The world is full of wonders, and there are many interesting things in our mathematics kingdom. For example, in my ninth exercise book, there is a thinking question that reads: "A bus goes from Dongcheng to Xicheng at a speed of 45 kilometers per hour and stops after 2.5 hours. At this time, it is just 18 km away from the center of the east and west cities. How many kilometers is it between East and West? When Wang Xing and Xiaoying solve the above problems, their calculation methods and results are different. Wang Xing's mileage is less than Xiao Ying's, but xu teacher said that both of them were right. Why is this? Have you figured it out? You can also calculate the calculation results of both of them. " In fact, we can quickly work out a method for this problem, which is: 45× 2.5 = 1 12.5 (km),112.5+18 =130.5 (. In fact, we have neglected a very important condition here, that is, the word "Li" mentioned in the condition is "just 18 km from the center of the east and west cities", and it does not say whether it has not yet reached the midpoint or exceeded the midpoint. If the distance from the midpoint is less than 18km, the formula is the previous one; If it is greater than 18km, the formula should be 45× 2.5 = 1 12.5 (km), 1 12.5-65448. Therefore, the correct answer should be: 45 × 2.5 = 1 12.5 (km),12.5+18 =130.5 (km),/kloc-. Two answers, that is to say, Wang Xing's answer and Xiaoying's answer are comprehensive.
In daily study, there are often many math problems with multiple solutions, which are easily overlooked in practice or examination. This requires us to carefully examine the problem, awaken our own life experience, scrutinize it carefully, and fully and correctly understand the meaning of the problem. Otherwise, it is easy to ignore other answers and make a mistake of generalizing.
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". 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.
For example, a triangle. A triangle is a plane figure composed of three line segments that are not on the same line. Through experiments and research, we know that the sum of the inner angles of the triangle is 180 degrees, and the sum of the outer angles is 360 degrees. The ground can be covered by six regular triangles.
Look at the regular quadrangle, which can be divided into two triangles. The sum of internal angles is 360 degrees, the degree of an internal angle is 90 degrees, and the sum of 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.
Hexagon can be divided into four triangles, the sum of internal angles is 720 degrees, the degree of one internal angle is 120 degrees, and the sum of external angles is 360 degrees. The ground can cover three regular quadrangles.
A heptagon can be divided into five triangles. The sum of internal angles is 900 degrees, the degree of internal angles is 900/7 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.
Not only can you cover the ground with a regular polygon, but you can also cover the ground with more than two or three kinds of graphics.
For example: regular triangle and square, regular triangle and hexagon, square and octagon, regular pentagon and octagon, regular triangle and square and hexagon. ...
In real life, we have seen all kinds of patterns composed of regular polygons. In fact, many patterns are often composed of irregular basic graphics.
References:
/question/45750 138.html？ Si=2 agrees with 3| Comments 2008-8-20/KLOC-0: 29 Yacht 168 | Level 1
The application of common factor and common multiple is closely related to life. To solve this kind of problem, we must first examine the meaning of the topic and understand its essence. On the basis of finding the greatest common factor and the least common multiple, do some in-depth research and strengthen comparative exercises to help students solve problems.
For example, (1) Xiaoming's study is 2.7 meters long and 2.25 meters wide. He is going to lay a square floor tile on the floor. How many tiles does he need at least? Idea: Many square tiles can be laid in a row along the length of the study, so the side length of the square tiles used is the factor of the length of Xiaoming's study, that is, the side length of the tiles must be the common factor of the length and width of the study. In the problem, the floor tiles should be as large as possible, that is, the greatest common factor of length and width is taken as the side length, and the minimum number of tiles required is: (270 ÷ 45) × (225 ÷ 45) = 30 (blocks).
(2) A floor tile is 25cm long and 20cm wide. Now I plan to use this tile to pave a square floor. How many tiles do I need at least? The side length of a big square paved with rectangular floor tiles is a multiple of the length and width of the floor tiles. The common multiple of 25 and 20 is100,200,300, ... So as long as the side length is more than cm, a square can be used to pave this kind of floor tile. The topic requires that the paved square has the smallest side length, and the side length must be the least common multiple of the floor tile's length of 25cm and width of 20cm, 100cm, (100 ÷ 25) × (100 ÷ 20) = 20 (block). Such a floor tile needs at least.
Contrast: The above two problems are all paved with floor tiles. The difference is that the problem (1) is to lay square bricks in a fixed area. In fact, it is to divide the big rectangle into small squares and concentrate on the word "fen". The greater the side length of the used floor tile, the less blocks are needed, and the maximum side length of the used floor tile is the greatest common factor of the length and width of this rectangle. Question (2) is to spell a square with several identical rectangles, and the key point is the word "spelling". The side length of the square is the common multiple of the length and width of the floor tile, and the side length of the square with the smallest area is the least common multiple of the length and width of the floor tile used.
& lt Mathematical problems of RMB.
One day, my mother and I went shopping. Mom went into the supermarket to buy things and let me stand at the place where I paid. I have nothing to do, just watch the assistant aunt collect money. After reading it, I suddenly found that the money collected by the assistant aunt was 1 yuan, 2 yuan, 5 yuan, 10 yuan, 20 yuan and 50 yuan. I feel very strange: Why isn't RMB from 3 yuan, 4 yuan, 6 yuan, 7 yuan, 8 yuan, 9 yuan or 30 yuan, 40 yuan or 60 yuan? I ran to ask my mother, and my mother encouraged me to say, "Think hard and calculate well. My mother believes you can figure out the reason." I calmed down and thought it over. After a while, I jumped up happily: "I know, because as long as you have 1 yuan, 2 yuan and 5 yuan, you can form 3 yuan, 4 yuan, 6 yuan, 7 yuan, 8 yuan and 9 yuan at will, and as long as you have 10 yuan, 20 yuan and 50 yuan, you can also form 30 yuan and 40 yuan." Why 2 yuan and 5 yuan? "I said," it is not convenient to use 1 yuan to form a larger number. "Now my mother showed a satisfied smile and praised me for observing more and thinking more. I am really more comfortable than eating my favorite ice cream.
Here, I also want to tell other children: in fact, there are math problems everywhere in life. As long as you pay more attention to observation and thinking, you will find many unexpected discoveries. Try it if you don't believe me!