Full name of the educational institution:
Municipal educational institution secondary school No. 3 of the village of Kochubeevskoye, Stavropol Territory
Subject area: mathematics
Lesson title: "Linear function, its schedule, properties.
Age group: 7th grade
Presentation title:Linear function, its graph, properties.
Number of slides: 37
Environment (editor) in which the presentation was made: Power Point 2010
This presentation
1 slide - title
2 slide-actualization of reference knowledge: definition of a linear equation, orally choose those that are linear from those proposed.
3 slide definition of a linear function.
4 slide recognition of a linear function from the proposed ones.
5 slide output.
6 slide ways to set the function.
7 slide-I give an example, I show.
8 slide - I give an example, I show.
9 slide task for students.
10 slide - checking the correctness of the task. I draw the attention of students to the relationship between the coefficients k and b and the location of the graphs.
11 slide conclusion.
12 slide - work with a graph of a linear function.
13 slide tasks for independent solution:construct graphs of functions (perform in a notebook).
14-17 slides show the correct execution of the task.
18-27 slides - oral and written assignments. I do not choose all tasks, but only those that are suitable for the level of preparedness of the classif there is time.
28 slide task for strong students.
29 slides - let's summarize.
30-31 slides - conclusions.
32-36 slides - historical background. (if there is time)
37 slide-Used literature
List of used literature and Internet resources:
1. Mordkovich A.G. and others. Algebra: a textbook for the 7th grade of educational institutions - M.: Education, 2010.
2. Zvavich L.I. and others. Didactic materials on algebra for grade 7 - M.: Enlightenment, 2010.
3. Algebra grade 7, edited by Makarychev Yu.N. et al., Education, 2010
4. Internet resources:www.symbolsbook.ru/Article.aspx%...id%3D222
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Slides captions:
Linear function, its graph, properties. Kiryanova Marina Vladimirovna, teacher of mathematics, secondary school No. 3 p. Kochubeevskoye, Stavropol Territory
Specify the linear equations: 1) 5y = x 2) 3y = 0 3) y 2 + 16x 2 = 0 4) + y = 4 5) x + y =4 6) y = -x + 11 7) + 0.5x – 2 = 0 8) 25d – 2m + 1 = 0 9) y = 3 – 2x 5
A function of the form y = kx + b is called linear. The graph of a function of the form y = kx +b is a straight line. Only two points are needed to construct a line, since only one line passes through two points.
Find equations of linear functions y =-x+0.2; y=12, 4x-5.7; y =- 9 x- 1 8; y=5.04x; y=-5.04x; y=1 26.35+ 8.75x; y=x -0, 2; y=x:8; y=0.005x; y=13 3 ,13 3 13 3 x; y= 3 - 10 , 01x ; y=2: x ; y=-0.0049; y= x:6 2 .
y \u003d kx + b - linear function x - argument (independent variable) y - function (dependent variable) k , b - numbers (coefficients) k ≠ 0
x x 1 x 2 x 3 y y 1 y 2 y 3
y \u003d - 2x + 3 is a linear function. The graph of a linear function is a straight line, to build a straight line, you need to have two points x - an independent variable, so we will choose its values ourselves; Y is a dependent variable, its value will be obtained by substituting the selected x value into the function. We write the results in the table: x y 0 2 If x \u003d 0, then y \u003d - 2 0 + 3 \u003d 3. 3 If x=2, then y = -2 2+3 = - 4+3= -1. - 1 Mark the points (0;3) and (2; -1) on the coordinate plane and draw a straight line through them. x y 0 1 1 Y \u003d - 2x + 3 3 2 - 1 we choose ourselves
Construct a graph of a linear function y \u003d - 2 x +3 Compose a table: x y 03 1 1 Construct points (0; 3) and (1; 5) on the coordinate plane and draw a line x 1 0 1 3 y through them
Option I Option II y=x-4 y =- x+4 Determine the relationship between the coefficients k and b and the location of the lines Draw a graph of a linear function
y=x-4 y=-x+4 I option II option x y 1 2 0 -4 x 1 2 0 4 y
x 0 y y = kx + m (k > 0) x 0 y y = kx + m (k 0, then the linear function y = kx + b increases if k
Using the graph of a linear function y \u003d 2x - 6, answer the questions: a) at what value of x will y \u003d 0? b) for what values of x will y 0? c) for what values of x will y 0? 1 0 3 y 1 x -6 a) y \u003d 0 for x \u003d 3 b) y 0 for x 3 at x 3 If x 3, then the line is located below the x-axis, which means that the ordinates of the corresponding points of the line are negative
Tasks for independent solution: build graphs of functions (perform in a notebook) 1. y \u003d 2x - 2 2. y \u003d x + 2 3. y \u003d 4 - x 4. y \u003d 1 - 3x Please note: the points you have chosen to build a straight line may be different, but the location of the graphs must necessarily match
Answer to task 1
Answer to task 2
Answer to task 3
Answer to task 4
Which figure shows the graph of a linear function y = kx ? Explain answer. 1 2 3 4 5 x y x y x y x y x y
The student made a mistake while plotting the function graph. In what picture? 1. y \u003d x + 2 2. y \u003d 1.5 x 3. y \u003d -x-1 x y 2 1 x y 3 1 x y 3 3
1 2 3 4 5 x y x y y x y x y In which figure is coefficient k negative? x
What is the sign of the coefficient k for each of the linear functions:
In which figure is the free term b in the equation of a linear function negative? 1 2 3 4 5 x y x y x y x y x y
Choose a linear function whose graph is shown in the figure y = x - 2 y = x + 2 y = 2 - x y = x - 1 y = - x + 1 y = - x - 1 y = 0.5x y = x + 2 y \u003d 2x Well done! Think!
