The first part: the teaching goal of selecting the model essay of the high school mathematics teaching plan template;
1。 By studying the optimization problems in life, we can understand the role of derivatives in solving practical problems and promote
Students fully understand the scientific value, applied value and cultural value of mathematics.
2。 Through the study of practical problems, students' ability of analyzing, solving and mathematical modeling is improved.
Teaching focus:
How to establish the objective function of practical problems is the focus and difficulty of teaching.
Teaching process:
First, the problem situation
Question 1 What is the length and width of each wire if it is a rectangle with a length of 60cm?
Question 2: Divide the iron wire with the length of 100cm into two sections, and each section is surrounded by a square. How to divide to minimize the area of two squares?
Question 3: What is the height of a square-bottomed uncovered water tank with a volume of 256L, and when is the most material-saving?
Second, the introduction of new courses.
Derivative is widely used in real life. By using derivative to find the maximum, we can find some maximum problems in real life
1。 Application of geometry (maximum area and volume, etc.). ).
2。 Applications in physics (maximum value of work and power).
3。 The application of economics (the most valuable in terms of profit).
Third, knowledge construction.
Example 1 Cut equal squares at the four corners of a square iron sheet with a side length of 60cm, and then fold its dotted line (as shown in the figure) to make a box with a square bottom and no cover. What is the side length of the bottom of the box and the largest volume of the bottom of the box? What is the maximum volume?
Description 1 There are generally four key steps in solving application problems: setting-column-solution-answer.
It shows that the derivative method is similar to the function extreme value method, with one step and several poles added.
Comparison between values and endpoint values.
When the volume of cylindrical metal beverage can is constant, how to choose its height, bottom and radius is the only way.
Can you use the least materials?
Variation When the surface area of a cylindrical metal beverage can is a constant value S, how to choose its height and bottom radius to minimize the materials used?
The function of 1 which shows that there is only one extreme value in the domain is called unimodal function.
It shows that finding the maximum value of unimodal function by derivative method can simplify the general method, and its steps are as follows:
S 1 column: lists the functional relationships.
S2: Find the derivative of the function.
S3 statement: It means that the function has only one maximum (minimum) value in the definition domain, so it can be judged as the maximum (minimum) value of the function and answered if necessary.
In the circuit shown in the figure, it is known that the internal resistance of the power supply is 0 and the electromotive force is 0. External resistance is
What is the age at which electricity is maximized? What is the maximum electric power?
It shows that we should pay attention to the condition of verifying the equality sign when seeking the maximum value, that is to say, the corresponding independent variable must have a solution when obtaining such a value.
Example 4 Two light sources A and B with intensity A and B respectively, and the distance between them is D. Q: Where is the minimum illumination on the line segment AB connecting the two light sources? Try to answer the above questions when a = 8, b = 1, d = 3 (illuminance is directly proportional to the intensity of light and inversely proportional to the square of the distance from the light source).
In economics, the cost of producing a unit product is called a cost function, which is recorded as follows: the income of selling a unit product is called an income function, which is recorded as; It is called the profit function, and is written as.
(1), how many units of products are produced, and the marginal cost is the lowest?
(2) Suppose, the unit price of the product, what kind of pricing can maximize the profit?
Fourth, classroom exercises.
1。 Divide the positive number A into two parts to minimize the cubic sum. These two parts should be divided into _ _ and _ _.
2。 An isosceles triangle is inscribed in a circle with radius r. When the cardinal number is high, its area is the largest.
3。 There is a rectangle with sides of 8 and 5 respectively. Cut out the same small square at each corner and fold the four sides to make a small box without a lid. To maximize the size of the carton, what should be the side length of the cut cubes?
4。 As shown in the figure, the cross section of the channel is isosceles trapezoid. When determining the section size, it is hoped that the wet circumference L = AB+BC+CD will be the smallest when the area of section ABCD is constant S, so that the water flow resistance will be small and the infiltration will be less, and the height H and bottom length B at this time can be obtained.
