Research-oriented learning project of mathematics 1, investigation of interest and profits and taxes on bank deposits 2, mathematical application problems in meteorology 3, how to develop problem-solving wisdom 4, discovery of polyhedral euler theorem 5, decision-making problem of house purchase loan 6, budget of house painting 7, paradox problem in daily life 8, application exploration of mathematical knowledge in physics 9, analysis and comparison of investment life insurance and investment banks 1 1 Extensive application of golden number, optimization algorithm in programming 12, application of cosine theorem in daily life 13, mathematics in securities investment 14, environmental planning and mathematics 15, how to calculate the difficulty and discriminant of a test paper 16, history of mathematics development. Kloc-0/8, Mathematical Problems in China's Sports Lottery 19, "Open Questions" and Their Thinking Countermeasures; Mathematics Learning Activities in Senior High School-Reflection after Solving Problems-Developing Problem-solving Wisdom 23. Mathematical problems in computer welfare lottery in China. Living conditions of middle school students in cities and towns. Composition and optimal design of urban and rural diet. How to place the military reconnaissance satellite 27? Scoring the relationship between people (friendship) 28, measuring successful buildings 29, looking for the law of people's emotional changes 30, how to save the money most cost-effectively 3 1, which supermarket is the cheapest 32, golden section in mathematics 33, investigation and statistics of communication network charges 34, optimization problems in mathematics 35, how to calculate the inflow of reservoirs 36, the influence of calculators on computing power 37, and the cultivation of mathematical inspiration. How to improve the efficiency of mathematics classroom? The application of quadratic function image features 40 pages. Monthly precipitation statistics 4 1. How to collect taxes reasonably? Composition of urban vehicles. Reasonable pricing of taxis. How much influence does the price, texture and brand of clothes have on consumers' ideas? 45. Problems and topics of research-based learning on the decision-making of house purchase loan (extracted from Ye Tingbiao's "Mathematical Herbs Garden") 1 "Setting up several parts" are often difficult to distinguish lines and lines, which generally appear in competitions. However, this kind of problem in several examples is not simple, mainly based on the basic properties of planes: the common point of two planes * * * line. Can this issue of equality be escalated? That is, turn it into several questions to answer. Question 2 Looking at mathematical problems from the point of view of movement and change will reveal the essence of the problems and their relationship, but a few years is not enough, so we can make a comprehensive study by sorting out and collecting materials in this field. As an example of dimension reduction, problem 3 can consider several transformations of straight line distance in different planes, such as line-surface distance, point-line distance, surface-surface distance and so on. Question 4: The distance of a straight line on a different plane is: the shortest line segment length between two moving points on a straight line on a different plane. So it can be solved from the perspective of function. That is, the distance function between two moving points is established, and the goal is achieved by finding the minimum value of the function. Many problems in Question 5 can be reduced to determining the projective position of a point on a plane. Such as point-to-surface distance, point-to-point distance, volume, etc. So it is very important to determine the projection of a point on the plane. Try to give a general method to determine it. Question 6: The plane angle of dihedral angle is the difficulty of setting several tables. Commonly used methods are: definition method, three perpendicular lines method and vertical plane method. Its essence is to locate a point, that is, when a point is on the edge of a dihedral angle, it is defined, when a point is on a half plane, it is vertical, and when a point is in space, it is vertical. The problem seems to have been solved. However, for more complex graphics, it is difficult to decide which point to use as an anchor because of the large number of points. Try to give the method and steps of making dihedral plane angle with straight line positioning. Question 7: Equal product transformation plays an important role in the establishment of several books, but unequal product transformation is its general situation and plays a greater role, but it is ignored by people. Using unequal product transformation can solve the problems of finding volume, distance and proving position relationship. Try to explore with the corresponding analogy method. Question 8 generalizes and extends the triple vertical theorem, that is, the sine and cosine theorem of the so-called trihedral angle and its special cases, and the sine and cosine theorem of the straight trihedral angle. To broaden your horizons. "Solving Several Problems" Question 9 For mathematical formulas, we should do three things: positive use, variable use and reverse use. If there are many formulas in the solution, such as the distance between two points, the distance from point to straight line, proportional point, slope formula and so on. We can get the proof of the construction method by considering its inverse use, and try to study the inverse use of various formulas in the solution to enrich the proof of the construction method. Question 10 We often use our aesthetic consciousness to look at any problem (including solving math problems) and adjust our action plan. In the solution of several problems, the theme of inspiring thinking with beauty is explored and collected, and it is sorted out and comprehensively studied. The problem 1 1 has some materials and special cases that are often ignored, which makes the solution of the problem incomplete, such as ignoring the existence of slope by point inclination method, ignoring the intercept of zero by intercept method and so on. Problem 12 realizes the evolution of propositions by the mutual transformation of angle parameters and distance parameters, so as to achieve the goal of taking points as surfaces and bypassing analogy. 