Grade 9 mathematics gives an example of three topics - function and geometry synthesis, the most comprehensive type of problem in junior high school, can be said to be the peak of difficulty, what kind of thinking spark can be collided between numbers and shapes?
1. Functions and triangles
The synthesis of functions and triangles uses images of primary, inverse proportional and quadratic functions as the carrier to explore the properties of isosceles triangles, right triangles, full equal triangles, and similar triangles. When solving such problems, mathematical ideas and methods such as the combination of numbers and shapes and classification discussions are often used.
Second, functions and quadrilaterals
The quads involved in the function and quad problem generally include parallelograms, rectangles, diamonds and squares, of which the comprehensive problem of parallelograms is the mainstay. In solving such problems, it is necessary to comprehensively apply the mathematical ideas of the pending coefficient method, the combination of numbers and shapes, and the classification discussion. For inquiry questions, it is generally necessary to make positive assumptions first, and then proceed from this, combined with known conditions for calculation and reasoning arguments: if a contradictory conclusion is drawn, the previous hypothesis can be rejected; if a reasonable conclusion is drawn, the assumption is affirmed to be correct, and the conclusion of the problem is finally drawn.
3. Functions and circles
The questions that put the circle in the Cartesian coordinate system and combine it with the image of the function have a strong openness and inquiry Many of the questions are novel in conception, strong in synthesis, have a certain degree of difficulty, and are often used as the finale questions in the examination. Solving such test questions requires not only solid mathematical knowledge, but also strong understanding of problems and problem-solving skills.