Knowledge and skills objectives:
Through the analysis of practical problems, students can further understand that equations are effective mathematical models to describe the real world, master the application problems of solving binary linear equations, and initially understand the basic ideas of solving binary linear equations.
Cultivate students' consciousness of solving practical problems by solving equations, and enhance students' mathematical application ability.
Process and method objectives:
Experience the process of solving practical problems with equations, and further understand that equations (groups) are effective mathematical models to describe the real world.
Emotional attitude and value goal;
1. further enrich students' successful experience in mathematics learning, stimulate students' curiosity in mathematics learning, and further form the consciousness of actively participating in mathematics activities and actively cooperating with others.
2. Through the "chicken and rabbit in the same cage", students are brought into the ancient mathematical problem scene, and students realize their "interest" in mathematics; Further emphasize the connection between classroom and life, highlight the practical value of mathematics teaching and cultivate students' humanistic spirit. Key points:
Experience and understand the process of solving practical problems with equations; Improve students' mathematical application ability.
Difficulties:
Establish equivalence relation and list correct binary linear equations.
Teaching process:
Review before class
Review: enumerate the general steps of solving application problems by linear equations of one variable.
Situation introduction
Exploration 1: There are chickens and rabbits in a cage today.
There are 35 heads on the table,
It's 94 feet below,
Seek chicken and rabbit geometry?
The problem of "pheasant rabbits in the same cage": Today, pheasant rabbits are in the same cage, with 35 heads on the top and 94 feet on the bottom. What are the geometric shapes of pheasant rabbits?
(1) drawing method
To represent the head, draw 35 heads first.
Think of all the heads as chickens and use legs to represent them. Draw 70 legs.
There are 24 legs left, two more on each head and two more on *** 12 head.
Four-legged rabbit (12) and two-legged chicken (23).
(2) one-dimensional linear equation method:
Chicken head+rabbit head = 35
Chicken feet+rabbit feet = 94
Suppose there are x chickens, then there are (35-x) rabbits.
2x+4(35-x)=94
Easier to understand than arithmetic.
Think about it: can we solve these problems in a simpler way?
Looking back at the binary linear equation learned last class, can you solve this problem?
(3) Binary linear equation method
Today, chickens and rabbits are in the same cage, with 35 heads above and 94 feet below. What are the geometric shapes of chickens and rabbits?
(1) has thirty-five heads, which means that chickens and rabbits have thirty-five heads.
There are 94 feet under it, which means that chickens and rabbits have 94 feet.
(2) If there are X chickens and Y rabbits, then there are (x+y) chickens and rabbits.
There are 2x chicken feet; There are 4y rabbits.
Solution: There are X chickens and Y rabbits in the cage.
Chicken and rabbit head xy35 feet 2x4y94
To solve this system of equations:
Exercise 1:
1. Let the number A be x and the number B be y, then "the sum of twice the number A and half the number B is 15" and the equation is _2x+05y= 15.
2. Xiaogang has five-cent coins and 1 yuan coins, each with a currency of six yuan and fifty cents. Let's say that there are x coins in a nickel and y coins in 1 yuan, and the equation is 05x+y=65.
Third, cooperative exploration.
Exploration 2: Logging with rope. If the rope is measured by three folds, the rope is five feet more; If the rope is 40% off, the rope is one foot more. What is the geometry of rope length and well depth?
Topic: measuring the depth of a well with a rope. If the rope is folded into three equal parts, the length of a rope is 5 feet longer than the depth of the well; If the rope is folded into four equal parts, the length of a rope is longer than the depth of the well 1 foot. How long is the rope and how deep is the well?
Find the equivalence relation:
Solution: If the rope is x feet long and the well is y feet deep, it is derived from the meaning of the question.
x=48
X=48y= 1 1。
So the length of the rope is 48 1 1 foot.
Think about it: find a simpler innovative solution?
Guide students to think of simpler methods step by step:
Find the equivalence relation:
(well depth +5)×3= rope length
(well depth+1)
Solution: If the rope is x feet long and the well is y feet deep, it is derived from the meaning of the question.
