Induction of junior high school mathematics application problems
Application of 1 equation
Equation application problem is a kind of problem-solving problem by enumerating algebraic equations, which runs through almost all junior high school algebra. The application problems of equations in junior high school algebra include enumerating the application problems solved by linear equations, linear equations, quadratic equations and fractional equations. The steps of solving equation application problems can be summarized in six words, namely, checking (examining questions), setting (setting unknowns), listing (listing equations), solving (solving equations), checking (testing) and answering. The content of the exam is mostly combined with some current hot topics, such as savings, per capita income, environmental protection, commodity discounts and so on.
Example 1. In order to encourage water conservation in a certain place, the monthly water fee is charged according to the following regulations: if the monthly water consumption of each household does not exceed 25 tons, the water fee per ton is charged at 1.25 yuan; If the monthly water consumption of each household exceeds 25 tons, the excess water fee per ton will be charged at 1.65 yuan. If the average water fee of a user in May is per ton 1.40 yuan, how much should the user pay in May?
Example 2, the state stipulates that the tax calculation method for individuals publishing articles or books is:
(1) The payment is not higher than that of 800 yuan; (2) If the contribution fee is not higher than 4,000 yuan in 800 yuan, a tax of 14% of the contribution fee in 800 yuan shall be paid; (3) The tax of 1 1% of the total contribution fee shall be paid if the contribution fee is higher than 4,000 yuan. A person once received a manuscript fee and paid personal income tax, 280 yuan. How much did this man get?
The application of inequality in solving practical problems is a new hot spot in the senior high school entrance examination in recent years. We call this kind of test questions inequality application problems. This question usually includes? Not less than? 、? No more than? 、? No more than? 、? Most? 、? At least? Keywords such as, are also often used to find integer solutions of inequalities.
Example: In order to improve the investment environment and residents' living environment, a city has transformed the old city. Now we need 500 thousand A and B tiles, all of which are completed by a brick factory. The factory has raw materials A6,543.8+0.8 million kilograms, raw materials B6,543.8+0.45 million kilograms, known bricks A6,543.8+0.0 million kilograms, raw materials A45,000 kilograms, raw materials B6,543.8+0.5 million kilograms, and the cost is 6,543.8+0.2 million yuan. To produce 654.38+00,000 B bricks, 20,000 kg of raw materials A and 50,000 kg of raw materials B were used, and the cost was 654.38+08,000 yuan. (1) With the existing raw materials, can the factory complete the task as required? If yes, according to the number of production blocks of A and B tiles, what kinds of production schemes are there? Please design (take 10000 block as 1 unit and integer).
② Try to analyze which production scheme you designed has the lowest total cost? What is the lowest cost?
3 function application problems
Function application problems mainly include linear function problems and quadratic function problems. Linear function problems can be roughly divided into: ① using image information to solve practical problems; (2) the resolution function in practical problems; ③ Comparison of schemes with economic accounting as the content; (4) Solve the maximum problem. Quadratic function problems are mainly divided into analytic function, maximum value, arch bridge or fountain design and so on.
A circular fountain will be built in the park. A column OA is installed in the center of the fountain perpendicular to the water surface, where O is right in the center of the water surface, and OA =1.25m. Water is ejected from the top of the column and flows in all directions along a parabola with the same shape. In order to make the shape of water flow more beautiful, it is required that the designed water flow reach the maximum height of 2.25m at the distance of OA1m. If other factors are not considered, the radius of the pool should be at least several meters to prevent the sprayed water from falling outside the pool; If the parabolic shape of the water jet is unchanged and the radius of the pool is 3.5 meters, how many meters should the maximum height of the water flow reach at this time to prevent the water flow from falling out of the pool?
Four statistical application problems
In recent years, there have been some applied problems involving the preliminary knowledge of statistics, which not only examine the basic knowledge of statistics, but also pay attention to the ability examination.
Example: A farmer planted 88 pomelo peach trees on the mountain, and now he is in the third harvest season. First, he randomly picked peaches from five fruit trees, and the peach yield on each fruit tree was as follows (unit: kg) 35, 35, 34, 39 and 37. According to the average sample, what is the total output of peaches this year? (2) If the price of pomelo peaches in the market is 5 yuan/kg, how much will farmers earn from selling pomelo peaches this year? ③ It is known that the income of this farmer from selling pomelo peaches in the first year is 1 1000 yuan. According to the above estimation, try to find the average annual growth rate of the income from selling pomelo peaches in the second and third years.
5 geometry application problems
Geometry comes from nature, and many problems are inseparable from reality. In recent years, there have been many new types of questions that use geometric knowledge to solve practical problems, which we call geometric application problems. Geometric application problems can be roughly divided into: ① height measurement and length measurement; (2) taking materials and cutting; ③ Scheme design problems; ④ The problem of pattern design.
