In short, the idea of combining numbers and shapes is a mathematical idea of combining numbers and shapes to solve mathematical problems in mathematics. The combination of numbers and shapes is to combine abstract mathematical language with intuitive graphics, to combine abstract thinking with image thinking, and to solve mathematical problems through the correspondence and transformation of numbers and shapes. There are three main types of solving mathematics problems in middle schools: changing "number" into "shape", changing "shape" into "number" and combining numbers with shapes.
Let's talk about the application of the idea of combining numbers with shapes in some problems in mathematics.
1, "number" to "shape"
Because "number" and "shape" are a kind of corresponding relationship, some of which are abstract and difficult for us to grasp, and "shape" has the advantages of image and intuition, which can express more specific thinking and play a qualitative role in solving problems, so we can find out the corresponding relationship between "number" and "shape" and use graphics to solve problems. We can identify a familiar "pattern" that meets the goal of the problem from the situation of the given problem. This pattern is a specific relationship or structure between exponent and shape. This method of transforming quantity problem into graphic problem and finally solving quantity problem through graphic analysis and reasoning is graphic analysis method. Visualization of quantitative problems is the condition for transforming quantitative problems into graphic problems. There are generally three ways to transform quantitative problems into graphic problems: applying plane geometry knowledge, applying solid geometry knowledge and applying analytic geometry knowledge. To solve a mathematical problem, generally speaking, we first analyze the structure of the problem and decompose it into known (conditions) and required (goals), and then compare the conditions and goals to find out the internal relationship between them. Therefore, for the problem of "number" changing into "shape", the basic idea to solve the problem is to clarify the conditions given in the problem and the goals sought. Starting from the known conditions or conclusions in the problem, first observe and analyze whether they are similar (the same) to the basic formula (theorem) or graphic expression learned, then make or construct a suitable graphic, and finally make use of the properties and geometric significance of the graphic that has been made or constructed.
Example 1: It is known that the three sides of a triangle are 5, 12 and 13 respectively. Find the area of this triangle.
Analysis: This question only gives the lengths of three sides of a triangle as 5, 12 and 13 respectively, but does not give the height of one side, so it seems impossible to find its area. Although I know that there is a Helen formula for calculating the area of three sides of a triangle, it is too troublesome. Here, if we can analyze this set of data and find out the relationship between 5, 12 and 13, it is easy to think of the inverse theorem of Pythagorean theorem-if a triangle with three sides A, B and C satisfies a2+B2 = C2；; This triangle is a right triangle. Because 52+ 122= 132, we can judge that a triangle with three sides of 5, 12 and 13 respectively is a right angle with 5, 12 as the right side and 13 as the hypotenuse. In this way, we turn this set of data 5, 12, 13 into a right triangle with 5, 12 as the right side and 13 as the hypotenuse through the inverse theorem of Pythagorean theorem. The transformation from "number" to "shape" is realized, and the triangle with three sides of 5, 12 and 13 is changed into a right triangle. Then the area of this triangle can be easily obtained. This is a typical combination of numbers and shapes using the inverse theorem of Pythagorean theorem.
2. Change "shape" into "number"
Although the shape has the advantages of visualization and intuition, it must be quantified by algebraic calculation. Especially for more complex shapes, we should not only correctly digitize, but also pay attention to the characteristics of the shape, explore the hidden conditions in the topic, make full use of the nature or geometric meaning of the shape, and correctly express the shape as a number for analysis and calculation.
The basic idea of solving problems: make clear the given conditions and expected goals, analyze the characteristics and properties of the given conditions and expected goals, understand the important geometric significance of the conditions or goals in the figures, correctly express the figures used in the stem of the questions with the knowledge learned, and then apply corresponding formulas or theorems according to the relationship between the conditions and conclusions.
Example 3: Enclose a rectangular area with a fence of a certain length. Xiao Ming thinks that the area around a square is the largest, but Xiao Liang doesn't think so. what do you think? (Selected from the third question of P30 exercise in the first volume of eighth grade mathematics of East China Normal University)
Analysis: The key to this question is "how to compare the size of square area and rectangular area when the perimeter is fixed", that is, how to represent square area and rectangular area with numbers and how to compare the size of square area and rectangular area. Let's assume that the length of the fence is L=4a, then the side length of the square is a, and according to the fact that the opposite sides of the rectangle are equal, one set of opposite sides is a-x, and the other set of opposite sides is a+x (x > 0), as shown below.
a a+x
a-x
Square rectangle
S square = A2, S rectangle = (A+X) (A-X) = A2-X2. Because x > 0, x2 > 0. Therefore, A2 > A2-X2 means s square > s rectangle. This is a typical practical application problem of constructing numbers from shapes.
3, "shape" and "number" change each other.
The mutual transformation of shape and number means that in some mathematical problems, it is not just a simple transformation from number or variable, but a mutual transformation of shape and number. We should not only think of changing the intuition of shape into the strictness of number, but also change the close connection between number and intuition of shape. To solve this kind of problems, it is often necessary to start from the known and the conclusion at the same time, carefully analyze and find out the mutual changes of the internal "shape" and "number". The general method is to look at "shape" and think about "number", and look at "number" and think about "shape". The essence is the combination of "number" and "shape" and "shape" and "number".
Example 5: there is a quadrilateral ABCD (as shown in the figure); ∠ABC = 90°； AB = 4mBC = 3mCD = 12m； DA = 13m。
Find the area of quadrilateral ABCD. (Selected from Question 7 of P63B Group of Grade 8 Mathematics in East China Normal University Edition)
Analysis: the result of this problem is to find the area of quadrilateral ABCD, if four sides C B.
ABCD is a special quadrilateral-right trapezoid, then we
The formula s = (upper bottom+lower bottom) /2 can be used. if
∠ bad = 90 You can use this formula and press the hook.
The inverse theorem of strand theorem requires BD2 = da2+AB2A.
But we can't find the length of BD,
Therefore, we can't find the degree of ∠ bad.
Starting with what is known ∠ ABC = 90, d
Ab = 4m，BC = 3m。 According to Pythagorean theorem, we can get AC = √ ab2+bc2 = √ 42+32 = 5m. In the triangle ACD, from AC = 5m, CD = 12m and DA = 13m, we get 52+ 122 = 62. In this way, we can turn the problem of finding the area of quadrilateral ABCD into the problem of finding the sum of the areas of two right triangles ABC and right triangle ACD. We solved the problem easily through its meaning.
Through the analysis of the results and known results, we first get the hypotenuse length AC by using the Pythagorean theorem through the right triangle ABC, that is, looking at the shape and thinking about the number; Then, according to AC = 5m, combined with the known CD = 12m and DA = 13m, we think that 52+ 122 = 132, that is, according to the inverse theorem of Pythagorean theorem, the AC2+Cd2 = da2 triangle is a right triangle, which belongs to the "number" thinking. Finally, the area of quadrilateral ABCD is converted into the sum of the areas of two right triangles ABC and right triangle ACD, and the problem is solved.
The combination of numbers and shapes is a common mathematical thinking method, which can simplify complex problems and concretize abstract problems. In order to improve students' ability to use the combination of numbers and shapes, teachers need to patiently and meticulously guide students to learn to contact, understand, use and master the combination of numbers and shapes.