Last class, we talked about "the whole idea of mathematics" in mathematics strategy. Its core idea is to use a certain relationship or condition in the problem as a whole, so as to achieve the purpose of solving the problem.

We know that no matter what kind of math problem, there are two parts. One part is known or hidden "conditions" to solve the problems in the problem, which we call "keys". The other part is the problem to be solved in the topic, which we call the "lock" to be opened.

In other words, any math problem is actually composed of a key and a lock. The process of solving problems is actually the process of finding the "key" to unlock.

Yes, those known or hidden conditions in math problems are a "key" to unlock. However, different locks require different keys to open. Therefore, some conditions need to be scattered to play the role of "key", and some conditions need to be taken as a whole to condense into the power of "key". But no matter whether it is dismantled or used as a whole, it is easy to solve the problem anyway. Be sure to use it flexibly according to the topic!

As we know, there are many types of application of "holistic thinking" in mathematics, that is to say, "holistic thinking" has many different forms of expression in solving mathematical problems. For example, last class, we talked about the "whole replacement method". Today we will talk about the "integral addition and subtraction" in the overall thinking of mathematics.

Last class talked about the "whole replacement method", so I won't go into details here. Interested friends can pay attention to "strategic research" and go to my home page to see the complete course.

By the way, I want to explain the second volume of junior high school mathematics in a systematic way recently. I want to help children tap the "bright spot" of "knowledge points" from the perspective of "strategy" to make it easier for children to learn mathematics. Interested friends can pay attention to me and my course will be sent to you as soon as possible!

Integer addition and subtraction

So, what is "integral addition and subtraction"? What is the difference between "integral addition and subtraction" and "integral substitution"?

The so-called "whole addition and subtraction" is a method that takes a certain relationship or condition in the problem as a whole and then "adds and subtracts" the whole to achieve the purpose of solving the problem.

There is a difference between "integral addition and subtraction" and "integral substitution". The key to integer addition and subtraction lies in addition and subtraction, which produces "key function" to open the lock. The whole substitution method focuses on substitution, which can solve the situation of "no way to start" and make the problem easy to solve.

In order to clarify the problem, let's give an example now:

explain

Example 1: if 3a+5b+8c=88 and 5a+3b=72, find a+b+c.

We know that if we can find the values of a, b and c respectively, then the problem will be solved naturally. However, it is not difficult to find that the conditions given in this question are far from achieving the purpose of finding A, B and C respectively, so we can only find another way to solve the problem.

We might as well use the "whole addition and subtraction" in the "whole idea of mathematics", take "3a+5b+8c" as a whole and "5a+3b" as another whole, and then carry out the "addition movement" on these two "whole", and you will find that the "magic effect" appears.

(3a+5b+8c)+(5a+3b)=88+72

8a+8b+8c= 160

8(a+b+c)= 160

a+b+c= 160÷8

a+b+c=20

Through this example, we can easily find that the focus of "integer addition and subtraction" includes two parts, namely "as a whole" and "addition and subtraction movement"

Let's give another example. This example uses "integral addition and subtraction" and "integral substitution". Carefully taste the differences between them, as follows:

Example 2: If x+3y+z =100; 3x+2y+z=200, find12x+15y+6z;

This problem can't find the values of x, y and z, so since the values of x, y and z can't be found, we can only find another way to solve the problem. Obviously, "integral addition and subtraction" is a good choice.

We regard "x+3y+z" as a whole and "3x+2y+z" as another whole. Through the addition movement of the two whole bodies, we get:

(x+3y+z)+(3x+2y+z)= 100+200

After merging similar projects:

4x+5y+2z=300

Then, we take "4x+5y+2z" as a "whole", then find the relationship between the problem and this "whole", and then substitute the "whole" to solve the problem:

12x+ 15y+6z

=3(4x+5y+2z)

=3 x 300

=900

Obviously, the whole addition and subtraction method and the whole substitution method are used to solve this problem. Because the whole method of substitution is the basic expression of the whole idea of mathematics, it is not surprising that the whole method of substitution often appears in other whole forms. However, we must make clear the difference between the two to avoid confusion.

At this point, it is estimated that some friends are vague about the difference between "integral addition and subtraction" and "integral substitution". It doesn't matter, I will continue to speak until you fully understand!

The core of "integral addition and subtraction" is "addition and subtraction movement". Since there is "addition and subtraction", it means that there are more than two whole participants. In other words, at least two conditions in the topic should be regarded as different "whole" respectively, and then "addition and subtraction movement" should be carried out to achieve "problem-solving effect"

The core of "holistic substitution method" is "substitution". To put it bluntly, it is to take a condition or relationship as a whole and then substitute it into it, so as to achieve the effect of solving the problem. If the object of its participation is "a whole", it can be realized.

Course summary

"Integral addition and subtraction" consists of two parts. Part is "whole" and part is "addition and subtraction". The ultimate goal of the whole "addition and subtraction movement" is to get a "desired effect", and then substitute this "effect" as a whole to solve the problem.

In other words, the "whole" in "whole addition and subtraction" cannot be directly substituted, and it can only be substituted after the desired effect is produced through "addition and subtraction movement". "Whole replacement method" can directly replace "whole"!

Ok, let's call it a day. The next class will talk about the "holistic transformation method" in the "holistic thinking of mathematics", so that we have it or not!