Deputy Director for Water Resources Management,
math teacher
Municipal educational institution "Secondary school No. 65 named after. B.P.Agapitova UIPMEC"
city of Magnitogorsk
y=kx + b
The graph of the equation y=kx + b is a straight line. When b=0, the equation takes the form y=kx, its graph passes through the origin.
1.y=3x-7 and y=-6x+2
3 is not equal to –6, then the graphs intersect.
2. Solve the equation:
3x-7=-6x+2
1-abscissa of the intersection point.
3. Find the ordinate:
Y=3x-7=-6x+2=3-7=-4
-4-ordinate of the intersection point
4. A(1;-4) coordinates of the intersection point.
Geometric meaning of the coefficient k
The angle of inclination of the straight line to the X axis depends on the values of k.
Y=0.5x+3
Y=0.5x-3.3
As /k/ increases, the angle of inclination to the X axis of straight lines increases.
k are equal to 0.5 and the angle of inclination to the X axis is the same for straight lines
The coefficient k is called the slope
From value b depends on the ordinate of the point of intersection with the axis Y .
b=4,(0,4)- dot
Y-Axis Intersections
b=-3,(0,-3)- Y-intercept point
1. The functions are given by the formulas: Y=X-4, Y=2x-3,
Y=-x-4, Y=2x, Y=x-0.5 . Find pairs of parallel lines. Answers:
A) y=x- 4 And y=2x b) y=x-4 And y=x-0.5
V) y=-x-4 And y=x-0.5 G) y=2x And y=2x-3
Slide 1
Algebra lesson in 7th grade “Linear function and its graph” Prepared by Tatchin U.V. mathematics teacher MBOU secondary school No. 3, SurgutSlide 2
Goal: developing the concept of “linear function”, the skill of constructing its graph using an algorithm. Objectives: Educational: - study the definition of a linear function, - introduce and study the algorithm for constructing a graph of a linear function, - practice the skill of recognizing a linear function using a given formula, graph, verbal description. Developmental: - develop visual memory, mathematically literate speech, accuracy, accuracy in construction, ability to analyze. Educational: - to cultivate a responsible attitude to academic work, accuracy, discipline, perseverance. - develop skills of self-control and mutual control
Slide 3
Lesson Plan: I. Organizational moment II. Update background knowledge III. Studying new topic IV. Consolidation: oral exercises, graphing tasks V. Solving entertaining tasks VI. Summing up the lesson, recording homework VII. Reflection
Slide 4
I. Organizational moment Having solved the words horizontally, you will learn keyword 1. An exact set of instructions describing the order of actions of the performer to achieve the result of solving a problem in a finite time 2. One of the coordinates of a point 3. The dependence of one variable on another, in which each value of the argument corresponds to a single value of the dependent variable 4. The French mathematician who introduced the rectangular coordinate system 5. An angle whose degree measure is greater than 900 but less than 1800 6. Independent variable 7. The set of all points of the coordinate plane, the abscissas of which are equal to the values of the argument, and the ordinates are equal to the corresponding values of the function 8. The road that we choose A L G O R I T M A B S C I S S A F U N C C I Y D E C A R T T U P O Y A R G U M E N T GRAPH I C P R Y M A Y
Slide 5
1. An exact set of instructions describing the order of actions of the performer to achieve the result of solving a problem in a finite time 2. One of the coordinates of a point 3. The dependence of one variable on another, in which each value of the argument corresponds to a single value of the dependent variable 4. The French mathematician who introduced the rectangular coordinate system 5. An angle whose degree measure is greater than 900 but less than 1800 6. Independent variable 7. The set of all points of the coordinate plane, the abscissas of which are equal to the values of the argument, and the ordinates are equal to the corresponding values of the function 8. The road that we choose A L G O R I T M A B S C I S S A F U N K C I A D E C A R T T U P O Y A R G U M E N T GRAPH I C P R Y M A Y
Slide 6
II. Updating basic knowledge Many real situations are described by mathematical models that represent linear functions. Let's give an example. The tourist traveled 15 km by bus from point A to point B, and then continued moving from point B in the same direction to point C, but on foot, at a speed of 4 km/h. At what distance from point A will the tourist be after 2 hours, after 4 hours, after 5 hours of walking? The mathematical model of the situation is the expression y = 15 + 4x, where x is the walking time in hours, y is the distance from A (in kilometers). Using this model, we answer the question of the problem: if x = 2, then y =15 + 4 ∙ 2 = 23 if x = 4, then y = 15 + 4 ∙ 4= 31 if x = 6, then y = 15 + 4 ∙ 6 = 39 The mathematical model y = 15 + 4x is a linear function. A B C
Slide 7
III. Studying a new topic. An equation of the form y=k x+ m, where k and m are numbers (coefficients) is called a linear function. To plot a linear function, you need to specify a specific x value and calculate the corresponding y value. Typically these results are presented in table form. They say that x is the independent variable (or argument), y is the dependent variable. 2 1 1 2 x x x y y x
Slide 8
Algorithm for constructing a graph of a linear function 1) Create a table for a linear function (associate each value of the independent variable with the value of the dependent variable) 2) Construct points on the coordinate plane xOy 3) Draw a straight line through them - graph of a linear function Theorem Graph of a linear function y = k x + m is a straight line.
