There are 6 line segments composed of 1 basic line segments, 5 line segments composed of 2 basic line segments, 4 line segments composed of 3 basic line segments, 3 line segments composed of 4 basic line segments, 2 line segments composed of 5 basic line segments, and 1 line segment composed of 6 basic line segments.
* * * There are line segments: 6+5+4+3+2+1 = (6+1) × 6 ÷ 2 = 21(strips) The answer (1) is1. (2) There are 2 1 line segments. This method of sorting first is called sorting method. This classification is not easy to be omitted and repeated. It can be inferred from the above example that if there are five points on a line segment, four basic line segments are formed, and the number of bus segments is the sum of four continuous natural numbers: 4+3+2+ 1. If there are n points, the total number of line segments is (n-1)+(n-2)+…+3+2+1= n× (n-1) ÷ 2 (strips). After finding this rule, we can use this formula to solve this kind of problem. In Example 2∠AOB (Figure 6-2), eight rays emitted from point O can form angles of various sizes. How many? Solving this problem is similar to the example 1, 10×9÷2=45 (pieces). There are 45 angles in the answer chart. Solution 3 Count, how many rectangles are there in Figure 6-3 * *? Analysis can count the number of rectangles in sequence, or find out the counting law of rectangles through analysis and research. A rectangle is made up of length and width. In the graph, * * * has three lengths (horizontal line segments) and three widths (vertical line segments), and there are nine rectangles in the graph with 3× 3 = 9. This kind of problem will be discussed later. Example 4 is shown in Figure 6-4. (1) For the shape shown above, if there are 1 1 triangles at the bottom, how many small triangles are there in this pile? (2) Now there are 169 small triangles in * * *, arranged as shown above, so how many triangles are there at the bottom? According to the chart, we can draw a rule that the relationship between the bottom and the total number is "2→4, 3→9, 4→ 16". And 22 = 4, 33=9, 44 = 16, that is, "the square of the base is exactly equal to the total". So we can get: (1) There are1small triangles in the lower layer, and * * There are 1 1 ×1(2). Exercise 1. There are 49 points on the line AB except the two ends. How many lines are there on this line segment? 2. How many triangles are there in the picture below? 3. Fold the rectangular cardboard with a length of 2 cm and a width of 1 cm layer by layer as shown below. (1) If five layers are stacked, the circumference is () cm. (2) If the perimeter is 120cm, * * * has () layers. One day, Xiao Ming said to some children, "Please name two numbers at will, and I'll work out their sum and subtract their difference!" " ""really? "Xiaoguang asked in surprise." Sure, please ask questions! "Xiao Ming said confidently. So, Xiaoguang wrote two questions: (348+256)-(348-256)-(7564+3125)-(7564-3125). As soon as Xiaoguang finished writing the second question, Xiaoming immediately said that the scores of the two questions were 5/kloc respectively. Add them together and the result is the same as Xiao Ming. Xiaolan wanted to understand the mystery of Xiao Ming's calculation, so she said the following four groups of numbers: 47 and 23, 400 and 278, 120 and 80, 16840 and 3020. As a result, Xiao Ming always tells the answer quickly. At this time, Xiao Ming asked Xiaolan, "Did you find the law?" "Haven't found it. However, I think the key lies in the smaller of the two numbers. "Xiaolan answered." Yes! You can understand by studying the relationship between numbers and smaller numbers! ""I see, this number is twice that of the smaller one! " Xiaoguang said excitedly. Xiao Ming explains to you: when we subtract the difference between two numbers from the sum of two numbers, we subtract a part of the larger number with more decimals from the sum of two numbers, and the result is the sum of two smaller numbers, that is, twice the smaller number. ""I see! " Everyone understands this. The characteristic of sum-multiple problem and multiple problem is to find out the application problem of two numbers by using the sum of two numbers and their multiples. The best assistant to solve and multiply application problems is to draw a line graph to express the quantitative relationship between two quantities, so as to find the method to solve the problem. If you don't believe me, please look at the following example. Example 1. Young Pioneers in Class Two, Class One, Grade Three did 360 good deeds, and the number of good deeds in Class Two