Induction of mathematical knowledge points in the last semester of senior two.
I. Pythagorean Theorem
1, Pythagorean theorem
The sum of squares of two right angles A and B of a right triangle is equal to the square of hypotenuse C, that is, a2+b2=c2.
2. Inverse theorem of Pythagorean theorem
If all three sides of a triangle have this relationship, then the triangle is a right triangle.
3. The number of Pythagoras
The three positive integers satisfied are called Pythagoras numbers.
Common Pythagorean arrays are: (3,4,5); (5, 12, 13); (8, 15, 17); (7,24,25); (20,2 1,29); (9,40,4 1); ..... (multiples of these pythagorean arrays or pythagorean numbers).
Second, prove
1. Sentences that judge things are called propositions. That is, a proposition is a sentence that judges one thing.
2. Theorem of the sum of the internal angles of a triangle: the sum of the three internal angles of a triangle is equal to 180 degrees.
(1) The idea of proving the theorem of the sum of internal angles of a triangle is to put three angles in the original triangle together to form a right angle. Generally need assistance.
(2) The outer angle of a triangle and its adjacent inner angle are complementary angles.
3. The relationship between the outer angle of a triangle and its non-adjacent inner angle.
(1) The outer angle of a triangle is equal to the sum of two non-adjacent inner angles.
(2) The outer angle of a triangle is larger than any inner angle that is not adjacent to it.
4. The basic steps to prove the proposition is true.
(1) Draw a picture according to the meaning of the question.
(2) according to the conditions and conclusions, combined with graphics, write the known and verified.
(3) Through analysis, find out the proof method of known derivation and write the proof process. It should be noted that: ① Under normal circumstances, the analysis process does not need to be written. (2) Every step of reasoning in the proof should have a basis. If both lines are parallel to the third line, then the two lines are also parallel to each other.
Mathematics review materials in the second volume of the eighth grade
Zero exponential power and negative integer exponential power
Important point: The nature of power (exponents are all integers) will be used in calculation and scientific notation to represent some numbers with smaller absolute values.
Difficulties: Understanding and applying the properties of integer exponential power.
First, review the exercises:
1、; =; =,=,=。
2. Calculation without calculator: ⊙(-2)2-2- 1+
Second, the scope of the index has been extended to all integers.
1, explore
Now, we have introduced zero exponential power and negative integer power, and the range of exponent has been extended to all integers. Then, is the essence of power learned in the operation of power still valid? Discuss with your classmates and judge whether the following formula is true.
( 1); (2)(a？ b)-3 = a-3 B- 3； (3)(a-3)2=a(-3)×2
2. Summary: After the range of exponent is extended to all integers, the arithmetic of power is still valid.
3. Example 1 calculates (2mn2)-3(mn-2)-5, and converts the result into a form containing only positive integer exponential powers.
Solution: The original formula = 2-3m-3n-6× m-5n10 = m-8n4 =
Exercise: Calculate the following categories and convert the results into a form that only contains positive integer exponential powers:
( 1)(a-3)2(ab2)-3； (2)(2mn2)-2(m-2n- 1)-3。
Third, scientific notation.
1. Memories: In previous studies, we used scientific notation to represent some numbers with large absolute values, that is, using the positive integer power of 10, we expressed the numbers with absolute values greater than 10 as a× 10n, where n is a positive integer,1≤ For example, 864000 can be written as 8.64× 105.
2. Similarly, we can use the negative integer power of 10 to express some numbers with smaller absolute values by scientific notation, that is, in the form of a× 10-n, where n is a positive integer,1≤∣∣∣∣.
3. Explore:
10- 1=0. 1
10-2=
10-3=
10-4=
10-5=
Induction: 10-n=
For example, 0.00002 1 in the above example 2(2) can be expressed as 2. 1× 10-5.
4. Example 2. The diameter of nanoparticles is 35 nanometers. How many meters is it equal to? Please use scientific symbols.
The analysis shows that:1nm = m. From = 10-9, we can know that1nm =10-9 m.
So 35 nm =35× 10-9 meters.
And 35×10-9 = (3.5×10 )×10-9.
