Back forward
Attention! Slide previews are for informational purposes only and may not represent all the features of the presentation. If you are interested this work, please download the full version.
Goals and objectives of the lesson:
- continue work on developing skills in solving systems of equations using the graphical method;
- conduct research and draw conclusions about the number of solutions to a system of two linear equations;
- develop interest in the subject through play.
DURING THE CLASSES
1. Organizing time(Planning meeting)- 2 minutes.
- Good afternoon! We are starting our traditional planning meeting. We are pleased to welcome everyone who is visiting us today in our laboratory (I represent the guests). Our laboratory is called: “WORK WITH interest and pleasure”(showing slide 2). The name serves as a motto in our work. “Create, Decide, Learn, Achieve with interest and pleasure" Dear guests, I present to you the heads of our laboratory (slide 3).
Our laboratory is engaged in the study of scientific works, research, examination, and works on the creation of creative projects.
Today the topic of our discussion is: “Graphical solution of systems linear equations" (I suggest writing down the topic of the lesson)
Program of the day:(slide 4)
1. Planning meeting
2. Extended academic council:
- Speeches on the topic
- Permission to work
3. Expertise
4. Research and discovery
5. Creative project
6. Report
7. Planning
2. Questioning and oral work (Extended Academic Council)- 10 min.
– Today we are holding an expanded scientific council, which is attended not only by the heads of departments, but also by all members of our team. The laboratory has just begun work on the topic: “ Graphic solution systems of linear equations". We must try to achieve the highest achievements in this matter. Our laboratory should be renowned for the quality of its research on this topic. As a senior researcher, I wish everyone good luck!
The results of the research will be reported to the head of the laboratory.
The floor for a report on solving systems of equations is... (I call the student to the board). I give the task a task (card 1).
And the laboratory assistant... (I give his last name) will remind you how to graph a function with a modulus. I give you card 2.
Card 1(solution to the task on slide 7)
Solve the system of equations:
Card 2(solution to the task on slide 9)
Graph the function: y = | 1.5x – 3 |
While the staff is preparing for the report, I will check how prepared you are to complete the research. Each of you must obtain permission to work. (We begin oral counting with writing down answers in a notebook)
Permission to work(tasks on slides 5 and 6)
1) Express at through x:
3x + y = 4 (y = 4 – 3x)
5x – y = 2 (y = 5x – 2)
1/2y – x = 7 (y = 2x + 14)
2x + 1/3y – 1 = 0 (y = – 6x + 3)
2) Solve the equation:
5x + 2 = 0 (x = – 2/5)
4x – 3 = 0 (x = 3/4)
2 – 3x = 0 (x = 2/3)
1/3x + 4 = 0 (x = – 12)
3) Given a system of equations:
Which of the pairs of numbers (– 1; 1) or (1; – 1) is the solution to this system of equations?
Answer: (1; – 1)
Immediately after each fragment of oral calculation, students exchange notebooks (with a student sitting next to them in the same section), the correct answers appear on the slides; The inspector gives a plus or minus. At the end of the work, department heads enter the results into the summary table (see below); 1 point is given for each example (it is possible to get 9 points).
Those who score 5 or more points are allowed to work. The rest receive conditional admission, i.e. will be required to work under the supervision of the department head.
Table (filled out by the boss)
(Tables are issued before the start of the lesson)
After receiving admission, we listen to the students’ answers at the blackboard. For the answer, the student receives 9 points if the answer is complete (the maximum number for admission), 4 points if the answer is not complete. Points are entered in the “admission” column.
If on the board correct solution, then slides 7 and 9 may not be shown. If the solution is correct, but not clearly executed, or the solution is incorrect, then the slides must be shown with explanations.
I always show slide 8 after the student’s answer on card 1. On this slide, conclusions are important for the lesson.
Algorithm for solving systems graphically:
- Express y in terms of x in each equation of the system.
- Graph each equation of the system.