x y 1 2 0 1 2 3 -1 -2 -1 -2 x y 1 2 0 1 2 3 -1 -2 -1 -2 y=2x y=2x+ 1 y=2x- 1 y=-2x+ 1 y = - 2x-1y=-2x
y=-0.5x+ 2 , y=-0.5x , y=-0.5x- 2 x y 1 2 0 1 2 3 -1 -2 -1 -2 3 4 5 6 -3 x y 1 2 0 2 3 -1 -2 -1 -2 3 4 5 6 -3 1 y=0.5x+ 2 y=0.5x- 2 y=0.5x y=-0.5x+ 2 y=-0.5x y=-0 .5x-2
y=x+ 1 y=x- 1 , y=x y 1 2 0 1 2 3 -1 -2 -1 -2 3 4 5 6 -3 x y 1 2 0 1 2 3 -1 -2 -1 -2 3 4 5 6 -3 x y=-x y=-x+ 3 y=-x- 3 y=x+ 1 y=x- 1 y=x
Write an equation for a linear function according to the following conditions:
summarize
Write the conclusions in a notebook We learned: * A function of the form y \u003d kx + b is called linear. * The graph of a function of the form y = kx + b is a straight line. *To draw a straight line, only two points are needed, since only one straight line passes through two points. *The k coefficient shows whether the line is increasing or decreasing. *Coefficient b shows at what point the line intersects the OY axis. *Condition of parallelism of two lines.
Wish you luck!
Algebra - this word comes from the title of the work of Muhammad Al-Khwarizmi "Al-jebr and Al-muqabala", in which algebra was presented as an independent subject
Robert Record is an English mathematician who in 1556 introduced the equal sign and explained his choice by the fact that nothing can be more equal than two parallel segments.
Gottfried Leibniz - German mathematician (1646 - 1716), who first introduced the term "abscissa" - in 1695, "ordinate" - in 1684, "coordinates" - in 1692.
Rene Descartes - French philosopher and mathematician (1596 - 1650), who first introduced the concept of "function"
References 1. Mordkovich A.G. and others. Algebra: a textbook for the 7th grade of educational institutions - M .: Education, 2010. 2. Zvavich L.I. and others. Didactic materials on algebra for grade 7 - M .: Education, 2010. 3. Algebra grade 7, edited by Makarychev Yu.N. et al., Enlightenment, 2010 4. Internet resources: www.symbolsbook.ru/Article.aspx%...id%3D222
Class: 7
The function occupies one of the leading places in the school algebra course and has numerous applications in other sciences. At the beginning of the study, in order to motivate, update the issue, I inform you that not a single phenomenon, not a single process in nature can be studied, no machine can be designed, and then operate without a complete mathematical description. One tool for this is a function. Its study begins in the 7th grade, as a rule, children do not delve into the definition. Particularly hard-to-reach concepts are such as domain of definition and domain of value. Using the known connections between the quantities in the problems of movement, the costs are shifting them into the language of the function, keeping the connection with its definition. Thus, in students the concept of function is formed at a conscious level. At the same stage, painstaking work is carried out on new concepts: domain of definition, domain of value, argument, value of a function. I use advanced learning: I introduce the notation D(y), E(y), introduce the concept of the zero of a function (analytically and graphically), when solving exercises with areas of constant sign. The earlier and more often students encounter difficult concepts, the better they are realized at the level of long-term memory. When studying a linear function, it is advisable to show the connection with the solution of linear equations and systems, and later with the solution of linear inequalities and their systems. At the lecture, students receive a large block (module) of new information, so at the end of the lecture, the material is "wrung out" and a summary is drawn up that students should know. Practical skills are developed in the process of performing exercises using various methods based on individual and independent work.
1. Some information about the linear function.
Linear function is very common in practice. The rod length is a linear function of temperature. The length of rails, bridges is also a linear function of temperature. The distance traveled by a pedestrian, train, car at a constant speed is a linear function of the time of movement.
A linear function describes a number of physical dependencies and laws. Let's consider some of them.
