As a rule, children begin to study algebra in elementary school. After mastering the basic principles of working with numbers, they solve examples with one or more unknown variables. Finding the meaning of an expression like this can be quite difficult, but if you simplify it using elementary school knowledge, everything will work out quickly and easily.
What is the meaning of an expression
A numerical expression is an algebraic notation consisting of numbers, parentheses and signs if it makes sense.
In other words, if it is possible to find the meaning of an expression, then the entry is not without meaning, and vice versa.
Examples of the following entries are valid numeric constructions:
- 3*8-2;
- 15/3+6;
- 0,3*8-4/2;
- 3/1+15/5;
A single number will also represent a numeric expression, like the number 18 from the above example.
Examples of incorrect number constructions that do not make sense:
- *7-25);
- 16/0-;
- (*-5;
Incorrect numeric examples are just a bunch of mathematical symbols and have no meaning. 
How to find the value of an expression
Since such examples contain arithmetic signs, we can conclude that they allow arithmetic calculations. To calculate the signs or, in other words, to find the meaning of an expression, it is necessary to perform the appropriate arithmetic manipulations.
As an example, consider the following construction: (120-30)/3=30. The number 30 will be the value of the numerical expression (120-30)/3.
Instructions:
Concept of numerical equality
A numerical equality is a situation where two parts of an example are separated by the “=” sign. That is, one part is completely equal (identical) to the other, even if displayed in the form of other combinations of symbols and numbers.
For example, any construction like 2+2=4 can be called a numerical equality, since even if the parts are swapped, the meaning will not change: 4=2+2. The same goes for more complex constructions involving parentheses, division, multiplication, operations with fractions, and so on.
How to find the value of an expression correctly
To correctly find the value of an expression, it is necessary to perform calculations according to a certain order of actions. This order is taught in mathematics lessons, and later in algebra classes in elementary school. It is also known as arithmetic steps.
Arithmetic steps:
- The first stage is the addition and subtraction of numbers.
- The second stage is where division and multiplication are performed.
- Third stage - numbers are squared or cubed.
By observing the following rules, you can always correctly determine the meaning of an expression:
- Perform actions starting from the third step, ending with the first, if there are no parentheses in the example. That is, first square or cube, then divide or multiply, and only then add and subtract.
- In constructions with brackets, perform the actions in the brackets first, and then follow the order described above. If there are several brackets, also use the procedure from the first paragraph.
- In examples in the form of a fraction, first find out the result in the numerator, then in the denominator, then divide the first by the second.
Finding the meaning of an expression will not be difficult if you acquire basic knowledge of elementary courses in algebra and mathematics. Guided by the information described above, you can solve any problem, even of increased complexity.
Find out the password from VK, knowing the login
LESSON TOPIC: Mathematical expressions. General lesson.
The purpose of the lesson: generalize and systematize all the knowledge children have about mathematical expressions, systematize and consolidate the corresponding skills.
List of knowledge and skills: the ability to distinguish mathematical expressions from other records; understanding the term “meaning of expression”; understanding the task “find the meaning of an expression”; knowledge of two types of mathematical expressions 9numerical expression, variable expression or literal expression; knowledge of two ways to calculate the value of expressions: performing actions in accordance with the rules of the order of actions and applying when calculating the rules of multiplying a sum by a number, dividing a sum by a number, etc., i.e., replacing a given expression with another based on the properties of arithmetic operations, identically equal to the given one; the ability to establish equality of expressions, relations 2more2, “less2; the ability to formulate an expression based on a problem and vice versa; the ability to determine the meaning of an expression (and its meaning) compiled for a task; ability to read an expression in different ways and write down expressions as they are read in different ways.
DURING THE CLASSES
(teacher) - The topic of today's lesson: mathematical expressions. The goal of your work in the lesson will be: to remember everything you know about mathematical expressions, repeat and consolidate everything you know how to do with them. First, select and read mathematical expressions from the data on the board.