- Find the coordinates of the intersection points of the graphs.
- Carry out a check (I draw students’ attention to the fact that the graphical method usually gives an approximate solution, but if the intersection of the graphs hits a point with whole coordinates, you can check and get an exact answer).
- Write down the answer.
3. Exercises (Examination)- 5 minutes.
Yesterday, serious mistakes were made in the work of some employees. Today you are already more competent in the matter of graphic solutions. You are invited to conduct an examination of the proposed solutions, i.e. find errors in solutions. Slide 10 is shown.
Work is going on in departments. (Photocopies of assignments with errors are given to each desk; in each department, employees must find errors and highlight them or correct them; photocopies must be handed over to the senior researcher, i.e. teacher). The boss adds 2 points to those who find and correct the mistake. Then we discuss the mistakes made and indicate them on slide 10.
Error 1
Solve the system of equations:
Answer: there are no solutions.
Students must continue the lines until they intersect and get the answer: (– 2; 1).
Error 2.
Solve the system of equations:
Answer: (1; 4).
Students must find the error in the transformation of the first equation and correct it on the finished drawing. Get another answer: (2; 5).
4. Explaining new material (Research and discovery)– 12 min.
I suggest that students solve three systems graphically. Each student solves independently in a notebook. Only those with conditional clearance can consult.
Solution
Without drawing graphs, it is clear that the straight lines will coincide.
Slide 11 shows the systems solution; It is expected that students will have difficulty writing down the answer in example 3. After working in the departments, we check the solution (the boss adds 2 points for a correct one). Now it's time to discuss how many solutions a system of two linear equations can have.
Students must draw conclusions on their own and explain them, listing the cases of the relative positions of lines on a plane (slide 12).
5. Creative project (Exercises)– 12 min.
The task is given for the department. The boss gives each laboratory assistant, according to his abilities, a fragment of his performance.
Solve systems of equations graphically:
After opening the brackets, students should receive the system:
After opening the parentheses, the first equation looks like: y = 2/3x + 4.
6. Report (checking the completion of the task)- 2 minutes.
After completing a creative project, students turn in their notebooks. On slide 13 I show what should have happened. The bosses hand over the table. The last column is filled out by the teacher and marked (the marks can be communicated to the students at the next lesson). In the project, the solution to the first system is assessed with three points, and the second – with four.
7. Planning (summarizing and homework)- 2 minutes.
Let's summarize our work. We did a good job. We’ll talk specifically about the results tomorrow at the planning meeting. Of course, all laboratory assistants, without exception, mastered the graphical method of solving systems of equations and learned how many solutions a system can have. Tomorrow each of you will have a personal project. For additional preparation: paragraph 36; 647-649(2); repeat analytical methods for solving systems. 649(2) and solve analytically.
Our work was supervised throughout the day by the director of the laboratory, Nouman Nou Manovich. He has the floor. (Showing the final slide).
Approximate grading scale
| Mark | Tolerance | Expertise | Study | Project | Total |
| 3 | 5 | 2 | 2 | 2 | 11 |
| 4 | 7 | 2 | 4 | 3 | 16 |
| 5 | 9 | 3 | 5 | 4 | 21 |
Lesson "Systems of linear equations with two variables"
Lesson motto:
"Activity is the only path to knowledge"
J. Bernard Shaw
Lesson objectives.
Didactic : Create conditions for the formation of the concept of “system of linear equations with two variables”, based on existing knowledge and life experience children.
Developmental : Continue the formation of abstract conceptual thinking based on the analysis of the relationship between systems of linear equations with two variables and their representation on a plane in the form of graphs. Based on deductive reasoning, help students draw up an algorithm for solving systems graphically and test it in independent work.
Educational : Contribute to the formation systems thinking And adequate self-esteem. Development of the ability to independently organize work; development of skills to find and use the necessary information on the Internet.
Stage 1. Preparing to perceive new material
A)Motivation
I want to tell you a riddle:
What is the fastest, but also the slowest.