1) l \u003d l o (1 + at) - linear expansion of solids.
2) v \u003d v o (1 + bt) - volumetric expansion of solids.
3) p=p o (1+at) - the dependence of the resistivity of solid conductors on temperature.
4) v \u003d v o + at - the speed of uniformly accelerated movement.
5) x= x o + vt is the coordinate of uniform motion.
Task 1. Define a linear function from tabular data:
| X | 1 | 3 |
| at | -1 | 3 |
Decision. y \u003d kx + b, the problem is reduced to solving the system of equations: 1 \u003d k 1 + b and 3 \u003d k 3 + b
Answer: y \u003d 2x - 3.
Problem 2. Moving uniformly and rectilinearly, the body passed 14 m in the first 8s, and 12 m in another 4s. Compose an equation of motion based on these data.
Decision. According to the condition of the problem, we have two equations: 14 \u003d x o +8 v o and 26 \u003d x o +12 v o, solving the system of equations, we get v \u003d 3, x o \u003d -10.
Answer: x = -10 + 3t.
Problem 3. A car leaving the city moving at a speed of 80 km/h. After 1.5 hours, a motorcycle drove after him, the speed of which was 100 km/h. How long will it take for the bike to overtake him? How far from the city will this happen?
Answer: 7.5 hours, 600 km.
Task 4. The distance between two points at the initial moment is 300m. The points move towards each other with speeds of 1.5 m/s and 3.5 m/s. When will they meet? Where will it happen?
Answer: 60 s, 90 m.
Task 5. A copper ruler at 0 ° C has a length of 1 m. Find the increase in its length with an increase in its temperature by 35 o, by 1000 o C (the melting point of copper is 1083 o C)
Answer: 0.6mm.
2. Direct proportionality.
Many laws of physics are expressed through direct proportionality. In most cases, a model is used to write these laws.
in some cases -
Let's take a few examples.
1. S \u003d v t (v - const)
2. v = a t (a - const, a - acceleration).
3. F \u003d kx (Hooke's law: F - force, k - stiffness (const), x - elongation).
4. E = F/q (E is the strength at a given point of the electric field, E is const, F is the force acting on the charge, q is the magnitude of the charge).
As a mathematical model of direct proportionality, one can use the similarity of triangles or the proportionality of segments (Thales' theorem).
Task 1. The train passed a traffic light in 5 seconds, and past a platform 150 m long, in 15 seconds. What is the length of the train and its speed?
Decision. Let x be the length of the train, x+150 be the total length of the train and the platform. In this problem, the speed is constant, and the time is proportional to the length.
We have a proportion: (x + 150): 15 = x: 5.
Where x = 75, v = 15.
Answer. 75 m, 15 m/s.
Problem 2. The boat went downstream 90 km in some time. In the same time, he would have passed 70 km against the current. How far will the raft travel in this time?
Answer. 10 km.
Task 3. What was the initial temperature of the air if, when heated by 3 degrees, its volume increased by 1% of the original.
Answer. 300 K (Kelvin) or 27 0 C.
Lecture on the topic "Linear function".
Algebra, 7th grade
1. Consider examples of tasks using well-known formulas:
S = v t (path formula), (1)
C \u003d c c (cost formula). (2)
Problem 1. The car, having driven away from point A at a distance of 20 km, continued its journey at a speed of 62 km/h. How far from point A will the car be after t hours? Compose an expression for the problem, denoting the distance S, find it at t = 1h, 2.5h, 4h.
1) Using formula (1), we find the path traveled by a car at a speed of 62 km/h in time t, S 1 = 62t;
2) Then from point A in t hours the car will be at a distance S = S 1 + 20 or S = 62t + 20, find the value of S:
at t = 1, S = 62*1 + 20, S = 82;
at t = 2.5, S = 62 * 2.5 + 20, S = 175;
at t = 4, S = 62*4+ 20, S = 268.
We note that when finding S, only the value of t and S changes, i.e. t and S are variables, and S depends on t, each value of t corresponds to a single value of S. Denoting the variable S for Y, and t for x, we get a formula for solving this problem:
Y= 62x + 20. (3)
Problem 2. A textbook was bought in a store for 150 rubles and 15 notebooks for n rubles each. How much did you pay for the purchase? Make an expression for the problem, denoting the cost C, find it for n = 5,8,16.
1) Using formula (2), we find the cost of notebooks С 1 = 15n;
2) Then the cost of the entire purchase is С= С1 +150 or С= 15n+150, we find the value of C:
at n = 5, C = 15 5 + 150, C = 225;
at n = 8, C = 15 8 + 150, C = 270;
at n = 16, C = 15 16+ 150, C = 390.