The following is written on the board:
1. 16·20·5-360:6 2. 63·756·0+ 8046=8046
3. (98-18 a):2+87 4. a=4
5. 50·37· 4= 50·4· 37=200· 37=7400
6. 1248 1 0 7. 98-14:2+5
Correct answer: (1, 3, 6, 7)
(students) - Mathematical expressions are records 1, 3, 6, 7. record 2 is an equality, on the left side of which is a numerical expression, and on the right side is the value of this expression (the product 63 756 and 0 is equal to zero, and the sum of zero and 8046 equal to 8046); entry 4 is equality; record 5 is a chain of equalities, a chain of expressions that are equal to each other, an expanded record of calculating a product based on the property of multiplication - Multiple numbers can be multiplied in any order.
Expressions 1, 6 and 7 are numerical expressions; 3 – letter expression.
(teacher) - Look at expressions 1, 6, 7. What task can you complete using them?
(students) -You can find the meaning of these expressions.
(teacher) - What rules do you need to remember?
(students) – Rules of procedure.
(teacher) - Find the meaning of expression 1, indicating the sequence of actions.
(students) – sequence (·, ·, :, ), 1540
(teacher) - Indicate the rational sequence for performing the multiplication action.
(students) – 20·5,100·16
(teacher) -Find the meaning of expression 6.
(students) – 0.
(teacher) - Consider the chain of equalities 5. Are the numbers multiplied in the order in which they are written in the first expression?
(students) - No.
(teacher) - What property of multiplication allows you to replace this expression with the second expression in the chain?
(students) – Rearranging the places of the factors does not change the product.
(teacher) - This means that the meaning of an expression can be found by performing actions strictly according to the rules of the order of actions. You can replace this expression with an equal one, using the properties of actions, and then perform the actions not in the order in which they should have been performed in the first expression, but in an order convenient for calculations.
(teacher) - Read the expressions using mathematical terms.
(teacher) - Open your notebooks, write down the number, “Class work,” the topic “Mathematical expressions.”
(teacher) - Write down expression 3 in your notebook, having read it first. To the right of it write the equality a=4. Skip down four squares. Write down expression 7. Open the textbooks on page 37. The tasks written on the cards given to you are designed in such a way that by choosing the correct expression for each task (from those written on the board and data in the textbook0 or a task and completing this task, you will consolidate the ability to find the meanings of expressions , using the rules of the order of actions, and repeat these rules themselves: the ability to find the meaning of letter expressions for a given value of the letter included in the expression; the ability to compare expressions, the ability to compose an expression for a problem and vice versa, to compose or find a corresponding problem in a textbook, the ability to determine the meaning of expressions, the ability to read and write expressions. After completing the tasks and checking yourself, you will be able to test yourself how well you know mathematical expressions and how to use this knowledge. Get to work, taking your remote control as your assistants and controllers.
TASKS ON CARDS
1. Find the value of the expression
2. Find the value of the expression, which is the sum of a particular expression containing the letter and numbers 2 and the number 87, with a=4.
Hint 1. the expression is written in your notebook
Hint 2.(9∙8 - 18∙a): 2+87
Consultation1. to find the value of an expression containing a letter, you need to mentally replace the letter in this expression with its value and calculate the value of the resulting numerical expression.
Consultation 2. First, the operations in parentheses are performed (first multiplication or division, and then addition or subtraction), then with the result of the calculation in parentheses, actions without parentheses: first multiplication or division, and then addition or subtraction.
3. Rewrite five times the expression in which the action signs are written in the following order: “-“, “:”, “+”. Calculate the value of this expression, first without placing parentheses, and then by placing parentheses in four different ways so that the values of the expression include the numbers 47, 96, 12, 86.
4. Find, among the expressions given in the exercises on page 37, an expression that is the difference of two products and an expression that is the sum of two quotients. Compare them. Write down the corresponding inequality in your notebook and on the remote control.
