However, in the big family of mathematics. I am deeply attracted by a pair of brothers. Their shape, their relationship and their universality make people feel that they are always around us and very close to us. They are axisymmetric figures.
Axisymmetric graphics are graphics that must be folded along a straight line, and the parts on both sides of the straight line overlap each other. The reason for their relationship is that they are always connected by a straight line, like a pair of inseparable brothers, and their relationship is very close. The straight line that pulls them together is their axis of symmetry. Of course, this axis of symmetry is like a fair judge. The length, area and size of the left and right sides are not bad at all, the only difference is the direction they face.
In the math textbook, we saw their figures, and also contacted and understood them. But what impressed me more was the graphics or things they played and constituted in their daily life.
1. Axisymmetric graphics in life
1, an axisymmetric figure in nature
When I walk in the street, I often see butterflies flying around. When a butterfly stays on a flower and spreads its wings, I find that if the midpoint of the butterfly's two tentacles is connected with its tail, the straight line where the connected line segments are located is its symmetry axis. The right wing is like a figure in which the left wing flips along the symmetry axis. There are many animals with axisymmetric figures like butterflies. Such as dragonflies, moths and so on. If it is autumn, looking at the rice fields from afar, I can't help but feel that it is another harvest season. In this happy season, I walked on the path near the field and picked up a golden leaf. After careful observation, I found that leaves also have symmetry axes. If we regard the meridian in the middle of the leaf as the symmetry axis of its left and right sides, then we will fold the right part of the leaf in half along this symmetry axis, which coincides with the left part of the leaf.
2. Axisymmetric graphics in trademarks
Once, my family and I went to China Bank to withdraw money, and accidentally found that the logo of China Bank was also an axisymmetric figure. There are two symmetry axes in this graph. The first line is formed by connecting two vertical lines in the icon, the other line is the midpoint of the line segment connected by the upper and lower horizontal lines in the box, and the straight line is its second symmetry axis. Such as China Bank, China Unicom, China Agricultural Bank and Mercedes-Benz. However, if you don't think the previous examples are usually noticed, then the examples mentioned below must be familiar to you. This example is a trademark. I will give you one first. My biggest interest is to eat snacks. So I am very familiar with the trademark "Want Want". I found that in the trademark Want Want, the midpoint of a line segment between the heels of two feet is the midpoint of its hair, and the straight line where the line segment to be connected is its axis of symmetry. It is this axis of symmetry that divides the icon Want Want into two equal parts. There are many symmetrical trademarks like Want Want. For example, the trademarks of Wuliangye, McDonald's, Converse and so on. These figures are common in our daily life, which does not tell us that mathematics is everywhere as long as we observe life carefully.
Second, the axisymmetric graphics in architecture
After talking about the common and common axisymmetric figures in life, we should also talk about the grand axisymmetric buildings in the building. Like Tiananmen Gate in China. If a line segment is used to connect the left and right sides of Tiananmen Gate, the straight line where the midpoint of this line segment is located is the axis of symmetry. Doesn't this axis of symmetry divide Tiananmen Gate into two identical parts? The Eiffel Tower in France is one of the landmark buildings in France. Its symmetry axis is the two sides connecting the bottom of the tower. The midpoint of the connected line segment is in a straight line with the spire. There are also some buildings that use axisymmetric methods. They built a large pool in front of the building, which made the building reflected in the water, thus forming an axisymmetric effect, increasing the space and making the original building more beautiful and spectacular. Like the Taj Mahal, it is the best example of the combination of architecture and axisymmetric graphics. On the other side of the earth, there is a building that deeply affects the history of the whole world. This building is the White House. This is the famous administrative building in Washington, USA. Behind the rise of the White House, axial symmetry has played an extremely important role. The symmetry axis of the White House is a straight line connecting the point at the top and the midpoint of the left and right line segments at the bottom. By the way, each of us will have a door at home, and some architects want to make the door look more solemn. The door is designed in this way, the left and right sides of the door are the same, and the doors of ancient yamen and some official residences are also designed in an axisymmetric form. Make the gate look more imposing and dignified. From this, we can easily find that as long as we understand the axisymmetric graphics and are good at using them, we can integrate them into all aspects.
3. Axisymmetric figures in the literature
1, axisymmetric figure in the text
As we all know, our Chinese nation has a long culture of 5000 years. So many years of culture have precipitated countless treasures. Paper-cutting is one of the oldest folk arts in China. Even in this work of art, there are many axisymmetric applications. Let me give you an example. I still remember when my grandmother taught me to cut the traditional Chinese character "Xi", I first folded the red paper in half and then swayed it on the paper with scissors for a while. When I opened the paper that had just been folded in half, the word "hi" appeared. I was happy and surprised after reading it, but I didn't know why. Now that I have grown up, I also know that in fact, in the process of cutting the word "Xi", axial symmetry is also used. There are many paper-cut works, and it is precisely because of the existence of axial symmetry that they are more exquisite and beautiful. Of course, in the simplified characters we are writing now, there is also axial symmetry. Such as "abundance", "eye" and "sharpness". The symmetry axis of characters is easier to find. Basically, you can find it horizontally and vertically. In fact, sometimes, the symmetry axis also has the function of replication. It can divide a word into two identical words, like "two". If its symmetry axis is regarded as the straight line where the line segment connected by the midpoint of the first horizontal line and the midpoint of the second horizontal line is located. Then, can't the patterns on the left and right sides be approximately regarded as two twos? At this time, axis symmetry has the function of replication, but in my eyes, it has another function. Take this "one" for example. As before, it is also the axis of symmetry of vertical painting. After painting, take this symmetry axis as the original words, and you will find it. "One" and this symmetry axis form a "ten". This is the second function of axisymmetric graphics in my eyes. Can change one word into another.
