Difficulty of calculating school examples. Engineering calculator
And when calculating the values of expressions, actions are performed in a certain order, in other words, you must observe order of actions.
In this article, we will figure out which actions should be performed first and which ones after them. Let's start with the most simple cases, when the expression contains only numbers or variables connected by plus, minus, multiply and divide signs. Next, we will explain what order of actions should be followed in expressions with brackets. Finally, let's look at the order in which actions are performed in expressions containing powers, roots, and other functions.
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First multiplication and division, then addition and subtraction
The school gives the following a rule that determines the order in which actions are performed in expressions without parentheses:
- actions are performed in order from left to right,
- Moreover, multiplication and division are performed first, and then addition and subtraction.
The stated rule is perceived quite naturally. Performing actions in order from left to right is explained by the fact that it is customary for us to keep records from left to right. And the fact that multiplication and division are performed before addition and subtraction is explained by the meaning that these actions carry.
Let's look at a few examples of how this rule applies. For examples, we will take the simplest numerical expressions so as not to be distracted by calculations, but to focus specifically on the order of actions.
Example.
Follow steps 7−3+6.
Solution.
The original expression does not contain parentheses, and it does not contain multiplication or division. Therefore, we should perform all actions in order from left to right, that is, first we subtract 3 from 7, we get 4, after which we add 6 to the resulting difference of 4, we get 10.
Briefly, the solution can be written as follows: 7−3+6=4+6=10.
Answer:
7−3+6=10 .
Example.
Indicate the order of actions in the expression 6:2·8:3.
Solution.
To answer the question of the problem, let's turn to the rule indicating the order of execution of actions in expressions without parentheses. The original expression contains only the operations of multiplication and division, and according to the rule, they must be performed in order from left to right.
Answer:
At first We divide 6 by 2, multiply this quotient by 8, and finally divide the result by 3.
Example.
Calculate the value of the expression 17−5·6:3−2+4:2.
Solution.
First, let's determine in what order the actions in the original expression should be performed. It contains both multiplication and division and addition and subtraction. First, from left to right, you need to perform multiplication and division. So we multiply 5 by 6, we get 30, we divide this number by 3, we get 10. Now we divide 4 by 2, we get 2. We substitute the found value 10 into the original expression instead of 5·6:3, and instead of 4:2 - the value 2, we have 17−5·6:3−2+4:2=17−10−2+2.
The resulting expression no longer contains multiplication and division, so it remains to perform the remaining actions in order from left to right: 17−10−2+2=7−2+2=5+2=7.
Answer:
17−5·6:3−2+4:2=7.
At first, in order not to confuse the order in which actions are performed when calculating the value of an expression, it is convenient to place numbers above the action signs that correspond to the order in which they are performed. For the previous example it would look like this: .
The same order of operations - first multiplication and division, then addition and subtraction - should be followed when working with letter expressions.
Actions of the first and second stages
In some mathematics textbooks there is a division arithmetic operations for the actions of the first and second stages. Let's figure this out.
Definition.
Actions of the first stage addition and subtraction are called, and multiplication and division are called second stage actions.
In these terms, the rule from the previous paragraph, which determines the order of execution of actions, will be written as follows: if the expression does not contain parentheses, then in order from left to right, the actions of the second stage (multiplication and division) are performed first, then the actions of the first stage (addition and subtraction).
Order of arithmetic operations in expressions with parentheses
Expressions often contain parentheses to indicate the order in which actions should be performed. In this case a rule that specifies the order of execution of actions in expressions with parentheses, is formulated as follows: first, the actions in brackets are performed, while multiplication and division are also performed in order from left to right, then addition and subtraction.
So, the expressions in brackets are considered as components of the original expression, and they retain the order of actions already known to us. Let's look at the solutions to the examples for greater clarity.
Example.
Follow these steps 5+(7−2·3)·(6−4):2.
Solution.
The expression contains parentheses, so let's first perform the actions in the expressions enclosed in these parentheses. Let's start with the expression 7−2·3. In it you must first perform multiplication, and only then subtraction, we have 7−2·3=7−6=1. Let's move on to the second expression in brackets 6−4. There is only one action here - subtraction, we perform it 6−4 = 2.
