Numerical inequalities 8. Development of an algebra lesson on the topic “Numerical inequalities” (8th grade)
Lesson topic:
Numerical inequalities.
Algebra 8th grade
Goals:
- repeat the rules for comparing different numbers;
- consolidate the concepts of “less” and “more”;
- get acquainted with the method of comparing any numbers and letter expressions;
- learn to use the comparison method when doing exercises
Compare the numbers:
11 and -13 7 and 2
Oral work
, =
17 -3 -17-(-3) 0
11,5 13,6 11,5-13,6 0
Conclusion: If a b, then a – b 0.
- And, conversely, if a – b 0, then a 0.
b, then a – b 0. And, conversely, if a – b 0, then a b "width="640"
Oral work
Compare the numbers. Compare the difference of these numbers with zero. , =
0,7 0,03 0,7-0,03 0
- Conclusion: If a b, then a – b 0.
- And, conversely, if a – b 0, then a b
Oral work
Compare the numbers. Compare the difference of these numbers with zero. , =
Conclusion: If a = b, then a – b = 0.
And, conversely, if a – b = 0, then a = b.
Compare numbers a and b if:
a – b = - 0.07, then a b
a – b = 0, then a b
a – b = 11.5, then a b
It is known that a b.
Can the difference a – b be expressed as 7.15? -12 ? 0 ?
A way to compare any numbers
Number a is greater than b , if the difference a – b – positive number
Number a is less than b , if the difference a – b - a negative number
Method for comparing numbers
To compare two numbers you need:
- find their difference;
- compare the difference with zero;
- draw a conclusion.
Working with the textbook
№ 726,
№ 730,
№ 731.
Reflection
When is the first number less than the second?
When is the first number greater than the second?
When is the first number equal to the second?
Formulate a way to compare numbers (literal expressions).
- I'm happy with the lesson, I really enjoyed it.
- I enjoyed the lesson, but there are gaps in my knowledge.
- I'm not happy with the lesson, I didn't understand anything and I don't know how to solve the examples.
Homework assignment
clause 28. def.; No. 728,
Inequality is a record in which numbers, variables or expressions are connected by a sign<, >, or . That is, inequality can be called a comparison of numbers, variables or expressions. Signs < , > , ⩽ And ⩾ are called inequality signs.
Types of inequalities and how they are read:
As can be seen from the examples, all inequalities consist of two parts: left and right, connected by one of the inequality signs. Depending on the sign connecting the parts of the inequalities, they are divided into strict and non-strict.
Strict inequalities- inequalities whose parts are connected by a sign< или >. Non-strict inequalities- inequalities in which the parts are connected by the sign or.
Let's consider the basic rules of comparison in algebra:
- Any positive number greater than zero.
- Any negative number is less than zero.
- Of two negative numbers, the one whose absolute value is smaller is greater. For example, -1 > -7.
- a And b positive:
a - b > 0,
That a more b (a > b).
- If the difference of two unequal numbers a And b negative:
a - b < 0,
That a less b (a < b).
- If the number is greater than zero, then it is positive:
a> 0, which means a- positive number.
- If the number is less than zero, then it is negative:
a < 0, значит a- a negative number.
Equivalent inequalities- inequalities that are a consequence of other inequalities. For example, if a less b, That b more a:
a < b And b > a- equivalent inequalities
Properties of inequalities
- If you add the same number to both sides of an inequality or subtract the same number from both sides, you get an equivalent inequality, that is,
If a > b, That a + c > b + c And a - c > b - c
It follows from this that it is possible to transfer terms of inequality from one part to another with the opposite sign. For example, adding to both sides of the inequality a - b > c - d By d, we get:
a - b > c - d
a - b + d > c - d + d
a - b + d > c
- If both sides of the inequality are multiplied or divided by the same positive number, then an equivalent inequality is obtained, that is,
- If both sides of the inequality are multiplied or divided by the same negative number, then the inequality opposite to the given one will be obtained, that is, therefore, when multiplying or dividing both parts of the inequality by a negative number, the sign of the inequality must be changed to the opposite.