Review and reflection on verb (abbreviation of verb)
(1) To solve the practical problem about the maximum and minimum values of functions, it is necessary to analyze the relationship between variables in the problem, find out the appropriate function relationship, and determine the definition interval of functions; The results obtained should conform to the practical significance of the problem.
(2) When judging the maximum value of a function according to the actual meaning of the problem, if the function has only one extreme point in this interval, then this extreme value is the maximum value, and there is no need to compare it with the endpoint value.
(3) Many practical problems about the maximum value are simply solved by derivative method.
Sixth, homework
Page 38 of the textbook 1, questions 2, 3 and 4.
The second part: The model essay of high school mathematics teaching plan template selects the interesting contest questions of high school mathematics (total 10 questions).
1, how many people lie?
There are five high school students who said the following in the face of the school news interview:
Love: "I haven't been in love yet." Shizuka: "Love is a lie."
Mary: "I have been to Kunming." Huimei: "Mary is lying."
Chiba: "Mary and Huimei are both lying." So, how many of these five people are lying?
2. Who are they?
There are angels, demons and people. Angels always tell the truth, and demons always tell lies. People sometimes tell the truth and sometimes tell lies.
The woman in black said, "I am not an angel." The woman in blue said, "I'm not human." The woman in white said, "I'm not a demon." So, who are these three people?
3. Half a kitten
I heard that grandpa's Persian cat gave birth to many kittens, so I came to grandpa's house happily. However, there are only 1 kitten left.
"How many kittens were born?" "Guess, if I guess right, I will give you the rest of the kitten. The nearby pet store heard about it and immediately came and bought all the kittens in half. " "Half?" "Yes, then, anyway, the neighbor's grandmother wanted it, so she gave her the remaining half and the other half. That's why there are only 1 kitten left. Then think about it. How many kittens have you given birth to?
4. Formula eaten by bugs
A bug that likes ink ate all the numbers in the following formula. Of course, it didn't eat the part without numbers (because there was no ink).
So, what does the original formula look like?
5. Clever collocation
Make five squares with 16 matches. Please move two matches to spell orthography 4.
6. Folding angle
When the regular triangle paper is folded in half as shown in the figure, the angle is? To what extent
7. Star Angle and
Find the sum of the stars' sharp angles.
8. ah Twin twins (plural of twin)
Before the husband died, he left a last note to his pregnant wife, saying that he would get 2/3 of his property if he gave birth to a boy, 2/5 if he gave birth to a girl, and the rest would be given to his wife.
As a result, twins were born. It's hard for a wife to break down. How do three people divide the property?
9. Which is better, giving away or reducing the price?
1 can price 100 yuan of coffee, which is better, "buy 5 cans for free 1 can" or "buy 5 cans for 20% cheaper"? Or are the two methods just as good?
10, converted into 15 degrees.
Origami is easy to make 45 degrees, right? Then, please fold it into 15 degrees, ok?
Part III: Selected model essays of high school mathematics teaching plan template 1. Course nature and tasks.
Mathematics is a science that studies the relationship between spatial form and quantity. It is the foundation of science and technology and an important part of human culture.
Mathematics is a compulsory public basic course for students in secondary vocational schools. The task of this course is to enable students to master the necessary basic knowledge of mathematics, have the necessary related skills and abilities, and lay the foundation for learning professional knowledge, mastering professional skills, continuing learning and lifelong development.
Second, the teaching objectives of the course
1. On the basis of nine-year compulsory education, enable students to further study and master the basic knowledge of mathematics necessary for professional posts and life.
2. Cultivate students' computing ability, using computing tools and data processing ability, and cultivate students' observation ability, spatial imagination ability, problem analysis and solving ability and mathematical thinking ability.
3. Guide students to gradually develop good study habits, practical awareness, innovative awareness and a scientific attitude of seeking truth from facts, and improve students' employability and entrepreneurial ability.
Third, the teaching content structure
The teaching content of this course consists of three parts: basic module, career module and expansion module.
1. Basic module is a compulsory basic content and basic requirement for students of all majors, and the teaching hours are 128 hours.
