Give a simple example, how to mark the corresponding points of irrational numbers on the number axis. Obviously, we can't mark it on the number axis by measuring, because no matter how accurate the scale is, it can only produce a limited decimal, but an infinite acyclic decimal. Therefore, we can only resort to other methods.
Is there any way to convert it into a finite decimal or integer relationship? At this time, we should remember the pythagorean theorem of right triangle: the sum of squares of two right-angled sides is equal to the square of hypotenuse, that is, a? +b？ =c？ If the lengths of both right angles are 1, then the length of the hypotenuse is.
Therefore, we can mark it on the number axis in this way.
After finding the length by Pythagorean theorem, the origin is taken as the center, the radius length is a circle, and the intersection of the circle and the number axis (to the right of the origin) is the corresponding point.
Another example is to find the minimum value of a function with absolute value, such as finding the minimum value of the function y=|x+ 1|+|x-2|+|x-3|.
Of course, we can discuss the values of "x+ 1", "x-2" and "x-3" respectively through classified discussion, so as to get rid of the absolute value to solve it. The problem is that eight situations are involved in solving the problem of removing the absolute value through classified discussion, and the process of solving the problem is more complicated.
If we use the geometric meaning of absolute value to transform the problem into a geometric problem through the number axis, it will be much more intuitive.
First of all, it should be clear that the absolute value represents the distance between two points on the number axis (the importance of thoroughly understanding the concept). For example, there are two points A and B on the number axis (A and B represent numerical values), and the distance between point A and point B is |a-b| or |b-a|.
Therefore |x+ 1|=|x-(- 1)|, which represents the distance between point "x" and point "-1"; |x-2| indicates the distance between point "x" and point "2"; |x-3| indicates the distance between point "x" and point "3". X is an unknown number, that is, point X is an uncertain point on the number axis. So the question can become: When point X moves from the left to the right of the number axis, where is the sum of the distances between point X and points "-1", "2" and "3" shortest?
Below, I use some cases to illustrate the application of the combination of numbers and shapes.
If we solve the problem by removing the root flag, it is difficult for us to solve the problem. But if we look at this function a little, we will find that it can actually be transformed into a formula for the distance between two points in a plane rectangular coordinate system (the importance of thoroughly understanding the formula).
Suppose the coordinates of point A are (0,2), point B is (2, 1), and point P is (x,0).
Then the problem is transformed into: in the plane rectangular coordinate system, point P is a point on the X axis, and the minimum value of the sum of the distances between point P and points A and B is the shortest PA+PB problem.
We know that Tan 45 = 1. It is proved that α+β = 45, which means that tan(α+β)= 1. How to construct the relationship between tan(α+β) and tanα and tanβ is obviously beyond junior high school knowledge.
Therefore, we should seek ways to prove it from other channels.
Students who have a thorough understanding of trigonometric functions should easily think of the meaning of trigonometric functions, and then look for verification methods from this perspective.
What is the meaning of trigonometric function? It represents the relationship between the angles and sides of a right triangle. For example, the tangent value in the condition given in this question represents the ratio of the opposite side to the adjacent side of an angle in a right triangle.
Knowing this, we can transform trigonometric function form into geometric form.
After transformation, the trigonometric function problem of this question becomes a geometric problem: BC=2, BD=3, AB=6, AB⊥CD, and verification ∠ CAD = 45.
You can try to write your own verification process. If you have no idea for the time being, you can refer to the following tips.
If you happen to be a junior high school student, you are confused or have difficulty in math learning. Welcome to add the following WeChat, communicate with me, or check out more articles I have written about learning.
Graduated from the Economics Department of Shenzhen University.
A literary lover who is engaged in stock investment, loves business model research and is addicted to education.
After graduating from college, I competed among hundreds of financial high flyers for the only stock research intern seat in a financial institution in Hong Kong.
I have been engaged in stock research and investment in Hong Kong and Shenzhen for many years, and have my own investment system and concept. Although it has experienced many market turmoil, it can still get a high return on investment. I selected a batch of bull stocks with the ratio of 10 for three years, five years and five years, and also avoided a batch of junk stocks that would bring great losses to the company.
I have also had many years of entrepreneurial experience in many industries. Because of my work experience, I deeply know the importance of learning ability, which is related to grades but far more than just reflected in grades. The learning ability cultivated by students in the stage is not only a sharp weapon to get into a good university, but also an essential ability to constantly improve themselves and build their competitiveness after work, so that they can overcome difficulties and establish solutions no matter what problems they encounter.
For more than ten years, I have read more than 50 books every year on average, and reading is the most important way of lifelong learning. In recent years, I began to do research in learning and education and read nearly 100 books.
Learning is a combination of attitude, thinking and time management. Anyone can become a schoolmaster as long as the above three points are well done. The whole learning process includes: contacting knowledge-absorbing concepts-exploring essence-summarizing application scope. Most students only do the first two steps and learn by rote, which is the main reason for their poor study.