We substitute the obtained values into the original expression: 5+(7−2·3)·(6−4):2=5+1·2:2. In the resulting expression, we first perform multiplication and division from left to right, then subtraction, we get 5+1·2:2=5+2:2=5+1=6. At this point, all actions are completed, we adhered to the following order of their implementation: 5+(7−2·3)·(6−4):2.
Let's write down a short solution: 5+(7−2·3)·(6−4):2=5+1·2:2=5+1=6.
Answer:
5+(7−2·3)·(6−4):2=6.
It happens that an expression contains parentheses within parentheses. There is no need to be afraid of this; you just need to consistently apply the stated rule for performing actions in expressions with brackets. Let's show the solution of the example.
Example.
Perform the operations in the expression 4+(3+1+4·(2+3)) .
Solution.
This is an expression with brackets, which means that the execution of actions must begin with the expression in brackets, that is, with 3+1+4·(2+3) . This expression also contains parentheses, so you must perform the actions in them first. Let's do this: 2+3=5. Substituting the found value, we get 3+1+4·5. In this expression, we first perform multiplication, then addition, we have 3+1+4·5=3+1+20=24. The initial value, after substituting this value, takes the form 4+24, and all that remains is to complete the actions: 4+24=28.
Answer:
4+(3+1+4·(2+3))=28.
In general, when an expression contains parentheses within parentheses, it is often convenient to perform actions starting with the inner parentheses and moving to the outer ones.
For example, let's say we need to perform the actions in the expression (4+(4+(4−6:2))−1)−1. First, we perform the actions in the inner brackets, since 4−6:2=4−3=1, then after this the original expression will take the form (4+(4+1)−1)−1. We again perform the action in the inner brackets, since 4+1=5, we arrive at the following expression (4+5−1)−1. Again we perform the actions in brackets: 4+5−1=8, and we arrive at the difference 8−1, which is equal to 7.
Also, each person had his own plot of land. There was a need to measure your plot of land.
A person had a need to calculate, measure everything around (stocks, livestock, products, land plot, building a house and so on.)
In addition to the above, a person learned to determine the shapes and sizes of surrounding objects, that is. it is round or square or oval... This means showing interest in the spatial forms of the real world.
Mathematics is so important in our world that there is not a single profession that does not require mathematics.
Carl Friedrich Gauss once said: “Mathematics is the queen of sciences, arithmetic is the queen of mathematics.”
Sign up for the course "Speed up mental arithmetic, NOT mental arithmetic" to learn how to quickly and correctly add, subtract, multiply, divide, square numbers and even extract roots. In 30 days, you'll learn how to use easy tricks to simplify arithmetic operations. Each lesson contains new techniques, clear examples and useful tasks.
Mathematician
A mathematician is, first of all, a specialist in mathematics. Both a teacher (teacher) of mathematics and a scientist who conducts his research in mathematics have the right to be called a mathematician. various areas mathematics.
The profession of mathematics is very complex and requires higher education at the university. Teaching mathematical skills is carried out, as a rule, in mathematics departments in higher educational institutions.
Mathematics classes (ranks and classes)
To make it easier for children, and not only children, to navigate numbers, a division of numbers into classes and ranks was invented.
Let's imagine the number 148951784296, and divide it by three digits: 148,951,784,296. So, from right to left: 296 is the class of units, 784 is the class of thousands, 951 is the class of millions, 148 is the class of billions. In turn, in each class 3 digits have their own digit. From right to left: the first digit is units, the second digit is tens, the third is hundreds. For example, the class of units is 296, 6 is ones, 9 is tens, 2 is hundreds.
This division is really very convenient and easy to remember. It is much easier when teaching children mathematics, when talking about some operation, to talk about how to fold in a column, for example. Because during the story you can name numbers by rank and class, and this will be much clearer to the student than simply calling them numbers.
Mathematics 1st grade
In the first grade they take a section of mathematics - arithmetic. Arithmetic is a branch of mathematics that works with numbers and calculations (operations with numbers).