2. Axisymmetric figures in the literature.
What I just said is the application of axial symmetry in this paper. What about sentences made up of words? Actually, if you think about it, there are. I remember playing a game with my classmates in the past, in which one person said a word and the other person immediately read it backwards. During the whole competition, I was deeply impressed by a sentence "Shanghai tap water comes from the sea". When we read this sentence backwards, we will find that it is exactly the same as the word order we are reading. Take a closer look, this is another application of axial symmetry. Let's just say that if we don't look at the word "Shanghai tap water comes from the sea", then the straight line where the midpoint of the word "Lai" is located can divide this sentence into two equal parts. Doesn't this prove that there are also axisymmetric applications in sentences? This series of examples also show us the achievements of axisymmetry in literature, which can make some works more perfect and have the function of making the finishing point. It can also change words and make sentences smooth. It brings more interest to words and sentences and adds a very beautiful stroke to literature.
4. Axisymmetric graphics in the Olympic Games
The 2008 Beijing Olympic Games is coming. In this grand gathering, people all over China are very excited, and people from all over the world participate in different forms. It is not difficult for us to find an axisymmetric figure-the Olympic five-ring flag.
We can connect the Olympic five-ring flag (as shown in figure 1). The point A where the yellow ring contacts the green ring and the point B on the black ring, and the symmetry axis is the straight line where the line segments A and B are located.
In the Olympic Games, there are five Olympic rings and mascots. The mascot of the 2008 Beijing Olympic Games is the Olympic Fuwa. A closer look at our Olympic Fuwa can't help but make people like it. Especially Fuwa Jingjing is charming. His simplicity and simplicity give people a sense of intimacy. Figure 2 is a picture of Fuwa Jingjing lifting weights. If you look at the picture in Figure 2, you will find that if you connect the point A in this picture with the point B at the bottom. Then the straight line where this line segment is located is the symmetry axis of Fuwa Jingjing. Surprisingly, the original Olympic Fuwa is also an axisymmetric figure.
Also at the Olympic Games, when the national flags of various countries slowly rose, it triggered my association with axisymmetric graphics. Like the British flag, its symmetry axis is a straight line connecting the midpoint of the upper and lower line segments of the national flag. There are many national flags like this. For example, the Canadian flag, the Italian flag and so on.
The ever-changing axisymmetric graphics make me dazzled and dizzy. In every change, you can find many surprises. Axisymmetric change is also ubiquitous, it exists in every corner, which also brings us a lot of convenience to study it.
In the process of studying axisymmetric figures, I learned that mathematics can only be discovered by careful observation. Only by understanding mathematics and using it well in life can mathematics be integrated into all aspects. Only by integrating mathematics into all aspects can we learn mathematics better.
In fact, the world of mathematics is really big. At this time, I really want to turn myself into a mountain and stand in the forest of mathematics. Into running water and mathematics, into floating white clouds in mathematics, into flying birds in mathematics.
I sincerely hope that everyone will discover mathematics with the eyes that discover beauty! Feel the math!
Cross-scientific trend of angle calculation in mathematics
There are many ways to calculate angles in mathematics. So far, we have learned the proof of congruence of triangle, equilateral triangle and isosceles triangle, and the content of parallel lines in the first chapter of the first volume of Grade 8. But I was bored when I was doing the 1 1 topic of the first chapter on objectives and evaluation!
1, original title:
In billiards, if the cue ball P hits point A near the table, bounces off the table and hits point B near the opponent's table, and then bounces off, is the route BC that the cue ball P passes parallel to PA?
Figure 1
As shown in figure 1, it is almost difficult to solve problems with conventional mathematical problem-solving ideas. I have pondered it for a long time and discussed it with several classmates, but there is no good way to solve it. Even we are wondering if this problem is wrong, so we confidently find the teacher and ask the solution of this problem. And the teacher told us that the method is:
Solution: According to the principle of plane mirror reflection in physics (the reflection angle is equal to the incident angle), it is known that ∠ 2 = ∠ 1, ∠ 4 = ∠ 3,
∠∠2 and ∠3 are complementary ∴∠ 1+∠ 2+∠ 3+∠ 4 = 180.
∵∠ 1+∠2+∠3+∠4+∠5+∠6=360
∴∠5+∠6= 180
∴PA‖CB (the inner angles on the same side are complementary and the two straight lines are parallel)
I was shocked. It is incredible that solving problems in mathematics should be based on the principle of plane mirror reflection in science. I asked the teacher if interdisciplinary knowledge could emerge in solving mathematical problems. The teacher said yes, and I was puzzled.