5. Find a word problem on pages 38 or 39 that can be solved by creating an expression that is the product of the sum of two two-digit numbers by 2 by 3. Write down this expression. Write down the solution to this problem step by step with an explanation in your notebook. Enter the number or value of the quantity resulting from the solution on the remote control, indicating the number of this task, the number of the word problem and then the number or value of the quantity.
6. Find problems that can be solved using the following expressions:
1) 20:5; 2) 8-5; 3) 8+5; 4)24∙3; 5) 108:24; 6) 50+45.
For each expression, indicate the number of the problem for which it was compiled. Give the number of expressions that make sense for this task. State what each one means.
RESULT OF THE LESSON
(teacher) –Using the “Control” key, check the correctness of each task. Assess your knowledge.
So, what do you know about mathematical expressions?
(students) - Mathematical expressions can be numeric or alphabetic.
To find the value of a numerical expression, you need to perform all the actions according to the rules of the order of actions. You can find the value of a numeric expression by using action properties.
To find the value of a literal expression for a given value of a letter, you need to replace the letter in the expression with its value and calculate the value of the resulting numeric expression.
Two numeric expressions can be compared. Of two numeric expressions, the one whose value is greater (less) is greater (less).
When solving word problems, expressions are composed, the value of the last of which (when writing a solution to actions) or the value of which (when writing a solution in the form of an expression and then an equality) gives the answer to the question of the problem.
(teacher) - What can you do with expressions?
We know how to find the value of a numerical expression using the rules of the order of actions and the properties of actions. We know how to compare expressions (to do this we need to calculate the value of each expression and compare them), we know how to determine the meaning of expressions compiled for a given task, we know how to compose expressions for tasks, we know how to find the meaning of a literal expression given the values of the letters included in it.
Note. For each answer, the teacher either offers to give a supporting example from the student himself, or he himself gives the corresponding task from those that were completed in the lesson.
(34∙10+(489–296)∙8):4–410. Determine the course of action. Perform the first action in the inner brackets 489–296=193. Then, multiply 193∙8=1544 and 34∙10=340. Next action: 340+1544=1884. Next, divide 1884:4=461 and then subtract 461–410=60. You have found the meaning of this expression.
Example. Find the value of the expression 2sin 30º∙cos 30º∙tg 30º∙ctg 30º. Simplify this expression. To do this, use the formula tg α∙ctg α=1. Get: 2sin 30º∙cos 30º∙1=2sin 30º∙cos 30º. It is known that sin 30º=1/2 and cos 30º=√3/2. Therefore, 2sin 30º∙cos 30º=2∙1/2∙√3/2=√3/2. You have found the meaning of this expression.
The value of the algebraic expression from . To find the value of an algebraic expression given the variables, simplify the expression. Substitute certain values for the variables. Complete the necessary steps. As a result, you will receive a number, which will be the value of the algebraic expression for the given variables.
Example. Find the value of the expression 7(a+y)–3(2a+3y) with a=21 and y=10. Simplify this expression and get: a–2y. Substitute the corresponding values of the variables and calculate: a–2y=21–2∙10=1. This is the value of the expression 7(a+y)–3(2a+3y) with a=21 and y=10.
note
There are algebraic expressions that do not make sense for some values of the variables. For example, the expression x/(7–a) does not make sense if a=7, because in this case, the denominator of the fraction becomes zero.
Sources:
- find the smallest value of the expression
- Find the meanings of the expressions for c 14
Learning to simplify expressions in mathematics is simply necessary in order to correctly and quickly solve problems and various equations. Simplifying an expression involves reducing the number of steps, which makes calculations easier and saves time.
