Express the expression as a power with a rational. Degree with rational exponent
Development of a math lesson in grade 11.
Lesson topic : "Degree with a rational exponent".
Lesson Objectives:
Educational:
- introduce the concept of degree with a rational exponent;
- primary consolidation of the acquired knowledge on the simplest tasks.
Educational: education of moral personality traits:
- purposefulness;
- perseverance in achieving the goal;
- independence, care;
The development of the ability to work in a team.
Educational : skill development
- mathematical speech;
- work independently and in pairs;
- mutual control and self-control.
Lesson type : Lesson learning new material.
Equipment : didactic material(cards with a specific color signal).
Lesson plan.
1. Organizational stage. (8min.)
2. Main stage. (30 min.)
3. Summing up. (2 minutes.)
- Organizational stage
Target: Create a favorable environment for working in the classroom, prepare students for the upcoming work, communicate the topic, purpose and plan of work.
Method: verbal.
Teacher activity | Student activities |
Hello guys. Who is absent? Are you familiar with the concept of “power of a number with an integer exponent”? For what a and n is it defined? List the properties of the degree with an integer indicator (students name the properties, the teacher writes on the blackboard, if they don’t call them, then you can write the left side on the blackboard, and let the students call the right one, the written properties remain on the blackboard). Let's do it orally. Simplify expressions: What is the difficulty in simplifying the last expression? So. Today we have several unusual lesson, today each of you will be in the role of a teacher. Try to formulate the theme of the lesson yourself. What is the purpose of the lesson? Let's start studying! | Expression , where n is an integer. Expression defined for all a and n, except for the case a=0 for. Answers: x 2; x 7; x 12; a 2 to 8 (a question arises when simplifying the last expression) The exponent is a fractional number. We are only familiar with the concept of "degree with an integer exponent" Degree with rational exponent and its properties. The study of the concept of degree with a rational indicator of its properties. Learn how to apply them to problem solving |
- main stage
Purpose: explanation of the algorithm for working on cards, the introduction of the concept of degree with a rational indicator; primary consolidation of the acquired knowledge on the simplest tasks.
Method: verbal.
Teacher activity | Student activities |
We will work as follows. Now each of you will receive a card with a certain color signal. Each card contains a theory, this definition and properties of a degree with a rational exponent. Also, in addition to the theoretical part, there is a practical part - tasks for self-fulfillment, and an obligatory part that you must complete. For completing additional tasks, you can get an additional grade. After you have familiarized yourself with the contents of your card, and also completed the task, you need to follow the route indicated on the board to find a partner according to the color signal. Having found it, you explain to each other in turn the theoretical material of your card, answer questions if any arise, then exchange cards and complete the practical part of the received card. Then, the person from whom you received the card checks the correctness of the task, if there are errors, corrects it. If there are no difficulties, continue to move along the route. If such a situation arises that you have already completed the task of your card, but your partner has not yet, then you proceed to the additional task. If you do not get a partner, then you can work in threes. For the lesson you need to go through the entire route. Within 8 minutes you get acquainted with the material of the received card, complete the task, then follow the route. In a notebook, write down the number and topic of today's lesson "Degree with a rational indicator", write down the theoretical material of the received card, and the solution of the practical part. In order not to get confused, write down the color signal in the fields. The score will be given for the correct performance of the work on all the cards. Who has questions about working with cards? If you have any questions during the work, you can contact me. the route is written on the board) | Students learn how to work in groups. Students receive a set of cards |
- Stage of independent work on cards(see Attachment)
- Debriefing stage
Purpose: to sum up the lesson.
Method: verbal.
Teacher activity | Student activities |
We are finishing work. How well you learned this material, we will check in the next lesson. Submit your notebooks for review now. What new concept did you learn in class today? Write on the board next to the properties of the degree of a number with an integer exponent, the properties of the degree with a rational exponent. Is it possible to represent a negative number as a power with a rational exponent? After the properties are written out: What can be said about these properties? (teacher points to blackboard) Reflection: - Did you enjoy being a teacher? What difficulties did you face? What pleasant sensations did you experience? Finish the sentence: “I would like to congratulate myself for…” Homework: At home, you need to learn the theoretical material of paragraph 34. No. 430, 431 (a, c), 437 (a, c), 444 Thank you all for your work, the lesson is over. | Degree with a rational exponent. One of the students writes the properties on the blackboard. No you can not. Students actively participate in the conversation |
Application
The purpose of the cards used:
Introduction of the concept and properties of a degree with a rational exponent;
Primary consolidation of acquired knowledge.