In the first grade, as a rule, they go through the first two simplest operations with numbers: addition, subtraction.
Addition is an arithmetic operation during which two numbers are added, and their result is a new one - the third.
a+b=c.
Subtraction is an arithmetic operation in which the second number is subtracted from the first number, and the result is the third.
The addition formula is expressed as follows: a - b = c.
Transactions are carried out in single digits. Double digits are rare. Because it is necessary for children to get used to it and understand the technique.
Examples for training:
Task No. 1:
Task No. 2:
Mathematics 2nd grade
The second class is more serious than the first. Operations are carried out with double digit numbers. In addition to addition and subtraction there is "greater than, less than or equal to" operation.
The essence of the operation “greater than, less than or equal to” is to compare two numbers.
Sign< означает «меньше», знак >means “more” and accordingly = equal.
For example, you need to compare two numbers 25 and 40
25 < 40, 25 меньше 40.
49 and 14. 49>14, 49 is more than fourteen.
It is set equal if the number on the left and right is the same, or the expression is equivalent.
Examples for training:
Task No. 1:
Task No. 2:
Mathematics 3rd grade
In third grade, students have an understanding of the four basic mathematical operations: addition, subtraction, multiplication, division.
And examples with problems are aimed at consolidating addition, subtraction and better mastery of multiplication and division.
Problems involving mental calculation of all four operations are popular. An example of this type may seem difficult at first. But once you think about it, the answer becomes obvious.
Also, the third class is performing actions in a column. You can find the method of counting in a column for each operation in our articles on the corresponding operations.
Examples for training:
Task No. 1:
Task No. 2:
Solve examples:
- 84 - 67 =
- 45 + 30 =
- 35: 5 =
- 37 + 14 =
- 23 + 53 =
- 16 * 7 =
- 9 * 6 =
- 72: 6 =
- 40 + 27 =
- 12 * 3 =
- 45: 9 =
- 59 + 36 =
- 0 * 19 =
- 88: 11 =
- 8 * 24 =
- 16 * 6 =
- 22 + 76 =
- 3 + 89 =
- 64: 8 =
- 96 - 54 =
Solve examples:
- (7 + 20) : 3 - 8 =
- (0 * 8 + 24) : 6 =
- (20: 2 + 40) : 5 =
- 48: 6 * 3 - 15 =
- (82 - 53 + 11) : 8 =
- (9 * 8 - 12) : 10 =
Calculate:
- 8 rubles 64 kopecks + 15 kopecks =
- 3 meters 45 cm + 16 meters 55 cm =
- 7 rub. 70 k. – 3 r. 84 k.
- 8 tons – 8 quintals =
- 5 km 400 m + 2 km 550 m
Solve the equations:
- x * 7 = 56
- x: 3 = 27
- x + 72 = 99 + 1
- 92 - x = 43 + 14
Problem 1
The school canteen uses 180 kg of bread per week. How many kilograms of bread are consumed in 2 days, if we assume that working week is 6 days?
Problem 2
At the carpentry workshop, the children made 87 birdhouses. They hung 11 birdhouses in a cool area, twice as many in a city park, and hung the rest of the birdhouses on the outskirts of the city. How many birdhouses have children hung on the outskirts of the city?
Solve examples
Solve examples
Compare
134 and 13 3-12
3(12-20:4) and 3 12-20:4
(63-27):9:5 and (63+27:9):5
Solve the problem
The length of the plot is 12 m, the width is 4 times less than the length. Find the perimeter and area of the plot.
Solve the problem
The girl read 24 pages of the book in three days. How many pages will she read in 5 days if she reads 2 more pages every day?
Translate
37 dec. 7 units = ... units
8 hundred. 2 dec. 8 units = ... units
6 dec. 7 units = ... units
5 hundred. 9 units = ... units
1 cell 4 units = ... units
33 dec. = ... units
Mathematics 4th grade
In the fourth grade, there is active work with units of measurement: length (cm, dts, m, km), mass (g, kg), time (s, h), speed (m/s, km/h). And also work with previous operations accordingly.
We are studying mathematical equations with one unknown.