Similarly, we note that C and n are variables, for each value of n there corresponds a single value of C. Denoting the variable C for Y, and n for x, we get the formula for solving problem 2:
Y= 15x + 150. (4)
Comparing formulas (3) and (4), we make sure that the variable Y is found through the variable x according to one algorithm. We considered only two different problems that describe the phenomena around us every day. In fact, there are many processes that change according to the obtained laws, so such a relationship between variables deserves to be studied.
Problem solutions show that the values of the variable x are chosen arbitrarily, satisfying the conditions of the problems (positive in problem 1 and natural in problem 2), i.e. x is an independent variable (it is called an argument), and Y is a dependent variable and there is a one-to-one correspondence between them , and by definition such a dependence is a function. Therefore, denoting the coefficient at x by the letter k, and the free term by the letter b, we obtain the formula
Y= kx + b.
Definition.View function y= kx + b, where k, b are some numbers, x is an argument, y is the value of the function, is called a linear function.
To study the properties of a linear function, we introduce definitions.
Definition 1. The set of admissible values of an independent variable is called the domain of the function definition (admissible - this means those numerical values x for which calculations y are performed) and is denoted by D (y).
Definition 2. The set of values of the dependent variable is called the range of the function (these are the numerical values that y takes) and is denoted by E(y).
Definition 3. The graph of a function is a set of points of the coordinate plane, the coordinates of which turn the formula into a true equality.
Definition 4. The coefficient k at x is called the slope.
Consider the properties of a linear function.
1. D(y) - all numbers (multiplication is defined on the set of all numbers).
2. E(y) - all numbers.
3. If y \u003d 0, then x \u003d -b / k, the point (-b / k; 0) - the point of intersection with the Ox axis, is called the zero of the function.
4. If x= 0, then y= b, the point (0; b) is the point of intersection with the Oy axis.
5. Find out in which line the linear function will line up the points on the coordinate plane, i.e. which is the graph of the function. To do this, consider the functions
1) y= 2x + 3, 2) y= -3x - 2.
For each function we will make a table of values. Let's set arbitrary values for the variable x, and calculate the corresponding values for the variable Y.
| X | -1,5 | -2 | 0 | 1 | 2 |
| Y | 0 | -1 | 3 | 5 | 7 |
Having built the resulting pairs (x; y) on the coordinate plane and connecting them for each function separately (we took the values of x with a step of 1, if you reduce the step, then the points will line up more often, and if the step is close to zero, then the points will merge into a solid line ), we notice that the points line up in a straight line in case 1) and in case 2). Due to the fact that the functions are chosen arbitrarily (build your own graphs y= 0.5x - 4, y= x + 5), we conclude that that the graph of a linear function is a straight line. Using the property of a straight line: a single straight line passes through two points, it is enough to take two points to construct a straight line.
6. It is known from geometry that lines can either intersect or be parallel. We investigate the relative position of the graphs of several functions.
1) y= -x + 5, y= -x + 3, y= -x - 4; 2) y= 2x + 2, y= x + 2, y= -0.5x + 2.
Let's build groups of graphs 1) and 2) and draw conclusions.
|
 |
Graphs of functions 1) are located in parallel, examining the formulas, we notice that all functions have the same coefficients at x.
Function graphs 2) intersect at one point (0;2). Examining the formulas, we notice that the coefficients are different, and the number b = 2.
In addition, it is easy to see that the lines given by linear functions with k › 0 form an acute angle with the positive direction of the Ox axis, and an obtuse angle with k ‹ 0. Therefore, the coefficient k is called the slope coefficient.
7. Consider special cases of a linear function, depending on the coefficients.
1) If b=0, then the function takes the form y= kx, then k = y/x (the ratio shows how many times it differs or what part is y from x).
A function of the form Y= kx is called direct proportionality. This function has all the properties of a linear function, its feature is that when x=0 y=0. The graph of direct proportionality passes through the origin point (0; 0).
2) If k = 0, then the function takes the form y = b, which means that for any values of x, the function takes the same value.
A function of the form y = b is called a constant. The graph of the function is a straight line passing through the point (0;b) parallel to the Ox axis, with b=0 the graph of the constant function coincides with the abscissa axis.
Abstract
1. Definition A function of the form Y= kx + b, where k, b are some numbers, x is an argument, Y is the value of the function, is called a linear function.
D(y) - all numbers.
E(y) - all numbers.
The graph of a linear function is a straight line passing through the point (0;b).
2. If b=0, then the function takes the form y= kx, called direct proportionality. The direct proportionality graph passes through the origin.
3. If k = 0, then the function takes the form y= b, is called a constant. The graph of the constant function passes through the point (0;b), parallel to the x-axis.
4. Mutual arrangement of graphs of linear functions.
The functions y= k 1 x + b 1 and y= k 2 x + b 2 are given.
If k 1 = k 2, then the graphs are parallel;
If k 1 and k 2 are not equal, then the graphs intersect.
5. See examples of graphs of linear functions above.
Literature.
- Textbook Yu.N. Makarychev, N.G. Mindyuk, K.I. Neshkov and others. "Algebra, 8".