Slide 9
Let's consider the use of an algorithm for constructing a graph of a linear function Example 1 Construct a graph of a linear function y = 2x + 3 1) Make a table 2) Construct points (0;3) and (1;5) in the xOy coordinate plane 3) Draw a straight line through them
Slide 10
If the linear function y=k x+ m is considered not for all values of x, but only for values of x from a certain numerical set X, then they write: y=k x+ m, where x X (is the sign of membership) Let's return to the problem In our situation, the independent the variable can take on any non-negative value, but in practice a tourist cannot walk at a constant speed without sleep and rest for any amount of time. This means that it was necessary to make reasonable restrictions on x, say, a tourist walks no more than 6 hours. Now let’s write down a more accurate mathematical model: y = 15 + 4x, x 0; 6
Slide 11
Let's consider next example Example 2 Construct a graph of a linear function a) y = -2x + 1, -3; 2 ; b) y = -2x + 1, (-3; 2) 1) Compile a table for the linear function y = -2x + 1 2) Construct points (-3;7) and (2;-3) on the coordinate plane xOy and Let's draw a straight line through them. This is a graph of the equation y = -2x + 1. Next, select a segment connecting the plotted points. x -3 2 y 7 -3
Slide 12
Slide 13
We plot the function y = -2x + 1, (-3; 2) How does this example differ from the previous one?
Slide 14
Slide 15
IV. Reinforce the topic you've learned Select which function is a linear function
Slide 16
Slide 17
Slide 18
Complete the following task: A linear function is given by the formula y = -3x – 5. Find its value at x = 23, x = -5, x = 0
Slide 19
Checking the solution If x = 23, then y = -3 23 – 5=-69 – 5 = -74 If x = -5, then y = -3 (-5) – 5= 15– 5 = 10 If x = 0 , then y = -3 0– 5= 0 – 5= -5
Slide 20
Find the value of the argument at which the linear function y = -2x + 2.4 takes the value equal to 20.4? Checking the solution When x = -9 the value of the function is 20.4 20.4 = - 2x + 2.4 2x =2.4 – 20.4 2x = -18 x= -18:2 x = -9
Slide 21
Next task Without performing any construction, answer the question: to which function does A (1;0) belong to the graph?
Slide 22
Slide 23
Slide 24
Slide 25
Name the coordinates of the points of intersection of the graph of this function with the coordinate axes With the OX axis: (-3; 0) Test yourself: With the OU axis: (0; 3)
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 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 number. 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.
The last example is a graph with a negative x coefficient. 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 both by teachers when teaching lessons according to the curriculum, and by schoolchildren when studying the material independently. The clarity of this presentation makes it easy to understand educational material on this topic.
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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B k b>0b0 y=kx+b (y=2x+1) I, III quarters y=kx+b (y=2x-1) I, III quarters y=kx I, III quarters Through the beginning of the coordinate K 0b0 y=kx+b (y=2x+1) I, III quarters y=kx+b (y=2x-1) I, III quarters y=kx I, III quarters Through the beginning of the coordinate K"> 0b0 y=kx +b (y=2x+1) I, III quarters y=kx+b (y=2x-1) I, III quarters y=kx I, III quarters Through the beginning of the coordinate K"> 0b0 y=kx+b (y =2x+1) I, III quarters y=kx+b (y=2x-1) I, III quarters y=kx I, III quarters Through the beginning of the coordinate K" title="b k b>0b0 y=kx +b (y=2x+1) I, III quarters y=kx+b (y=2x-1) I, III quarters y=kx I, III quarters Through the beginning of the coordinate K"> title="b k b>0b0 y=kx+b (y=2x+1) I, III quarters y=kx+b (y=2x-1) I, III quarters y=kx I, III quarters Through the beginning of the coordinate K"> !}
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Lesson Information Card:
Academic subject: algebra
Subject:"Linear function and its graph"
Lesson type: explanation of new material
Lesson location in curriculum: third lesson in the “Functions” section. A linear function is learned after students have learned the concepts of a function and its graph, can answer questions about domain and domain, can find the value of a function from a graph, and can find the argument corresponding to the value of the function. They know how to define a function. In this lesson, students should learn the definition of a linear function and learn how to plot its graph. Determine the location of the graph depending on the numbers k and b. The main content of the material being studied is set curriculum And mandatory minimum content of education in mathematics.