was twice that of Class One. How many good deeds did the Young Pioneers of Class One and Class Two in Grade Three do? Analysis: Draw a line segment. As can be seen from the above figure, if we take the number of good deeds of Class 1 as 1 time, "the number of good deeds of Class 2 is twice that of Class 1", then the sum of the number of good deeds of Class 1 and Class 2 is equivalent to three times that of Class 1, and it can also be understood that the number of good deeds of Class 3 is 360. If we count the number of good people and deeds, we can also count the number of good people and deeds in Class One. Solution: The first category: 360÷(2+ 1)= 120 (parts) The second category: 360- 120=240 (parts) or 120×2=240 (parts). Example 2. My sister has 20 extracurricular books and my sister has 25 extracurricular books. How many extra-curricular books did my sister give her? The number of extra-curricular books is twice that of my sister? Analysis: The key to the problem of drawing line segments is to find out which quantity is variable and which quantity is invariant. From the known conditions, it is concluded that no matter how many books the elder sister gives to the younger sister and how many books the younger sister gets, the sum of the books of the elder sister and the younger sister is the same. If we regard my sister's remaining books as 1, then my sister's extra-curricular books can be regarded as two extra-curricular books equal to my sister's remaining books, that is, some multiples of the two are equivalent to three times my sister's remaining books. According to the method to solve the problem of doubling, we first find out how many extra-curricular books my sister has, and then compare them with the original extra-curricular books to find out how many books my sister gave her. Solution: 1 The number of books after class is: 20+25=45 (this). 2. After the sister gave the sister some books, some multiples of the two sisters were: 2+ 1=3 (times). 3. The number of books left by my sister is: 45÷3= 15 (copies). 4. The number of extracurricular books given by my sister to my sister is: 25- 15 = 15. Example 3. Two grain depots, A and B, originally stored 320 tons of rice. Later, Grain Depot A shipped 40 tons and Grain Depot B shipped 20 tons. At this time, the rice in Warehouse A was twice as much as that in Warehouse B, so how many tons did these two grain depots originally store? Analysis: According to "320 tons of rice were originally stored in the two grain depots A and B, and then 40 tons were transported from the warehouse A and 20 tons were transported to the warehouse B", we can find out how many tons of rice were stored in the two grain depots A and B at this time. According to "at this time, the rice in warehouse A is twice as much as that in warehouse B", if the rice in warehouse B is regarded as 1, then the rice in warehouses A and B is three times as much as that in warehouse B, so that we can find out how many tons of rice in warehouse B, then how many tons of rice in warehouse B, and then how many tons of rice in warehouse A, and the solution: 1. 40 tons are shipped from warehouse A and 20 tons are shipped from warehouse B. At this time, the tonnage of rice stored in two grain depots is: 320-40+20=300 (tons). At this time, the tonnage of rice stored in grain depot B is 300÷(2+ 1)= 100 (ton). 3. The tonnage of rice stored in grain depot B is a comprehensive formula: warehouse B (320-40+20) ÷ (2+1)-20 = 80 (ton) Warehouse A 320-80=240 (ton) Answer: warehouse A originally stored 240 tons of rice, and warehouse B originally stored 80 tons of rice. Example 4. The fruit shop sends 380 kilograms of fruit, of which 40 kilograms is less than three times that of pears. How many kilograms of apples and pears did the fruit shop send? Analysis: Considering that the number of pears is 1, because apples are 40 kilograms less than pears and three times as much as pears, if the total amount of fruit transported is 380 kilograms plus 40 kilograms, it is equal to four times the weight of pears. Line chart: solution: 1. The weight of delivered pears is: (380+40) ÷ (3+1) =105 (kg). 