What exactly is mathematics? We say that mathematics is a science that studies the relationship between spatial form and quantity in the real world. It is widely used in modern life and production, and it is an indispensable basic tool for studying and studying modern science and technology, and mathematics is also indispensable in life.

In real life, we will see all kinds of patterns composed of regular polygons. For example, we will see ceramic tiles at home, shops, central squares, hotels, restaurants and many other places. They usually have different shapes and colors. Actually, there is a math problem.

On the tiled floor or wall, adjacent floor tiles or tiles are evenly attached together, and there is no gap on the whole floor or wall. Why can these shapes of floor tiles or tiles cover the ground without gaps?

For example, a triangle. A triangle is a plane figure composed of three line segments that are not on the same line. As we know, the sum of the inner angles of a triangle is 180 degrees, and the sum of the outer angles is 360 degrees. The ground can be covered by six regular triangles.

Looking at the regular quadrangle, it can be divided into two triangles, the sum of the internal angles is 360 degrees, the degree of one internal angle is 90 degrees, and the sum of the external angles is 360 degrees. The ground can cover four regular quadrangles.

What about regular pentagons? It can be divided into three triangles, the sum of internal angles is 540 degrees, the degree of one internal angle is 108 degrees, and the sum of external angles is 360 degrees. It cannot cover the ground.

……

From this, we come to the conclusion. An N-polygon can be divided into (n-2) triangles, and the sum of internal angles is (n-2)* 180 degrees, the degree of an internal angle is (n-2)* 180÷2 degrees, and the sum of external angles is 360 degrees. If (n-2)* 180÷2 can be divisible by 360, then it can be used to pave the way; If not, it can't be used to pave the road.

Ceramic tiles, such ordinary things, have such interesting mathematical mysteries, not to mention other things in life?

As for literature, art and sports, mathematics is essential. We can see from CCTV's literary and art grand prix program that when an actor is graded, it is often "to remove a highest score" and then "to remove a lowest score". Then, the average score of the remaining scores is calculated as the actor's score. Statistically speaking, "the highest score" and "the lowest score" have the lowest credibility, so they are removed.

As Mr. Hua said: In the past 65,438+000 years, mathematics has developed by leaps and bounds. It is no exaggeration to use "ubiquitous mathematics" to summarize the wide application of mathematics in various aspects, such as the vastness of the universe, the tiny particles, the speed of rockets, the cleverness of chemical industry, the change of the earth, the mystery of biology and the complexity of daily use. It can be predicted that the more advanced science is, the wider the scope of applied mathematics will be.

It can be asserted that there are only departments that can't apply mathematics, and in principle, they will never find areas that can't apply mathematics.

About "0"

0, can be said to be the earliest human contact number. Our ancestors only knew nothing and existence at first, and none of them was 0, so 0 isn't it? I remember the primary school teacher once said, "Any number minus itself is equal to 0, and 0 means there is no number." This statement is obviously incorrect. As we all know, 0 degrees Celsius on the thermometer indicates the freezing point of water (that is, the temperature of ice-water mixture at standard atmospheric pressure), where 0 is the distinguishing point between solid and liquid water. Moreover, in Chinese characters, 0 means more as zero, such as: 1) fragmentary; A small part. 2) The quantity is not enough for a certain unit ... At this point, we know that "no quantity is 0, but 0 not only means no quantity, but also means the difference between solid and liquid water, and so on."

"Any number divided by 0 is meaningless." This is a "conclusion" about 0 that teachers from primary school to middle school are still talking about. At that time, division (primary school) was to divide a copy into several parts and figure out how many there were in each part. A whole cannot be divided into 0 parts, which is "meaningless". Later, I learned that 0 in a/0 can represent a variable with zero as the limit (the absolute value of a variable is always smaller than an arbitrarily small positive number in the process of change) and should be equal to infinity (the absolute value of a variable is always larger than an arbitrarily large positive number in the process of change). From this, another theorem about 0 is obtained: "A variable whose limit is zero is called infinitesimal".

"Room 203 105 in 2003", although all of them are zeros, they are roughly similar in appearance; They have different meanings. 0 indicator vacancy of 105 and 2003 cannot be deleted. 0 in Room 203 separates "Building (2)" from "House Number". (3) "(that is, Room 8 on the second floor) can be deleted. 0 also means that ...

Einstein once said: "I always think it is absurd to explore the meaning and purpose of a person or all living things." I want to study all the numbers of "existence", so I'd better know the number of "non-existence" first, so as not to become what Einstein called "absurd". As a middle school student, my ability is limited after all, and my understanding of 0 is not thorough enough. In the future, I hope (including action) to find "my new continent" in the "ocean of knowledge".

Mathematics in life

There is a riddle: there is something invisible and intangible, but it is everywhere. What is this? The answer is: air. And mathematics, like air, is invisible and intangible, but it exists all the time around us.

