1. The concept of equality and inequality
2. Properties of equalities and inequalities. Examples of solving equalities and inequalities
Numerical Equalities and Inequalities
Let be f and g- two numeric expressions. Let's connect them with an equal sign. We will receive an offer f= g which is called numerical equality.
Take, for example, the numerical expressions 3 + 2 and 6 - 1 and connect them with the equal sign 3 + 2 = 6-1. It is true. If we connect 3 + 2 and 7 - 3 with an equal sign, then we get a false numerical equality 3 + 2 = = 7-3. Thus, from a logical point of view, numerical equality is a statement, true or false.
Numeric equality is true if the values of the numeric expressions on the left and right sides of the equality are the same.
Properties of equalities and inequalities
Let us recall some properties of true numerical equalities.
1. If we add the same numerical expression that makes sense to both sides of the true numerical equality, then we also get the true numerical equality.
2. If both sides of a true numerical equality are multiplied by the same numerical expression that makes sense, then we also get a true numerical equality.
Let be f and g- two numeric expressions. Let's connect them with a ">" (or "<»). Получим предложение f > g(or f < g), which is called numerical inequality.
For example, if you combine the expression 6 + 2 and 13-7 with the ">" sign, we get the true numerical inequality 6 + 2> 13-7. If you connect the same expressions with the sign "<», получим ложное числовое неравенство 6 + 2 < 13-7. Таким образом, с логической точки зрения числовое неравенство - это высказывание, истинное или ложное.
Numerical inequalities have a number of properties. Let's consider some.
1. If we add the same numerical expression that makes sense to both sides of a true numerical inequality, then we also get a true numerical inequality.
2. If both sides of a true numerical inequality are multiplied by the same numerical expression that has meaning and a positive value, then we also get a true numerical inequality.
3. If both sides of a true numerical inequality are multiplied by the same numerical expression that has meaning and a negative value, and also change the sign of the inequality to the opposite, then we also get a true numerical inequality.
Exercises
1. Determine which of the following numeric equalities and inequalities are true:
a) (5.05: 1/40 - 2.8 5/6) 3 + 16 0.1875 = 602;
b) (1/14 - 2/7): (-3) - 6 1/13: (-6 1/13)> (7- 8 4/5) 2 7/9 - 15: (1/8 - 3/4);
c) 1.0905: 0.025 - 6.84 3.07 + 2.38: 100< 4,8:(0,04·0,006).
2. Check if the numerical equalities are true: 13 93 = 31 39, 14 82 = 41 28, 23 64 = 32 46. Can you say that the product of any two natural numbers will not change if you rearrange the numbers in each factor?
3. It is known that x> y - true inequality. Will the following inequalities be true:
a ) 2x> 2y; v ) 2x-7< 2у-7;
b) - x/3<-y/ 3; G ) -2x-7<-2у-7?
4. It is known that a< b - true inequality. Replace * with ">" or "<» так, чтобы получилось истинное неравенство:
a) -3.7 a * -3,7b; G) - a/3 * -b/3 ;
b) 0.12 a * 0,12b; e) -2 (a + 5) * -2(b + 5);
v) a/7 * b/ 7; f) 2/7 ( a-1) * 2/7 (b-1).
5. Given inequality 5> 3. Multiply both sides by 7; 0.1; 2.6; 3/4. Is it possible, on the basis of the results obtained, to assert that for any positive number a inequality 5a> 3a is it true?
6. Complete the assignments that are intended for primary school students, and draw a conclusion about how the concepts of numerical equality and numerical inequality are interpreted in the elementary mathematics course.
Two numerical mathematical expressions connected by the "=" sign are called equality.
For example: 3 + 7 = 10 - equality.
Equality can be right or wrong.
The point of solving any example is to find the meaning of the expression that turns it into a true equality.
To form ideas about true and false equalities in the 1st grade textbook, examples with a window are used.
For example:
Using the selection method, the child finds suitable numbers and checks the correctness of the equality by calculation.
The process of comparing numbers and designating relationships between them using comparison signs leads to inequalities.
For example: 5< 7; б >4 - numerical inequalities
Inequalities can also be true and false.
For example:
Using the selection method, the child finds suitable numbers and checks the correctness of the inequality.
Numeric inequalities are obtained by comparing numeric expressions and numbers.
For example:
When choosing a comparison sign, the child calculates the value of the expression and compares it with a given number, which is reflected in the choice of the corresponding sign:
10-2> 7 5 + K7 7 + 3> 9 6-3 = 3
Another way of choosing the comparison sign is possible - without reference to the calculation of the value of the expression.