Examples for training:
Task No. 1:
Task No. 2:
A man on a bicycle covered the distance from the city to the village, equal to 60 km, in 4 hours. On the way back he slowed down by 3 km/h. How long did the cyclist spend on the train?
The plane's 16-hour journey is 4,150 km long. The plane flew for 3 hours at a speed of 660 km/h and another 2 hours at a speed of 730 km/h. How far does the plane have to travel in the last hour?
In 5 hours, the corn farmer flew 220 km. What distance will the corn truck cover if the speed is increased by 7 km/h?
Mathematics 5th grade
In the fifth grade, students begin to study topics such as fractions and mixed numbers. You can find information about operations with these numbers in our articles on the relevant operations.
Fractional number is the ratio of two numbers to each other or the numerator to the denominator. A fractional number can be replaced by division. For example, ¼ = 1:4.
Mixed number– this is a fractional number, only with a highlighted whole part. The integer part is allocated provided that the numerator is greater than the denominator. For example, there was a fraction: 5/4, it can be transformed by highlighting the whole part: one whole and ¼.
Examples for training:
Task No. 1:
Task No. 2:
Mathematics 6th grade
In 6th grade, the topic of converting fractions to lowercase notation appears. What does it mean? For example, given the fraction ½, it will be equal to 0.5. ¼ = 0.25.
Examples can be compiled in the following style: 0.25+0.73+12/31.
Examples for training:
Task No. 1:
Task No. 2:
Task No. 3:
There were a total of 92 chairs in the two classrooms. 16 chairs were moved from the first class to the second class and then their number was equalized. How many chairs were there in first and second class initially?
There were 240 kg of apples in two boxes. 18 kg of apples were transferred from the second box to the first. Afterwards, the number of apples in the first and second boxes was equal. How many kilograms of apples were initially in the first and second box?
The motorist left the city for the village at a speed of 11.5 km/h. After 2.4 hours, a bus left from the same place and in the same direction at a speed of 46 km/h. How long will it take for the bus to catch up with the car?
Games for developing mental arithmetic
Special educational games developed with the participation of Russian scientists from Skolkovo will help improve mental arithmetic skills in an interesting game form.
Game "Quick Count"
The game "quick count" will help you improve your thinking. The essence of the game is that in the picture presented to you, you will need to choose the answer “yes” or “no” to the question “are there 5 identical fruits?” Follow your goal, and this game will help you with this.
Game "Quick addition"
The game "Quick Addition" develops thinking and memory. The main point games to select numbers whose sum is equal to a given number. In this game, a matrix from one to sixteen is given. A given number is written above the matrix; you need to select the numbers in the matrix so that the sum of these digits is equal to the given number. If you answered correctly, you score points and continue playing.
Game "Guess the operation"
The game “Guess the Operation” develops thinking and memory. The main essence of the game is to choose mathematical sign so that the equality is true. There are examples on the screen, look carefully and put the right sign"+" or "-" so that the equality is true. The “+” and “-” signs are located at the bottom of the picture, select the desired sign and click on the desired button. If you answered correctly, you score points and continue playing.
Game "Mathematical matrices"
"Mathematical Matrices" is great brain exercise for kids, which will help you develop his mental work, mental calculation, quick search for the necessary components, and attentiveness. The essence of the game is that the player has to find a pair from the proposed 16 numbers that will add up to a given number, for example in the picture below the given number is “29”, and the desired pair is “5” and “24”.
Visual Geometry Game
The game "Visual Geometry" develops thinking and memory. The main essence of the game is to quickly count the number of shaded objects and select it from the list of answers. In this game, blue squares are shown on the screen for a few seconds, you need to quickly count them, then they close. Below the table there are four numbers written, you need to select one correct number and click on it with the mouse. If you answered correctly, you score points and continue playing.
Game "Simplification"
The game “Simplification” develops thinking and memory. The main essence of the game is to quickly perform a mathematical operation. A student is drawn on the screen at the blackboard, and a mathematical operation is given; the student needs to calculate this example and write the answer. Below are three answers, count and click the number you need using the mouse. If you answered correctly, you score points and continue playing.