3(y+5)=x
4(y+ 1)=x
x=48
y= 1 1
So the rope is 48 feet long and the well depth is 1 1 foot.
Exercise 2: A and B race. If B runs first 10 meter, A can catch up with B in 5 seconds. If B runs 2 seconds first, A can catch up with B in 4 seconds. Let the speed of A be x m/s and the speed of B be y m/s, then the equation can be listed as (b).
Induction:
Enumerate the general steps of solving practical problems by binary linear equations;
Review: Review the equivalence relation in the topic.
Hypothesis: Hypothesis is unknown.
Column: According to the equivalence relation, list the equations.
Solution: Solve the equations and find the unknown.
Answer: Check whether the unknown meets the meaning of the question and write the answer.
Fourth, think independently.
Exploration 3: Using rectangular and square cardboard as edges and bottoms, we can make vertical and horizontal open cartons as shown in the figure. At present, there are 1000 square cardboard and 2000 rectangular cardboard in the warehouse. How many cartons are made for each of the two models, just to make the cardboard in stock run out?
Solution: Let's make X vertical cartons and Y horizontal cartons. According to the meaning of the question, you must
x+2y= 1000
4x+3y=2000
Solve this equation set X=200.
y=400
A: There are 200 vertical boxes and 400 horizontal boxes, which just makes the stock of cardboard run out.
Exercise 3: If there are 500 square cardboard and 100 1 rectangular cardboard in the above question, can you just use up the cardboard after making several vertical boxes and several horizontal boxes?
Solution: Let's make X vertical cartons and Y horizontal cartons according to the meaning of the question.
Y is not a natural number, so it doesn't matter, so it's impossible to make several cartons, and just used up the cardboard without stock.
Induction:
V. Standard evaluation
1. Solve the following application problems
(1) Buy some 4-cent and 8-cent stamps, 80 cents in 6 yuan. It is known that there are 40 8-cent stamps more than 4-cent stamps, so how many stamps did you buy each?
Solution: There are four stamps X and eight stamps Y, which are derived from the meaning of the question:
4x+8y=6800①
y-x=40②
So, there are 540 4-cent stamps and 580 8-cent stamps.
(2) If a project is all sunny, it can be completed in 15 days; If it rains, it can only be done on rainy days.
Workload. It is now known that there are 3 more rainy days than sunny days during the construction period. How many days will it take to complete this project?
Analysis: As the total workload is unknown, we set it as the unit of 1.
It will be finished on a sunny day.
It can be done in rainy days.
Solution: Assuming that it is sunny for x days and rainy for y days, the total workload is 1.
Total days: 7+ 10= 17.
Therefore, *** 17 days can complete the task.
Sixth, application improvement.
The school bought 232 pencils, ballpoint pens and pens, which cost 300 yuan. Among them, the number of pencils is four times that of ballpoint pens. Known pencil 0.60 yuan, a ballpoint pen 2.7 yuan, a pen 6.3 yuan. How many pens are there in each of the three types?
Analysis: the number of pencils+ballpoint pens+pens =232.
Number of pencils = number of ballpoint pens ×4
Pencil price+ballpoint pen price+pen price =300
Solution: equipped with X pencil, Y ballpoint pen and Z pen. According to the meaning of the problem, we can get the ternary linear equations:
Substitute ② into ① and ③ to get binary linear equations.
4y+y+z=232④
0.6×4y+2.7x+6.3z=300⑤
solve
So there are 175 pencils, 44 ballpoint pens and 12 pens.
Seven, experience the harvest
1. Solve the problem of chickens and rabbits in the same cage.
2. Solve the problem of cable logging
3. General steps to solve application problems
Seven. distribute
Textbook 1 16 exercise on pages 2 and 3.
x+y=35
2x+4y=94
x=23
y= 12
One third of rope length-well depth =5
Quarter of rope length-well depth = 1
-y=5①
①-②, yes
-y= 1②
-y=5①
-y=5①
-y=5①
X=540
Y=580
y-x=3②
x=7
y= 10
x+y+z=232①
x=4y②
0.6x+2.7y+6.3z=300③
X= 176
Y=44
Z= 12
Solution of Binary Linear Equations —— Substitution Teaching Content: P96, Section 2, Chapter 8 of Seventh Grade Mathematics, People's Education Press.