Nappimep:
The sum of the numbers 7 and 2 will certainly be greater than the number 7, which means that 7 + 2> 7.
The difference between the numbers 10 and 3 will certainly be less than the number 10, which means that 10 is 3< 10.
Numeric inequalities are obtained by comparing two numeric expressions.
To compare two expressions is to compare their values. For example:
When choosing a comparison sign, the child calculates the values of the expressions and compares them, which is reflected in the choice of the corresponding sign:
Another way of choosing the comparison sign is possible - without reference to the calculation of the value of the expression. For example:
For setting comparison signs, one can carry out the following reasoning:
The sum of 6 and 4 is greater than the sum of 6 and 3, since 4> 3, which means 6 + 4> 6 + 3.
The difference between the numbers 7 and 5 is less than the difference between the numbers 7 and 3, since 5> 3, which means 7 - 5< 7 - 3.
The quotient of 90 and 5 is greater than the quotient of 90 and 10, because when dividing the same number by a larger number, the quotient is smaller, which means 90: 5> 90:10.
To form ideas about true and false equalities and inequalities in the new edition of the textbook (2001), tasks of the form are used:
For verification, the method of calculating the value of expressions and comparing the resulting numbers is used.
Inequalities with a variable are practically not used in the latest editions of the stable mathematics textbook, although they were present in earlier editions. Inequalities with variables are actively used in alternative mathematics textbooks. These are inequalities of the form:
+ 7 < 10; 5 - >2; > 0; > O
After the introduction of a letter to denote an unknown number, such inequalities take on the usual form of inequality with a variable:
a + 7> 10; 12-d<7.
The values of the unknown numbers in such inequalities are found by the selection method, and then each matched number is checked by substitution. The peculiarity of these inequalities is that several numbers can be selected that fit them (giving the correct inequality).
For example: a + 7> 10; a = 4, a = 5, a = 6, etc. - the number of values for the letter a is infinite, any number a> 3 is suitable for this inequality; 12 - d< 7; d = 6, d = 7, d = 8, d = 9, d = 10, d = 11, d = 12 - количество значений для буквы d конечно, все значения могут быть перечислены. Ребенок подставляет каждое найденное значение переменной в выражение, вычисляет значение выражения и сравнивает его с заданным числом. Выбираются те значения переменной, при которых неравенство является верным.
In the case of an infinite set of solutions or a large number of solutions to an inequality, the child is limited to selecting several values of the variable for which the inequality is true.
In this lesson, you and the frog will become familiar with the mathematical concepts of "equality" and "inequality", as well as comparison signs. Use fun and interesting examples to learn how to compare groups of shapes using pairing and compare numbers using a number ray.
Theme:Familiarity with basic concepts in mathematics
Lesson: Equality and Inequality
In this lesson we will get acquainted with mathematical concepts: "equality" and "inequality".
Try to answer the question:
There are tubs against the wall
Each has exactly one frog.
If there were five tubs,
How many frogs would there be? (fig. 1)
Rice. 1
The poem says that there were 5 tubs, in each tub there is 1 frog, no one was left without a pair, which means the number of frogs is equal to the number of tubs.
Let's denote the tubs by the letter K, and the frogs by the letter L.
We write the equality: K = L. (Fig. 2)
Rice. 2
Compare the number of two groups of shapes. There are many figures, they are of different sizes, arranged without order. (fig. 3)
Rice. 3
Let's make pairs of these figures. We connect each square with a triangle. (fig. 4)
Rice. 4
Two squares were left without a pair. This means that the number of squares is not equal to the number of triangles. Let us denote the squares by the letter K, and the triangles by the letter T.
We write the inequality: K ≠ T. (Fig. 5)
Rice. 5
Output: You can compare the number of items in two groups by pairing. If all the elements have enough pairs, then the corresponding numbers are equal, in this case we put between numbers or letters =... This entry is called equality... (fig. 6)
Rice. 6
If there is not enough pair, that is, extra items remain, then these numbers not equal... We put between numbers or letters unequal sign... This entry is called inequality.(fig. 7)
Rice. 7
The elements left without a pair show which of the two numbers is greater and by how much. (fig. 8)
Rice. eight
The method of comparing groups of shapes using pairing is not always convenient and takes a lot of time. You can compare numbers using the number ray. (fig. 9)
Rice. nine
Compare these numbers using the number beam and put a comparison sign.
We need to compare the numbers 2 and 5. Let's look at the number beam. The number 2 is closer to 0 than the number 5, or they say the number 2 on the number ray is more to the left than the number 5. This means that 2 is not equal to 5. This is an inequality.