2. The weight of delivered apples is: 105×3-40=275 (kg). Example 5. The school library bought *** 1000 story books, twice as many as science books, and 20 fewer than literature books 12. How many story books, science books and literature books do you want the school to buy? Analysis: According to the conditions, 12 science and technology books are twice as many as story books, and 20 literature and art books are less than story books. It can be seen that they are all compared with storybooks, and the amount is 1 based on the number of storybooks. It is known that the total number of three types of books is 1000. If we add 20 books to literature books, there will be as many as story books. If you subtract 12 from science books, it will be twice as much as story books, and the total will become1000+20-12 =100. Line chart: solution: 1. Number of story books: (1000-12+20) ÷ (1+1+2) = 252 (this) 2. Number of scientific and technological books: 250. Or 1000-252-5 16=232 (Ben) answer: the school library bought 252 story books, 5 16 science books and 232 literature books. Summary: As can be seen from the above examples, the key points of solving the application problem of harmony and multiplication are: harmony (multiple+1)= decimal (smaller number, that is, multiple of 1) ×××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× If the second barrel of oil is injected into the first barrel of 40kg, then the oil in the first barrel is exactly three times that of the second barrel. How many kilograms were originally stored in the first and second barrels? There are 340 peach trees and pear trees in the orchard, among which the number of peach trees is 20 times more than that of pear trees. How many trees are there? Lingling's father's salary is twice that of her mother. Her father spent 360 yuan to buy a bicycle, which is exactly half of Lingling's parents' combined salary. What is Lingling's father's monthly salary? There are two piles of cement, the first pile has 87 bags and the second pile has 69 bags. How many bags can I take from the first pile to the second pile, so that the cement in the second pile is three times that of the first pile? Answer 1. Yuan Yuan: 84(2+ 1)= 28 (Ben) Fang Fang: 28×2=56 (Ben) 2. Original barrel b: 240 ÷ (3+1)+40 = ÷ (3+1) = 80 (peach tree): 340-80=260 (tree) 4. Mom: (360×2)÷(2+ 1)=240 (yuan) Dad: 240× 2. Let's first look at a simple topic of drawing-dividing squares. Divide the square into a "well" shape with four line segments, and you can divide the square into nine equal-sized blocks. This figure is usually called nine squares. A square can also be divided into 10 blocks with four line segments, but the size of each block is not necessarily equal. So, how to divide a square into 10 blocks with four line segments? Please think about it first, and draw while thinking. Hands and brains can participate in activities at the same time, which can make up for each other's shortcomings and find the answer faster. In fact, it is not difficult to divide a square into 10 blocks. There are two ways to divide it. Think about it, can a square be divided into 1 1 blocks with four line segments? How should I divide it? Please draw a picture. "One stroke" law [topic] Can you draw every figure without leaving the paper? Give it a try. To solve this problem correctly, we must find out the characteristics of a stroke diagram. As early as the18th century, the famous Swiss mathematician Euler discovered a law. Euler believes that a graph that can draw strokes must be a connected graph. Connected graph means that all parts of a graph are always connected by edges, and all three graphs in this problem are connected graphs. However, not all connected graphs can be drawn with one stroke. Whether a stroke can be made depends on the number of odd points and even points in the graph. What is odd or even? Points connected by odd (odd) edges are called singularities; A point connected with an even edge is called an even point. As shown in figure 1, ① and ④ are singular points, ② and ③ are even points. What is the law of mathematician Euler seeking strokes? 1. Any connected graph composed of even points can be drawn with one stroke. When drawing, you can start from any even point, and finally you can finish drawing with this point as the end point. For example, Figure 2 is an even number of points, and the lines drawn can be: ①→→→→→→→→ ② ②. Any connected graph with only two singularities (the rest are even points) can be drawn with one stroke. When drawing, one singularity must be the starting point and the other singularity must be the end point. For example, the line of the graph 1 is: ①→→→→→→→ ③→→→→ ①→ ④ 3. Other graphics cannot be drawn in one stroke. Please try: 1. Draw the diagram 1 and other lines in Figure 2. 2. Can you draw a stroke in Figure 3? How many lines are there? The picture below is the symbol of the International Olympic Games. Can you draw a picture? Please draw it if you can. Share to