- Didactic materials on algebra for grade 8 / V.I. Zhokhov, Yu.N. Makarychev, N.G. Mindyuk. - M .: Education, 2006. - 144 p.
- Supplement to the newspaper September 1 "Mathematics", 2001, No. 2, No. 4.
"linear function". 7th grade
Goals:
Educational:
Repeat, generalize, consolidate, test knowledge and skills on the topic "Linear function";
To form the ability to synthesize and generalize the knowledge gained in the lessons of mathematics and physics.
Developing:
Development of skills for plotting function graphs y \u003d kx + b;
Development of logical thinking, initiative, independence;
Development of skills to analyze and draw conclusions.
Educational:
To cultivate accuracy, graphic culture, speech culture;
Develop the ability to work in groups, listen to the opinion of a partner.
Equipment:
Handout;
Multimedia - projector;
A computer.
Lesson type: generalizing.
Work form: frontal
DURING THE CLASSES.
1. Organizational moment. (Slide #2)
The teacher announces the topic of the lesson.
2. Setting goals and objectives of the lesson. (Slide #3)
The teacher together with the students formulate the goals and objectives of the lesson.
3. Reflection. (Slide number 4).
Teacher: Choose from the proposed drawings the one that matches your mood at the beginning of the lesson and mark it.
If you feel good, you are ready to learn new material and you think that all the questions will be clear to you, then choose a happiness emoticon.
If you are worried that you are not ready enough to learn new material and are worried that not all questions will be clear to you, then choose a sadness emoticon.
If you are worried that you are not at all ready to learn new material and most of the questions will be incomprehensible to you, then choose a crying emoticon.
CHECK HOMEWORK
4. Oral repetition of key questions of algebra.
Frontal work with the class . (Slide number 5).
What is a linear function?
Its scope?
Under what condition does a linear function become a direct proportionality?
What is the graph of a linear function and direct proportionality?
How to plot a linear function (direct proportionality)?
What is the reason for the difference in the graphs of these functions?
What kinds of linear function y = kx + b do you know? (Slide number 6)
5. Independent work.
Students are asked to complete the following tasks in writing in the form of a test. (Slides #7 - 15)
When taking the test, students fill out an answer sheet. (See Attachment).
Which function graph is redundant? (Slide number 8)
In which figure is the coefficient k in the equation of a linear function negative? (Slide number 9)
In which figure is the free term b in the equation of a linear function positive?
(Slide number 10)
Make up the equations of the lines shown in the figures. (Slide number 11)
Which figure shows a graph of direct proportionality y \u003d kx? Explain answer.
(Slide number 12)
The student made a mistake when plotting a graph of one function. In what picture?
(Slide number 13)
The figure shows the function graphs: y \u003d 3x, y \u003d - 3x, y \u003d x - 3. What number shows the graph of the function y \u003d -3x? (Slide number 14)
Set the formula to a linear function whose graph is parallel to the straight line y \u003d -8x + 11 and passes through the origin. (Slide number 15)
The work performed is checked. (Slides #16-24))
6. Work with the class.
Make a mathematical model to solve the problem. (Slide number 25)
In the human body there is always a certain number of bacteria, there are about 10 thousand of them. During an influenza epidemic, if the patient does not take antibiotics, the number of bacteria in the body increases by 50,000 every day.
How many bacteria will be in the human body after 3 days, after 4 days?
Write the formula in your notebook and answer the following questions:
Will this relationship be linear?
What can you say about the behavior of the graph of this function?
Make this chart in your notebook.
Students complete this task on their own. After that, the decision is discussed with all students. (Slide number 26)
WORK WITH CARDS
7. Mathematics is an applied science and now you will consider the application of a linear function in other sciences and areas of our life.
Class work.
Problems for the application of a linear function in physics are considered. (Slides #27 - 32)
Tasks are considered in
Anatomy (Slides No. 47 - 48).
Psychology (Slides No. 49 - 51).
PHYSICAL MINUTE
WORK IN PAIRS
Criminology (Slides No. 52 - 54).
Economics (Slides No. 55 - 56).
In everyday life (Slides No. 57 - 58).
Conclusion .
So, today in the lesson we examined the use of a linear function in various sciences and fields of activity (Slide No. 59)
9. Expanding horizons - the report of one of the children
Students are invited to think about the following task: What happens inside when you open the door lock? (Slide number 60 - 61)
(This task is offered to students as a home task for a group of strong students)
After that, one of the students in this group talks about the ongoing process.
It turns out that arithmetic operations can be applied to functions according to certain rules and under certain conditions. I will give a very clear example where the need to apply actions to functions occurs.
Look at the drawing. Do you know how this key opens the door? What happens inside when you open the door lock? To open the lock, you need to turn the drum in which the keyhole is made. But this is prevented by the pins, standing closely inside the well, sliding up and down. Each of the pins must be raised to such a height that their upper ends are flush with the surface of the drum. This makes the key.