Wonderful "golden numbers"

Take a line segment, find a point on the line segment, and let this point divide the line segment into two parts, one is long and the other is short, and the ratio of long segment to short segment is exactly equal to the ratio of whole segment to long segment. This point is the golden section of this line segment. This ratio is: 1: 0.6 18… and 0.6 18… This number is called "golden number".

Interestingly, this number can be seen everywhere in life: the navel is the golden section of the total length of the human body; The angle between two adjacent leaves on the stems of some plants is exactly the angle between the two radii that divide the circumference into 1: 0.6 18 …. According to research, this angle has the best effect on ventilation and lighting of the factory building.

Architects especially prefer the number 0.6 18 ... Whether it is the pyramids of ancient Egypt, Notre Dame de Paris or the modern Eiffel Tower, the number 0.6 18 ... It is also found that most of the subjects of some famous paintings, sculptures and photography are at the 0.618 ... Musicians think that putting the bridge at 0.618 of the strings will make the sound softer and sweeter.

The number 0.6 18… also makes the optimization method possible. Optimization method is a method to solve the optimization problem. If it is necessary to add a chemical element to increase the strength of steel during steelmaking, it is assumed that the amount of a chemical element to be added per ton of steel is between 1000-2000g. In order to find the most suitable dose, we usually take the midpoint of the interval for experiments, and then compare it with the experimental results of 1000 g and 2000 g respectively, select two points with higher intensity as the new interval, and then take the midpoint of the new interval for experiments until the most ideal effect is obtained. However, this method is inefficient. If the measuring point is taken at the interval of 0.6 18, the efficiency will be greatly improved. This method is called "0.6 18 method". Practice has proved that the former method can be realized by using "0.6 18 method" to test 16.

There are so many examples and applications of "golden number" in life. Perhaps, there are more mysteries waiting for us to explore, so that it can serve us better and solve more problems for us.

Wonderful axial symmetry

If a straight line can be found on a graph, the graph can be aligned along this straight line, so that the two sides can completely overlap. Such a figure is called an axisymmetric figure, and this straight line is called an axis of symmetry.

Careful observation shows that the plane is a standard axisymmetric object. Looking down, its wings, fuselage and tail are all symmetrical. Axial symmetry makes its flight more stable. If the plane is not axisymmetric, it will fly stagger. At that time, who wanted to fly?

Looking closely, it is not difficult to find that many works of art are also axisymmetric. Take the simplest example: the bridge. It is the most common work of art in life. Take the bridges in Jinhua as examples: Tongji Bridge, Jinhongqiao, Shuanglong Bridge and Riverside Bridge. They are all axisymmetric. The ancient buildings in China are even more obvious. Ancient palaces were basically symmetrical. Another famous: the layout of Beijing. This is the most typical axisymmetric layout. It is symmetrical with the Forbidden City, Tiananmen Square, Monument to the People's Heroes and Qianmen as the central axis. The application of axial symmetry in art can make art look more beautiful.

Axisymmetry is also a biological phenomenon: human ears, eyes and limbs all grow symmetrically. The axial symmetry of the ear makes the sound we hear have a strong three-dimensional sense and can also determine the position of the sound source; The symmetry of the eyes allows us to see objects more accurately. It can be seen that our life is inseparable from axial symmetry.

Mathematics is very close to us, and it is reflected in all aspects of life. We can't live without mathematics, which is everywhere. These are just two very common examples, and there are countless such examples. I think mathematics in life can bring more discoveries to people.

But now I guess it's useless. There are so many words to write at this point.

Junior high school mathematics is a whole. The second grade is the most difficult, and the third grade has the most test sites. Relatively speaking, although there are many knowledge points in junior high school mathematics, they are all relatively simple. Many students feel no pressure when studying at school, and gradually accumulate a lot of minor problems. These problems are highlighted after entering the second day of junior high school and encountering difficulties (such as increasing the number of subjects and deepening the difficulty).

At present, some freshmen in the second grade of the senior high school entrance examination network just don't pay enough attention to the mathematics in the first grade. After entering the second grade, they found that they couldn't keep up with the teacher's progress and found it more and more difficult to learn mathematics. I hope to join our remedial class to make up for it. The main reason for this problem is that we don't pay enough attention to the math foundation of junior one. Here are a few common problems in senior one mathematics learning:

1, the understanding of knowledge points stays at the level of a little knowledge;

2. We can never master the key mathematical skills of solving problems, treat each problem in isolation, and lack the ability to draw inferences from others;

3. When solving a problem, there are too many small mistakes, and the problem can never be completely solved;

4. The problem-solving efficiency is low, and a certain number of problems cannot be completed within the specified time, which is not suitable for the examination rhythm;

5. I haven't formed the habit of summarizing and summarizing, and I can't habitually summarize the knowledge points I have learned;

If these problems can't be solved well in the first grade, students may have a decline in their grades in the polarization stage of the second grade. On the contrary, if we can lay a good foundation of mathematics in grade one, the study in grade two will only increase the number and difficulty of knowledge points, and students will easily adapt to the learning methods.

