1, investigate bank deposit interest and profits and taxes.

2. Mathematical application in meteorology.

3. How to cultivate problem-solving wisdom

4. The discovery of polyhedral euler theorem.

5. Purchase loan decision.

6. Budget for painting the house

7. Paradoxes in daily life

8. Exploration on the application of mathematical knowledge in physics.

9. Analysis and comparison of investment life insurance and investment banks

10, the wide application of golden number

1 1, optimization algorithm in programming

Application of Cosine Theorem in Daily Life

13, securities investment mathematics

14, environmental planning and mathematics

15. How to calculate the difficulty and discrimination of a test paper?

16, History of Mathematics Development

17, talking about "providing for the aged"

Mathematical problems in China's sports lottery.

19, "open questions" and its thinking countermeasures

20. Thinking methods to solve application problems

2 1, learning activities of high school mathematics-problem solving analysis a) from trial to rigor, b) from one to another.

22, high school mathematics learning activities-reflection after solving problems-developing problem-solving wisdom.

23. Mathematical problems in China computer welfare lottery.

24. Living conditions of middle school students in cities and towns

25, urban and rural diet structure and optimization design

26. How to place military reconnaissance satellites?

27. Score the relationship (friendship) between people.

28. Measuring successful buildings

29, looking for people's emotional changes.

30. What is the most cost-effective deposit?

3 1, which supermarket is the cheapest?

32, the golden section in mathematics

33, communication network tariff survey statistics

34, the optimization problem in mathematics

35. How to calculate the inflow of the reservoir?

36, the influence of calculator on computing power

37, the cultivation of mathematical inspiration

38. How to improve the efficiency of mathematics classroom?

39, quadratic function image feature application

40, statistical monthly precipitation

4 1. How to collect taxes reasonably

42. Composition of urban vehicles

43. Reasonable pricing of taxi fare

44. How much does the price, texture and brand of clothes affect consumers' thinking?

45, the purchase loan decision-making issues

Problems and topics of research-based learning (from Ye Tingbiao's Mathematics Vanilla Garden)

Set up several parts

Question 1

Usually, the points * * * and lines * * * that appear in the competition are often difficult to level. 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

If we look at mathematical problems from the viewpoint of movement and change, we will find the essence of the problems and the relationship between them, but this aspect is not enough in the process of establishing several books, and we can conduct a comprehensive study by sorting out and collecting this information.

Question 3: As an example of dimension reduction, we can consider several transformations of straight line distance in different planes, such as the distance from straight line to plane, the distance from point to point, and the distance from plane to plane.

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.

Question 5

Many problems in establishing several projective geometries 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

It is difficult to find the plane angle of dihedral angle, and the 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 non-equal product transformation is its general situation, which 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.

Solve several parts

Question 9

For mathematical formulas, we should do three things: use, change and reverse. 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) to 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.

Question 16

Solving the ellipse problem is not as easy as a circle. Can we simplify the problem first, that is, the rounding of ellipse, and then study the rounding of conic curve (including its degradation 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 the conic 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.

Functional part

Question 23 An empty set 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 them "dressing function", 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 the equation (inequality) with parameters, if the solution is known to determine the range of parameters, we usually use the function idea and the combination of numbers and shapes to separate parameters, try to summarize the types of questions and summarize the 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.

Triangular part

Question 34 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. Try to explore the role of the combination of 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) Still preconceived.

Thus it is transformed into the slope of the connecting line between the moving point (cosx.sinx) and the fixed point. And consider whether the background connection of various construction methods 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 triangle, there are always two transformations when there are mixed edges and corners, that is, using sine and cosine theorems to transform into angle relationship or edge relationship, and discuss the enlightenment of one of them to the other solution.

Inequality part

Question 4 1

If a mathematical proposition starts from the front, there are many classifications and a large amount of calculation, and even it can't be solved. At this time, it may be better to consider its opposite side to solve 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: Explore absolute inequalities and physical simulation methods.

If you have any related topics, please ask your colleagues to raise them.

References:

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