From the point of view of mathematics, all this mechanics is nothing but the operation of adding two functions. One of them is the profile of the key, the other is the line that outlines the top ends of the pins when the lock is locked. The secret of the door lock is that as a result of adding two functions, a constant function is obtained, the constant value of which is equal to the diameter of the drum.
10. Summing up the lesson. (Slides No. 62 - 63).
Teacher: Let's repeat it again.
What did you learn new?
What have you learned?
What did you find especially difficult?
11. Homework. (Slide number 64).
12. Reflection:
Teacher: With what mood you leave the lesson, you show by choosing an emoticon. (Slide number 65)
Teacher: Lesson is over! All the best!
Thank you for the lesson. (Slide number 66)
13. Literature:
Textbook "Algebra - 7", Yu.N. Makarychev, N.G. Mindyuk, K.I. Neshkov, S.B. Suvorov, Moscow, Enlightenment, 2009.
Textbook "Physics - 7", N.V. Peryshkin, Moscow, "Drofa" 2009.
"Collection of tasks in physics for grades 7 - 9", V.I. Lukashik, E.V. Ivanova, Moscow, Enlightenment, 2008.
Frontal laboratory classes in physics in grades 7-11, Moscow, Enlightenment,
2008
Internet resources.
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Lesson summary
Certified teacher: Sindeeva Elena Nikolaevna ___________________________________________________
Subject: Algebra______________________________ Grade 7______________________________________
Lesson topic: "Graphs of linear functions." _________________________________________________________
The objectives of the study of the topic:
Metasubject (developing):
Communicative: create conditions for the development of communication skills;
Regulatory: create conditions for the development of skills to analyze, compare, draw a conclusion; to show initiative and independence;
Cognitive: create conditions for the formation of the skill of working with ready-made tests;
Subject (educational): to promote the assimilation of the mutual arrangement of graphs of linear functions;
create conditions for the formation of skills to apply the acquired knowledge.
Personal (educational): to promote the development of a positive attitude towards academic work; skill
express your point of view and listen to someone else's.
Lesson objectives:
Check homework.
Repeat the theoretical material on the previous topic.
Strengthen the ability to work according to ready-made schedules.
Develop the ability to observe, analyze, draw conclusions.
Check understanding of the material.
Lesson type: primary consolidation of new knowledge.
Educational and didactic support of the lesson and teaching aids:, tests, individual cards, tables, presentation.
| Stages of work | (to be completed by the teacher) |
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| Organizing time, including: setting a goal that should be achieved by students at this stage of the lesson (what should be done by students so that their further work in the lesson is effective) description of the methods of organizing the work of students at the initial stage of the lesson, the mood of students for learning activities, the subject and topic of the lesson (taking into account the real characteristics of the class with which the teacher works) | Teacher: Hello guys! Today we will continue to work on the study of the relative position of the graphs of linear functions. We must study the relative position of the graphs of linear functions and be able to apply them in practice. The purpose of the stage of the lesson: To promote the development of a positive attitude to educational work, the ability to express one's point of view and listen to someone else's. Didactic tasks of the lesson stage: Get involved in the business rhythm, prepare for work, develop communication skills, develop the ability to analyze the action plan. Method of organizing the work of students: Oral communication of the teacher. Form of organization of educational activities: Conversation. Teacher: Today we are working using images on the TV screen, please follow the rules of conduct in the lesson. Everyone has a sheet with a lesson plan on the table, where you will make your suggestions. Try to be active. At the end of the lesson, please indicate your attitude to the lesson and indicate your mood. Teacher activity: Voices the topic, plan and purpose of the lesson. Student activities: Analyze and comment on the lesson plan. Teacher: Guys, here is the lesson plan, analyze it and make your suggestions. Lesson plan: oral work. Card work. Checking homework. Oral performance of tasks on the topic, according to ready-made schedules. Independent work on options in pairs. Test execution. Summarizing. Homework. Result: Students analyze the lesson plan, make their suggestions. |
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| Survey of students on the material given at home, including: determination of the goals that the teacher sets for the students at this stage of the lesson (what result should be achieved by the students); definition of goals and objectives that the teacher wants to achieve at this stage of the lesson; description of methods that contribute to the solution of the goals and objectives; description of the criteria for achieving the goals and objectives of this stage of the lesson; determination of possible actions of the teacher in case he or students fail to achieve their goals; description of methods for organizing joint activities of students, taking into account the characteristics of the class with which the teacher works; description of methods of motivating (stimulating) the educational activity of students during the survey; description of methods and criteria for evaluating student responses during the survey. | Teacher: 3 people work at the blackboard, solve examples from homework: I: y=-4x-1 and y=2x+5 II: y=-2x+3 and y=x-6 A) parallel to the graph of the function B) parallel to the graph of the function and passes through the origin C) intersects with the graph of the function D) intersects with the graph of the function at point A (0; -42) 2 people work on cards. (Appendix 1) The purpose of the stage of the lesson: To create conditions for the development of skills to analyze, compare, draw a conclusion, for the manifestation of initiative and independence. Didactic tasks of the lesson stage: Reveal the level of knowledge on homework, identify typical mistakes, correct knowledge. Method of organizing the work of students: Self-analysis, self-assessment. Form of organization of educational activities: Individual cards, work at the blackboard, conversation. Teacher activities: Offers tasks on cards, organizes a conversation using previously studied material. Student activity: Solve the task on the card, answer the questions of the teacher and students. Result: Students find the coordinates of the intersection points of the graphs of linear functions, explaining what additional knowledge was used. The rest of the guys correct the mistakes and