This property can be used to change the signs of all terms of an inequality by multiplying both sides by -1 and changing the sign of the inequality to the opposite:
-a + b > -c
(-a + b) · -1< (-c) · -1
a - b < c
Inequality -a + b > -c tantamount to inequality a - b < c
Municipal budgetary educational institution "Kachalinskaya secondary school No. 2"
Ilovlinsky district, Volgograd region
Developing a lesson using an interactive whiteboard
algebra for 8th grade students
on this topic"Numerical Inequalities"
Mathematic teacher
Postoeva Zh.V.
Stanitsa Kachalinskaya
2009
A lesson on the topic “Numerical inequalities” was developed for 8th grade students based on the textbook “Algebra” by Yu.N. Makarychev.
Goals:
Continue to improve your skills in using abbreviated multiplication formulas. Derive a method for comparing numbers and literal expressions. To achieve from students the ability to apply knowledge to perform tasks of a standard type (training exercises), reconstructive-variative type, creative type;
Development of skills in applying knowledge in a specific situation; development of logical thinking, skills to compare, generalize, correctly formulate tasks and express thoughts; development of independent activity of students.
Cultivating interest in the subject through the content of educational material, nurturing such character qualities as communication when working in a group, perseverance in achieving goals.
Lesson type: learning new material.
Form: lesson - research.
Equipment:
Interactive whiteboard and multimedia equipment
Lesson structure
Lesson stage
Screenshot of the program window Notebook
To work in class, students are seated in groups of 3-4 people.
Lesson topic message
Communicating the goals and objectives of the lesson.
Activation of students' knowledge and skills necessary to perceive new knowledge.
Using examples, the formulas for abbreviated multiplication and comparison of different numbers are repeated:
Decimal fractions,
Common fractions with like numerators,
Common fractions with different denominators,
Proper and improper fractions.
Natural
Decimals
Common fractions
first the number was less second, and the difference was negative .
Oral work on comparing different numbers:
Natural
Decimals
Common fractions
and comparing the resulting differences with zero.
For comparison, the following numbers are taken so that first the number was more second, and the difference was positive .
Behind the curtain is a conclusion that students must come to on their own.
Oral work on comparing different numbers:
Decimals
Common fractions
and comparing the resulting differences with zero.
For comparison, the following numbers are taken so that first the number was equals second, and the difference was equal to zero .
Behind the curtain is a conclusion that students must come to on their own.
The teacher suggests doing an oral exercise to compare numbers if their difference is known.
If students find it difficult to answer, there is a hint behind the screen that they can use.
This exercise is also performed orally. Students must justify their answer.
Teacher: who can formulate: when one number is greater than another;
when one number is less than another
when two numbers are equal.
Who can tell me what to do to compare two numbers?
Hidden behind the curtain is a statement of how to compare numbers, which is revealed after students answer.
An example is provided to prove it - comparing two literal expressions. The proof is carried out together with the students, while the teacher gradually opens the curtain.
The teacher once again returns to the formulation of the method for comparing numbers.
Exercise No. 728 is given to apply knowledge. Students perform tasks a) and b) exercises in notebooks and on the board with comments on the solution. Tasks c) and d) are performed independently in groups.
The teacher reviews the solutions in groups and answers students’ questions.
The task a) students solve on the board and in notebooks, b) they are asked to solve it orally with comments, c) - independently.
Students perform tasks a) and b) in groups. The teacher reviews the solutions, while one of the group explains the solution.
Task d) is performed on the board with comments.
To reinforce new material, students are asked questions, and after answering them, rules for repeated visual perception are pulled out from behind the screen.
Lesson summary: comments on students’ work in class, grading, recording homework in diaries.
Lesson on the topic “Numerical inequalities”
Goals:
- Educational: introduce the definition of the concepts “more” and “less”, numerical inequalities, teach how to apply them to proving inequalities;
- Developmental: develop the ability to use theoretical knowledge when solving practical problems, the ability to analyze and summarize the data obtained; develop cognitive interest in mathematics, broaden your horizons;
- Educational: to form positive motivation for learning.
