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Participants: 8th grade of a correctional school (or 7th grade of a general education school).

Lesson time: 1 academic hour (35 minutes).

Lesson Objectives:

1. Strengthen knowledge and skills on the topic “Function y=kx”;
2. Learn to build a graph of a linear function;
3. Develop a desire for independence research activities;
4. Continue to develop the ability to work with drawing tools (ruler).

Lesson Objectives:

1. Conduct a comparative analysis of the functions y=kx and y=kx+b;
2. Introduce students to the concept of “Linear function” and its graph;

Equipment for the lesson:

1. Textbook Sh.A. Alimova “Algebra 7”;
2. Presentation on the topic “Linear function and its graph”;
3. Computer;
4. Touch screen;
5. Cards with images of graphs of functions y=2x and y= – 2x ( Annex 1);
6. Cards with tasks for constructing a graph of a linear function ( appendix 2);
7. Card “Rectangular coordinate system” ( Appendix 3);
8. Cards for research work"Similarities and differences" ( Appendix 4);
9. Card “Definition of a linear function” ( Appendix 5).

Lesson Plan:

1. Organizational moment – ​​2 min;
2. Updating knowledge – 5 min;
3. Explanation of new material – 15 min;
4. Problem solving – 10 min;
5. Summing up the lesson – 2 min;
6. Homework- 1 min.

During the classes

I. Organizational moment

Checking compliance with the orthopedic regimen of students; recording the date of the lesson, lesson topic; familiarizing students with the goals and objectives of the lesson.

II. Updating knowledge

Exercise 1: graph the function y=2x.

To complete the task, students with severe damage to the musculoskeletal system are given the “Rectangular coordinate system” card.

If students do not cope with the task, analyze the task together with the students.

Job Analysis:

• This function belongs to the function y=kx. What object is the graph of this function?
• Through how many points can a straight line be drawn unambiguously?
• This means that in order to construct a graph of the function y=2x, it is necessary to construct two points in the coordinate system that belong to this function. How to find the coordinates of a point that belongs to the graph of a function given by the formula?

After the analysis, students independently construct a graph.

Task 2: Let's consider the properties of the constructed function.

• Is this function increasing or decreasing?
• Name the values ​​of x for which the function is positive.
• Name the values ​​of x for which the function is negative.

So, we repeated the plotting of the function y=kx and its properties. Today we will get acquainted with another type of function, which is related to the function y=kx. We will conduct a comparative analysis of the two functions to clarify their relationship. If someone is the first to see similarities and differences and draw conclusions, write them down on a card (give out a “Similarities and Differences” card).

III. Explanation of new material

A linear function is a function of the form y=kx+b, where k and b are given numbers. (slide 2)

Task 3: Functions are written on the board. Name the coefficients k and b in the linear functions indicated on the board (Figure 1):

Task 4: Orally complete 579 on page 140. Students take turns naming the function and giving a detailed answer to the question.

1. y=-x-2 – is a linear function. The coefficient before x is -2, the free term is -2.
2. y=2x2+3 – is not a linear function, since x is to the second power.
3. y=x/3- is a linear function, since the coefficient of x is 1/3, the free term is 0. Help from the teacher in case of difficulty: what number is the independent variable x multiplied by, if written x/3=x*1/3 ? What is the free term if it is not in the record?
4. y=250 is a linear function, since the coefficient of x is 0, the free term is 250. Teacher help in case of difficulty: by what number can the independent variable x be multiplied if the product kx is missing?
5. y=3/x+8 – is not a linear function, since division by x is performed, not multiplication. Teacher help in case of difficulty: When multiplying a fraction by a number, is this number multiplied by the numerator or denominator?
6. y=-x/5+1 – is a linear function, since the coefficient of x is 1/5, the free term is 1. Teacher help in case of difficulty: When multiplying a fraction by a number, is this number multiplied by the numerator or denominator?

Let's continue studying the linear function.

Let us show that the graph of a linear function, just like the graph of the function y=kx, is a straight line. To do this, we define a linear function, for example, y=x+1, in the form of a table for a certain number of points.

So, the function is given by the formula y=x+1. What are the coefficient k and the free term b of this function? Which variable is the independent one?