complete the answers. Those who answer at the blackboard get a mark. Teacher: While the guys are solving problems on the board, we will repeat the main points learned in the last lesson, answer the questions orally. The purpose of the stage of the lesson: To activate the knowledge of students necessary to complete the test work. Didactic tasks of the lesson stage: repeat the concepts of a function, a graph of a function, fixing the geometric meaning of the coefficient k and b functions y = kx + b; mutual arrangement of graphs of linear functions. The teacher's activity: asks questions, controls the correctness of the answer, corrects the wrong answers together with the students. Student activities: Answer the questions: (Appendix 2. Presentation. Slides 5,6,7) Method of organizing the work of students: Partial-search. Form of organization of educational activities: Frontal work. What is a linear function? What is the graph of a linear function? How many points must be marked on the plane to draw a line? How to plot a linear function? What is a direct proportional function? What is a direct proportional graph? In what coordinate quarters is the graph of the function y=k x for k0‚k What is k called? What depends on k on the graph? What is the relative position of two straight lines in a plane?
Outcome: Answer questions. Teacher: let's check the correctness of the homework (Slide 9,10,11), work on the cards, well done guys, they did everything right. And now we will solve the following task together. Write down the number 1.11.13, class work and the topic of the lesson: Generalization of the topic - the relative position of the graphs of a linear function. Task: (Appendix 1. Presentation. Slide 13)
Among the functions given by the formulas y=x+0.5 (1) ; y \u003d -0.5x + 4 (2); y=5x-1 (3); y \u003d 1 + 0.5x (4); y=2x-5 (5); y=0.5x-2 (6) name those whose graphs a) parallel to the graph of the function y \u003d 0.5x + 4 b) intersects with the graph of the function y \u003d 2x + 3 c) coincides with the graph of the function y \u003d 4-0.5x The purpose of the stage of the lesson: To form a cognitive motive. Education of personal qualities of students (kindness, attention, helping those in need). Didactic tasks of the lesson stage: Organize students to accept the cognitive task. Method of organizing the work of students: Creating a problem situation. Form of organization of educational activities: Problem-dialogue. Teacher activity: Creates a problem situation to find the correct answer to the question asked. Student activities: Analyze the task, outline a plan for completing the task, |
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| Physical education minute. Purpose: To prevent fatigue. | The purpose of the stage of the lesson: To create conditions for the prevention of fatigue. Without turning your head, look up-down-right-left and close your eyes. "YES" - stretch your arms up "NO" - stretch your arms forward “I don’t know” - stretch your arms to the sides. Are the following statements true: 1. The graph of direct proportionality passes through the origin, 2. The function argument is a dependent variable, 3. To build a graph of a linear function, two points are enough, 4. If k 1 \u003d k 2, then the graphs of linear functions intersect, 5. The formula y=6/x defines a linear function. |
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| Consolidation of educational material, assuming: setting a specific educational goal for students (what result should be achieved by students at this stage of the lesson); definition of goals and objectives that the teacher sets for himself at this stage of the lesson; a description of the forms and methods for achieving the set goals in the course of consolidating new educational material, taking into account the individual characteristics of the students with whom the teacher works. description of the criteria to determine the degree of assimilation by students of new educational material; Description of possible ways and methods of responding to situations when the teacher determines that some students have not mastered the new educational material. | The purpose of the lesson stage: To promote the development of a positive attitude towards educational work, create conditions for the development of skills to analyze, compare, draw a conclusion, to show initiative and independence, to form the skills to apply the acquired knowledge. Didactic tasks of the lesson stage: Reveal the level of assimilation of the material, correct knowledge, organize activities for the application of knowledge in a changed situation, analyze the success of assimilation of the material. Method of organizing the work of students: Independent work in the form of a test. (Appendix 3) Form of organization of educational activities: individual work, work in pairs. The teacher's activity: advises students on the test, organizes the verification of the exercises, focuses the attention of students on the final results of the activity, asks questions to achieve the goal of the lesson, sums up the lesson. Student activities: perform a test, carry out mutual verification, correction of knowledge, using the theory of this paragraph of the textbook, analyze the work of comrades, answer the teacher's questions when summing up the lesson. Result: Students complete the test, rate their neighbor in the desk, sort out all the questions and problems that arise. Teacher: !. What did we learn in class today? 2. Why do we need to know the relative position of the graphs of linear functions? 3. When will we need it? The result of the lesson: summing up, achieving the goal of the lesson, marking. |
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| Homework, including: setting goals for independent work for students (what students should do in the course of doing homework); determining the goals that the teacher wants to achieve by setting homework; defining and explaining to students the criteria for successful completion of homework. | The purpose of the lesson stage: Together with the students, determine a plan for completing homework, give the necessary explanations, check the corresponding entry in the diaries. Didactic tasks of the lesson: Understand the content and methods of doing homework. Method of organizing the work of students: Verbal. Form of organization of educational activities: Consultation. Teacher activity: Gives comments on homework. Student activity: Write the task in the diary. Homework: Having a list of 10 tasks on the topic of the chapter and not only (in 2 versions), (Appendix 4) the task of the students is, having an idea of the upcoming test, to complete those of the proposed tasks that, in the opinion of the students, are most necessary for them to prepare. Result: Write down the task in the diary, listen to the teacher's comments, ask questions. |
APPENDIX#1
CARD #1
1. The equation of a straight line has the form y \u003d kx + v. for the function y \u003d 8 + 2x, write down what are equal to k and in?
