Summary of important knowledge points of mathematics in the first volume of the eighth grade
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, A? +b？ =c？ .
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.
Third, data analysis.
1, average
(1) In general, for n numbers x? x？ ... x n, let's put (x? +x？ +? +x n) is called the arithmetic average of these n numbers, and the average for short is recorded as.
② In practical problems, the "importance" of each data in a set of data may be different, so when calculating the average of this set of data, each data is often given a weight, which is called weighted average.
2. Median and mode
① Median: generally, n data are arranged in order of size, and the data in the middle position (or the average of the two data in the middle) is called the median of this group of data.
② The data with the highest frequency in a group of data is called the pattern of this group of data.
③ Average, median and mode are all statistics that describe the trend in data set.
(4) When calculating the average, all the data participate in the operation, which can make full use of the information provided by the data, so it is commonly used in real life, but it is easily influenced by extreme values.
⑤ Median has the advantage of simple calculation and little influence by extreme value, but it can't make full use of all data information.
⑥ When the number of repetitions of each data is roughly equal, the pattern often has no special meaning.
3. Analyze the concentration trend of data from the statistical chart.
4. Degree of data dispersion
① In real life, people not only pay attention to the concentration trend of data, but also pay attention to the degree of dispersion of data, that is, the degree of deviation from the concentration trend. The difference between the data in a set of data and the minimum data (called range) is a statistic that describes the degree of data dispersion.
② Mathematically, the dispersion degree of data can also be described by variance or standard deviation.
③ Variance is the average of the square of the difference between each data and the average.
(4) where x 1, the mean value of x ... xn, s2 is the variance, and the standard deviation is the arithmetic square root of the variance.
⑤ Generally speaking, the smaller the range, variance or standard deviation of a set of data, the more stable it is.
Expanding reading: remedial measures for junior high school mathematics
The promotion method of weakness in algebra;
Starting with the rational number operation in the first stage, if the previous one falls too much. You can selectively do real number operations. Basic computing power cannot be left behind.
Multiplication formula and power operation in the second stage of junior high school. Use the existing algebraic formula, and then simplify it according to the topic, mainly by means of calculation model and certain topic training.
Methods of Improving Weaknesses in Application Problems
The biggest criticism of students' application problems is that they can't understand them. Mathematics comes from life, and their reading comprehension ability is strong, just like playing word games. Some students always ignore some conditions of the topic and are eager to write, so the error rate is particularly good. Correct examination of questions stems from sensitivity to numbers.
Training methods of mathematical thinking;
(1) Conduct targeted special training to avoid blind crowd tactics.
(2) Establish a mathematical knowledge network to achieve mastery through a comprehensive study and exchange needed goods.
(3) Diligence can make up for it, properly consolidate old knowledge and review and expand. Can effectively exercise mathematical thinking.