Therefore, many people's mathematics learning is always miserable, mastering mathematics knowledge, but losing points because of carelessness; Some students are impatient when solving problems, and they are not careful enough, so they often "miss" the conditions, which leads to unsatisfactory problem solving and so on.
From here, we can see that if a person wants to improve his math score, he needs to improve his comprehensive math ability in all aspects, be strict with himself, not ignore every problem, and persevere, so as to finally improve his math score.
Mathematics learning and improving the comprehensive ability of mathematics can not be completed in a day or two, but through repeated learning (that is, review) to gradually improve their mathematical ability. Some students always regard review as too simple or even insignificant. Review is nothing more than rereading the knowledge in the book, reciting the relevant concepts, theorems and formulas in the textbook, and reviewing and studying in such a state of mind, and even thinking that doing problems is review. How to be effective?
Review refers to re-learning, remembering what you have forgotten before, repeating what you have learned, making yourself more impressed and staying in your mind for a longer time. To put it bluntly, everyone needs to review their math study. By reviewing this circular learning model, we can become more familiar with knowledge and improve our ability to use knowledge.
So, is your review effective? Is your review useful? It depends on whether it can help you improve your knowledge, improve your comprehensive ability in mathematics, improve your grades and so on.
On the other hand, if you think review is useless, it can only show that there are still some misunderstandings in your understanding of review, such as not really recognizing the characteristics of mathematics and not distinguishing it from other disciplines in review methods. Mathematics itself is a highly applied subject. Learning mathematics is learning to solve problems, but you just want to improve your academic performance by asking questions and tactics. This way of learning mathematics is also wrong, because you ignore the accumulation of basic knowledge, methods and skills and the refinement of mathematical ideas.
Typical case analysis 1:
AB and CD are two chords of ⊙ O. The straight lines AB and CD are perpendicular to each other, and the vertical foot is point E, which connects AD. The point B is BF⊥AD, and the vertical foot is point F. The straight line BF and the straight line CD intersect at point G. 。
(1) As shown in figure 1, when point E is outside ⊙O, connect BC, and prove that BE is equal to ∠ GBC;
(2) As shown in Figure 2, when the point E is within ⊙O, connect AC and AG, and verify that AC = Ag.
(3) As shown in Figure 3, under the condition of (2), connect BO and extend the intersection point AD to point H. If BH bisects ∠ABF, AG=4, and tan∠D=4/3, find the length of line segment AH.
Test center analysis:
A comprehensive problem of a circle.
Stem analysis:
(1) We can get ∠D=∠EBC by using the properties of quadrilateral inscribed in a circle, and then get ∠GBE=∠EBC by using the complementary relation, and then we can get it.
(2) Firstly, get ∠D=∠ABG, then get △ BCE △ BGE (ASA) by using the judgment and properties of congruent triangles, and then get it by using the properties of isosceles triangle;
(3) Find the length of CO first, then tan∠ABH=NH/NB=3a/6a= 1/2, and then find the value of A with OP2+PB2=OB2.
Thinking about solving problems:
This topic mainly examines the composition of circles, the relationship between Pythagorean theorem and acute trigonometric function, congruent triangles's judgment and knowledge of nature, and draws the conclusion that tan∠ABH=OP/PB= 1/2 is the key to solving problems.
What will you do next after taking such a comprehensive math problem for the senior high school entrance examination? For the answer, some students can't wait to enter the next one when they see that the answer is correct; If the answer is wrong, look at the correct answer flow immediately, and then simply find out why you did it, and can't wait to enter the training of the next topic. The above two problem-solving habits are the biggest learning behaviors of many students. There seems to be no problem, but the potential problem is huge. Without reflecting and summing up the problem, can you promise to do it right next time? If you make a mistake, can you promise not to make a mistake next time?
When solving math problems, I am most afraid of doing one problem and throwing another. I'm tired of doing it every day, and I don't know anything after I finish it. I don't even know if some of my classmates' questions will help me. How can such math learning be effective?
Mathematics learning is not to solve problems but to solve problems, but to test our learning effect by solving problems and find out the shortcomings in learning, so as to improve and improve. So the summary after solving the problem is very important, which is a great opportunity for us to learn. For example, for a solved mathematical problem, we can reflect on the problems existing in knowledge, which concepts, theorems, formulas and other basic knowledge are not well mastered, and how to apply these knowledge in the process of solving problems; What problem-solving methods and skills are used, and whether they can be mastered and used skillfully; Can you sum up the problem-solving process into several steps? Can you summarize the questions and master the general problem-solving methods of such questions?
When solving mathematical problems, we should carefully choose topics, so as to be few and precise. Only by solving those high-quality and representative problems can we achieve the effect of improving mathematics scores. But the reality is that most students do not have the ability to distinguish and analyze the quality of questions, and they will only solve problems blindly. This requires everyone to ask the teacher for help and choose exercises suitable for their own learning situation. Only what suits you is the best.
Typical example analysis 2:
As shown in figure 1, in the square ABCD and the square CGEF, three points B, C and G are on the same straight line, m is the midpoint of the line segment AE, and the extension line of DM intersects with EF at point N, connecting FM, which is easy to prove: DM=FM, DM⊥FM (without writing the proof process).
(1) As shown in Figure 2, when B, C and F are on the same straight line, the extension line of DM intersects with N, and other conditions remain unchanged. What is the relationship between DM and FM? Please write a guess and prove it;
(2) As shown in Figure 3, when points E, B and C are on the same straight line, the extension line of DM and CE intersect at point N, and other conditions remain unchanged, what is the relationship between DM and FM? Please write a guess directly.
Test center analysis:
Quadrilateral synthesis problem.
Stem analysis:
(1) connects △DFN and NF, and quadrilateral ABCD and CGEF are squares, thus obtaining AD∨BC, BC∨GE, thus obtaining AD∨GE, obtaining ∠DAM=∠NEM, proving △ MAD △ Men, and obtaining DM.
(2) Connect △DFN, NF, get AD∨BC from the square of quadrilateral ABCD, get AD∨CN from points E, B and C on the same line, get ∠DAM=∠NEM, prove △ MAD △ MEN, and get DM=MN.
Thinking about solving problems:
This question examines congruent triangles's judgment, the nature of a square and the judgment and nature of an isosceles right triangle. The difficulty of this problem is the method of auxiliary line, and the key to solve this problem is to make auxiliary line and find the right direction.
Some people can't solve math problems correctly, not because they don't have a good grasp of relevant math knowledge, but because they lack the spirit of taking exams seriously. Before we solve any math problem, we must first examine the problem, make clear the meaning of the problem and analyze the problem. Especially for math problems, careful examination is particularly important.
Solving mathematical problems is actually to build a bridge between known conditions and conclusions to be solved, that is, to reduce and eliminate these differences on the basis of analyzing the differences between known conditions and conclusions to be solved. If you don't analyze all the conditions in the topic clearly and sort out the order, you may not be able to connect the conditions of the topic with the conclusions you want to seek.
When we straighten out the subject conditions, it will test a person's proficiency, understanding and flexible application ability of mathematical methods. For example, to solve the problem of geometric synthesis of functions, we should not only master the basic concepts and properties of functions, images and so on. And I know all kinds of geometric figures like the back of my hand.