Development of phenomenal mental arithmetic
We have looked at only the tip of the iceberg, to understand mathematics better - sign up for our course: Accelerating mental arithmetic - NOT mental arithmetic.
From the course you will not only learn dozens of techniques for simplified and quick multiplication, addition, multiplication, division, and calculating percentages, but you will also practice them in special tasks and educational games! Mental arithmetic also requires a lot of attention and concentration, which are actively trained when solving interesting problems.
Speed reading in 30 days
Increase your reading speed by 2-3 times in 30 days. From 150-200 to 300-600 words per minute or from 400 to 800-1200 words per minute. The course uses traditional exercises for developing speed reading, techniques that speed up brain function, methods for progressively increasing reading speed, the psychology of speed reading and questions from course participants. Suitable for children and adults reading up to 5000 words per minute.
Development of memory and attention in a child 5-10 years old
The course includes 30 lessons with useful tips and exercises for children's development. In every lesson useful advice, several interesting exercises, an assignment for the lesson and additional bonus at the end: an educational mini-game from our partner. Course duration: 30 days. The course is useful not only for children, but also for their parents.
Super memory in 30 days
Remember necessary information quickly and for a long time. Wondering how to open a door or wash your hair? I’m sure not, because this is part of our life. Easy and simple exercises for memory training can be made part of your life and done a little during the day. If eaten daily norm meals at a time, or you can eat in portions throughout the day.
Secrets of brain fitness, training memory, attention, thinking, counting
The brain, like the body, needs fitness. Exercise strengthen the body, mentally develop the brain. 30 days useful exercises and educational games to develop memory, concentration, intelligence and speed reading will strengthen the brain, turning it into tough nut to crack.
Money and the Millionaire Mindset
Why are there problems with money? In this course we will answer this question in detail, look deep into the problem, and consider our relationship with money from psychological, economic and emotional points of view. From the course you will learn what you need to do to solve all your financial problems, start saving money and investing it in the future.
Knowledge of the psychology of money and how to work with it makes a person a millionaire. 80% of people take out more loans as their income increases, becoming even poorer. On the other hand, self-made millionaires will earn millions again in 3-5 years if they start from scratch. This course teaches you how to properly distribute income and reduce expenses, motivates you to study and achieve goals, teaches you how to invest money and recognize a scam.
Mathematical-Calculator-Online v.1.0
The calculator performs the following operations: addition, subtraction, multiplication, division, working with decimals, root extraction, exponentiation, percent calculation and other operations.
Solution:
How to use a math calculator
| Key | Designation | Explanation |
|---|---|---|
| 5 | numbers 0-9 | Arabic numerals. Entering natural integers, zero. To get a negative integer, you must press the +/- key |
| . | period (comma) | Separator to indicate a decimal fraction. If there is no number before the point (comma), the calculator will automatically substitute a zero before the point. For example: .5 - 0.5 will be written |
| + | plus sign | Adding numbers (integers, decimals) |
| - | minus sign | Subtracting numbers (integers, decimals) |
| ÷ | division sign | Dividing numbers (integers, decimals) |
| X | multiplication sign | Multiplying numbers (integers, decimals) |
| √ | root | Extracting the root of a number. When you press the “root” button again, the root is calculated from the result. For example: root of 16 = 4; root of 4 = 2 |
| x 2 | squaring | Squaring a number. When you press the "squaring" button again, the result is squared. For example: square 2 = 4; square 4 = 16 |
| 1/x | fraction | Output in decimal fractions. The numerator is 1, the denominator is the entered number |
| % | percent | Getting a percentage of a number. To work, you need to enter: the number from which the percentage will be calculated, the sign (plus, minus, divide, multiply), how many percent in numerical form, the "%" button |
| ( | open parenthesis | An open parenthesis to specify the calculation priority. A closed parenthesis is required. Example: (2+3)*2=10 |
| ) | closed parenthesis | A closed parenthesis to specify the calculation priority. An open parenthesis is required |
| ± | plus minus | Reverses sign |
| = | equals | Displays the result of the solution. Also above the calculator, in the “Solution” field, intermediate calculations and the result are displayed. |
| ← | deleting a character | Removes the last character |
| WITH | reset | Reset button. Completely resets the calculator to position "0" |
Algorithm of the online calculator using examples
Addition.