The sign "≠" (not equal) only fixes the inequality of numbers, but does not indicate which of them is greater and which is less.
Of the two numbers on the number ray, the smaller is to the left and the larger to the right. (fig. 10)
Rice. ten
This inequality can be written differently, using less sign "< » or greater than sign ">" :
On the number ray, the number 7 is to the right than the number 4, therefore:
7 ≠ 4 and 7> 4
The numbers 9 and 9 are equal, so we put the = sign, this is equality:
Compare the number of dots and the number and put the appropriate sign. (fig. 11)
Rice. eleven
In the first picture, we need to put the sign = or ≠.
Compare two points and the number 2, put an = sign between them. This is equality.
We compare one point and the number 3, on the number ray the number 1 is to the left of the number 3, we put the sign ≠.
Compare the four points and 4. Put the = sign between them. This is equality.
Compare three points and number 4. Three points - this is number 3. On the number beam it is to the left, put the sign ≠. This is inequality. (fig. 12)
Rice. 12
In the second figure, between the dots and numbers, you need to put the signs =,<, >.
Let's compare five points and the number 5. Between them we put an = sign. This is equality.
Let's compare three dots and the number 3. Here you can also put the = sign.
Let's compare five points and the number 6. On the number ray, the number 5 is to the left of the number 6. Put the sign<. Это неравенство.
Let's compare two points and one, the number 2 is to the right of the number ray than the number 1. We put the> sign. This is inequality. (fig. 13)
Rice. 13
Insert a number in the box to make the equality and inequality correct.
This is inequality. Let's look at the number beam. Since we are looking for a number less than the number 7, then it must be to the left of the number 7 on the number ray. (fig. 14)
Rice. fourteen
Several numbers can be inserted into the window. The numbers 0, 1, 2, 3, 4, 5, 6 are suitable here. Any of them can be substituted in the window and get several correct inequalities. For example 5< 7 или 2 < 7
On the number ray, find numbers that are less than 5. (Fig. 15)
Rice. 15
These are the numbers 4, 3, 2, 1, 0. Therefore, any of these numbers can be substituted in the window, we get several correct inequalities. For example, 5> 4, 5> 3
You can substitute only one number 8.
In this lesson, we got acquainted with the mathematical concepts: "equality" and "inequality", learned how to correctly place comparison signs, practiced comparing groups of figures using pairing and comparing numbers using a number ray, which will help in the further study of mathematics.
Bibliography
- Alexandrova L.A., Mordkovich A.G. Grade 1 mathematics. - M: Mnemosina, 2012.
- Bashmakov M.I., Nefedova M.G. Maths. 1 class. - M: Astrel, 2012.
- Bedenko M.V. Maths. 1 class. - M7: Russian Word, 2012.
- Igraem.pro ().
- Slideshare.net ().
- Iqsha.ru ().
Homework
1. What comparison signs do you know, in what cases they are used? Write down the comparison signs for the numbers.
2. Compare the number of items in the figure and put the sign “<», «>"Or" = ".
3. Compare the numbers by putting the sign “<», «>"Or" = ".
Municipal budgetary educational institution of the city of Irkutsk, secondary school number 23
The lesson was developed by: .
Lesson type: a lesson in the discovery of new knowledge.
Lesson construction technology: technology for the development of critical thinking. System-activity approach, health-saving technologies.
Lesson topic: True and False Equality and Inequality.
Lesson objectives: learn to find (recognize) true and false equalities and inequalities.
Strengthen the ability to write equality and inequality using symbols. To form the ability to compare, analyze, generalize on various grounds, to model the choice of methods of activity, to group.
Develop the ability to ask, be interested in other people's opinions and express their own; enter into dialogue.
Basic terms, concepts: equality, inequality, true, false, comparison., signs "greater than", "less", "equal".
Planned results:
- students should be aware of true and false inequalities;
- students should have a general understanding of true and false equalities;
- students must recognize true and false equalities and true and false inequalities;
- students should be able to analyze the proposed situation;
- students should be able to reproduce the knowledge gained.
Personal UUD:
- to define common rules of conduct for all;
- define the rules for working in pairs;
- to evaluate the assimilated content of the educational material (based on personal values);
- to establish a connection between the purpose of the activity and its result.
Regulatory UUD:
- determine and formulate the goal of the lesson;
- formulate educational tasks, draw conclusions;
- work according to the proposed plan, instructions;
- express your assumptions on the basis of educational material;
- to distinguish a correctly completed task from an incorrect one.
