The seventh grade mathematics absolute value paper 1 on the analysis of absolute value in junior high school mathematics.
From the beginning of junior high school mathematics learning, the number field learned in primary school has been expanded, and at this time a very important mathematical concept has become inevitable, that is, absolute value. Absolute value is both important and difficult for junior high school mathematics learning and senior high school mathematics learning. Especially junior high school students, the understanding and application of the concept of absolute value is too superficial, and the understanding of this concept is not deep enough, which leads to mistakes in solving problems. Therefore, teachers should attach great importance to it in mathematics teaching to promote students' deep understanding of the concept of absolute value.
First, the relationship between the concept of absolute value and the comparison of rational numbers.
First of all, we must understand the geometric meaning of absolute value. It is a distance, a nonnegative quantity, and it is nonnegative, that is |a|? 0; Secondly, we should understand the nature of absolute value, which reveals the meaning of absolute value from three aspects of the nature of numbers: the absolute value of a positive number is itself, the absolute value of zero is zero, and the absolute value of a negative number is its inverse.
For example, the positions of three points A, B and C on the number axis are shown in the following figure. Try to find: | A+B |+B+C |+A-C |.
Solution: according to the number axis, c>0, a |c| >|b|,
? a+b & lt； 0，b+c & gt； 0，a-c & lt； 0
? The original formula =-(a+b)+(b+c)-(a-c) =-a-b+b+c-a+c = 2c-2a.
It is precisely because of the concept of absolute value that the comparison of two negative numbers can be transformed into the comparison of positive numbers that students are familiar with through the relationship of absolute values, rather than being expressed on the number axis, which can be restored to the knowledge that students have mastered.
Second, the relationship between absolute value and rational number addition and subtraction.
For the addition and subtraction of rational numbers, it is the sharp weapon of absolute value that is finally unified into the addition and subtraction learned in primary school, adding two numbers with the same symbol, taking their own symbols and adding their absolute values; Add two different symbols with unequal absolute values, take the symbol with larger absolute value, and subtract the symbol with smaller absolute value from the symbol with larger absolute value.
For example, find a number x so that its distance to -3 is equal to 7.
Solution: According to the distance formula between two points on the same number axis:
|x-(-3)|=7？ |x+3|=7？ x+3=？ 7 ? X=4 or x=- 10.
With this conclusion, it will be widely used to find the length, area and circumference of a line segment in the study of functions in the future, and the distance formula between two points on the plane will be easier to understand.
Third, the relationship between absolute value and quadratic root
In the quadratic form =|a|, because a2 is non-negative and the meaningful intervals of A are all real numbers, the essence of the problem returns to the operation of absolute value. This kind of operation appears frequently in the correlation operation of quadratic form, and it is also an error-prone point for students to solve problems. Still emphasize the positive and negative judgment of numbers. It can be seen that the application of absolute value is by no means ordinary, and teachers need to strengthen and deepen it in daily teaching, seize the connection and deepen it. At the same time, the organic combination of absolute value nonnegative and square relation nonnegative, square root nonnegative also often appears, in most cases in the form of nonnegative and zero. At this time, we can't make full use of the sum of several non-negative numbers and zeros to cancel each other out, and the inverse of zero is zero, so that each non-negative number is zero respectively. Only on this premise can we solve the problem.
For example, A, B and C are three sides of a triangle, and +|b-4|+(c-5)2=0. Try to find the perimeter of the triangle.
Because =|a-6|, there are |a-6|+|b-4|+(c-5)2=0, and |a-6|? 0，|b-4|？ 0，(c-5)2？ 0, so a-6=0, b-4=0, c-5=0, so a=6, b=4, c=5, and the perimeter of the triangle is a+b+c=6+4+5= 15.
Fourthly, the relationship between absolute value and inequality.
Know and understand the geometric meaning of absolute value and solve inequality |x|? A and |x|? a(a & gt； 0), relatively easy to understand; Right |x|? For a, it can be understood as a point whose distance from the origin is greater than a, so it must be the point on the right of the number A or the point on the left of the number -a, so the solution set is X >；; A or x