Task goals.
The first task: the formation of the ability to represent an expression in the form of a degree with a rational indicator, using the definition of a degree with a rational indicator.
The second task: the formation of the ability to represent an expression as a root from a number, using the definition of a degree with a rational exponent.
The third task: the formation of the ability to find numerical values, factorize, and compare numbers using the definition and properties of a power with a rational exponent.
red card
Definition . degree of number with a rational), is called the number. So by definition.
Example 1
where r,s- rational numbers, , .
Example 2
Task 1. Imagine
Task 2. .
Task 3. .
Additional tasks.Find the value of an expression.
blue card
Definition . degree of number with a rational, where m is an integer and n is a natural number (), is called the number. So by definition.
Example 1
Power properties with rational exponent,where r,s are rational numbers,, .
Example 2
Task 1. Imagine in the form of a degree with a rational exponent.
Task 2. Express as a root of a number.
Task 3. Multiply.
Additional tasks.Multiply.
green card
Definition . degree of number with a rational, where m is an integer and n is a natural number (), is called the number. So by definition.
Example 1
Power properties with rational exponent,where r,s are rational numbers,, .
Example 2
Task 1. Imagine in the form of a degree with a rational exponent.
Task 2. Express as a root of a number.
Task 3. Find the value of a numeric expression.
Additional tasks.Find the value of an expression.
orange card
Definition . degree of number with a rational, where m is an integer and n is a natural number (), is called the number. So by definition.
Example 1
Power properties with rational exponent,where r,s are rational numbers,,
Comment. At the rational power of a is not defined.
Example 2. Compare numbers.
Task 1. Imagine in the form of a degree with a rational exponent.
Task 2. Express as a root of a number.
Task 3. Compare numbers.
Additional tasks.Find the value of an expression.
The purpose of the lesson:
- Introduce the concept of a degree with a rational exponent; to teach how to translate a degree with a rational indicator to the root and vice versa; calculate powers with a rational exponent.
- The development of memory, thinking.
- Formation of activity.
Lesson type: Explanation of new material.
Equipment: Computer, interactive whiteboard, interactive resources, use of DER.
"What we know is limited, and what we do not know is infinite."
P. Laplace
During the classes
I.Actualization.
Teacher:
1. Remember the definition of degree with natural indicator?
Student:
Answer. degree of number A with an integer n>0, is called the product n multipliers, each of which is equal to A.
Example: 5 3 = 5 5 5
Teacher:
2. Defining an exponent with a negative integer exponent?
Student:
Answer. a - n = 1/a n where
Example: 10 -4 = 1/10 4 ; 3 -8 \u003d 1/3 8; (1/5) -2 = 5 2.
Teacher:
3. The expression a n is defined for all a and n except..
Student:
Answer. Case a = 0 for n ≤ 0
Teacher:
4. What can replace =
Student:
Answer. (Root n- from the number A equals A to the extent 1/ n)= a 1/n
Teacher:
5. List the properties of degrees with an integer exponent.
Student:
Answer. For anyone A≠ 0 and any integers m and n have the properties
1. a m a n = a m + n
2. a m ÷ a n = a m-n
3. (am) n = a mn
For any a ≠ 0 and b ≠ 0 and any n, the properties
4. (ab) n = a n b n
5 .(a/b) n = a n/ b n
6. Oral work. Express the root as a power:
Express as a positive exponent:
7 -3 ; 2 -2 ; 6 -3
Express as a negative exponent:
(1/4) 5 ; (1/21) -3 ;
II. Explanation of new material.
Using a collection of digital educational resources.
DER No. 30. A degree with a rational exponent and its properties.
I will explain with specific examples.
Note: When a< 0 рациональная степень числа, а не определена.
Let's explain this with an example. Consider (-64) 1/3 = 3 √-64 = -4. On the other hand: 1/3 = 2/6 and then (-64) 1/ 3 = (-64) 2/6 = 6 √(-64) 2 = 6√64 2 = 6 √4 6 = 4. We get contradiction.
III. Consolidation of new material.
CER No. 31. Practice.
1. Express as a root expression.
2. Express the expression as a power with a rational exponent.
Control.
CER No. 32. Practice. Find the value of a numeric expression.
Control.
IV. Lesson results.
We have studied a degree with a rational exponent and its properties, but where can they come in handy?
Representing an expression as a power ....
Express the expression as the root 5 3/6 = ...
Calculate powers with a rational exponent.