Addition of integers natural numbers { 5 + 7 = 12 }
Addition of whole natural and negative numbers { 5 + (-2) = 3 }
Adding decimals fractional numbers { 0,3 + 5,2 = 5,5 }
Subtraction.
Subtracting natural integers ( 7 - 5 = 2 )
Subtracting natural and negative integers ( 5 - (-2) = 7 )
Subtracting decimal fractions (6.5 - 1.2 = 4.3)
Multiplication.
Product of natural integers (3 * 7 = 21)
Product of natural and negative integers ( 5 * (-3) = -15 )
Product of decimal fractions ( 0.5 * 0.6 = 0.3 )
Division.
Division of natural integers (27 / 3 = 9)
Division of natural and negative integers (15 / (-3) = -5)
Division of decimal fractions (6.2 / 2 = 3.1)
Extracting the root of a number.
Extracting the root of an integer ( root(9) = 3)
Extracting the root from decimals( root(2.5) = 1.58 )
Extracting the root of a sum of numbers ( root(56 + 25) = 9)
Extracting the root of the difference between numbers (root (32 – 7) = 5)
Squaring a number.
Squaring an integer ( (3) 2 = 9 )
Squaring decimals ((2,2)2 = 4.84)
Conversion to decimal fractions.
Calculating percentages of a number
Increase the number 230 by 15% ( 230 + 230 * 0.15 = 264.5 )
Reduce the number 510 by 35% ( 510 – 510 * 0.35 = 331.5 )
18% of the number 140 is (140 * 0.18 = 25.2)
The free program LoviOtvet is a functional calculator for solving examples and equations. The Lovi Answer program automatically solves mathematical examples and equations with the output of actions and stages of their solution.
Why is such a program needed? The Catch the Answer program is a kind of mathematical solver that displays the answer, with a step-by-step solution to the completed task.
The Catch the Answer program will be of interest to schoolchildren and their parents. With this program, parents can check the homework completed by the student. Also, schoolchildren and students can solve examples and equations using this mathematical calculator.
Adults who no longer remember much from the school course, as well as students, will be able, using this program, to quickly solve a mathematical example of any degree of complexity.
In the LoviOtvet program you can perform the following mathematical operations:
- Perform operations with natural numbers.
- Perform operations with fractions (decimal, ordinary, mixed).
- The program will allow you to simplify expressions and perform operations with polynomials.
- Solve linear and quadratic equations.
Examples and equations will be solved in the Lovi Answer program step by step, with sequential actions. Visually, in the program window, you will see the solution to an example or equation. Answer and step by step actions to solve it, they will be written down on a kind of notebook sheet. All stages of the solution can be recorded in the program in a column.
You can download the LoviOtvet program from the manufacturer’s official website. The program is available for use on computers with operating systems Windows system. There are versions of the program for devices running the operating system Android system, for Apple devices (iPad, iPhone/iPod), for mobile phones(java, java-mini).
Catch the Answer download
After downloading, you can install the program on your computer.
Installing the Lovi Answer program
Start the installation process of the LoviOtvet program on your computer.
Be careful when installing the program! Uncheck the boxes where you are prompted to install additional programs in order to avoid installing third-party software on your computer.
Once the installation of the program on your computer is complete, the main window of the LoviOtvet program will open.
Overview of the Lovi Answer program
At the top of the program window there are menu buttons for controlling the program.
Using the "Edit" menu button, you can copy the solution to your computer by selecting the desired copy option from the context menu. From the "Settings" menu you can select the sheet size, cells, clear history. Here you can change the display color of the program window by moving the slider to the desired location along the color scale.
Below the menu bar there is a field in which the task is entered.
On the left side of the window there are buttons and switches for entering data. Here is the main and additional panel.
The additional panel can be hidden using the “Hide additional panel” button. From here, if necessary, you can change the sheet size and the size of the cells in the work area.
The rest of the program window is occupied by the work area, in which the solution to the task will be displayed.