Instructions
Learn to calculate powers of c. When multiplying powers c, a number is obtained whose base is the same, and the exponents are added b^m+b^n=b^(m+n). When dividing powers with the same bases, the power of a number is obtained, the base of which remains the same, and the exponents of the powers are subtracted, and the exponent of the divisor b^m is subtracted from the exponent of the dividend: b^n=b^(m-n). When raising a power to a power, the power of a number is obtained, the base of which remains the same, and the exponents are multiplied (b^m)^n=b^(mn) When raising to a power, each factor is raised to this power. (abc)^m=a^m *b^m*c^m
Factor polynomials, i.e. imagine them as a product of several factors - and monomials. Take the common factor out of brackets. Learn the basic formulas for abbreviated multiplication: difference of squares, squared difference, sum, difference of cubes, cube of sum and difference. For example, m^8+2*m^4*n^4+n^8=(m^4)^2+2*m^4*n^4+(n^4)^2. These formulas are the main ones in simplification. Use the method of isolating a perfect square in a trinomial of the form ax^2+bx+c.
Abbreviate fractions as often as possible. For example, (2*a^2*b)/(a^2*b*c)=2/(a*c). But remember that you can only reduce multipliers. If the numerator and denominator of an algebraic fraction are multiplied by the same number other than zero, then the value of the fraction will not change. You can convert expressions in two ways: chained and by actions. The second method is preferable, because it is easier to check the results of intermediate actions.
It is often necessary to extract roots in expressions. Even roots are extracted only from non-negative expressions or numbers. Odd roots can be extracted from any expression.
Sources:
- simplification of expressions with powers
Trigonometric functions first emerged as tools for abstract mathematical calculations of the dependences of the values of acute angles in a right triangle on the lengths of its sides. Now they are very widely used in both scientific and technical fields of human activity. For practical calculations of trigonometric functions of given arguments, you can use different tools - several of the most accessible ones are described below.
Instructions
Use, for example, the calculator program installed by default with the operating system. It opens by selecting the “Calculator” item in the “Utilities” folder from the “Standard” subsection, placed in the “All programs” section. This section can be opened by clicking on the “Start” button to the main operating menu. If you are using the Windows 7 version, you can simply type “Calculator” into the “Search programs and files” field of the main menu, and then click on the corresponding link in the search results.
Count the number of steps required and think about the order in which they should be performed. If this question is difficult for you, please note that the operations enclosed in parentheses are performed first, then division and multiplication; and subtraction are done last. To make it easier to remember the algorithm of the actions performed, in the expression above each action operator sign (+,-,*,:), with a thin pencil, write down the numbers corresponding to the execution of the actions.
Proceed with the first step, following the established order. Count in your head if the actions are easy to perform verbally. If calculations are required (in a column), write them down under the expression, indicating the serial number of the action.
Clearly track the sequence of actions performed, evaluate what needs to be subtracted from what, divided into what, etc. Very often the answer in the expression is incorrect due to mistakes made at this stage.
A distinctive feature of the expression is the presence of mathematical operations. It is indicated by certain signs (multiplication, division, subtraction or addition). The sequence of performing mathematical operations is corrected with brackets if necessary. To perform mathematical operations means to find .
What is not an expression
Not every mathematical notation can be classified as an expression.
Equalities are not expressions. Whether mathematical operations are present in the equality or not does not matter. For example, a=5 is an equality, not an expression, but 8+6*2=20 also cannot be considered an expression, although it contains multiplication. This example also belongs to the category of equalities.
The concepts of expression and equality are not mutually exclusive; the former is included in the latter. The equal sign connects two expressions:
5+7=24:2
This equation can be simplified:
5+7=12
An expression always assumes that the mathematical operations it represents can be performed. 9+:-7 is not an expression, although there are signs of mathematical operations here, because it is impossible to perform these actions.
There are also mathematical ones that are formally expressions, but have no meaning. An example of such an expression:
46:(5-2-3)
The number 46 must be divided by the result of the actions in brackets, and it is equal to zero. You cannot divide by zero; the action is considered prohibited.
Numeric and algebraic expressions
There are two types of mathematical expressions.
