Keywords: "the ideal source" of mathematics test questions
Naturally, the most ideal and basic "ideal source" of the life system of college entrance examination mathematics questions is the theorems and formulas involved in the current high school mathematics textbooks, the variants and deduction of examples and exercises in the textbooks, the college entrance examination questions in previous years, the mathematics competitions in previous middle schools, and the excellent simulation questions in various places. This is an ocean of test questions. Many students can freely master the series, combination, networking and block diagram of knowledge, but how to control the ocean of test questions with the least amount of test questions is still at a loss. In this paper, the law of college entrance examination questions in recent years is explored, and the "ideal source" of mathematical questions that can activate mathematical knowledge and improve mathematical ability is sought.
First, as the concept of "source"
1. If positive pure decimal is "source"
Analysis: According to the parallelogram law of vector addition, OP is the diagonal of the parallelogram, and the quadrilateral should take the opposite extension lines of OA and OB as two adjacent sides, so the maximum value of Y is 0, at this time =x, so the value range of X is [0, 1].
2. Quadratic function y=ax +bx+c(a≠0) is undoubtedly the highlight of middle school mathematics and college entrance examination. Its definition, range, opening, symmetry axis and monotonicity are so familiar, but it makes our students feel strange and terrible. If we introduce absolute value, there will be y=ax +b|x|+c, then the quadratic function will be equivalently transformed into a piecewise function y = ax+bx+c (x > 0) ax-bx+c (x < 0), and the definition domain, range, opening, symmetry axis and monotonicity at this time will not be as simple as the quadratic function. Finding the minimum value of y=x +|x- 1|+2, and then introducing parameters to find the minimum value of y=x +|x-a|+2, is not so simple, and becomes a college entrance examination question.
(3) Almost every mathematical knowledge point can be triggered and transformed into a mathematical test. For example, y=log is monotonic in its domain. After introducing the absolute value, there will be y=log |x|, and there will be symmetry. If you continue to transform y=log |x- 1|, you will lose the properties of even function, and the symmetry axis will become x= 1, and then.
Logarithmic function is monotonous, linear function is monotonous, and every knowledge point is easy to master. What would it be like if we started with a simple form? For example, x ∈ [0, 1], find the maximum (minimum) value of y =log (ax- 1). This topic examines the domain ax- 1 > 0 of logarithm, the monotonicity law of compound function, classification and so on. Logarithmic function is an "ideal source" with a strong "reproduction rate", which can be combined with quadratic function, unary rational fraction function and y=x+ to get a good test question for examining mathematical knowledge and ability.
Third, "analogy" is the best test "source"
Case 1: Let the function f(x)=, and get the value of f(-4)+…+f(0)+…+f(5)+f(6) by using the method of deducing the sum formula of the first n items of arithmetic progression in the textbook.
Analysis: In this question, arithmetic progression's first n terms are added with the formula in reverse order, and the characteristics of each factor are observed, trying to calculate f(x)+f( 1-x).
Case 2: There is Pythagorean theorem in plane geometry: "Let AB and AC on both sides of △ABC be perpendicular to each other, then AB +AC =BC." Extending to space, by analogy with Pythagorean theorem of plane geometry, the relationship between the side area and the bottom area of a triangular pyramid can be studied, and a correct conclusion can be drawn: "Let ABC, ACD and ADB of the triangular pyramid A-BCD be perpendicular to each other, then? Shake it? Shake it? Shake it? Shake it? Shake it. "
Analysis: With regard to the analogy between spatial problems and plane problems, we can usually compare the following geometric features:
Note: "Please read the original text in PDF format for the charts, notes and formulas involved in this article."
This article is the full text of the original. Users who don't have a PDF browser should download and install the original text first.