Annotation: This lesson is aimed at 7th grade students with in-depth study of mathematics using the textbook “Algebra 7”, authors Yu.N. Makarychev, N.G. Mindyuk, K.I. Neshkov, I.E. Feoktistov. The lesson follows a script multimedia presentation, which saves time that the teacher spends on constructing on the board. The presentation is made using colorful illustrations, animation and sound effects. If necessary, the stage of the lesson where difficulties arose can be repeated. The lesson used materials not included in mandatory standards education.
Objective of the lesson: introduce the concept of a linear function and its graph. Test students' ability to read a graph.
Lesson objectives:
teach apply acquired knowledge to solve practical problems;
develop creativity;
intensify attention of students through the use of multimedia;
bring up interest in the subject, confidence in a positive learning outcome.
Equipment:
multimedia;
Methods:
information and development;
visual;
reproductive;
partly - search engines.
| Lesson stage | Time (min) |
||
| Organizational moment. | Creating conditions for successful joint activities | ||
| Checking homework. | Frontal and individual verification, creating a working atmosphere for the lesson. Frontal testing of theoretical material. Repetition. | ||
| Statement of the problem | Creation of a mathematical model of the problem. Formulating the purpose of the lesson. | ||
| The main part of the lesson consists of several stages | Definition of a linear function. Graph of a linear function. Methods for specifying a linear function. | ||
| First stage | Introduction of the concept of linear function. | ||
| Second stage | Graphing a Linear Function | ||
| Third stage | Location of the graph of a linear function | ||
| Summing up | Testing students' skills through independent work. Reflection. Grading. | ||
| Homework | Introducing students to homework. |
Lesson progress
Organizational moment
Hello guys. Sit down.
Checking homework
Define a function. What is the name of the independent variable? How can I define a function? What is a graph of a function?
3. Statement of the problem. The famous Polish mathematician Hugo Steinhaus jokingly claims that there is a law that is formulated as follows: a mathematician will do it better. Namely, if you entrust two people, one of whom is a mathematician, to perform any work unfamiliar to them, then the result will always be the following: the mathematician will do it better. Imagine the problem: There were 500 tons of coal in the warehouse. They began to haul away 30 tons of coal every day. How many tons of coal will be in the warehouse in x days? Let's create a mathematical model for solving this problem. (Slide No. 1)
y = 500 – 30x
Let's calculate the value for x=2 and x=5 (Slide No. 2)
Let's create a table of values in increments of 1 for x and y (Slide No. 3)
Additional questions: 1) How much coal will remain in the warehouse if it takes 7 days to remove it? 2) Will there be enough coal for 20 days?
Let's show the dependence of y on x on the coordinate plane (Slide No. 4) What did we get?
Today we will study functions that can be specified by a formula of the form y = kx+b, where k and b are some numbers other than zero. Such functions are called linear. The graph of a linear function is a straight line.
4. The main part of the lesson. Tell me, is the function y = 2x+1 linear? What will her schedule be? How many points are needed to construct a straight line? Let's conclude: To build a graph of a linear function, you need to select two argument values and find the value of the function for these argument values. Construct points on the coordinate plane. Draw a straight line through these points. So, we build a graph of the function y = 2x+1 (Slide No. 6, No. 7)
Intermediate reflection: Select linear functions (Slide No. 8)
Graph the function y = 3x-4. Check using slide number 9
Let us introduce the concept of domain of definition and domain of value of a linear function.
Let's consider the dependence of the location of the graph of a linear function on the numbers k and
b. Look at the graphs on slide No. 11 and draw a conclusion.
Schematic graphs (Slide No. 12)
Reflection: (slide number 13)
Which function is called linear? What is her schedule?
At what angle (acute or obtuse) is the straight line inclined to the x-axis if
1) k ˃0 2) k ˂ 0
What is the domain of a linear function?
What is the range of a linear function?
Independent work according to options with random checking.
No. 1063 (b, d)
Homework: No. 1065 (a, e), No. 1066, 1068 (b, d)
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