2. The vocational module is a limited elective content to meet the needs of students studying related majors. Each school chooses to arrange teaching according to the actual situation, and the teaching hours are 32~64 hours.
3. The expansion module is an optional content to meet the needs of students' personality development and continuous learning, and the teaching hours are not uniformly stipulated.
Fourth, the teaching content and requirements
(a) the statement of the teaching requirements of this syllabus 1. Cognitive requirements (divided into three levels)
Understanding: knowing the meaning of knowledge and its simple application.
Understanding: Understand the concepts and laws of knowledge (definitions, theorems, laws, etc. ) and links with other related knowledge. Proficiency: Be able to apply the concepts, definitions, theorems and laws of knowledge to solve some problems. 2. Skills and abilities training requirements (divided into three skills and four abilities)
Calculation skills: according to laws, formulas or certain operation steps, correctly solve the operation. Skills of using calculation tools: correctly use scientific calculators and commonly used mathematical tools and software. Data processing skills: process data (data table) and extract relevant information as needed. Observation ability: according to data trends, quantitative relationships or graphs and charts, describe their laws.
Spatial imagination: according to the description of words and languages, or simple geometry and its combination, imagine the corresponding spatial graphics; Be able to find out the basic elements and their positional relationships in basic graphics, or draw graphics according to conditions.
Ability to analyze and solve problems: be able to analyze simple problems related to mathematics in work and life and use appropriate mathematical methods to solve them.
Mathematical thinking ability: according to the learned mathematical knowledge, using analogy, induction, synthesis and other methods, thinking, judging, reasoning and solving mathematics and its application problems in an orderly manner; According to different problems (or requirements), the appropriate model (mode) will be selected.
(2) Teaching content and requirements 1. Basic module (128 class hours)
Unit 1 (10 class hour)
Unit 2 Inequality (8 hours)
Unit 6 Series (10 class hour)
Unit 7 Plane Vector (Vector) (10 class hour)
Unit 8 Equations of Lines and Circles (18 hours)
Unit 10 probability statistics (16 class hours)
2. Career module
Unit 2 Coordinate Transformation and Parameter Equation (12 class hours)
Chapter four: the teaching goal of selecting the model essay of the high school mathematics teaching plan template;
1, understand and master the concept of tangent of a curve at a certain point;
2. Understand and master the definition of the slope of the curve tangent at a point and the solution of the tangent equation;
3. Understand the practical background of tangent concept, cultivate students' ability to solve practical problems and cultivate students' transformation.
The ability to solve problems and the idea of combining numbers and shapes.
Teaching focus:
Understand and master the definition of the slope of the curve tangent at a point and the solution of the tangent equation.
Teaching difficulties:
Understanding the slope of tangent at a certain point with the idea of "infinite approximation" and "local straight line replacing curve"
Teaching process:
First, the problem situation
1, problem situation.
How to accurately describe the changing trend of a point on the curve?
If you enlarge the curve near point P, you will find that the curve looks a bit like a straight line near point P.
If you enlarge the curve near point P again, you will find that the curve looks almost straight near point P. In fact, if we continue to enlarge, the curve will approach a straight line near point P, which is the closest straight line to the curve among all the straight lines passing through point P.
So we can use this straight line to replace the curve near point P, that is, near point P, the curve can see a straight line (that is, use a straight line to replace the curve in a small range).
2. Inquiry activities.
As shown in the figure, straight lines l 1 and l2 are two straight lines passing through a point p on the curve.
(1) Try to determine which straight line near point P is closer to the curve;
(2) Can you draw a straight line l3 near point P that is closer to the curve than l 1 and l2?
(3) Can you make a straight line near point P that is closer to the curve than l 1, l2, l3?
Second, architectural mathematics
Definition of tangent: as shown in the figure, let q be a point different from p on curve C, and the straight line PQ is called the secant of the curve. When point q moves along curve c to point p, secant PQ approaches curve c near point p. When the point Q is infinitely close to the point P, the straight line PQ eventually becomes a straight line L passing through the point P, which is also called the tangent of the curve at the point P ... This method is called secant approaching tangent.