Cognitive UUD:
- to navigate in a textbook, notebook;
- to navigate in their knowledge system (to determine the boundaries of knowledge / ignorance);
- find answers to questions using your knowledge;
- to analyze the educational material;
- make the comparison, explaining the comparison criteria.
Communicative UUD:
- listen and understand the speech of others;
- learn to express your thoughts with sufficient completeness and accuracy, to prove your opinion.
Organization of space
Forms of work: frontal, work in pairs, individual.
DURING THE CLASSES
Organizing time.
Invented by someone
Simple and wise
When meeting, say hello:
"Good morning!"
Good morning, my dear students! Good morning everyone present!
We are glad to have guests at our lesson. After all, it is not for nothing that folk wisdom says: "Guests in the house are joy to the owners!" Let's turn to our respected teachers, say hello to them, nod our heads. Well done, you showed yourself to be polite, well-mannered students.
Pupil:
We were expecting guests today
And they greeted with excitement:
Are we good at
And write and respond?
Do not judge too harshly
After all, we did not study much.
Teacher: We are starting a math lesson, which means important discoveries await us. What qualities will be useful to you in a math lesson? (H observance, resourcefulness, attentiveness, accuracy, accuracy, etc.).
Stage 1. "Call".
Teacher: Let's start with exercising the mind. (One answers, and the children honk).
2. The sum of the numbers 3 and 3?
3. Decreased 7, subtracted 4, the value of the difference?
4.1 term 1, the second term 6, the value of the sum?
5. Difference between numbers 6 and 4?
6. 5 increase by 1?
7. Decrease 6 by 6?
8. 4, is it 2 and?
9. The number of the previous number 7?
10. The number following the number 9?
11. 7 candles were burning, 2 candles were extinguished. How many candles are left? (Two candles.)
12. Kolya's portfolio fits in Vasya's portfolio, and Vasya's portfolio can be hidden in Seva's portfolio. Which of these portfolios is the largest?
13. (Diagram on the board). More people live in China than in India, and more people live in India than in Russia. Which of these countries has the largest population?
2 ultrasound. Take a close look at the board.
5…9 8 … 8 7-1 … 4 8 – 4 … 3 + 1
What groups can be divided into everything that is depicted, written on the board?
Children's answers: - Objects of wildlife, mathematical notes, geometric shapes; - Equality and inequality, etc.
The children formulate the lesson topic: Equality and Inequality.
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(On the desk)
In your workbook, write down equalities in 1 column. (1 child at the blackboard). Write down the inequalities in the second column. (1 child at the blackboard, children do not see the record).
Examination. Output.
Physiotherapy for the eyes.
Methodical reception: plus - minus - a question. Teacher: - guys, everyone has table number 1 on their desk. What assignment do you think I can offer you? (Children's options). In column 3, you need to mark each statement with a sign: "+" if the statement is correct, "-" - if it is wrong, and "?" - if you find it difficult to answer. We always put the icons in pencil. To whom everything is clear, you can get to work. (Pause). And with the guys who have doubts, I propose to start working together.
Table No. 1.
*Equality? | ||
*Inequality? | 3 + 4 = 7 | |
**Equality? | 6 = 4 + 2 | |
**Equality? | 6 < 7 | |
Equality? | ||
Equality? | 2 + 3 + 1 = 2 + 4 | |
Inequality? | 9 > 7 | |
Inequality? | 6 <3 | |
Equality? | ||
Equality? | ||
Inequality? | 2 - 1 < 8 | |
Inequality? | 8 > 4 + 4 | |
Equality? | 5 – 3 = 2 | |
Equality? | 8 – 3 = 2 + 3 | |
Inequality? | 9 > 9 |
Was the task easy to complete? What difficulties did you face?
Fizminutka
1. How many points are in this circle,
raise our hands so many times.
2. How many green Christmas trees,
so many bends
3. How many circles are there,
so many jumps.
4. Together we count the stars
so much squat together.
Reception: Z-H-U.
So what do I know ?! Fill in 1 column of the table.
Table No. 2.
- What would you like to learn in class today? (Answers of children). Fill in the 2nd column of the table. (Children formulate the topic of the lesson on their own).
Stage 2. Comprehension.
Welcome. Insert(text marking system (mat. records)).
Guys, how do you think we can know if we reasoned correctly or not? (Possible children's answers: Find an answer on the global Internet, ask adults, ask a teacher, in a textbook).
