Sections: Mathematics
Goals: consolidate knowledge of the properties of a function when performing exercises, test the skills and abilities of students and the degree of their assimilation of the studied material during independent work, repeat previously studied material.
Tasks: encourage students to self-control, mutual control, and self-analysis of their educational activities. Develop creative and mental thinking.
Method of work in the lesson:
Students work in pairs. Each desk is a separate option. It is advisable to seat the children next to the weaker student and the stronger one.
An envelope with 1) an assessment sheet, 2) a sheet for oral work, 3) a “Loto” task + a rebus is distributed to each desk.
In the previous lesson, you can assign independent homework according to the following options:
Task 1. Construct a figure bounded by the graphs of functions.
Option 1.
Option 2. 
Stage 1. Organizational moment (3 min) Greeting. Report topic. State the lesson plan. The work consists of three stages. Students record the results of each stage on individual assessment sheets. (distribute the assessment sheet from Appendix 2)
Stage 2. Checking homework (5 min)
Students exchange their notebooks with the next desk.
1 student at the board shows solution No. 350 Slide 3
Checking homework No. 1. Slide 4
We calculate the number of points: for correctly completed number 350 - 1 point, for correctly completed independent work we set points as follows: for each correctly constructed graph 1 point, 1 point for a correctly designated figure. Result – 5 points for completing 2 tasks correctly. We put points on the score sheet. Slide 6
Stage 3. Oral work (Repetition of theory) (5 min) Slide 6
Distribute to students a sheet with a task for oral work (see Appendix 2)
2 minutes . For checking. Verification with mutual control (we change answers again). Slide 7
Stage 4. Practical part (20 min) Slide 10-13
Goal: to be able to determine the identity of a point without constructing a graph, compare numbers using the properties of a function graph, promote teamwork and develop the cognitive process with the help of puzzles.
On their desks, students have a card with a task, an envelope with answer options (9 cards with different answers, but 3 have correct ones) and a blank card with the task number for composing a rebus.
The tasks are designed in such a way that the first two letters are solved by one student, and the second two letters are solved by the second student, and only No. 3 is solved together.
“Loto” – differentiated independent work(performed according to options and in pairs)
Exercise 1. Solve 3 tasks from the option written on the card, find cards with the correct answers and cover the corresponding tasks with them, then you will get a rebus on the top side of them.
Task 2. Solve the puzzle by answering the question.
IN 1. What is another name for the arithmetic square root?
AT 2. What mathematician once remarked that: “A mathematical theory can be considered perfect only when you have made it so clear that you undertake to explain its content to the first person you meet?
"Lotto" Option 1 |
|
| No. 1. At what point does the graph of a function and a straight line intersect? | |
| a) y = 2; b) 2у = 3 | c) y = -2; d) y = 4. |
| C (1600;40), N (900;-30) | E (0.81; 0.9); P (0.5, 0.25) |
| No. 3. Compare the numbers A) ; b) ; V) ; G) ; d). |
|
"Lotto" Option 2 |
|
| No. 1. At what point does the graph of a function and a straight line intersect? | |
| a) y = 3; b) 2у = 5 | c) y = -3; d) y = 6. |
| No. 2. Which points belong to the graph of the function | |
| A (2500;50), C (400;-20) | B (0.64; 0.8); P (0.3, 0.09) |
| No. 3. Compare the numbers A) ; b) ; V) ; G) ; d). |
|
Answer card:
2. Write down differentiated homework
“3” – 357
“4” – 357 + 351 (b, d)
“5” – 357 + 351 (b, d) + 456
Individual homework for strong students:
Construct graphs of functions in one coordinate system and draw conclusions about what happens to the graph of the function. (graph conversion has not been studied yet).
Republic of Tatarstan, Cheremshansky district, village. Cheremshan
MBOU "Cheremshansky Lyceum"
Lesson topic: “Function y = √x, its properties and graph”
Sakhabieva Elvira Maratovna
Mathematic teacher
MBOU "Cheremshansky Lyceum",
With. Cheremshan
2015-2016
Function y = √x, its properties and graph
Lesson type: Lesson on introducing new material.
Lesson type: combined.
Grade: 8
The purpose of the lesson:
Tasks:
Educational
Educational
Developmental
Equipment for the lesson:Power point presentation
UMK: Algebra 8th grade, Yu.N.Makarychev, N.G. Mindyuk, K. I. Neshkov, S.B. Suvorov, 2nd ed.-M.: Education, 2014.-287 p.
During the classes
Slide 1 .Welcoming students, Lesson motto... Mathematics must then be taught, because it puts the mind in order... M.V.Lomonosov
Frontal work with the class:
Slide 2. 1). Guys, let's remember the definition of arithmetic square root(The arithmetic square root of a is a non-negative number whose square is equal to a)
So the important condition here is a>0
2) Oral work
Slide 3. a) Is it true that: = 0.3; (Student answer: yes)= 0.5; (Student answer: no)
= 4?
(Student answer: no), (Student answer: yes)
Slide 4. b) Choose an irrational number among the numbers ; (=0.8 rational number, etc.)