During the classes:
1. Preparation and motivation.
Today we begin to study the important and relevant topic “Numerical inequalities”. If we slightly change the words of the great Chinese teacher Confucius (who lived more than 2400 years ago), we can formulate the task of our lesson: “I hear and forget. I see and remember. I do and I understand.”Let's formulate the purpose of the lesson together. (Students formulate a goal, the teacher complements).
Study numerical inequalities and their definition, and learn how to apply them in practice.
In practice, we often have to compare values. For example, the area of Russia ( 17 098 242 ) and the area of French territory ( 547 030 ) , the length of the Oka River (1500 km) and the length of the Don River (1870 km).
2.Updating basic knowledge.
Guys, let's remember everything we know about inequalities.
Guys, look at the board and compare:
3.6748 and 3.675 36.5810 and 36.581 | And 0.45 5.5 and | 15 and -23 115 and -127 |
What is inequality?
Inequality - a relationship between numbers (or any mathematical expression capable of taking on a numerical value) indicating which one is greater or less than another.
Inequality signs (›; ‹) appeared for the first time in 1631, but the concept of inequality, like the concept of equality, arose in ancient times. In the development of mathematical thought, without comparing quantities, without the concepts of “more” and “less,” it was impossible to reach the concept of equality, identity, or equation.
What rules were used to compare numbers?
a) of two positive numbers, the one whose modulus is greater is greater;
b) of two negative numbers, the one whose modulus is smaller is greater;
c) any negative number is less than a positive number;
d) any positive number greater than zero;
e) any negative number is less than zero.
What rule do we use to compare numbers located on a coordinate line?
(On a coordinate line, a larger number is represented by a point lying to the right, and a smaller number by a point lying to the left.)
Note that depending on the specific type of numbers we used one or another comparison method. It is not comfortable. It would be easier for us to have a universal way of comparing numbers that would cover all cases.
3. Studying new material.
Arrange the numbers in ascending order: 8; 0; -3; -1.5.
What is the smallest number? What is the largest number?
What numbers can be substituted for a and b?
a – b =8
a – b =-3
a – b =-8
a – b =1.5
a – b = 0
Please note that when you subtract a smaller number from a larger number, you get a positive number; When you subtract a larger number from a smaller number, you get a negative number.
A universal way to compare numbers is based on the definition of numerical inequalities: Number a is greater than number b if the difference a – b is a positive number; number a is less than number b if the difference a – b is a negative number. Note that if the difference a – b = 0, then the numbers a and b are equal.
4. Consolidation of new material.
Compare numbers a and b if:
A) a – b = - 0.8 (a is less than b, since the difference is a negative number)
B) a – b = 0 (a = b)
B) a – b = 5.903 (a is greater than b, since the difference is a positive number).
Solve with explanation at the board No. 724, 725 (orally), 727 (if time permits), 728 (a, d), 729 (c, d), 730, 732.
5. Lesson summary. D/z.learned def. No. 726, 728 (a, d), 729 (c, d), 731.
Guys, today in class we repeated previously studied material on inequalities and learned a lot about inequalities.
1) What is “inequality”?
2) How to compare two numbers?
3) Guys, raise your hands, who had difficulties in the lesson?
Preview:
a) of two positive numbers, the one whose modulus is greater is greater; b) of two negative numbers, the one whose modulus is smaller is greater; c) any negative number is less than a positive number; d) any positive number greater than zero; e) any negative number is less than zero.
What numbers can be substituted for a and b? a – b = 8 a – b =-3 a – b =- 8 a – b =1.5 a – b = 0 Arrange the numbers in ascending order: 8; 0; -3; -1.5.
The number a is greater than the number b if the difference a – b is a positive number; number a is less than number b if the difference a – b is a negative number. Note that if the difference a – b is 0, then the numbers a and b are equal.
Compare numbers a and b if: A) a – b = - 0.8 B) a – b = 0 C) a – b = 5.903
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