If an expression contains only numbers and symbols of mathematical operations, such an expression is called numeric. If in an expression, along with numbers, there are variables denoted by letters, or there are no numbers at all, the expression consists only of variables and symbols of mathematical operations, it is called algebraic.
The fundamental difference between a numerical value and an algebraic value is that a numerical expression has only one value. For example, the value of the numerical expression 56–2*3 will always be equal to 50; nothing can be changed. An algebraic expression can have many values, because any number can be substituted. So, if in the expression b–7 we substitute 9 for b, the value of the expression will be 2, and if 200, it will be 193.
Sources:
- Numeric and algebraic expressions
Goals: improve skills in composing expressions and calculating their meanings; continue to develop skills in solving complex problems; develop attention and reasoning skills.
During the classes
I. Organizational moment.
II. Verbal counting.
1. Mathematical dictation.
a) The number was reduced by 8 and we got 20. Name this number.
b) The number was increased by 6 and we got 15. Name this number.
c) If the number is increased by 5 times, it becomes 30. What number is this?
d) If the number is reduced by 4 times, it becomes 8. What number is this?
2. Geometry on matches.
a) How many squares are there in the drawing? How many other polygons? What are these polygons?
b) Remove one stick so that 3 squares remain. Find several solutions and compare them.
c) Remove one stick so that 4 squares remain. Find several solutions and compare them.
d) Remove two sticks so that 4 squares remain.
3. Compare the time shown on the clock. Using the same rule, draw the hands on the last clock.
III. Lesson topic message.
IV. Work on the topic of the lesson.
Task No. 5(p. 74).
Students read the assignment.
– How many parts does the expression consist of?
– What action will be performed last?
– Write down the expression and calculate its value.
Task No. 6(p. 74).
- Read the text. Is he a task?
– What is known? What do you need to know?
– Write down briefly the conditions of the problem.
It was 25 liters. and 14 l.
Used - 7 liters.
Left - ? l.
1) How many sheets were there?
25 + 14 = 39 (l.).
2) How many sheets are left?
39 – 7 = 32 (l.).
Answer: 32 sheets.
V. Repetition of the material covered.
1. Work according to the textbook.
Task No. 13(p. 75).
– Look at the drawing.
– What are these figures called?
– What is the area of the shaded part of the figure?
– How many cells are in the yellow figure? (28 cells.)
– How many cells are in the blue figure? (24 cells.)
– How many cells form 1 cm2? (4 cells.)
– How to calculate the area in this case?
28: 4 = 7 (cm 2).
24: 4 = 6 (cm 2).
Task No. 14(p. 75).
Students create “machine” diagrams and answer the questions in the assignment.
Task No. 15(p. 75).
Students work independently. Peer testing in pairs.
2. Work using cards.
Task No. 1.
Write down expressions and calculate their values.
a) From the number 90, subtract the sum of the numbers 42 and 8.
b) Increase the difference between the numbers 58 and 50 by 7.
c) From the number 39, subtract the difference between the numbers 17 and 8.
d) Reduce the sum of numbers 13 and 7 by 9.
e) From the number 38, subtract the difference between the numbers 17 and 9.
f) Reduce the sum of numbers 7 and 6 by 10.
g) To the number 8 add the difference between the numbers 75 and 70.
h) Increase the difference between numbers 13 and 4 by 20.
Task No. 2.
There were as many apples in the vase as there were on the plate. 5 more apples were put into the vase, and there were 14 apples in it. How many apples are there on the plate and in the vase together? Find an expression to solve the problem and calculate its value.
VI. Lesson summary.
– What new did you learn in the lesson?
– Name the components of all arithmetic operations.
Homework: No. 139 (workbook).
Lesson 108
Corner. right angle
Goals: introduce students to the concept of “angle”; teach how to perform a right angle model; learn to identify right and indirect angles in a drawing; improve computing skills; develop attention and eye.
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