The biggest, but also the smallest.
The longest, but also the shortest.
The most expensive, but also cheaply valued by us?
It's time, guys. We only have 40 minutes, but I would really like for them not to drag on, but to fly by. They were not spent in vain, but were spent usefully.
b) Introductory conversation
In our Everyday life we have to decide how simple tasks“Tanya, go to the store,” and complex “Tanya, go V shop, do laundry, cook soup, learn homework, etc.. ”, this requires the simultaneous fulfillment of several conditions.
In mathematics, there are also simple problems: “The sum of two numbers is 15. Find these numbers,” a little more complicated: “The difference of two numbers is 5. Find these numbers,” and complex ones, requiring the simultaneous fulfillment of two or more conditions. It is one of these problems that we will get acquainted with in today's lesson.
Consider the solution to this problem: on the board
“The sum of two numbers is 15, and their difference is 5. Find these numbers.” Determine the type of task: simple or complex. How many conditions must be met at the same time? Let's combine these two conditions with a curly brace (integer symbol). What is the complexity of the solution? It’s true, finding a solution will take a lot of time, and we don’t know any other way yet. What should I do? - Get acquainted with a new way to solve such problems.
b) Working with terms (slide)
Let's remember what concepts you know:
Linear equation with two variables -…
Graph of a Linear Equation with 2 Variables -...
The algorithm for constructing a graph is...
The relative position of the graphs is ...
System - …
System of linear equations with 2 variables - ...
The system solution is...
Methods for solving systems - ...
State the wording of the terms you know (check D.Z .)
Which terms are unfamiliar to you? Which term appeared several times? Indeed, the key term of our lesson is “system”.
Stage 2. Learning new material
a) The concept of a system
It turns out that the proposed problem can be solved faster if we use such a concept as a system. Are you familiar with this word? How do you understand it? The dictionary of foreign words gives 9 interpretations of this word. Listen to some of them. (I read selectively .) from Greek . - , compiled from parts ; compound ) , totalityelements, locatedin a relationshipAndconnectionsFriendWithfriend, whichformsdefined. , unity.
System (from σύστημα - a whole made up of parts; connection) - being in relationships and connections with each other, which forms a certain integrity, .Reducing the multitude to one is the fundamental principle of beauty.
In everyday practice, the word "system" can be used in different meanings, in particular :
theory , for example, system;
classification , For example, D. I. Mendeleev;
completed method of practical activity , For example, ;
way of organization mental activity , For example, ;
set of natural objects , For example, ;
some property of society , For example, , and so on.;
a set of established norms of life and rules of behavior , For example, or system values;
pattern (“a system can be traced in his actions”);
design (“weapons of the new system”);
Which options are best for us? Why?
“ System (Greek word) - ... a whole made up of parts; compound.
Symbol (sign);
Form for recording the simultaneous fulfillment of two or more conditions”
What do you think is the topic of the lesson?
Lesson topic
Systems of linear equations with two variables
( We write down the topic of the lesson in a notebook and on the board )
b) Goal setting
What is your goal in the lesson? - We must understand what a system of linear equations is and how it is used to solve problems, what is the solution to the system, how to solve it, ways to solve the system. Apply this knowledge in independent work.
All I can do is wish you successful achievement of your goal and help each of you, if possible.
c) Solution of a system of equations
( A symbolic record of the system, the design of the conditions and solutions to the problem appear on the board and in notebooks in the process of solving the problem .)
Let's return to the problem statement and executea short description of the condition :
Let x be the first number, y the second number. According to 1 condition, their sum is 15. This means x+y=15. We got 1 equation with two variables. According to condition 2, their difference is 5. This means x-y=5. We got 2 equation with two variables.
How to answer the task question?
To answer the question of the problem, it is necessary to find such values of the variables x and y that turn each of the equations into true equality, i.e. find general solutions these two equations - you need to solve a system of two equations with two variables.