To solve the example, use the appropriate buttons to enter the expression, and then click on the “Answer” button. The solution can be displayed in several versions: standard solution, common fractions, solution “in a column”.
After clicking on the triangle on the far right side of the field in which an example or equation is entered, an additional field will open in which the history of calculations will be displayed. In this field you can clear the payment history.
You can read more about how to use the mathematical calculator on the official website of the manufacturer of the LoviOtvet program, on the “How to use” website page.
Catch Answer online
The manufacturer launched online version LoviOtvet program, which is available at the following address: https://calc.loviotvet.ru/.
According to the manufacturer, the version of Lovi Answer online is less functional than the program that is installed on a computer or mobile device. But anyway online calculator may be useful in some cases to complete assigned tasks.
Conclusions of the article
The free program Catch the Answer is a mathematical solver and calculator that helps schoolchildren, students and parents complete or check solutions to examples and equations of any degree of complexity.
LoviOtvet - a program for solving examples and equations (video)
This lesson discusses in detail the procedure for performing arithmetic operations in expressions without and with brackets. Students are given the opportunity, while completing assignments, to determine whether the meaning of expressions depends on the order in which arithmetic operations are performed, to find out whether the order of arithmetic operations is different in expressions without parentheses and with parentheses, to practice applying the learned rule, to find and correct errors made when determining the order of actions.
In life, we constantly perform some kind of action: we walk, study, read, write, count, smile, quarrel and make peace. We perform these actions in different orders. Sometimes they can be swapped, sometimes not. For example, when getting ready for school in the morning, you can first do exercises, then make your bed, or vice versa. But you can’t go to school first and then put on clothes.
In mathematics, is it necessary to perform arithmetic operations in a certain order?
Let's check
Let's compare the expressions:
8-3+4 and 8-3+4
We see that both expressions are exactly the same.
Let's perform actions in one expression from left to right, and in the other from right to left. You can use numbers to indicate the order of actions (Fig. 1).
Rice. 1. Procedure
In the first expression, we will first perform the subtraction operation and then add the number 4 to the result.
In the second expression, we first find the value of the sum, and then subtract the resulting result 7 from 8.
We see that the meanings of the expressions are different.
Let's conclude: The order in which arithmetic operations are performed cannot be changed.
Let's learn the rule for performing arithmetic operations in expressions without parentheses.
If an expression without parentheses includes only addition and subtraction or only multiplication and division, then the actions are performed in the order in which they are written.
Let's practice.
Consider the expression
This expression contains only addition and subtraction operations. These actions are called first stage actions.
We perform the actions from left to right in order (Fig. 2).
Rice. 2. Procedure
Consider the second expression
This expression contains only multiplication and division operations - These are the actions of the second stage.
We perform the actions from left to right in order (Fig. 3).
Rice. 3. Procedure
In what order are arithmetic operations performed if the expression contains not only addition and subtraction, but also multiplication and division?
If an expression without parentheses includes not only the operations of addition and subtraction, but also multiplication and division, or both of these operations, then first perform in order (from left to right) multiplication and division, and then addition and subtraction.
Let's look at the expression.
Let's think like this. This expression contains the operations of addition and subtraction, multiplication and division. We act according to the rule. First, we perform in order (from left to right) multiplication and division, and then addition and subtraction. Let's arrange the order of actions.
Let's calculate the value of the expression.
18:2-2*3+12:3=9-6+4=3+4=7
In what order are arithmetic operations performed if there are parentheses in an expression?
If an expression contains parentheses, the value of the expressions in the parentheses is evaluated first.
Let's look at the expression.
30 + 6 * (13 - 9)
We see that in this expression there is an action in parentheses, which means we will perform this action first, then multiplication and addition in order. Let's arrange the order of actions.
30 + 6 * (13 - 9)
Let's calculate the value of the expression.
30+6*(13-9)=30+6*4=30+24=54
How should one reason to correctly establish the order of arithmetic operations in a numerical expression?
Before starting calculations, you need to look at the expression (find out whether it contains parentheses, what actions it contains) and only then perform the actions in the following order:
1. actions written in brackets;
2. multiplication and division;
3. addition and subtraction.