2. Cross-scientific problems in the mathematical angle operation of the senior high school entrance examination:
Why does physical knowledge appear in the calculation of mathematical angle? I started to investigate and search, but I was still surprised. It turns out that the proposition of the senior high school entrance examination has a trend of interdisciplinary comprehensive questions.
Figure 2
two
(1) As shown in Figure 2, the light L shines on the plane mirror I and then reflects back and forth between the plane mirrors I and II. It is known that ∠ α = 55 and ∠ γ = 75, so what is ∠β?
Solution: According to the principle of plane mirror reflection in physics (the reflection angle is equal to the incident angle), we can get:
∠BAC=∠α=55，∠CBA=∠γ=75
∴∠bca= 180-∠BAC-∠CBA = 180- 130 = 50
We can get ∠ ACN = ∠ BCN = ∠ CAN = 25 from the "normal" knowledge in physics.
And ∵∠ BCN+∠ β = 90.
∴∠β=90 -∠BCN=65
② As shown in Figure 3, the angle of intersection between plane mirror α and β is θ, the incident light AO is parallel to β, and the reflected light O'b is parallel to α after two reflections. What is ∠ θ?
Solution: ∫BO '‖α
∴∠∠ 1 = ∠ 2 (two straight lines are parallel and have the same angle)
And ∠ 3 = ∠ 4 (two straight lines are parallel and the internal dislocation angles are equal).
∫AOβ
∴∠∠ 1 = ∠ 5 (two straight lines are parallel with the same included angle),
According to the principle of plane mirror reflection in physics (reflection angle equals incident angle):
∠2=∠3,∠5=∠6,
∴ Get: ∠ 1 = ∠ 2 = ∠ 3 = ∠ 4 = ∠ 5 = ∠ 6.
∵∠4+∠5+∠6= 180 ∴∠4=∠5=∠6=60
∴∠ 1=∠2=∠3=∠4=∠5=∠6=60
∵∠3+∠6+∠θ= 180 ∴∠θ= 180 -∠3-∠6=60
From the process of solving the above problems, we can easily find that both the angle calculation in ordinary life and the angle calculation in the senior high school entrance examination mathematics have partially infiltrated the scientific content, especially the optical knowledge, so that the problems that were difficult to solve with pure mathematical knowledge can be successfully solved with the help of science. Yes, this shows that cross-professional comprehensive topics have now become a new trend in the proposition of the senior high school entrance examination.
3. Analyze the reasons and their influence on modern students:
Why is there such a comprehensive question? It is actually very simple to think about it carefully, because solving practical problems with mathematical knowledge is the starting point of learning mathematics. When practical problems are difficult to be solved by pure mathematics, the connectivity of disciplines naturally becomes the inevitable path to solve the problems. It is not difficult to imagine how common and important it will be to solve more practical problems across disciplines in a more complex world in the future.
However, this trend is undoubtedly a great new challenge for us students. The continuity of disciplines and the chain of thinking are what modern students need more than their former students. This will be a challenge and the mentality will be an extinction. For example, if a student only wants to solve three typical examples with pure mathematical thinking, it is quite difficult and the consumption of time is fatal. On the other hand, if you can properly master the knowledge of this topic and use it well, then this kind of topic will become extremely simple.
4. Summarize and put forward my opinions and suggestions:
From the calculation of the angle in the textbook to the calculation of the angle in today's senior high school entrance examination, you will encounter strange things that use scientific knowledge to solve mathematical problems. At first, I was at a loss. Through search and analysis, I finally realized that this has become a trend in the proposition of the senior high school entrance examination. This is also a new idea and method of solving problems arising from the improvement of the application scope of mathematics in life.
I was surprised and happy with my discovery. Fortunately, I found such a problem. I believe I will be more careful in solving math problems in the future. But what if I didn't misuse the knowledge of different subjects, resulting in unnecessary points? This is a great pity, but it is indeed a realistic problem we are facing now, so I put forward the following suggestions and my views:
The all-round development of (1) discipline has encountered interdisciplinary comprehensive problems, and partial subjects are absolutely not allowed. Only students who are all-round in discipline will have a greater winning rate. After all, the knowledge of two or more subjects is only the score of one subject, which is a pity because another subject is missing.
Do more questions, accumulate experience and do more questions, you will become sensitive to these types of questions, and your thinking will be unimpeded, so experience is very important. If you do too much, you will naturally think of which subjects to use when you see the comprehensive questions.
(3) Although we should pay attention to such problems, we should not abuse them. Some students begin to use knowledge of different subjects because they are nervous and sensitive. In this way, they will be deducted a lot of points. This is not right. Facing the exam, we should try our best to relax. First of all, we should think about how to solve obstacles, and use them when we find that we can solve them with the knowledge of other disciplines, so as to ensure that we don't lose points.
(4) Now the middle angle operation in mathematics has a cross-scientific trend, which is the result of the development of knowledge. I believe that there will be more updated comprehensive questions under this trend. I only hope that we can follow the trend and make progress together!