In order to participate in the activities of hosting the 2008 Olympic Games in Beijing. ① One class got the task of making 240 colorful flags, but the students of 10 failed to participate in the production for some reason, so the rest of the class had to make 4 more colorful flags than originally planned to complete the task. How many students are there in this class? ② If both sides are 1 and a (a >; 1), cut into three rectangular colored flags (no cloth left), so that the aspect ratio of each colored flag is the same as that of the precursor, draw the schematic diagrams of two different cutting methods, and write the corresponding value of A (without writing the calculation process).
From the perspective of applied mathematics, it is self-evident that if we can examine the harmony of problem structure, pursue the simplicity, singularity and novelty of problem solving, tap the unity of proposition and conclusion, guide students into the kingdom of mathematics and cultivate their spiritual sentiments, then it is self-evident to arouse students' curiosity, stimulate their interest in learning, improve their learning efficiency and cultivate their creative thinking ability.
Knowledge points of junior high school mathematics application problems
First, the travel problem.
Analysis of the main points of travel problems
Basic concept: Travel problem is to study the movement of objects, and it studies the relationship between the speed, time and travel of objects.
Basic formula: distance = speed? Time; Distance? Time = speed; Distance? Speed = time
Key question: determine the position in the journey.
Meeting questions: speed and? Meeting time = meeting distance (please write another formula)
Pursuit question: Pursuit time = distance difference? Speed difference (write other formulas)
Running water problem: smooth journey = (ship speed+water speed)? Shunshui time
Upstream stroke = (ship speed-water speed)? Backward time
Downstream speed = ship speed+current speed = ship speed-current speed.
Still water velocity = (downstream velocity+upstream velocity)? 2
Water velocity = (downstream velocity-upstream velocity)? 2
Running water problem: the key is to determine the speed of the object, refer to the above formula. get through
Bridge problem: the key is to determine the moving distance of the object, refer to the above formula.
Basic question type: Given any two quantities of distance (encounter problem and pursuit problem), time (encounter time and pursuit time) and speed (speed sum and speed difference), find the third quantity.
Second, the issue of profit.
Profit per commodity = selling price-buying price
Gross profit = sales-expenses
Profit rate = (selling price-purchase price)/purchase price * 100%
Third, the basic formula for calculating interest
The basic formula for calculating the interest of savings deposits is: interest = principal? Duration? interest rate
Interest rate conversion:
The conversion relationship among annual interest rate, monthly interest rate and daily interest rate is:
Annual interest rate = monthly interest rate? 12 (month) = daily interest rate? 360 (days);
Monthly interest = annual interest rate? 12 (month) = daily interest rate? 30 (days);
Daily interest rate = annual interest rate? 360 (days) = monthly interest rate? 30 days
Pay attention to the consistency with the deposit period when using interest rates.
Profit formula and discount problem
Profit = selling price-cost
Profit rate = profit? Cost? 100%= (price? Cost-1)? 100%
Several main application problems in junior high school and their quantitative relations
1, trip problem
? Basic quantities and relations: distance = speed? time
? Equal relationship in the problem of meeting: the distance of one party+the distance of the other party = the distance between them.
? The equivalence relation in the pursuit problem: the distance of the pursuer-the distance of the pursued = the distance apart.
? Drive against the wind (current)
Forward speed =V static+wind (water) speed
Inverse speed =V still wind (water) speed
2. Sales problem? Basic quantity:
Upper and lower amount = principal? Percentage of increase and decrease
Discount = actual selling price? Original price? 100% (discount
Interest = principal? Interest rate? time
Interest after tax = principal? Interest rate? Time? ( 1-20%)
3. Engineering problems? Basic quantity and relationship: total work = work efficiency? working hours
4. Distribution problem
Generally, this problem has invariants, and invariants are an essential equality relationship of sequence equations.
Fourth, the concentration problem.
Solute weight+solvent weight = solution weight.
The weight of solute? The weight of the solution? 100%= concentration
The weight of the solution? Concentration = weight of solute
The weight of solute? Concentration = solution weight
Verb (abbreviation for verb) growth rate
If the percentage of average increase (decrease) is X, the number before increase (or decrease) is A, and the number after increase (or decrease) is B, then their quantitative relationship can be expressed as: a( 1+x)n =b or a( 1-x) =bn.
Cost (purchase price), selling price (actual selling price), profit (loss amount) and profit rate (loss rate)
? Basic relationship: profit = selling price-cost, loss = cost-selling price, profit = cost? Loss of profit margin = cost? loss rate
Problem-solving skills of junior high school mathematics application problems
1. Examination: Find out the meaning of the question and the known and unknown numbers in the question;
2. Find equivalence relation: find one (or several) equivalence relation that can express all the meanings of the application problem;
3. Set the unknown: according to the found equation, choose to set the unknown directly or indirectly.
4. List equations (groups): List equations according to the established equivalence relation.
5. Solve the equation (or equations) and find the value of the unknown quantity;
6. Inspection: necessary inspection of the results;
7. Answer: Complete answer, including company name.
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