Partially we answered today.
How to apply a degree with a rational exponent when converting and simplifying expressions, finding the values of expressions, we will learn in the next lessons.
V. Homework.
Degree with rational exponent
Khasyanova T.G.,
mathematics teacher
The presented material will be useful to teachers of mathematics when studying the topic "Degree with a rational indicator".
The purpose of the presented material: disclosure of my experience in conducting a lesson on the topic "Degree with a rational indicator" work program discipline "Mathematics".
The methodology of the lesson corresponds to its type - a lesson in the study and primary consolidation of new knowledge. The basic knowledge and skills were updated on the basis of previously gained experience; primary memorization, consolidation and application of new information. Consolidation and application of new material took place in the form of solving problems of varying complexity that I tested, giving a positive result in mastering the topic.
At the beginning of the lesson, I set the following goals for the students: educational, developing, educational. At the lesson, I used various methods of activity: frontal, individual, steam room, independent, test. The tasks were differentiated and made it possible to identify, at each stage of the lesson, the degree of assimilation of knowledge. The volume and complexity of tasks corresponds to the age characteristics of students. From my experience - homework, similar to the tasks solved in the classroom, allows you to securely consolidate the acquired knowledge and skills. At the end of the lesson, reflection was carried out and the work of individual students was evaluated.
The goals have been achieved. The students studied the concept and properties of a degree with a rational exponent, learned how to use these properties in solving practical problems. Behind independent work grades are announced in the next lesson.
I believe that the methodology used by me for conducting classes in mathematics can be applied by teachers of mathematics.
Lesson topic: Degree with a rational indicator
The purpose of the lesson:
Identification of the level of mastering by students of a complex of knowledge and skills and, on its basis, the application of certain solutions to improve the educational process.
Lesson objectives:
Tutorials: to form new knowledge among students of basic concepts, rules, laws for determining the degree with a rational indicator, the ability to independently apply knowledge in standard conditions, in changed and non-standard conditions;
developing: think logically and implement Creative skills;
educators: to form an interest in mathematics, to replenish the vocabulary with new terms, to obtain additional information about the world around. Cultivate patience, perseverance, the ability to overcome difficulties.
Organizing time
Updating of basic knowledge
When multiplying powers with the same base, the exponents are added, and the base remains the same:
For example,
2. When dividing powers with the same bases, the exponents are subtracted, and the base remains the same:
For example,
3. When raising a degree to a power, the exponents are multiplied, and the base remains the same:
For example,
4. The degree of the product is equal to the product of the powers of the factors:
For example,
5. The degree of the quotient is equal to the quotient of the powers of the dividend and the divisor:
For example,
Solution Exercises
Find the value of an expression:
Solution:
In this case, none of the properties of a degree with a natural exponent can be applied explicitly, since all degrees have different bases. Let's write some degrees in a different form:
(the degree of the product is equal to the product of the degrees of factors);
(when multiplying powers with the same base, the exponents are added, and the base remains the same; when raising a degree to a power, the exponents are multiplied, but the base remains the same).
Then we get:
IN this example the first four properties of the degree with a natural exponent were used.
Arithmetic square root
is a non-negative number whose square isa,
. At
- expression
not defined, because there is no real number whose square is equal to a negative numbera.
Mathematical dictation(8-10 min.)
Option
II. Option
1. Find the value of the expression
A)
b)
1. Find the value of the expression
A)
b)
2. Calculate
A)
b)
IN)
2. Calculate
A)
b)
V)
Self test(on the lapel board):
Response Matrix:
№ option/task
Task 1
Task 2
Option 1
a) 2
b) 2
a) 0.5
b)
V)
Option 2
a) 1.5
b)
A)
b)
at 4
II. Formation of new knowledge
Consider the meaning of the expression, where - positive number – fractional number and m-integer, n-natural (n>1)
Definition: degree of number a›0 with rational exponentr = , m-whole, n- natural ( n›1) a number is called.
So:
For example:
Notes:
1. For any positive a and any rational r, the number positively.
2. When
rational power of a numberanot defined.
Expressions such as
don't make sense.
3.If fractional positive number
.
If fractional negative number, then -doesn't make sense.
For example: - doesn't make sense.
Consider the properties of a degree with a rational exponent.
Let a>0, в>0; r, s - any rational numbers. Then a degree with any rational exponent has the following properties:
1.
2.
3.
4.
5.
III. Consolidation. Formation of new skills and abilities.
Task cards work in small groups in the form of a test.
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