1. Table information type
Example 1 (Ordos, 2006) The state implements the policy of "two exemptions and one subsidy" for students in the nine-year compulsory education stage. The following table is part of the free textbook subsidy provided by a middle school in our city.
A total of 789 people
Free subsidy amount per person (RMB) 1 10 90 50
Number of people (persons)
Total free subsidy (RMB) 4,000-26,200 yuan.
If you want to know the data in the blank, you can set the number of students in the seventh grade to, the number of students in the eighth grade to, and list the equation as () according to the meaning of the question.
A.B.
C.D.
To evaluate the application of this kind of graphic information, we should be good at mining information from charts, finding some hidden information, constructing corresponding mathematical models, and flexibly using the knowledge we have learned to solve practical problems.
2. New types of situational applications
Example 3. (Hainan, 2006) A shopping center is selling two Olympic goods, Fuwa toys and badges, which are the mascots of the 2008 Beijing Olympic Games. According to the information provided in the picture below, what is the price of a box of Fuwa toys and a badge?
Solution. Let a box of Fuwa toys and a badge cost RMB and RMB respectively.
According to the meaning of the question, get
To solve this system of equations, you must
A: A box of Fuwa toys and a badge cost RMB and RMB respectively.
(Note: Other solutions are simulated according to the above grading standards. )
Example 4. (Xiangxi, 2006) In the construction of a new socialist countryside, a village actively responded to the call of the Party, vigorously mobilized farmers to expand the planting area of tobacco leaves and vegetables, and achieved good economic benefits. This year, the planting area of tobacco and vegetables in the village increased by 800 mu compared with last year, among which the planting area of tobacco and vegetables increased, bringing the planting area of tobacco and vegetables in the village to 4200 mu. Ask the village.
Solution: Suppose that the area planted with tobacco and vegetables in this village last year was mu, mu,
According to the meaning of the question:
Solution:
A: Last year, the area planted with tobacco leaves and vegetables in this village was 2,200 mu, 1200 mu.
Commenting on new situational application questions is an examination of integrating theory with practice. In recent years, it has become a compulsory content in the senior high school entrance examination papers all over the country, which embodies the direction of mathematics education reform. The emphasis is on analysis, the examination of questions is the key, the understanding of core words is the foundation, the format is the core, and the writing format must be complete and accurate.
3. Image information types
Example 4 (Xiangtan, 2006)
Xiao Gang and Xiao Ming went to a boutique stationery store and bought the same pen and the same exercise book. Answer the questions according to the following dialogue:
Xiao Gang: Aunt, I want to buy three pens and two exercise books. How much will it cost?
Shop assistant: Just 19 yuan.
Xiaoming: Aunt, how much does it cost to buy 1 pen and 3 exercise books?
Shop assistant: It only costs 1 1 yuan.
How much does (1) 1 pen and 1 exercise book cost?
(2) Xiaoming has money from 20 yuan and needs to buy 1 pen and some exercise books. How many exercise books can he buy at most?
Solution: (1) Suppose it costs RMB to buy a pen and RMB to buy an exercise book.
According to the meaning of the question:
Get a solution
(2) Set the purchased exercise book as one.
rule
Because it is a non-negative integer, the maximum value of is.
Answer: (1) You need 5 yuan to buy 1 pen and 2 yuan to buy 1 exercise book. (2) Xiao Ming can buy at most 7 exercise books.
(Note: (2) If used, the integer 7 can also be used for further analysis. )
The key to solving practical problems by commenting on equations is to find out the equivalent relationship in practical problems. This question presents a shopping situation and gives relevant information in the form of a conversation between two people. When solving problems, we should analyze their dialogue, find out the equivalent relationship contained in it, set unknown numbers and set equations.
Example 6 (Loudi, 2006). Xiaoying and Xiao Qiang meet to go to the supermarket to buy their favorite walkman and schoolbag. Can you find out the unit price of their favorite walkman and schoolbag according to their conversation (pictured)?
Solution: Suppose the unit price of their favorite schoolbag is RMB, and the unit price of the walkman is RMB.