=35× 10 1+(-9)=3.5× 10-8,
So the diameter of this nanoparticle is 3.5× 10-8 meters.
Step 5 practice
(1) expressed in scientific notation:
( 1)0.00003; (2)-0.0000064; (3)0.00003 14; (4)20 13000.
② Fill in the blanks with scientific symbols:
(1) 1 sec is 1 microsecond 100000 times, then 1 microsecond = _ _ _ _ _ _ _ sec;
(2) 1mg = _ _ _ _ _ _ _ kg；
(3) 1 micron = _ _ _ _ _ _ meters; (4) 1 nanometer = _ _ _ _ _ _ micron;
(5) 1 cm2 = _ _ _ _ _ m2; (6) 1 ml = _ _ _ _ _ cubic meters.
Review methods of mathematics syllabus for junior two.
First, overcome psychological fatigue.
First, we should have a clear learning purpose. Learning is like pumping water from a river. The more power, the greater the water flow. Motivation comes from the purpose, and only by establishing the correct learning purpose can we have a strong learning motivation; Second, we should cultivate a strong interest in learning. The formation of interest is related to the excitement center of cerebral cortex, accompanied by pleasant, cheerful and positive emotional experience. Psychological fatigue is caused by the negative emotion of cerebral cortex resistance. Therefore, cultivating one's interest in learning is the key to overcome psychological fatigue. With interest, learning will have enthusiasm, consciousness and initiative, and psychology will be in a good competitive state; Third, we should pay attention to the diversification of learning. Book learning itself is boring and monotonous. If you study a course or a chapter repeatedly, it is easy to suppress the cerebral cortex, resulting in psychological saturation and boredom. Therefore, candidates may wish to review each course alternately.
Second, overcome the plateau phenomenon.
The plateau phenomenon in review refers to the phenomenon that when reviewing for a certain period, it often stagnates, not only making no progress, but also regressing. The platform period is not that there is no progress in learning, but that some progress and some retrogression are balanced with each other, which makes the review effect not fundamentally changed and makes people frustrated and disappointed. When candidates encounter plateau in the process of reviewing for the exam, don't be impatient or lose confidence, but find out the reasons for learning methods and enthusiasm. Adjust the review progress in time, make more efforts to use your brain scientifically and improve the review efficiency.
Third, pay attention to reviewing "mistakes"
If you are not good at coming out of mistakes in review, there will be more and more defects and loopholes. If left unchecked, the ant nest will eventually burst. During the preparation period, in order to reduce the error rate, in addition to timely revision and comprehensive and solid review, the key issue is to find out the reasons and constantly review the mistakes. That is, read the wrong questions regularly, recall the reasons for the mistakes, and sort out all kinds of wrong questions and reasons. For those problems that are repeatedly wrong, you can consider doing it again to avoid "future troubles." The reasons for the mistakes are: problems in conceptual understanding, problems caused by carelessness, and illusions caused by sloppy writing, so as to effectively avoid making similar mistakes in the exam.
Fourth, grasp the psychological characteristics and do a good job in reviewing before the exam.
Practice has proved that a person's temperament, personality, psychological stability and other factors will also affect the review before the exam. In the process of preparing for the exam, candidates should make an exam review plan according to their own psychological characteristics, adjust the review progress according to their own mentality, choose and use the review methods well, and make their exam review achieve the expected results.
1, textbooks can not be ignored.
For junior two students, they are all learning new lessons, and textbooks are important review materials that are easy to ignore. Usually, everyone takes notes in class at school, and basically doesn't read textbooks. Students are advised to read and understand the knowledge points repeatedly according to the textbook while reading notes, and to think, ponder and integrate exercises in after-school exercises to deepen their understanding of the knowledge points. We should also focus on memorizing the key contents and key examples in the textbook.
2. Wrong title
I believe that students with good study habits should have a wrong book, copy down the wrong questions in each exercise, homework and exam, make the answers clear, find out the reasons for the mistakes, find out the weak points in their knowledge and ability, and take them out and look at them often. When you encounter repeated mistakes, you should take the initiative to discuss with your classmates and ask the teacher to thoroughly understand the problem and avoid making similar mistakes again.
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