2. Construct graphs of functions y = 3-x and y = -x in one coordinate system.
CARD #2
What is the name of the function y \u003d 2x - 3?
Construct graphs of functions y = x + 2 and y = x in one coordinate system.
APPENDIX#3
1 OPTION
a) y=2x-1 and y=2x+3
A) intersect
B) parallel
B) match
b) y=3x+2 and y=2x-3
A) intersect
B) parallel
B) match
c) y=0.5x+ and y=0.75 +x
A) intersect
B) parallel
B) match
a) y \u003d 12x -8 and y \u003d? x + 4 intersected
c) y \u003d 12x - 8 and y \u003d ?x - ? matched.
OPTION 2
1. Without constructing, determine the relative position of the function graphs:
a) y=6x-1 and y=4x+5
A) intersect
B) parallel
B) match
b) y=x-0.5 and y=-+0.6x
A) intersect
B) parallel
B) match
c) y \u003d 0.5x + 2 and y \u003d 0.5x -4
A) intersect
B) parallel
B) match
2. Select and insert such a number instead of the question mark so that the graphs of the functions:
a) y \u003d -27x + 1 and y \u003d? x -9 intersected
c) y \u003d -27x + 1 and y \u003d? x -? matched.
3. Compose a function for the graph shown in the figure:
APPENDIX#4
Option I
1. Reduce the fraction:
a B C)
2. Plot Equation 3 X + at+1 = 0. Does the point A (; -3) belong to it?
3. Plot the linear function graph y = -2x + 1.
Use the chart to find:
a) the largest and smallest values of the function on the segment [-1; 2];
b) variable values X, at which at = 0, at
4. Transform Equation 2 X – at– 3 = 0 to the form of a linear function y=kx + m. What are equal k and m?
5. Find the largest and smallest values of the linear function 2 X – at– 3 = 0 on the segment [-1; 2].
3X + 2at- 6 = 0 with coordinate axes;
b) determine whether the point K (; 3.5) belongs to the graph of this equation.
at = 3 - X and at = 2X.
y=kx +
m k and m?
y=kx formula if it is known that its graph is parallel to the line -3 X + at – 4 = 0.
10. At what value R solution of equation 5 X + RU – 3R= 0 is a pair of numbers (1;1)
Option II.
1. Reduce the fraction:
a B C)
2. Plot Equation 2 X - at– 3 = 0. Does the point A (; 2) belong to it?
3. Graph the linear function y = 2x - 3.
Use the chart to find:
a) the largest and smallest values of the function on the segment [-2; one];
b) variable values X, at which at = 0, at 0.
4. Transform Equation 3 X + at– 2 = 0 to the form of a linear function y=kx + m. What are equal k and m?
5. Find the largest and smallest values of the linear function 3 X + at– 2 = 0 on the segment [-1; one].
6. a) Find the coordinates of the point of intersection of the graph of the linear equation
2X - 5at- 10 = 0 with coordinate axes;
b) determine whether the point M (-; -2.6) belongs to the graph of this equation.
7. Find the coordinates of the point of intersection of the lines at = - X and at = X - 2.
8. The figure shows a graph of a linear function y=kx +
m. What are the values of the coefficients k and m?
9. a) Define a linear function y=kx formula if its graph is known to be parallel to the line 4 X + at + 7 = 0.
b) Determine whether the given function is increasing or decreasing. Explain the answer.
10. At what value R solution of the equation - px + 2y + R= 0 is a pair of numbers (-1;2)
