tree planting problem is a typical "average score" problem, which helps to understand the meaning of division . It is derived from in-class knowledge and is higher than in-class knowledge. It contains rich ways of thinking and mathematical thinking methods. For in-class thinking training is a good supplement. In the process of researching the tree planting problem, through the analysis of actual situational problems, the essence of the tree planting problem was summarized to improve students' abstract thinking ability. In the
tree planting problem, we can abstract trees into points and abstract the distance between trees into line segments. By observing the real scene, we can find that the distance between trees is equal.
1. Knowledge exploration
For a line segment, which represents the "distance" between two points, there are two endpoints.
We can observe: 1 line segments, 2 endpoints
The number of endpoints m is 1 more than the number of line segments n, that is, m=n+1.
Based on the above line segments, starting from the end, continue to draw line segments of equal length.
We observe: 2 line segments, 3 endpoints
The number of endpoints m is 1 more than the number of line segments n, that is, m=n+1.
Repeat the above operation,
We observe: 3 line segments, 4 endpoints
The number of endpoints m is 1 more than the number of line segments n, that is, m=n+1.
Through the above experiment, can increase the number of line segments and endpoints infinitely, but the quantitative relationship between the number of endpoints m and the number of line segments n remains unchanged, that is, m=n+1.
2. Capacity improvement
"How many trees" endpoint number m, "distance between two trees" line segment length b
When it is known that the total length of a tree to be planted a and the distance between two trees b, the line segment can be calculated Count (a÷b=number of line segments n). According to the quantitative relationship summarized above, you can easily calculate the number of planted trees m, that is, a÷b+1=m.
In the same way, according to the number of planted trees m, line segments If the length is b, you can also calculate the distance between two trees n, and the total length of the tree planted is a. The meaning of multiplication and division is learned, and students' reverse thinking and abstract thinking are cultivated.
To summarize, the tree planting issue involves
total length a: the full length of the tree planting route.
distance b: the distance between two numbers.
Number of segments n: How many trees are planted in the total length
Number of trees m: The total number of trees planted
Number of segments n = Total length a÷ Tree spacing b
trees m = Total length a÷ Tree spacing b+1
Total length a = Tree distance b × (tree m-1)
Tree distance b = Total length a÷ (number of trees m-1)
More quantitative relationships are not repeated, please summarize by yourself.
3. Application innovation
1. According to the tree planting situation at the endpoints, it can be divided into the following three categories
(1) Trees should be planted at both ends of the road
(2) There are no trees planted at both ends of the road
(3) Trees should be planted at one end of the road. No trees are planted at the other end
2. It is divided into unilateral tree planting and bilateral tree planting
3. According to the shape of the road, it can be divided into linear type and circular type
. In addition, in real life, the tree planting problem itself changes a lot, and in different situations Under the situation, more tree planting problems are derived, such as sawing wood, climbing stairs, planting flags, etc. The key to is to grasp the core "average score" of the tree planting problem, clearly distinguish the corresponding meanings and quantitative relationships of "total distance", "interval length", and "number of trees", so as to remain unchanged in response to changes.
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