According to the meaning of the question
solve
Answer: (omitted)
Commenting on such topics is a new highlight of examination questions in recent years, which makes many topics more vivid and closer to life. We should pay attention to analyze their conversation from the image and find out the equivalent relationship.
4. Decision-making
Example 7 (Dingxi City, 2006). Company A has two brands of calculators, brand A has three different models and brand B has two different models. Xinhua Middle School wants to buy a calculator of one model from two brands.
(1) Write all options (represented by tree diagram or list method);
(2) If there is the same possibility of choosing various options in (1), what is the probability of choosing a model calculator?
(3) It is known that Xinhua Middle School bought ***40 calculators of brand A and brand B (the price is shown in the figure), and just spent 1 1,000 yuan, of which the calculator of brand A is a model calculator. How many models of calculators do you want to buy?
Solution: (1) The tree diagram is as follows:
first
The list is as follows:
There are six possible outcomes:.
Note: If the purchase plan is expressed in other ways and is correct, only 1 point will be given.
(2) Because there are two schemes to choose the model calculator, that is, the probability that the model calculator is selected is.
(3) As can be seen from (2), there are two purchase models and model calculators when choosing a scheme.
According to the meaning of the problem, you can get a solution
After investigation, it does not meet the meaning of the question and is abandoned;
When selecting a scheme, set the purchase model and model calculator to two.
According to the meaning of the problem, you can get a solution
So Xinhua Middle School bought five models of calculators.
To comment on such questions, we need to understand the meaning of the questions and get correct information. It is to transform the actual decision-making problems into binary linear equations, that is, to solve the practical application problems by constructing mathematical models, which is a hot issue in the senior high school entrance examination.
I. Form types
Example 1 (Foshan, Guangdong) There are triple rooms and double rooms in the housekeeping department of a hotel. The charging data are as follows.
Ordinary (RMB/room/day) Deluxe (RMB/room/day)
Triple room 150 300
Double room 140 400
In order to attract tourists, group accommodation is offered a 50% discount. A 50-member tour group stayed in this hotel during the discount period, and stayed in some ordinary rooms with three people and two people. If each room is just full, and the accommodation fee is 15 10 yuan for one day, how many ordinary rooms for three people and two people have the tour group stayed in?
Analysis: This question provides some information in tabular form, and mainly examines the ability to solve practical problems by using binary linear equations.
Solution: a three-person ordinary room and a double-person ordinary room live in separate rooms, depending on the meaning of the question.
Answer: A little.
Smart you, how can you set an unknown number to answer this question?
Second, look at the picture type.
Example 2 (Nanning, Guangxi) According to the information provided on the right, the unit price of each tennis racket is _ _ yuan, and the unit price of each table tennis racket is _ _ yuan.
Analysis: According to the conditions of graphic capture and problem solving, two branches can be found by observing the graphic.
Tennis rackets and 1 table tennis rackets * * 200 yuan, 2 table tennis rackets and 1.
Tennis racket * * * 160 yuan. According to the obtained conditions, the equations can be solved.
Solution: Set each tennis racket yuan, each table tennis racket yuan, according to the meaning of the question, get.
solve
Third, look at the types of picture dialogues.
Example 3 (Maoming, Guangdong)
Solution: Suppose 1 notebook, 1 pen. According to the meaning of the question, you get
Answer: A little.
Example 4 (Zhejiang Quzhou Middle School Examination) is on May Day.
During the Golden Week, Xiaoming, Liang Xiao and other students were all with their families.
Visit Jianglang Mountain. Below are photos of Xiao Ming and his wife when they bought tickets.
His father's conversation:
Question: (1) How many adults did Xiao Ming go to? How many students?
(2) Please help Xiao Ming think about which way to buy tickets is more economical? And explain why.
Solution: (1) Suppose a * * went to an adult and a student.
solve
(2) If you buy a group ticket, you need (yuan), which can save 14 yuan, so you can save money by buying a group ticket.
Clever, can you provide them with a more economical way to buy tickets? (Hint: Xiaoming and his party went to 12 people).