Thinking: As shown above, P is a point on the known curve C. How to find the tangent equation of P?
Third, the application of mathematics.
Example 1 Try to find the tangent slope at point (2,4).
Analytical solution 1: Let p (2 2,4), Q(xQ, f(xQ)),
The slope of secant PQ is:
When Q is close to point P along the curve, secant PQ is close to the tangent of point P, so secant slope is close to tangent slope.
When the abscissa of point Q is infinitely close to the abscissa of point P, that is, when xQ is infinitely close to 2, kPQ is infinitely close to constant 4.
Therefore, the tangent slope of curve f (x) = x2 at point (2,4) is 4.
Solution 2 Let p (2 2,4) and Q(xQ, xQ2), then the slope of secant PQ is:
What time? When x infinitely approaches 0, kPQ infinitely approaches the constant 4, so that the curve f (x) = x2 and the tangent slope at point (2,4) is 4.
Try to find the tangent slope at x = 1
Solution: Let p (1, 2), q (1+δ x, (1+δ x) 2+ 1), then the slope of secant PQ is:
What time? When x infinitely approaches 0, kPQ infinitely approaches the constant 2, so the tangent slope of curve f (x) = x2+ 1 at x = 1 is 2.
Summarize the general steps to find the tangent slope of a point on a curve:
(1) Find the coordinates of fixed point P and set the coordinates of dispatching point Q;
(2) Find the slope of secant PQ;
(3) At this time, the secant is close to the tangent, so the secant slope is close to the tangent slope.
Think about the above picture, P is a point on the known curve C, how to find the tangent equation of P?
Not set
Therefore, when infinity approaches 0, infinity approaches the slope of the tangent at that point.
Variant training
1。 Known, find the tangent slope and tangent equation of the curve at;
2。 Known, find the tangent slope and tangent equation of the curve at;
3。 It is known to find the tangent slope and tangent equation of the curve at the point.
class exercise
It is known to find the tangent slope and tangent equation of the curve at the point.
Fourth, review summary
The tangent of point P on the 1. curve is the straight line closest to the curve near point P among all the straight lines passing through point P, so the change trend of point P can be reflected by the tangent of this point (local curves are replaced by straight lines).
2. According to the definition, the tangent slope and equation of the curve at a point can be obtained by using the method of secant approaching tangent.
V. Homework
Chapter 5: Selected high school mathematics teaching plan template 1 model essay. Teaching objectives:
Master the concept, coordinate representation and operational properties of vectors to achieve mastery. The related properties of vectors can be applied to solve problems such as plane geometry and analytic geometry.
Second, the teaching focus:
Properties of vectors and comprehensive application of related knowledge.
Third, the teaching process:
(1) Main knowledge:
1. Master the concept, coordinate representation and operational properties of vectors to achieve mastery. The related properties of vectors can be applied to solve problems such as plane geometry and analytic geometry.
(B) Case analysis: omitted
Fourth, summary:
1, further proficient in vector operation and proof; Can use the knowledge of triangle solution to solve related application problems,
2. Infiltrate the idea of mathematical modeling and effectively cultivate the ability to analyze and solve problems.
Verb (short for verb) Homework:
leave out
Part VI: Selected model essays of high school mathematics teaching plan template I. Teaching objectives.
Knowledge and skills
Master the monotonicity and range of trigonometric functions.
Process and method
Experience the monotonicity exploration process of trigonometric function and improve the logical reasoning ability.
Emotional attitude values
In the process of guessing, improve the interest in learning mathematics.
Second, the difficulties in teaching
Teaching focus
Monotonicity and range of trigonometric functions.
Teaching difficulties
The process of exploring monotonicity and range of trigonometric functions.
Third, the teaching process
(A) the introduction of new courses
Ask a question: How to study the monotonicity of trigonometric functions?
(2) Summarize the homework
Question: What did you learn today?
Guide students to review: basic inequalities and the process of derivation and proof.
Homework after class:
Think about how to compare the values of trigonometric functions with monotonicity.