This paper lists some defined test questions in the 2009 senior high school entrance examination paper, and makes a classification analysis to attract readers' attention.
I. Definition of "Concept"
The commentary defines the unary quadratic equation in the new operation, which is more innovative and refreshing than directly giving the solution of the unary quadratic equation.
Third, define "point"
Example 3 (Taizhou, 2009) defines that a point with the same distance from one set of opposite sides of a convex quadrilateral and the same distance from another set of opposite sides is called a quasi-interior point of a convex quadrilateral. As shown in figure 1, PH=PJ, PI=PG, then point p is the quasi-interior point of quadrilateral ABCD.
Figure 1
Figure 2
(1) As shown in Figure 2, the bisectors FP and EP of ∠AFD and ∠DEC intersect at point P, which proves that point P is the quasi-interior point of quadrilateral ABCD;
(2) Draw the quasi-interior points of parallelogram and trapezoid respectively;
(3) Judge the truth of the following propositions:
① Any convex quadrilateral must have a quasi-interior point.
② Any convex quadrilateral must have only one quasi-interior point.
③ If P is the quasi-interior point of ABCD of any convex quadrilateral, then PA+PB=PC+PD or PA+PC=PB+PD.
Analysis (1) as shown in Figure 2, the points P are PG⊥AB, PH⊥BC, PI⊥CD, PJ⊥AD,
PJ=PH, because EP is equal to ∠ dec.
Similarly PG=PI.
So point p is the quasi-interior point of quadrilateral ABCD.
Figure 3
(2) The intersection of AC and BD on the diagonal of parallelogram is a quasi-interior point, as shown in Figure 3( 1). Or take the intersection of the midpoint lines of two opposite sides of the parallelogram as the quasi-interior point, as shown in Figure 3 (2);
The intersection of the bisector of the angle between the two waists of the trapezoid and the midline of the trapezoid is the quasi-interior point, as shown in Figure 4.
Figure 4
(3)① True; 2 true; 3 fake.
Explaining this question requires students to understand the definition of "quasi-interior point" accurately, and can draw and prove it with the new definition to test students' computing ability and deductive reasoning ability. The questions are proof, drawing and filling in the blanks, and the knowledge points are angular bisector, central symmetry and trapezoidal midline, which is very comprehensive.
Fourth, define "line"
(1) Find the value of b;
(2) Find the analytical expression of parabola passing through this point (expressed by algebraic expression with d);
(3) Definition: If the triangle formed by the vertex of a parabola and the two intersections of the X axis is a right triangle, then this parabola is called a "beautiful parabola".
Inquiry: When the size of d (0 < d < 1) changes, is there a beautiful parabola in this group of parabolas? If it exists, please find the corresponding value of d.
(3) There is a beautiful parabola.
According to the symmetry of parabola, a right triangle must be an isosceles right triangle with the vertex of parabola as the right vertex.
The height on the hypotenuse of this isosceles right triangle is higher than half of the hypotenuse, 0 < d < 1,
The length of hypotenuse of isosceles right triangle is less than 2,
So the height on the hypotenuse of the isosceles right triangle must be less than 1,
That is, the ordinate of the parabola vertex must be less than 1.
Because when x= 1,
The commentary defines the unary quadratic equation in the new operation, which is more innovative than directly giving the solution of the unary quadratic equation.
This problem is the synthesis of linear function and quadratic function. It is difficult to judge that a right triangle must be an isosceles right triangle with the vertex of the parabola as the right vertex by using the symmetry of the parabola, and then use the midline property on the hypotenuse of the right triangle and the trinity property of the isosceles triangle.
Verb (abbreviation for verb) defines "form"
Example 5 (Yiwu, 2009) It is known that point A and point B are moving points on the X axis and Y axis respectively, and point C and point D are points on the function image. When the quadrilateral ABCD (points A, B, C and D are arranged in sequence) is a square, it is called the companion of this function image. For example, as shown in fig. 6, the square ABCD is one of the adjoint squares of the image and has a linear function y=x+ 1.
Figure 6
(1) If the function is a linear function y=x+ 1, find the side lengths of all partner squares of its image;
(2) If a function is an inverse proportional function, its partner is ABCD, and point D(2, m) (m < 2) is on the inverse proportional function image, find the value of m and the inverse proportional decomposition function;
(3) If the function is a quadratic function, its image partner is ABCD, and the coordinates of a point in C and D are (3,4). Write the coordinates of the other vertex of the partner on the parabola, and write one of the parabolic analytical expressions that meets the meaning of the question, and judge whether the number of partners in the parabola you write is odd or even.
So the companion number of parabola is even.
This topic comprehensively examines the three types of functions that students have learned in junior high school, and also examines important mathematical ideas such as classified discussion, combination of numbers and shapes, equation thinking and function thinking. The proponent is really original.
The newly defined questions are presented in different forms, and students' existing mathematical knowledge is examined from different angles. Based on various situations, it comprehensively examines students' application ability in new situations, including important mathematical ideas such as classified discussion thought, combination of numbers and shapes, equation thought and function thought. Based on the foundation, not rigidly adhering to the curriculum standards, it will bring new guidance to daily teaching and new vitality to future teaching.