How to record a system? With what symbol? (I listen to everything answer versions )
Indeed, it is customary to write a system of equations using a curly bracket, only the bracket is placed on the left. (I record the system in general view, next to the system for the task .)
System of linear equations with 2 variables is called...record
What does it mean to solve a system? How to do it?
We can pick up pairs of numbers. (Selecting a solution )
Let's check your solution by substituting this pair of numbers into the system: 10 and 5
Both equalities are true, which means the pair of numbers (10;5) is a solution to the system. (Write down the answer ) Answer: (10;5)
Is selecting a pair of numbers a universal way to solve systems? Why? What are your assumptions? Let's get acquainted with other ways to solve systems of equations, but to do this you need to know what the solution to the system is.
Let's consider a system of two equations with two variables. (I point to the system written down in general form .)
Formulate what is called a solution to the system. Compare your version with the textbook definition. (Working with the textbook definition .) Whose version was confirmed?
System solution linear equations with two variables is called a pair of variable values(a pair of numbers ), reversingeach equation of the system into the correct equality.
Work with the definitionBy known to youalgorithm : read, highlight keywords, we recite the definition in pairs.
Let's check how we understand: - What does it mean to “solve an equation”?
What is the solution to the first (second) equation?
Are these two different pairs of numbers?
What does it mean to “solve the system”? Formulate a definition and test yourself in a similar way. (Working with definition by algorithm )
Solve the system equations - means finding all its solutionsor prove that there are no solutions.
Let's check how we understand:How many solutions can the system have: 0,1,2 or more? You can check the correctness of your answer by reading the paragraph to the end.
Stage 3. Primary consolidation of new knowledge
Let's solve No. 1056 (orally) Who understood?
Who can solve a similar number. Which? Choose either of the two: No. 1057 or No. 1058.
Emotional pause. Anyone curious? Look under your chair. There is nothing? Strange. What did you want to see? What did I want to see? That's right, I wanted to seeways looking under the chair. Demonstrate it again and let others watch it too. What is all this for? This word is in the title of the next stage of our lesson:
Stage 4. Gaining new knowledge
a) Methods for solving systems...
We already talked about their existence at the beginning of the lesson. How many are there? What are their names?
It's great to have curious people in your class. What is the difference between curious and inquisitive?
Let's look through the textbook and find the answer to the question about methods. (Scrolling or watching to table of contents ). Let's write down methods for solving systems on the board and in a notebook.
Methods for solving systems linear equations with two variables: graphical method; substitution method; addition method.
- Let's consider a method for solving systems that is based on the material from the previous lesson.Let me remind you that the result of the group independent work there were graphs of the relative positions of linear equations with two variables. In addition, we made several conclusions about relative position graphs, their wording you wrote down in your notebook.
- There is a hint hidden in the very name of the method. What is this method? Let's write it down.
Graphic method.
At the beginning of the lesson we remembered a number of terms. (Returning to the list of terms )
What knowledge do we need now? (Student answers ):
The graph of a linear equation with 2 variables is a straight line.
The system contains two such equations, which means you need to construct two straight lines.
Two straight lines on a plane can intersect, not intersect, or coincide.(I lead the children to the conclusion about the essence of the graphic method)
Did I understand you correctly?essence graphic method solutions of systems is that: The graphical solution of a system of linear equations with two variables is reduced to findingcoordinates of common points graphs of equations (i.e. straight lines).
How to do it? (I appeal to everyone, listen to all versions, supporting those who are on the right path - creating an algorithm.).
The graphs of two linear equations of a system are two straight lines; Each one requires two points to construct. If the lines intersect, then there will be one common point (one solution to the system), if the lines do not intersect, there are no common points (no solutions to the system), and if the lines coincide, all points will be common (infinitely many solutions to the system).