We will take arbitrary values ​​of the independent variable x, located close to each other on the coordinate axis:

Let's plot the found points in the coordinate system (click the mouse to display the coordinate system). We mark the points we found (click the mouse to plot the found points). Connect the constructed points (click the mouse to construct a straight line). It really turns out straight. If necessary, you can further select values ​​of the independent variable to obtain a more accurate construction.

So, the graph of a linear function is a straight line (slide 3).

How many points are enough to construct so that a straight line can be unambiguously drawn through them?

This means that to build a graph of a linear function, it is enough to (click the mouse to display the algorithm):

1. choose two convenient values ​​for the independent variable x;
2. find the value of the function from the selected x values;
3. Mark the found points on the coordinate plane;
4. Draw a straight line through the constructed points.

Task 5: in the rectangular coordinate system constructed for task 1, construct a graph of the function: y=2x+5, y=2x+3, y=2x-4, y=2x-2, y=2x+1. Give students task cards (Appendix 3). Each student constructs one of the functions (at the teacher’s discretion). When constructing a graph, try to answer the questions on the “Similarities and Differences” card yourself.

Let's check the function graphs you have built (slide 4). First, students name their chosen points.

We build a graph of the function y=2x+5 (click the mouse): take convenient points (-2;1) and (0;5), draw a straight line through them (click the mouse).

We build a graph of the function y=2x+3 (click the mouse): take convenient points (0;3) and (1;5), draw a straight line through them (click the mouse).

We build a graph of the function y=2x+1 (click the mouse): take convenient points (0;1) and (1;3), draw a straight line through them (click the mouse).

We build a graph of the function y=2x-2 (click the mouse): take convenient points (0;-2) and (1;0), draw a straight line through them (click the mouse).

We build a graph of the function y=2x-4 (click the mouse): take convenient points (0;-4) and (2;0), draw a straight line through them (click the mouse).

Previously, you plotted the function y=2x (click the mouse). Now each of you has built one more graph y=2x+5, y=2x+3, y=2x-4, y=2x-2, y=2x+1.

Last opportunity to fill out the “Similarities and Differences” cards yourself.

What do the formulas of the linear functions you constructed have in common? After receiving the answer, click the mouse.

How did the similarities show up in their graphs? After receiving the answer, click the mouse.

Why did this happen? What is the k coefficient responsible for?

Each of the constructed functions has k = 2, therefore the angles between the graphs and the Ox axis are equal, which means the lines are parallel (click the mouse).

How do the formulas of the constructed linear functions differ? After receiving the answer, click the mouse.

How did the difference show up on their graphs? After receiving the answer, click the mouse to display the coefficient b of each function and display it on the graph.

What do you think the free term b is responsible for?

What conclusion can you draw? How are the graphs of the functions y=kx and y=kx+b related to each other?

1. the graph of the function y=kx+b is obtained by shifting the graph of the function y=kx by b units along the ordinate axis (slide 5);
2. graphs of functions with identical values ​​of coefficient k are parallel lines.

Let's look at other examples:

1. The graphs of the functions y=-1/2x+1 and y=-1/2x (click the mouse) are parallel. One from the other is obtained by shifting by one unit along the Oy axis.
2. The graphs of the functions y=3x-5 and y=3x (click the mouse) are parallel. One from the other is obtained by shifting by five units along the Oy axis.
3. The graphs of the functions y=-3/7x-3 and y=-3/7x (click the mouse) are parallel. One from the other is obtained by shifting by three units along the Oy axis.

After summing up the comparison, fill out the “Similarities and Differences” cards. Provide individual assistance to students as needed.

IV. Problem solving

Task 6: construct a rectangular coordinate system with a unit segment equal to two notebook cells. In the coordinate system, construct the graphs of the functions indicated in 581. Students with severe damage to the musculoskeletal system are given a ready-made coordinate system.

V. Summing up the lesson

What function did you get acquainted with today? After receiving the answer, click the mouse and say the definition of a linear function again.

Which object is the graph of a linear function? After receiving the answer, click the mouse and once again talk about the method of constructing a graph of a linear function.

How are the graphs of the functions y=kx+b and y=kx related to each other? After receiving the answer, click the mouse and once again talk about the similarities and differences of the functions y=kx and y=kx+b.