Please open the textbook on page 38 (3, 8), no. 96 (9, 6). And find a boy and a girl who, just like you, coped with the task. “Katya and Sasha performed the same tasks. Look what they did. " With which icons we can comment on the answer. In the textbook, we put "+" if it is correct, "-" if it is wrong. We work in pairs.
Well done! Raise your hands for those who learned new things in the math lesson (Children's answers: equalities and inequalities are true (correct entry) and incorrect (entry with errors). Can we fill in the 3 column of the table? (Children fill out).
The method of "subtle questions".
(1 student at the blackboard, the rest of the children work in pairs).
Handout: "Equal", "inequality", "true", "true", "incorrect", "incorrect", "9> 3", "5 + 1< 8», «6 < 4», «7 >5 + 4 "," 5 - 1 = 4 "," 9 = 4 + 2 "," 6 = 6 "," 3 = 8 ".
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Stage 3. Reflection.
Guys, continue the phrase:
“Today, at a math lesson, I learned….”;
"It was interesting to me…";
"Now I can ...".
Thank you for the lesson! In the lesson, we tried to think, answer correctly, proving your opinion, which means you will achieve great success in mathematics! Well done!
Two numerical mathematical expressions connected by the "=" sign are called equality.
For example: 3 + 7 = 10 - equality.
Equality can be right and wrong.
The point of solving any example is to find the meaning of the expression that turns it into a true equality.
To form ideas about true and false equalities in the 1st grade textbook, examples with a window are used.
For example:
Using the selection method, the child finds suitable numbers and checks the correctness of the equality by calculation.
The process of comparing numbers and designating relationships between them using comparison signs leads to inequalities.
For example: 5< 7; б >4 - numerical inequalities
Inequalities can also be true and false.
For example:
Using the selection method, the child finds suitable numbers and checks the correctness of the inequality.
Numeric inequalities are obtained by comparing numeric expressions and numbers.
For example:
When choosing a comparison sign, the child calculates the value of the expression and compares it with a given number, which is reflected in the choice of the corresponding sign:
10-2> 7 5 + K7 7 + 3> 9 6-3 = 3
Another way of choosing the comparison sign is possible - without reference to the calculation of the value of the expression.
Nappimep:
The sum of the numbers 7 and 2 will certainly be greater than the number 7, which means that 7 + 2> 7.
The difference between the numbers 10 and 3 will certainly be less than the number 10, which means that 10 is 3< 10.
Numeric inequalities are obtained by comparing two numeric expressions.
To compare two expressions is to compare their values. For example:
When choosing a comparison sign, the child calculates the values of the expressions and compares them, which is reflected in the choice of the corresponding sign:
Another way of choosing the comparison sign is possible - without reference to the calculation of the value of the expression. For example:
For setting comparison signs, one can carry out the following reasoning:
The sum of 6 and 4 is greater than the sum of 6 and 3, since 4> 3, which means 6 + 4> 6 + 3.
The difference between the numbers 7 and 5 is less than the difference between the numbers 7 and 3, since 5> 3, which means 7 - 5< 7 - 3.
The quotient of 90 and 5 is greater than the quotient of 90 and 10, because when dividing the same number by a larger number, the quotient is smaller, which means 90: 5> 90:10.
To form ideas about true and false equalities and inequalities in the new edition of the textbook (2001), tasks of the form are used:
For verification, the method of calculating the value of expressions and comparing the resulting numbers is used.
Inequalities with a variable are practically not used in the latest editions of the stable mathematics textbook, although they were present in earlier editions. Inequalities with variables are actively used in alternative mathematics textbooks. These are inequalities of the form:
+ 7 < 10; 5 - >2; > 0; > O
After the introduction of a letter to denote an unknown number, such inequalities take on the usual form of inequality with a variable:
a + 7> 10; 12-d<7.
The values of the unknown numbers in such inequalities are found by the selection method, and then each matched number is checked by substitution. The peculiarity of these inequalities is that several numbers can be selected that fit them (giving the correct inequality).
For example: a + 7> 10; a = 4, a = 5, a = 6, etc. - the number of values for the letter a is infinite, any number a> 3 is suitable for this inequality; 12 - d< 7; d = 6, d = 7, d = 8, d = 9, d = 10, d = 11, d = 12 - количество значений для буквы d конечно, все значения могут быть перечислены. Ребенок подставляет каждое найденное значение переменной в выражение, вычисляет значение выражения и сравнивает его с заданным числом. Выбираются те значения переменной, при которых неравенство является верным.
In the case of an infinite set of solutions or a large number of solutions to an inequality, the child is limited to selecting several values of the variable for which the inequality is true.