Stage 5. Primary consolidation of new material
Let's try the method of solving systems that you discovered on the problem that you solved by selection at the beginning of the lesson, because we already know its answer. The solutions may be different, but the answer is the same. (We solve the system graphically, commenting on the solution with phrases from which we will later compose an algorithm.)
Algorithm for solving a system of linear equations with two variables graphically
Leaflets with a graphic solution of the system are attached to the board.
Stage 6. Consolidation and primary control of knowledge
a) Drawing up an algorithm ( Group work )
Briefing : Unite in groups of 4 people, take an envelope with an algorithm for solving systems cut into pieces graphically. You need:
1) assemble the algorithm on a piece of paper, numbering its parts.
2) use a ready-made algorithm when solving the system proposed to you (No. 1060, 1061)
3) check the correctness of the tasks - on the slide
Time to complete the task for a group is 10 minutes (after completing the task, the group checks the algorithm and solution of the system, evaluates the group’s work, commenting on its assessment ).
The result of the group’s work will be a assembled algorithm of the following form:
Algorithm for solving a system of linear equations with two variables graphically:
1. We build in the coordinate planegraphs of each equation systems, i.e.two straight (based on an algorithm for plotting a linear equation with 2 variables).
2. Findingintersection point graphs. Let's write it downcoordinates .
3. We draw a conclusion aboutnumber of system solutions .
4. Write it downanswer .
This method of solving systems is called graphical. It has one drawback. What disadvantage? we're talking about?
Summing up the work of the groups, we once again talk through the stages of the algorithm (handing out reminders with the algorithm )
Laptops (lesson-research)
b) Solution with comments No. 1060, a, b, c, d and 1061 a), b) – by groups).
Who understands how such tasks are performed?( Self-assessment )
Stage 7. Solve systems of equations graphically and study them using the specified algorithm
when solving a system of equations, express the variable in each of the equationsythroughxand build graphs in one coordinate system);
compare for each system the ratio of coefficients atx, at
Then the system has no solutions
Then the system has many solutions
Stage 8. Homework
(Appendix 3.)
1.Solve test tasks and fill out the table:
Job number
Possible answer
1.Which pair of numbers is the solution to the system of equations: has infinitely many solutions? . Compose another equation so that together with the given one it forms a system:
a) having infinitely many solutions;
b) has no solutions.
Answer: a) b)
The ability to formulate the same statements in both geometric and algebraic languages is given to us by a coordinate system, the invention of which, as you already know, belongs to Rene Descartes, a French philosopher, mathematician and physicist. It was he who created the foundations of analytical geometry, introduced the concept of geometric quantity, developed a coordinate system, and established the connection between algebra and geometry.
As an additional assignment, you are asked to prepare a message and presentation about the life and work of Rene Descartes. Your presentation may contain historical information, scientific facts. You can devote it to any one task or problem related to Rene Descartes. The main requirement is that your message should not exceed 10-12 minutes. Deadline of this assignment- 1 Week. I wish you success!
Criteria by which the presentation will be assessed:
criteria for the content of the presentation (5-7 points);
criteria for presentation design (5-7 points);
compliance with copyright (2-3 points).
9 stage. Summing up the lesson
- Let's remember the key points of the lesson - new terms (acceptance of unfinished sentences: I I start a phrase, and the children finish it ) system, solutions...
Reflection - leaves. Post-test scores
Epigraph-result. Watching your neighbor solve math problems will never teach you how to solve it yourself.
In this lesson we will look at solving systems of two equations in two variables. First, let's look at the graphical solution of a system of two linear equations and the specifics of the set of their graphs. Next, we will solve several systems using the graphical method.
Topic: Systems of equations
Lesson: Graphical method for solving a system of equations
Consider the system
A pair of numbers that is simultaneously a solution to both the first and second equations of the system is called solving a system of equations.
Solving a system of equations means finding all its solutions, or establishing that there are no solutions. We have looked at the graphs of the basic equations, let's move on to considering systems.