VI. Homework

Know the definition of a linear function, 582 – to plot a graph of a linear function and to determine the values ​​of the variables x and y from the graph, 589 (oral) – give a complete answer to the question (with explanation).

Thank you for the lesson(slide 7) !

Lesson objectives: formulate a definition of a linear function, an idea of ​​its graph; identify the role of parameters b and k in the location of the graph of a linear function; develop the ability to build a graph of a linear function; develop the ability to analyze, generalize, and draw conclusions; develop logical thinking; formation of independent activity skills

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Full name of educational institution:

Municipal educational institution secondary school No. 3 in the village of Kochubeevskoye, Stavropol Territory

Subject area: mathematics

Lesson title: “Linear function, its graph, 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

Slide 2 - updating of background knowledge: definition of a linear equation, orally select those that are linear from those proposed.

Slide 3 - definition of a linear function.

4 slide recognition of a linear function from those proposed.

5 slide - conclusion.

6 slides - ways to set a function.

Slide 7 I give an example and show.

Slide 8 - I give an example and show it.

9 slide task for students.

Slide 10 - checking the correctness of the task. I draw students’ attention to the relationship between the coefficients k and b and the location of the graphs.

11 slide output.

Slide 12 - working with the graph of a linear function.

13 slide-Tasks for independent solution:build graphs of functions (do it in a notebook).

Slides 14-17 - showing the correct execution of the task.

Slides 18-27 are oral and written tasks. I don’t choose all tasks, but only those that are suitable for the level of readiness of the class.if there is time.

28 slide task for strong students.

29 slides - let's summarize.

30-31 slides - conclusions.

Slides 32-36 - historical background. (subject to time availability)

Slide 37 - Used literature

List of used literature and Internet resources:

1.Mordkovich A.G. and others. Algebra: textbook for 7th grade of general education institutions - M.: Prosveshchenie, 2010.

2. Zvavich L.I. and others. Didactic materials on algebra for grade 7 - M.: Prosveshchenie, 2010.

3. Algebra 7th grade, edited by Makarychev Yu.N. et al., Education, 2010.

4. Internet resources:www.symbolsbook.ru/Article.aspx%...id%3D222

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Slide captions:

Linear function, its graph, properties. Kiryanova Marina Vladimirovna, mathematics teacher, Municipal Educational Institution Secondary School No. 3, village. 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. To construct a straight line, only two points are needed, since only one straight line passes through two points.

Find equations of linear functions y =-x+0.2; y= 1 2 , 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=0.00 5x; y=13 3 ,13 3 13 3 x; y= 3 - 1 0 , 01x ; y=2: x ; y = -0.004 9; y= x:6 2 .

y = 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 U 1 U 2 U 3

y = - 2x + 3 – linear function. The graph of a linear function is a straight line, to construct 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 is obtained by substituting the selected value of x into the function. We write the results in the table: x y 0 2 If x = 0, then y = - 2 0 + 3 = 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= - 2x+3 3 2 - 1 we choose ourselves

Construct a graph of the linear function y = - 2 x +3 Let's make a table: x y 03 1 1 Let's construct points (0; 3) and (1; 5) on the coordinate plane and draw a line through them x 1 0 1 3 y

I option II option y=x-4 y =- x+4 Determine the relationship between the coefficients k and b and the location of the lines Plot 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 the linear function y = 2x - 6, answer the questions: a) at what value of x will y = 0? b) at what values ​​of x will y  0? c) at what values ​​of x will y  0? 1 0 3 y 1 x -6 a) y = 0 at x = 3 b) y  0 at x  3 If x  3, then the straight line is located above the x axis, which means the ordinates of the corresponding points of the straight line are positive c) y  0 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 (do it in a notebook) 1. y = 2x – 2 2. y = x + 2 3. y = 4 – x 4. y = 1 – 3x Please note: the points you choose to construct a straight line may be different, but the location of the graphs must match

Answer to task 1

Answer to task 2

Answer to task 3

Answer to task 4

Which figure shows the graph of the linear function y = kx? Explain the answer. 1 2 3 4 5 x y x y x y x y x y

The student made a mistake when graphing a function. In what picture? 1. y =x+2 2. y =1.5x 3. y =-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 picture is the coefficient k negative? x

State 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

Select the 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 = 2x Well done! Think about it!

