This lesson is intended for schoolchildren who have already become familiar with the concept of square roots in an algebra course. To retrieve the information, they can go back and review previous presentations.
Now it's time to study the properties of square roots. Knowing their properties, schoolchildren will be able to work with them and transform them. This presentation is intended both for schoolchildren (for independent study) and also for teachers (for taking notes and teaching a lesson).
slides 1-2 (Presentation topic "Properties of square roots", theorem)
This lesson begins with the square root theorem of the product. An algebraic representation of this theorem is demonstrated. The proof is given below because it is easy for students to understand. The theorem is proven in a simple way. The square root, consisting of the product of two factors, can be replaced by one notation - x, that is, a change. Thus, the theorem is proved by the replacement method. After the replacement, the idea of the proof becomes clear and simple. In order for schoolchildren to remember and better understand the theorem, you can offer them practical examples.
slides 3-4 (theorems)
The next slide is devoted to a more detailed analysis of the proof. It is presented in the form of a table consisting of three columns. The first column demonstrates the replacement. The second column consists of an algebraic representation of the theorem itself, but using new notation. And finally, the third column provides a proof of the new theorem obtained.
After re-explaining this theorem and its proof, two remarks are made in which different cases are considered for different radical expressions (in this case, numbers). So, the first theorem talks about the first property of square roots. We can call this property the property of multiplying roots.
The second property is demonstrated on the next slide. It is also given in the form of a theorem. Its essence is that the division that is performed under the root can be written as a division of the roots of the numerator and denominator. Because we're talking about about fractions, then it is necessary to check the condition that the denominator is not equal to zero. Also, it is necessary to check the condition that the cut expression, as a whole, is not equal to zero.
This theorem is proven in a similar way. The proof is given on the same slide where the theorem is stated, in the table. Before showing students a proof, you can have them do it themselves.
slides 5-6 (examples)
Further, the next slide provides examples. They can be solved based on the study of properties. Students will be able to see that knowing the properties will make the solution simpler than it would otherwise be. One of the examples is based on knowledge of the multiplication property, the other - the division theorem. In the second example, a mixed fraction is given as a fraction. In order to apply the property, it is necessary to convert a mixed fraction into a common fraction. Schoolchildren already know how this can be done.
slides 7-8 (examples)
The next example, which is shown on the next slide, asks students to remember the formula for the difference of squares, applying which, the solution will become simpler than in the opposite case. Of course, you can square numbers and perform subtraction, however, in this case, schoolchildren will be faced with unnecessary operations.
Below are two more examples with solutions for securing the material.
Further, on the next slide, students can learn how to extract square roots without using a table of squares. This is not as difficult as it may seem at first glance. Often, when tests Using a calculator is prohibited. In order for students to be able to handle examples like these, it is very important to understand the contents of this slide.
slide 9 (example)
And finally, the last final slide leads practical example extracting the square root of a four-digit number.
As you can see, this presentation contains important theorems that talk about some properties of square roots, as well as clear examples that relate to theorems. Proofs of the theorems are also given. Showing a presentation during an algebra lesson can help students better understand the topic.
Let us recall the basic properties of arithmetic roots (sometimes called the properties of nth roots). All of them are true for square roots, since: for. This is the most important property that allows you to move from roots to rational powers. After such a transition, you can use all the properties of degrees.properties of degrees
Solution. 1) Let us first simplify the first part of the expression. Using property 3, we get. Now let's apply properties 6 and 5:. Let us now apply property 1: As a result, we get: By property 5: By property 5:. Let's substitute the results of calculations from 1) and 2) into the expression. Here we used the properties of 2 and 8 arithmetic roots. Answer: 2.
Progress of solving the equation Replacement: a; 0.5a,2a = 2a; - 1.3a = - 13; a = 10; Reverse replacement: 10; 5x = 1000; x = 200. Answer: 200.
Generalization of the material 1. What mathematical concept did we work with today, the root of the nth degree 2. What did we use to calculate the root of the nth degree of the property of the root of the nth degree 3. How many roots does the equation x n = a have, if n is an odd number (for example: x 7 = 5) one root 4. How many roots does the equation x n = a have, if n is an even number (for example: x 12 = a) depends on a: if a is negative, then there are no roots; if a = 0, then one root; if a is positive, then there are two roots.