(This needs to be decided at the board)
Slide 5. c) Calculate:
7; there is no decision. =
3. Generalization and systematization of knowledge. (From your seat optional)
Slide 6 . Now let's calculate the area of a square with a side equal to
Let's remember what is the area of a square?, S= . =18)
Here calculate the area of a rectangle with sides and
Let's remember the area of the rectangle (S=a*b, S= . =14*5=70)
Let's calculate the area of a right triangle whose legs
4. Testing students' knowledge and skills to prepare for a new topic.
Slide 7. Guys, please look at the formulas.
Who remembers the name of this function. (linear, quadratic).
Let's remember what is the graph of this function? (line and parabola)
What are the independent variables (they are located inside the formula) and the dependent variables (they are located separately)?
Slide 8. - Today we will look at a new feature y =
(Let's define an independent variable and a dependent variable and what values do they take?)
Slide 9.- Lesson topic: Function y = , its properties and graph.
Slide 10. Objective of the lesson:- We must study the properties and graph of the function y =.
Slide 11. To do this, we will define several values of this function and build a table.
Connect the dots with a smooth line (the hand goes from left to right)
Slide 12. Look at what points the graph passes through?
In which quarters will the graph of the function y = be located??
The graph should be viewed from left to right, the graph goes up, which means the function is increasing.
5. Consolidation of knowledge
Slide 13.
Orally find the meaning of the functions on the slide
No. 355 (Using the graph in the textbook on p. 85, fig. 17, find the valueand make a table)
Municipal educational institution
secondary school No. 1
Art. Bryukhovetskaya
municipal formation Bryukhovetsky district
Mathematic teacher
Guchenko Angela Viktorovna
year 2014
Function y =
, its properties and graph
Lesson type: learning new material
Lesson objectives:
Problems solved in the lesson:
teach students to work independently;
make assumptions and guesses;
be able to generalize the factors being studied.
Equipment: board, chalk, multimedia projector, handouts
Timing of the lesson.
Determining the topic of the lesson together with students -1 min.
Determining the goals and objectives of the lesson together with students -1 min.
Updating knowledge (frontal survey) –3 min.
Oral work -3 min.
Explanation of new material based on creating problem situations -7min.
Fizminutka –2 minutes.
Plotting a graph together with the class, drawing up the construction in notebooks and determining the properties of a function, working with a textbook -10 min.
Consolidating acquired knowledge and practicing graph transformation skills –9min .
Summing up the lesson, providing feedback -3 min.
Homework -1 min.
Total 40 minutes.
During the classes.
Determining the topic of the lesson together with students (1 min).
The topic of the lesson is determined by students using guiding questions:
function- work performed by an organ, the organism as a whole.
function- possibility, option, skill of a program or device.
function- duty, range of activities.
function character in a literary work.
function- type of subroutine in computer science
function in mathematics - the law of dependence of one quantity on another.
Determining the goals and objectives of the lesson together with students (1 min).
The teacher, with the help of students, formulates and pronounces the goals and objectives of this lesson.
Updating knowledge (frontal survey – 3 min).
Oral work – 3 min.
Frontal work.
(A and B belong, C does not)
Explanation of new material (based on creating problem situations – 7 min).
Problem situation: describe the properties of an unknown function.
Divide the class into teams of 4-5 people, distribute forms for answering the questions asked.
Form No. 1
y=0, with x=?
The scope of the function.
Set of function values.
One of the team representatives answers each question, the rest of the teams vote “for” or “against” with signal cards and, if necessary, complement the answers of their classmates.
Together with the class, draw a conclusion about the domain of definition, the set of values, and the zeros of the function y=.
Problem situation : try to build a graph of an unknown function (there is a discussion in teams, searching for a solution).
The teacher recalls the algorithm for constructing function graphs. Students in teams try to depict the graph of the function y= on forms, then exchange forms with each other for self- and mutual testing.
Fizminutka (Clowning)
Constructing a graph together with the class with the design in notebooks – 10 min.
After a general discussion, the task of constructing a graph of the function y= is completed individually by each student in a notebook. At this time, the teacher provides differentiated assistance to students. After students complete the task, the graph of the function is shown on the board and students are asked to answer the following questions:
Conclusion: Together with the students, draw a conclusion about the properties of the function and read them from the textbook:
Consolidating acquired knowledge and practicing graph transformation skills – 9 min.
Students work on their card (according to the options), then change and check each other. Afterwards, graphs are shown on the board, and students evaluate their work by comparing it with the board.
Card No. 1
Card No. 2
Conclusion: about graph transformations
1) parallel transfer along the op-amp axis
2) shift along the OX axis.
9. Summing up the lesson, providing feedback – 3 min.
SLIDES – insert missing words
The domain of definition of this function, all numbers except ...(negative).
The graph of the function is located in... (I) quarters.
When the argument x = 0, the value... (functions) y = ... (0).