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- 1 y =-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

Create an equation for a linear function using the following conditions:

summarize

Write down your conclusions in your notebook. We learned: *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. *To construct a straight line, only two points are needed, since only one straight line passes through two points. *Coefficient k shows whether the straight line is increasing or decreasing. *Coefficient b shows at what point the straight line intersects the OY axis. *Condition of parallelism of two lines.

I wish you success!

Algebra - this word comes from the title of the work of Muhammad Al-Khorezmi “Aljabr and Almuqabala”, 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 was a German mathematician (1646 – 1716), who was the first to introduce the term “abscissa” in 1695, “ordinate” in 1684, and “coordinates” in 1692.

Rene Descartes - French philosopher and mathematician (1596 - 1650), who first introduced the concept of “function”

Used literature 1. Mordkovich A.G. and others. Algebra: textbook for 7th grade of general education institutions - M.: Prosveshchenie, 2010. 2. Zvavich L.I. and others. Didactic materials on algebra for grade 7 - M.: Education, 2010. 3. Algebra 7th grade, edited by Makarychev Yu.N. and others, Education, 2010. 4. Internet resources: www.symbolsbook.ru/Article.aspx %...id%3D222

The presentation for 7th grade on the topic “Linear function and its graph” talks about the concept of “linear function”. During the work, students will need to convey the main idea that a linear function should contain the necessary conditions when constructing its graph.

slides 1-2 (Presentation topicand "Linear function and its graph", example)

The first slide shows the formula by which each linear formula is built. Accordingly, any function that takes the form of this formula will be linear. Students should learn this formula so that in the future they can build a graph of a linear function using it.

slides 3-4 (examples)

In order for schoolchildren to more or less understand how to use this formula, it is necessary to look at several examples that clearly show exactly how to obtain data from a specific problem and then substitute them instead of the variables of this formula. This is why the first example is given.

In the second example, a different task is given with different meanings so that students have the opportunity to consolidate the knowledge they have just acquired on this topic.

slides 5-6 (example, definition of a linear function)

The next slide shows the results of two examples, namely two equations of a linear function, compiled using the appropriate formula. Below it is broken down into its individual components. That is, it is important to convey to schoolchildren that a linear function consists of two important elements, or rather the coefficients of the binomial. If you go by the formula, then they are the variables k and b.

Next, students should carefully examine the definition of the linear function itself. In his formula, x is the independent variable, while k and b can be any numbers. In order for the linear function itself to exist, some condition must be met. It states that the number b must be equal to the condition that the number k, on the contrary, must not be equal to zero.

slides 7-8 (examples)

For greater clarity, the next slide shows an example of constructing a graph, compiled using the formula in two ways. That is, during the construction, two conditions were taken into account: first, coefficient b is equal to the number 3, second, coefficient b is equal to zero. Using the presentation, you can see that these graphs differ only in the location of the straight line along the Y axis.

In the second example of constructing a graph of a linear function, students should understand the following: firstly, the graph with a coefficient k equal to zero passes through the origin of coordinates, and secondly, the coefficient k is responsible, depending on its value, for the degree of slope of the resulting graph along the Y axis.

slides 9-10 (example, graph of a linear function)

The next slide shows an example of a special graph, where the coefficient k is equal to zero, and the function itself is equal to the value of the coefficient b.

So, having conveyed the above material to the students, the teacher must now explain that a graph constructed using a linear function is always a line, that is, a straight line.

Now you should look at several examples of plotting graphs in order to understand the dependence of the conditions for the value of the coefficients, and also learn how to determine the coordinates of points on the graph.

slides 13-14 (examples)

In example number 4, 7th grade students must independently determine the coordinates of the graph in accordance with the condition.

The following example was created to make it as clear to schoolchildren as possible how to construct a graph of a linear function with a positive coefficient x, on which the location of the line on the X axis directly depends.

slides 15-16 (examples)

For the same reason, the presentation provides an example of plotting a graph with a negative value of the coefficient x.

As last example a graph with a negative coefficient x appears. To complete it, students must determine the coordinates of the specified graph and construct a graph based on these coordinates. This slide ends the presentation.

This material can be used by both teachers when conducting lessons on curriculum, and by schoolchildren when studying the material independently. The clarity of this presentation makes it easy to understand educational material on this topic.