Teaching objectives
(1) Basic knowledge and skill goal: Simple binary linear equations will be solved by substitution elimination method.
(2) Process and Method Purpose: To explore the process of solving binary linear equation by substituting elimination method, and to understand the reduction thinking method embodied in the basic idea of substituting elimination method.
(3) Emotion, attitude and values: attract students' attention and stimulate their interest in learning by providing appropriate situational materials; Learn to communicate and cooperate in cooperative discussion, cultivate good mathematical thoughts, and gradually infiltrate the consciousness of analogy and reduction.
The key of teaching emphasis and difficulty
Teaching emphasis: solving binary linear equations by substitution elimination method
Teaching difficulties: explore how to solve binary linear equations by substituting elimination method and feel the idea of "elimination"
The key to teaching: deform one equation in the equation set, and future generations will enter another equation, eliminate an unknown and become a linear equation. Students are seventh-grade students in ethnic minority areas, and their basic knowledge is weak, especially the content of linear equations with one variable is not thoroughly mastered. In addition, they are tired of learning and have poor ability of unity and cooperation. In this lesson, basketball games and common disinfectants are designed to study binary linear equations, which can not only stimulate their interest in learning, but also solve the problems involved in this lesson, paving the way for further study of binary linear equations.
Teaching content analysis: The main content of this section is to learn the first method of solving equations-substitution elimination method on the basis of the concepts of binary linear equation (group) and binary linear equation (group) in the previous section. And understand the basic idea of "elimination method" for solving binary linear equations. The solution of binary linear equation not only uses the solution of univariate linear equation learned before, but also reviews and improves the knowledge learned in the past. At the same time, it also lays a foundation for solving practical problems by using equations in the future. Through the application of binary linear equations in practical problems, students' awareness of learning and using mathematics is further enhanced, and the value and significance of learning mathematics are realized. There are two solutions to binary linear equations in junior high school: substitution elimination method and addition and subtraction elimination method. The teaching materials are arranged in the order of solving first and then using. This arrangement can not only learn the solution of the previous section, but also consolidate the previous knowledge in the application of the latter section. However, there are fewer corresponding exercises in the textbook, but it also gives students more room to play.
Teaching aid preparation teacher preparation: ppt multimedia courseware projector
Teaching methods This class adopts the teaching method of "problem introduction-inquiry and solution-induction and reflection" and insists on heuristic teaching.
teaching process
(1) Create a situation and introduce a new basketball league. Every game has to be decided. Each team wins 2 points and loses 1 point. In order to get a better ranking, baoan middle school team wants to score 40 points in all 22 matches. What are the winning and losing games of this team?
(2) Cooperation and exchange, exploring new knowledge. The first step is to get a preliminary understanding of method of substitution 1. In addition to solving the above problems with linear equations of one variable, two unknowns can be set to list the student activities of linear equations of two variables: linear equations of one variable and linear equations of two variables are listed respectively. The number of fields played by two students is X, and the number of negative fields is Y..
x+y=22
2x+y=40
② Let the winning field number be X and the negative field number be 22-X..
2x+(22-x)=40
2, independent exploration, group discussion, then how to solve the binary linear equations? What is the relationship between binary linear equations and linear equations?
3. Students sum up and teachers add the above solutions. The first step is to express an unknown number in one equation of binary linear equations with a formula containing another unknown number, and then substitute it into another equation to realize elimination, and then get the solution of this binary linear equations. This method is called substitution elimination method, or substitution method for short.
The second step is to solve the equations with method of substitution, and write the following equations as a formula with X (1) 2x-y = 5 (2) 4x+3y-1= 0. Student activities: Try to finish it yourself, and the teacher corrects the thinking: Can X be expressed as a formula with Y?
