Basic Mathematics Thought Reflected in Junior Middle School Mathematics Textbooks

Mathematical thinking method is the essence of mathematics discipline and one of the important contents of mathematical literacy. Only by fully mastering and understanding can we effectively use knowledge and form ability. So, what is mathematical thought? Mathematical thinking means that the spatial form and quantitative relationship of the real world are not reflected in people's consciousness, but produce results through thinking activities, which is the essential understanding of mathematical facts and theories.

There are more than 30 kinds of mathematical ideas involved in the whole set of junior high school mathematics textbooks. Several main mathematical ideas are summarized here.

First, the idea of using letters to represent numbers is one of the basic mathematical ideas.

This idea is mainly reflected in the first chapter "Basic Knowledge of Algebra" in the first volume of Algebra. For example:

Let A be a and B be represented by an algebraic expression: (1) The sum of A and B is twice: 2 (a+b) (2) The difference between 1/3 of A and 1/2 of B is 1/3a-65438+.

Second, the idea of combining numbers with shapes.

The combination of numbers and shapes is one of the most important and basic thinking methods in mathematics, and it is an effective idea to solve many mathematical problems. The following contents in practical mathematics textbooks reflect this idea.

1, the one-to-one correspondence between points on the number axis and real numbers.

2. One-to-one correspondence between points on the plane and ordered real number pairs.

3. The relationship between function and image.

4. The sum, difference, multiplication and division of line segments (angles) should make full use of numbers to reflect shapes.

5. Solving triangles, finding angles and side lengths, and introducing trigonometric functions are how to solve problems by algebraic methods. 6. In the chapter of "Circle", the definition of greetings, the positional relationships between points and circles, straight lines and circles, and circles are all treated as quantitative relationships.

7. The second statistical method in preliminary statistics is to draw statistical charts, which are used to reflect the distribution and development trend of data. In fact, it is through the "shape" to reflect the data dressing situation, development trend and so on. In fact, it is to embody the characteristics of numbers through "shape", which is a direct application of the idea of combining numbers and shapes in practice.

Third, change ideas.

In the whole junior middle school mathematics, the idea of transformation has been running through it. Transforming thinking is to turn an unknown (to be solved) problem into a solved or easily solved problem, which is one of the basic thinking methods of mathematics. The following contents reflect this idea:

1, the solution of the fractional equation is to transform the fractional equation into the quadratic equation that I learned before. Here, the new problem to be solved has become a solved problem, which embodies the transformation idea.

2. Solve the right triangle; Turn the non-right triangle problem into a right triangle problem; Turn practical problems into mathematical problems.

3. In the chapter of "circle", the proof of the fillet theorem is analyzed: the idea of proving the tangent angle theorem is to find the tangent length of two circles. These transformations are all done through auxiliary lines.

4. Solve the problem of a line segment or area in a triangle or polygon as a similarity ratio problem.

Fourth, the idea of classification.

The classification of sets, rational numbers, algebraic expressions, real numbers, angles, triangles, quadrilaterals, the positional relationship between points and circles, the positional relationship between straight lines and circles, and the positional relationship between circles are all discussed through classification.

V. Special and General Ideas

1. In the chapter of "Circle", the theorem of circumferential angle and tangent angle are proved by special to general methods, while the theorem of intersecting chords and its inference are general to special ideological applications.

2. In the chapter "Algebraic Multiplication and Division", firstly, the general operation properties of power are abstractly summarized in the special case of sum of numbers. For example:103x103 = (10x10) (10x10) =10x/kloc-0.

a3？ A3 =a3+2 am? An am+n

The derivation of multiplication formula is a process from general to special.

Sixth, analogical thinking.

The properties of 1. inequality and the solution of one-dimensional inequality are often compared with the properties of equality and the solution of nonlinear sum.

2. Through the reciprocal, absolute value and operation law of rational numbers, we can obtain real and sensitive knowledge about reciprocal, absolute value and operation law.

3.

In the addition and subtraction of quadratic roots, it is pointed out that "merging similar quadratic roots is similar to merging similar terms" Therefore, the addition and subtraction of quadratic roots can be compared with the addition and subtraction of algebraic expressions.

4.

"Measurement of angles, comparison of angles, sum and difference lines of angles" can be compared with the relevant knowledge of line segments. The operation of degrees, minutes and seconds can be compared with the operation of hours, minutes and seconds.

5. Compare the attributes of similar polygons with those of similar triangles.

Seven. Universality of numbers

Whether the similar operation with numbers has such a property, such as the generality of numbers, is inferred from the nature of numbers in the chapter of multiplication and division of algebraic expressions; Deduce a formula from the reduction of a number.

Eight, similar merger ideas

The concrete embodiment of this idea in the chapter "Addition and subtraction of algebraic expressions" is to merge similar terms. In the chapter "Roots", similar roots are merged.

Nine, there is no approximate idea

The idea of infinite approximation is embodied in infinite acyclic decimal and irrational number approximation.

Ten, symmetrical transformation thought

In radical multiplication and radical division, √a2 =a(a=0) and so on. Equivalent transformation, symmetric transformation and inverse formula are used many times.