The greatest value of the function... (does not exist), smallest value - …(equals 0)
10. Homework (with comments – 1 min).
According to the textbook- §13
According to the problem book– No. 13.3, No. 74 (repetition of incomplete quadratic equations)
Basic goals:
1) form an idea of the feasibility of a generalized study of the dependencies of real quantities using the example of quantities related by the relation y=
2) to develop the ability to construct a graph y= and its properties;
3) repeat and consolidate the techniques of oral and written calculations, squaring, extracting square roots.
Equipment, demonstration material: handouts.
1. Algorithm:
2. Sample for completing the task in groups:
3. Sample for self-test of independent work:
4. Card for the reflection stage:
1) I understood how to graph the function y=.
2) I can list its properties using a graph.
3) I did not make mistakes in independent work.
4) I made mistakes in independent work (list these mistakes and indicate their reason).
During the classes
1. Self-determination for educational activities
Purpose of the stage:
1) include students in educational activities;
2) determine the content of the lesson: we continue to work with real numbers.
Organization of the educational process at stage 1:
– What did we study in the last lesson? (We studied the set of real numbers, operations with them, built an algorithm to describe the properties of a function, repeated functions studied in 7th grade).
– Today we will continue to work with a set of real numbers, a function.
2. Updating knowledge and recording difficulties in activities
Purpose of the stage:
1) update educational content that is necessary and sufficient for the perception of new material: function, independent variable, dependent variable, graphs
y = kx + m, y = kx, y =c, y =x 2, y = - x 2,
2) update mental operations necessary and sufficient for the perception of new material: comparison, analysis, generalization;
3) record all repeated concepts and algorithms in the form of diagrams and symbols;
4) record an individual difficulty in activity, demonstrating at a personally significant level the insufficiency of existing knowledge.
Organization of the educational process at stage 2:
1. Let's remember how you can set dependencies between quantities? (Using text, formula, table, graph)
2. What is a function called? (A relationship between two quantities, where each value of one variable corresponds to a single value of another variable y = f(x)).
What is the name of x? (Independent variable - argument)
What is the name of y? (Dependent variable).
3. In 7th grade did we study functions? (y = kx + m, y = kx, y =c, y =x 2, y = - x 2,).
Individual task:
What is the graph of the functions y = kx + m, y =x 2, y =?
3. Identifying the causes of difficulties and setting goals for activities
Purpose of the stage:
1) organize communicative interaction, during which the distinctive property of the task that caused difficulty in learning activities is identified and recorded;
2) agree on the purpose and topic of the lesson.
Organization of the educational process at stage 3:
-What's special about this task? (The dependence is given by the formula y = which we have not yet encountered.)
– What is the purpose of the lesson? (Get acquainted with the function y =, its properties and graph. Use the function in the table to determine the type of dependence, build a formula and graph.)
– Can you formulate the topic of the lesson? (Function y=, its properties and graph).
– Write the topic in your notebook.
4. Construction of a project for getting out of a difficulty
Purpose of the stage:
1) organize communicative interaction to build a new method of action that eliminates the cause of the identified difficulty;
2) fix a new method of action in a symbolic, verbal form and with the help of a standard.
Organization of the educational process at stage 4:
Work at this stage can be organized in groups, asking the groups to build a graph y =, then analyze the results. Groups can also be asked to describe the properties of a given function using an algorithm.
5. Primary consolidation in external speech
The purpose of the stage: to record the studied educational content in external speech.
Organization of the educational process at stage 5:
Construct a graph of y= - and describe its properties.
Properties y= - .
1.Domain of definition of a function.
2. Range of values of the function.
3. y = 0, y> 0, y<0.
y =0 if x = 0.
y<0, если х(0;+)
4.Increasing, decreasing functions.
The function decreases as x.
Let's build a graph of y=.
Let's select its part on the segment. Note that we have = 1 for x = 1, and y max. =3 at x = 9.
Answer: at our name. = 1, y max. =3
6. Independent work with self-test according to the standard
The purpose of the stage: to test your ability to apply new educational content in standard conditions based on comparing your solution with a standard for self-test.
Organization of the educational process at stage 6:
Students complete the task independently, conduct a self-test against the standard, analyze, and correct errors.
Let's build a graph of y=.
Using a graph, find the smallest and largest values of the function on the segment.
7. Inclusion in the knowledge system and repetition
The purpose of the stage: to train the skills of using new content together with previously studied: 2) repeat the educational content that will be required in the next lessons.
Organization of the educational process at stage 7:
Solve the equation graphically: = x – 6.
One student is at the blackboard, the rest are in notebooks.
8. Reflection of activity
Purpose of the stage:
1) record new content learned in the lesson;
2) evaluate your own activities in the lesson;
3) thank classmates who helped get the result of the lesson;
4) record unresolved difficulties as directions for future educational activities;
5) discuss and write down your homework.
Organization of the educational process at stage 8:
- Guys, what was our goal today? (Study the function y=, its properties and graph).
– What knowledge helped us achieve our goal? (Ability to look for patterns, ability to read graphs.)
– Analyze your activities in class. (Cards with reflection)
Homework
paragraph 13 (before example 2) № 13.3, 13.4
Solve the equation graphically:
Construct a graph of the function and describe its properties.