13 summarizes the problems and solutions related to the midpoint, so that it can be applied to the corresponding bifurcation problems and methods. Problem 14 the relationship between coordinate transfer method and parameter method in solving trajectory problem is studied. In the simple solution of the problem 15 about the symmetry of a special straight line with a slope of 1, the solution strategy with wider application scope is summarized. Problem 16 Solving the elliptic problem is not as easy as a circle. Can we reduce the problem to the rounding of ellipse, and then study the rounding of conic curve (including its degeneration such as two intersecting straight lines and parallel lines, etc.) )? Problem 17 sorts out the problems related to focal radius and makes them "pure algebra", then studies its "pure algebraic solution" and explores new methods. The problem 18 extends the point difference method to solve the problem of strings, so that it can solve the problem of "strings have fixed points" In the problem of finding the trajectory of the problem 19, a simple judgment of purity. Question 20: There is a "projective thought" in the derivation of the formulas of the fixed point, the chord length and the distance from the point to the straight line, which expands the position or function of this thought in solving several problems. Question 2 1 summarizes the problem-solving function of translation transformation. Question 22 The problem of determining the parameter range in a conic curve related to the midpoint chord is often solved by establishing inequalities. In various methods, the condition that the point is in the curve is taken as the standard. Try to extend this method to the case of constant score chords. The empty set of function part problem 23 is a subset of all sets, but this fact is often ignored when solving the closed set problem. Try to sort out all kinds of problems in this regard. Question 24: Sort out the rules and types of domain (especially the types of compound functions). Question 25: When solving the range, monotone interval and minimum positive period of a function, it is often hoped that the independent variables will appear in one place, so the principle of variable concentration provides the direction for solving the problem. Try to learn all types related to the principle of variable concentration (such as collocation, division with remainder, etc.). ). Question 26 summarizes the related methods of finding the function value domain, and discusses the general situation of discriminant method-the condition of using real root distribution in the evaluation domain. Question 27: Use the geometric background of conditional maximum to evolve and classify propositions. Question 28: Looking back at solving exponential and logarithmic equations (inequalities), we call it "dressing functions", so that we can change the equations (inequalities) at will. Can you use this to make up some good questions? Question 29: Explore all "inverse function is itself" functions. Therefore, a class of equations with abstract functions can be solved and all types of such equations can be summarized. Question 30 odd function is defined at the origin, and its implicit condition is f(0)=0. Try to write and develop a proposition according to this fact. Question 3 1 Put two mirrors opposite each other. If you are in it, you will see that many portrait positions are periodic. Can you make this fact mathematical? What happens when the axial symmetry is changed to central symmetry? Question 32 For an equation (inequality) with parameters, if the solution is known to determine the range of parameters, we usually use the idea of function and the idea of combining numbers and shapes to separate parameters, trying to summarize the types of questions and methods of separating parameters. Question 33: Change the principal component of an equation (inequality) with parameters and the position of parameters to evolve the proposition. Explore the role of changing principal components. The combination of numbers and shapes is one of the important thinking methods in mathematics, but the trigonometric function line in the unit circle has been forgotten. This paper attempts to explore its function of combining numbers and shapes in solving triangular problems. Question 35 summarizes the value range of x when sinx+cosx=a, and the implied conclusion when this condition involves the problem condition. Question 36 combs the types of triangle substitution and what kind of problems it can solve. Question 37. In the method of constructing the maximum value of triangle, the type can be transformed into: 1) the slope of the connecting line between the moving point (ccosx.asinx) and the fixed point (-d, -b); 2) Or it can be transformed into the slope of the connecting line between the moving point (cosx.sinx) and the fixed point. Consider the relationship between the backgrounds of various construction methods and whether this relationship can be used to solve geometric problems. Question 38: A trigonometric formula can not only be used, but also reversed and changed. Try to sort out the latter. Question 39 summarizes the common methods of proving trigonometric identity with first chord, higher chord and tangent. Question 40: When judging the shape of a triangle, there are always two transformations for the conditions with mixed edges and corners, that is, the sine and cosine theorems are used to transform into angle relations or edge relations, and the enlightenment of one of them to the other solution is explored. Problem 4/kloc-0 in inequality/If a mathematical proposition is classified from the front, it needs a lot of calculation and even can't be solved. At this time, it may be better to consider its negative side to get the solution set, and then take its complement set to get the solution of the original proposition. We call it "complementary set method" and try to sort out the common types of complementary set method. Question 42 summarizes the skills of "rounding" when using mean inequality to find the maximum value, as well as the skills of splitting and adding terms. Question 43: Observing the structural features of the formula, such as analyzing the exponents and coefficients in the formula, reveals the direction of the problem. Question 44: Explore this famous inequality (such as Cauchy inequality, rank inequality, etc. ) and various proofs, and find its background to deepen the understanding of inequality. Question 45 sorted out the commonly used substitutions (triangle substitution, mean substitution, etc.). ) and explore its role in proposition transformation. Question 46 considers the change of mean inequality and the background significance of the inequality after the change. Question 47: The rotationally symmetric inequality with polynomial denominator is often difficult to prove because it is difficult to participate in the total score. Explore a substitution, the denominator is that polynomials become monomials. Question 48: Exploring Absolute Inequality and Physical Simulation If you have any related topics, please ask your colleagues to raise them.

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