Example 1 Solving equations by substitution method X-Y = 3 13x-8Y = 14②.
Idea: First, observe which coefficient in this equation group is smaller, and find that the coefficient of X in ① is 1, so you can determine that it is easier to eliminate X. First, use the algebraic expression containing Y to represent X, and then substitute it into ② to eliminate it.
Solution: X=y+3③ is transformed from ①.
Substituting ③ into ② gives 3 (y+3)-8y = 14.
Solve this equation and you get y =- 1.
Substitute y =- 1 into ③ to get X=2.
So the solution of this system of equations is x = 2y =- 1.
How to check whether the result is correct? Student activities: oral test.
The third step is to use method of substitution to solve the equations in real life.
According to market research, the sales volume of a disinfectant in large bottles (500g) and small bottles (250g) is 2:5. A factory produces 22.5 tons of this disinfectant every day. How many bottles should these disinfectants be divided into large bottles and small bottles? Thinking: This problem is a practical application problem, which can be solved by using binary linear equations as a tool, so it is necessary to establish a model and find two equivalent relationships. From the meaning of the question, we can know that the number of large bottles: the number of small bottles = 2: 5; Large bottle of disinfectant+small bottle of disinfectant = total output (the process of solving problems is omitted) Teacher's activities: inspire and guide students to establish a binary linear equation model. Student activities: Try to divide these disinfectant solutions into X large bottles and Y small bottles, and get 5x=2y500x+250y=22500000, and calculate x=20000y=50000.
The fourth step, group discussion, get the steps of students' activities: According to the problem-solving process of Example 1 and Example 2, can you sum up the steps of solving binary linear equations with method of substitution? Discuss in groups. Students conclude, teachers supplement, and summarize method of substitution's steps to solve the binary linear equation: ① Select a binary linear equation with simple coefficients for deformation, and use an algebraic expression containing an unknown number to represent another unknown number; (2) Substitute the deformed equation into another equation, eliminate an unknown number, and get a linear equation (when substituting, be careful not to substitute into the original equation, only substitute into another equation without deformation, so as to achieve the purpose of elimination); ③ Solve this one-dimensional linear equation and get the unknown value; (4) Substituting the obtained unknown quantity into the deformation equation in (1) to obtain the value of another unknown quantity; ⑤ Simultaneous two unknowns with "{"are the solutions of equations; ⑥ Finally, check whether the result is correct (substitute into the original equations and check whether the equations satisfy left = right).
(3) Competition in groups to consolidate new knowledge In order to stimulate students' interest and consolidate what they have learned, I divided the whole class into four groups and designed the exercises on page P98 of the book into several independent, informative and interesting sections. The exercises are presented in the form of group competition from easy to difficult and from simple to deep, which not only improves students' enthusiasm, cultivates team spirit, but also gives full play to students' various abilities.
(4) Summary, knowledge review 1. What have you gained through the learning activities of this class? 2. What do you think should be paid attention to when using method of substitution to solve binary linear equation?
(5) Assign homework 1, homework: P 103 Page 65438 +0, 2, 4 Questions 2. Thinking: Put forward practical problems that can be solved by using binary linear equations in daily life. Design says that the elimination method of Ming Dynasty embodies the thinking method of "turning the unknown into the known" in mathematics learning, and the principle of transformation is to turn unfamiliar problems into familiar ones to solve new problems. Based on this understanding, this course is designed according to the idea of "introducing mathematical problems around us-finding the solution of linear equations with one variable-exploring the substitution and elimination method of linear equations with two variables-typical examples-general steps of inductive substitution method". Give full play to students' subjective initiative and teachers' leading role, and adhere to heuristic teaching. Teachers create interesting situations, arouse students' enthusiasm for consciously participating in learning activities, integrate the process of knowledge discovery into interesting activities, and attach importance to the process of knowledge occurrence. By comparing the solving process of unknown linear equations with that of binary linear equations, the substitution (elimination) solution of binary linear equations can be obtained. This comparison enables students to master new knowledge while reviewing old knowledge.