discussion are: clear classified objects, unified standards, no omission, no repetition, hierarchical discussion and no leapfrog discussion.
When a problem has multiple situations or the derivation result is not unique, it is often discussed in categories first, and then summarized. For example, to remove the absolute value sign of |a|, we need to discuss the sign of the internal formula of absolute value. There are three situations to remove the absolute value sign. There are also some classified discussions about the position relationship in mathematics and geometry.
Example 5: Two people, A and B, ride bicycles in opposite directions from two places 75km apart. The speed of A is 15km/n, and the speed of B is10 km/n. How many hours have passed, the distance between them is 25km.
Analysis: A and B will be 25km apart before and after meeting. Answer in two situations.
Example 6: Draw the bisector of ∠AOB = 60, ∠COB = 50, OD ∠AOB and OE ∠COB in the same picture, and calculate the degree of ∠DOE.
Brief analysis: The general drawing of ∠COB in ∠AOB can be divided into internal and external situations.
Fourth, the idea of transformation and transformation.
When solving some mathematical problems, if it is difficult to solve them directly, we can use appropriate mathematical methods to transform them into a new problem (a relatively familiar problem) through observation, analysis, analogy and association, and solve the original problem by solving the new problem. We call this thinking method "transformation and transformation thinking method". Transformation is the process of transforming mathematical propositions from one form to another. Transformation is to reduce the problems to be solved to a class of problems that have been solved or are relatively easy to solve through a certain transformation process. The thought of conversion is the most basic way of thinking in middle school mathematics.
The idea of transformation refers to transforming a problem into a solved or easily solved problem by means of observation, association and analogy according to the existing knowledge and experience. For example, the solution of binary linear equations, ternary linear equations is essentially to solve the learned linear equations. If the total number of times several people shake hands (single handshake) is called "handshake problem", then it is like there is no three-point connection between n points; * * * The number of included angles of endpoint rays (less than a right angle); The number of line segments formed by several points on a line segment; The single round robin match between football teams can be transformed into a "handshake problem".
Example 7: Make six squares of the same size with matches of the same length, with at least one match.
Analysis: these six squares with the same size can be regarded as six faces of a cube, so that,
Use matching at least. (It is actually 12 sides of a cube).
Example 8: How many triangles of the same size can be placed with six matchsticks of the same length?
Analysis: Six equal-length matchsticks can be regarded as three sides of a regular triangular pyramid, so at most, they can
Put four triangles.
Fifth, the idea of inverse transformation.
The idea of inverse transformation refers to the inverse application of some definitions, theorems, formulas and rules and the inverse analysis of problem-solving ideas. Such as addition and subtraction, function, general sum reduction, bracket removal and bracket addition are all reciprocal transformations.
Example 9: Calculation:
Analysis: Reverse the law of multiplication and distribution.
Example 10:
Analysis: inverse power operation rule.
Example11:| a-| a | =-2a when a=
Analysis: Using reverse analysis, such as 12, first look at the absolute result, and according to the nonnegativity of the absolute value, it is concluded that -2a≥0, then a≤0.
Sixth, the idea of function and equation.
Function thought refers to a corresponding thought between variables. The idea of equation refers to transforming the quantitative relationship between known quantities and unknown quantities in mathematical problems into equations or mathematical models such as equations. When the function value is zero, the function problem is transformed into an equation problem. Equation can also be regarded as the problem of finding independent variables when the function value is zero.
Example 12: The triple complementary angle of an angle and its complementary angle are complementary angles. Find the degree of this angle. Analysis: Setting equations (groups) in geometry problems will solve problems.
Example 13: an engineering team wants to recruit 700 workers, divided into two jobs: A and B.
The monthly wages of this type of workers are 800 yuan and 1200 yuan respectively. At present, the number of B-type workers is required to be not less than 3 times that of A-type workers. When asked how many people are recruited in A-type workers and B-type workers, can the monthly salary be the least?
Analysis: establish the function relationship, determine the range of independent variables, and use the monotonicity (increase or decrease) of linear functions to solve problems.
In short, in mathematics teaching, we should effectively master the above-mentioned typical mathematical thinking methods, and at the same time pay attention to the infiltration process. According to the textbook content and students' understanding level, it should be infiltrated in a planned and step-by-step way from junior high school, making it a link from knowledge to ability and a magic weapon to improve students' learning efficiency and mathematics ability.