The seventh grade mathematical inequality of Beijing Normal University and its basic nature teaching plan.
1. Understand and master the concepts and properties of inequalities; (key)
2. Inequalities can be used to express the quantitative relationship of simple problems. (Key points and difficulties)
First, situational introduction
One day, a group of monkeys went to pick peaches together. When you divide peaches, if you divide each monkey into three parts, there are 59 left. If each monkey gets five, then the last monkey gets less than five peaches. Do you know how many monkeys and peaches there are?
Second, cooperative exploration.
Detection point 1: inequality
The concept of inequality of the first kind
In the following categories: ①-3
A.5 B.4 C.3 D. 1
Analysis: ③ is an equation, ④ is an algebraic expression, and there is no inequality, so it is not an inequality. There are 1256 inequalities and ***4 inequalities, so choose B.
Method summary: This question examines the judgment of inequality. Inequalities are generally expressed by inequalities. The key to solve this kind of problem is to identify the common inequalities: >, <,? ,? ,? If there is no such inequality in the formula, it is not inequality.
Variant training: see "Excellence in learning and practice" in this lesson? Standardized classroom training? Question 1
Type 2 expresses quantitative relations with inequalities.
List inequalities according to the following quantitative relations:
(1) The sum of x and 2 is negative;
(2) The sum of antonyms of m and 1 is nonnegative;
(3) The difference between A and -2 is not more than 3 times;
(4) The sum of squares of numbers A and B is not less than twice the product.
Analysis: (1) negative number is less than 0; (2) Non-negative number is greater than or equal to 0; (3) Not greater than or less than; (4) Not less than or equal to.
Solution: (1) x+2
(2)m- 1？ 0;
(3)a+2？ 3a；
(4)a2+b2？ 2ab。
Variant training: see "Excellence in learning and practice" in this lesson? Standardized classroom training? Question 5
Inequalities in the third kind of practical problems
Liang Liang plans to buy a student tablet computer with his pocket money. He has 55 yuan now, and plans to deposit in 20 yuan every month from now on. Knowing that he needs at least 350 yuan, the inequality that can be used to calculate the number of months that X needs is ().
A.20x-55？ 350 B.20x+55？ 350
C.20x-55？ 350 D.20x+55？ 350
Analysis: Inequality in this question: Now there is 55 yuan, who intends to save 20 yuan every month from now on, knowing that he needs at least 350 yuan. List inequalities 20x+55? 350. therefore, choose B.
Method summary: When inequality is used to represent quantitative relations in practical problems, we should find out two quantities representing unequal relations in the stem of the question and express them by algebra; Correctly understand the key words in the question, such as greater than, not greater than, less than, not less than, insufficient, not greater than, at least, at most, etc.
Variant training: see "Excellence in learning and practice" in this lesson? Standardized classroom training? Question 4
Exploration point 2: the essence of inequality
Type one compares the size of algebra.
According to the nature of inequality, the following deformation is correct ()
A. ac2 & gtbc2 is obtained from a>b.
B. Author AC2 >；; Bc2 gets a>b.
C.by- 12a >； 2 get an a
D. pass 2x+1>; X gets X.
Analysis: a > in a; When B and c=0, ac2=bc2, so A is wrong; Both sides of the inequality in b are multiplied or divided by the same positive number, and the sign of the inequality remains unchanged, so b is correct; Both sides of the inequality in C are multiplied or divided by the same negative number. If the direction of inequality changes, the right side will be multiplied by -2, so C is wrong. Both sides of the inequality in D add and subtract the same algebraic expression, and the direction of the inequality remains the same, so D is wrong. So choose B.
Method Summary: This question examines the essence of inequality. Note that when both sides of the inequality are multiplied or divided by the same negative number, the direction of the inequality changes.
Variant training: see "Excellence in learning and practice" in this lesson? Consolidate and improve after class? Question 2
Type 2 converts inequality into? X> answer? Or? X< answer? In the form of
Turn the following inequality into? X> answer? Or? x