Homework If you have completed it completely, study paragraph 33, analyze examples 1 and 3 from the textbook, complete 417 (a, b), 419 (a, b). If errors are made in the additional part of the work 394, 410 (a) If errors are made in the mandatory part of the work (a, b)
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Independent work 1.a) 13 b) 6 2. 
Properties of roots of degree n
Teacher: Ivashko Marina Firsovna
MBOU "Lyceum No. 8"
Sosnovy Bor
Leningrad region.
Theorem 1. For natural numbers m, n (m ≥ 2, n ≥2) non-negative numbers and the equalities are valid
Theorem 1.
Proof.
N we get equal numbers:
Theorem 1.
Proof.
Let us raise the left and right sides of the equality separately to the power mn we get equal numbers:
Theorem 1.
Proof.
Let us raise the left and right sides of the equality separately to the power mn we get equal numbers:
Theorem 1. For natural numbers m, n (m ≥ 2,n ≥2) non-negative numbers and the equalities are valid
Comment. If m, n are odd, then Theorem 1 is valid for all A Є R.
Theorem 2. and the equality is true
Proof.
Let a Є R be an arbitrary number. Then
Therefore, by virtue of equality
Example 2.
Comment. For natural number m and real number and the equality is true
Theorem 3. Let a be a positive number, p an integer,
n – natural number (n ≥2). Then the equality is true
Theorem 3.
Proof.
If p Є N, then the equality has already been proven.
If p=0, then
If pn from a positive number we get:
Literature
- Textbook for 10th grade of general education institutions.
S. M. Nikolsky, M. K. Potapov,
N.N. Reshetnikov, A.V. Shevkin.
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Slide captions:
Grade 11 Lesson topic: “Properties of the nth root”
The objectives of the lesson are repetition, generalization and systematization of the topic material. formation of skills to apply techniques: comparison, generalization, highlighting the main thing, transferring knowledge to a new situation. development of mathematical outlook, thinking and speech, attention and memory. nurturing interest in mathematics and its applications, communication skills, and general culture. students’ awareness of the importance of the topic “Properties of the nth degree root” when choosing the profession of a pilot engineer and aviation specialist.
Flight planning – Organizing time, topic, goals and objectives of the lesson Air reconnaissance - activation of knowledge on the topic, oral exercises, Preparation for flights - game - competition Performing independent sorties - independent work Debriefing - summing up, homework Strategic plan
hard to learn, easy to fight!
Task: continue the formulation 1. The n-degree root (n = 2,3,4,5, ...) of the product of two non-negative numbers is equal to... the product of the n-degree roots of these numbers: = Example: = = 2*3=6 Air intelligence service
2. If a ≥ 0, b> 0 and n =2,3,4,5,... then the equality is true = Example: = =
3. If a ≥ 0, n =2,3,4,5,... and k is any natural number, then the equality is true Example:
4. If a ≥ 0, n and k are natural numbers greater than 1, then the equality is true Example:
5. If the indicators of the root and radical expression are multiplied or divided by the same natural number, then... the value of the root will not change Example:
Game - competition “Preparation for flights” Questions for the crews I. Find the value of the numerical expression 1, 5 2) 6 2 3) 4) 2 5) 2 1)
II. Bring the radicals to the same root exponent and compare them 2) > 1)
III. Convert the given expression to the form 2) 1) 4) * * 3)
Results of pre-flight preparation Evaluation criteria 5-7 points - rating “satisfactory” 8-10 points - rating “good” 11-15 points - rating “excellent” 1 crew 2 crew 3 crew Number of points Maximum number of points 15 Crew rating
Carrying out independent flights Independent work Group A assignment rated “satisfactory” Group B and C assignment additionally graded “good” and “excellent”
Debriefing 1 crew assessment 2 crew assessment 3 crew assessment
Homework §3, No. 3.5, 3.6, 3.16, 3.23, 3.24 Textbook: Mathematics. Grade 11. A.G. Mordkovich and others - M.: Mnemozina, 2009.
Thanks for the lesson!
Annex 1