Example 1. Solve the system 
Solution:
These are linear equations, the graph of each of them is a straight line. The graph of the first equation passes through the points (0; 1) and (-1; 0). The graph of the second equation passes through the points (0; -1) and (-1; 0). The lines intersect at the point (-1; 0), this is the solution to the system of equations ( Rice. 1).
The solution to the system is a pair of numbers. Substituting this pair of numbers into each equation, we obtain the correct equality.
We got the only solution linear system.
Recall that when solving a linear system, the following cases are possible:
the system has a unique solution - the lines intersect,
the system has no solutions - the lines are parallel,
the system has an infinite number of solutions - the straight lines coincide.
We have reviewed special case systems when p(x; y) and q(x; y) are linear expressions of x and y.
Example 2. Solve a system of equations 
Solution:
The graph of the first equation is a straight line, the graph of the second equation is a circle. Let's build the first graph by points (Fig. 2).
The center of the circle is at point O(0; 0), the radius is 1.
The graphs intersect at point A(0; 1) and point B(-1; 0).
Example 3. Solve the system graphically
Solution: Let's build a graph of the first equation - it is a circle with a center at t.O(0; 0) and radius 2. The graph of the second equation is a parabola. It is shifted upward by 2 relative to the origin, i.e. its vertex is point (0; 2) (Fig. 3).
Graphs have one common point- t. A(0; 2). It is the solution to the system. Let's plug a couple of numbers into the equation to check if it's correct.
Example 4. Solve the system
Solution: Let's construct a graph of the first equation - this is a circle with a center at t.O(0; 0) and radius 1 (Fig. 4).
Let's plot the function This is a broken line (Fig. 5).
Now let's move it 1 down along the oy axis. This will be the graph of the function
Let's place both graphs in the same coordinate system (Fig. 6).
We get three intersection points - point A(1; 0), point B(-1; 0), point C(0; -1).
We looked at the graphical method for solving systems. If you can plot a graph of each equation and find the coordinates of the intersection points, then this method is quite sufficient.
But often the graphical method makes it possible to find only an approximate solution of the system or answer the question about the number of solutions. Therefore, other methods are needed, more accurate, and we will deal with them in the following lessons.
1. Mordkovich A.G. and others. Algebra 9th grade: Textbook. For general education Institutions.- 4th ed. - M.: Mnemosyne, 2002.-192 p.: ill.
2. Mordkovich A.G. and others. Algebra 9th grade: Problem book for students educational institutions/ A. G. Mordkovich, T. N. Mishustina and others - 4th ed. - M.: Mnemosyne, 2002.-143 p.: ill.
3. Makarychev Yu. N. Algebra. 9th grade: educational. for general education students. institutions / Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, I. E. Feoktistov. — 7th ed., rev. and additional - M.: Mnemosyne, 2008.
4. Alimov Sh.A., Kolyagin Yu.M., Sidorov Yu.V. Algebra. 9th grade. 16th ed. - M., 2011. - 287 p.
5. Mordkovich A. G. Algebra. 9th grade. In 2 hours. Part 1. Textbook for students of general education institutions / A. G. Mordkovich, P. V. Semenov. — 12th ed., erased. - M.: 2010. - 224 p.: ill.
6. Algebra. 9th grade. In 2 parts. Part 2. Problem book for students of general education institutions / A. G. Mordkovich, L. A. Aleksandrova, T. N. Mishustina and others; Ed. A. G. Mordkovich. — 12th ed., rev. - M.: 2010.-223 p.: ill.
1. College.ru section on mathematics ().
2. Internet project “Tasks” ().
3. Educational portal“I WILL SOLVE THE USE” ().
1. Mordkovich A.G. and others. Algebra 9th grade: Problem book for students of general education institutions / A. G. Mordkovich, T. N. Mishustina, etc. - 4th ed. - M.: Mnemosyne, 2002.-143 p.: ill. No. 105, 107, 114, 115.