Then how can we lay a good foundation for mathematics in senior one?

(1) Explore concepts and formulas carefully.

Many students pay insufficient attention to concepts and formulas. This problem is reflected in three aspects: first, the understanding of the concept only stays on the surface of the text, and the special situation of the concept is not paid enough attention. For example, in the concept of algebraic expression (an expression expressed by letters or numbers is algebraic expression), many students ignore that "a single letter or number is also algebraic expression". Second, concepts and formulas are blindly memorized and have nothing to do with practical topics. The knowledge learned in this way can't be well connected with solving problems. Third, some students do not pay attention to the memory of mathematical formulas. Memory is the basis of understanding. If you can't memorize the formula, how can you skillfully use it in the topic?

Our suggestions are: be more careful (observe special cases), go deeper (know the common test sites in the topic), and be more skilled (we can use it freely no matter what it looks like).

2) Summarize similar topics.

This work is not only for teachers, but also for our classmates. When you can summarize the topics, classify the topics you have done, know which types of questions you can do, master the common methods of solving problems, and which types of questions you can't do, you will really master the tricks of this subject and truly "let it change, I will never move." If this problem is not solved well, after entering the second and third grades, students will find that some students do problems every day, but their grades will fall instead of rising. The reason is that they do repetitive work every day, and many similar problems are repeated, but they can't concentrate on solving the problems that need to be solved. Over time, the problems that can't be solved have not been solved, and the problems that can be solved have also been messed up because of the lack of overall grasp of mathematics.

Our suggestion is that "summary" is the best way to reduce the number of topics.

(3) Collect your typical mistakes and solve the problems that you can't solve.

The most difficult thing for students is their own mistakes and difficulties. But this is precisely the problem that needs to be solved most. There are two important purposes for students to do problems: First, to practice the knowledge and skills they have learned in practical problems. The other is to find out your own shortcomings and make up for them. This deficiency also includes two aspects, mistakes that are easy to make and contents that are completely unknown. However, the reality is that students only pursue the number of questions and deal with their homework hastily, rather than solving problems, let alone collecting mistakes. We suggest that you collect your typical mistakes and problems that you can't do, because once you do, you will find that you thought you had many small problems before, but now you find this one is recurring; You thought you didn't understand many problems before, but now you find that these key points have not been solved.

Our suggestion is: doing problems is like digging gold mines. Every wrong question is a gold mine. Only by digging and refining can we gain something.

(4) Ask and discuss questions that you don't understand.

Find problems you don't understand and actively ask others for advice. This is a very common truth. But this is what many students can't do. There may be two reasons: first, insufficient attention has been paid to this issue; Second, I'm sorry, I'm afraid of asking teachers to be trained and asking students to be looked down upon by them. With this mentality, you can't learn anything well. "Building a car behind closed doors" will only make your problems more and more. Knowledge itself is coherent, the previous knowledge is unclear, and it will be more difficult to understand later. When these problems accumulate to a certain extent, you will gradually lose interest in the subject. Until I can't keep up.

Discussion is a very good learning method. A difficult topic, after discussion with classmates, may get good inspiration and learn good methods and skills from each other. It should be noted that it is best to discuss with your classmates at the same level, and everyone can learn from each other.

Our suggestion is that "diligence" is the foundation and "thirst for knowledge" is the key.

(5) Pay attention to the cultivation of actual combat (examination) experience.

Examination itself is a science. Some students usually get good grades. Teachers ask questions in class, and they can do anything. I can also do problems after class. But when it comes to the exam, the results are not ideal. There are two main reasons for this: first, the test mentality is not bad, and it is easy to be nervous; Second, the examination time is tight and it can never be completed within the specified time. Bad mentality, on the one hand, we should pay attention to our own adjustment, but at the same time we also need to exercise through large-scale examinations. Every exam, everyone should find a suitable adjustment method and gradually adapt to the rhythm of the exam with the passage of time. The problem of slow problem solving needs students to solve in their usual problem solving. Doing homework at ordinary times can limit time and gradually improve efficiency. In addition, in the actual exam, we should also consider the completion time of each part to avoid unnecessary panic.

Our suggestion is: treat "homework" as an exam and "exam" as homework.

Above, we give some suggestions for the problems that often appear in junior one mathematics, but one thing to emphasize is that the most important thing of any method is to be effective. Students must avoid formalization and pursue practical results in their study. Any exam is a test of people's minds, and it is by no means a test of whether everyone's notes are clear and whether the plan is comprehensive.

Rational number (what is rational number; Several classification methods of rational numbers: the embodiment of rational numbers in life ...)

Number axis (what is number axis; What can the number axis do? What's the use of counting axes in life ...)

Prism (the definition of prism; Where can I see a prism in my life? What kinds of prisms are there ...)

Pyramids (ibid.);

Tangram (how tangram is formed; Clever use of jigsaw puzzles; How many convex polygons can a jigsaw puzzle spell, and how to prove it ...);

Three views (three views in different situations ...)