The diagram will help you remember this simple rule (Fig. 4).
Rice. 4. Procedure
Let's practice.
Let's consider the expressions, establish the order of actions and perform calculations.
43 - (20 - 7) +15
32 + 9 * (19 - 16)
We will act according to the rule. The expression 43 - (20 - 7) +15 contains operations in parentheses, as well as addition and subtraction operations. Let's establish a procedure. The first action is to perform the operation in parentheses, and then, in order from left to right, subtraction and addition.
43 - (20 - 7) +15 =43 - 13 +15 = 30 + 15 = 45
The expression 32 + 9 * (19 - 16) contains operations in parentheses, as well as multiplication and addition. According to the rule, we first perform the action in parentheses, then multiplication (we multiply the number 9 by the result obtained by subtraction) and addition.
32 + 9 * (19 - 16) =32 + 9 * 3 = 32 + 27 = 59
In the expression 2*9-18:3 there are no parentheses, but there are multiplication, division and subtraction operations. We act according to the rule. First, we perform multiplication and division from left to right, and then subtract the result obtained from division from the result obtained by multiplication. That is, the first action is multiplication, the second is division, and the third is subtraction.
2*9-18:3=18-6=12
Let's find out whether the order of actions in the following expressions is correctly defined.
37 + 9 - 6: 2 * 3 =
18: (11 - 5) + 47=
7 * 3 - (16 + 4)=
Let's think like this.
37 + 9 - 6: 2 * 3 =
There are no parentheses in this expression, which means that we first perform multiplication or division from left to right, then addition or subtraction. In this expression, the first action is division, the second is multiplication. The third action should be addition, the fourth - subtraction. Conclusion: the procedure is determined correctly.
Let's find the value of this expression.
37+9-6:2*3 =37+9-3*3=37+9-9=46-9=37
Let's continue to talk.
The second expression contains parentheses, which means that we first perform the action in parentheses, then, from left to right, multiplication or division, addition or subtraction. We check: the first action is in parentheses, the second is division, the third is addition. Conclusion: the procedure is defined incorrectly. Let's correct the errors and find the value of the expression.
18:(11-5)+47=18:6+47=3+47=50
This expression also contains parentheses, which means that we first perform the action in parentheses, then from left to right multiplication or division, addition or subtraction. We check: the first action is in parentheses, the second is multiplication, the third is subtraction. Conclusion: the procedure is defined incorrectly. Let's correct the errors and find the value of the expression.
7*3-(16+4)=7*3-20=21-20=1
Let's complete the task.
Let's arrange the order of actions in the expression using the learned rule (Fig. 5).
Rice. 5. Procedure
We don't see numerical values, therefore we will not be able to find the meaning of the expressions, but we will practice applying the learned rule.
We act according to the algorithm.
The first expression contains parentheses, which means the first action is in parentheses. Then from left to right multiplication and division, then from left to right subtraction and addition.
The second expression also contains parentheses, which means we perform the first action in parentheses. After that, from left to right, multiplication and division, after that, subtraction.
Let's check ourselves (Fig. 6).
Rice. 6. Procedure
Today in class we learned about the rule for the order of actions in expressions without and with brackets.
References
- M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 1. - M.: “Enlightenment”, 2012.
- M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 2. - M.: “Enlightenment”, 2012.
- M.I. Moro. Math lessons: Methodical recommendations for the teacher. 3rd grade. - M.: Education, 2012.
- Regulatory document. Monitoring and evaluation of learning outcomes. - M.: “Enlightenment”, 2011.
- "School of Russia": Programs for primary school. - M.: “Enlightenment”, 2011.
- S.I. Volkova. Mathematics: Test work. 3rd grade. - M.: Education, 2012.
- V.N. Rudnitskaya. Tests. - M.: “Exam”, 2012.
- Festival.1september.ru ().
- Sosnovoborsk-soobchestva.ru ().
- Openclass.ru ().
Homework
1. Determine the order of actions in these expressions. Find the meaning of the expressions.
2. Determine in what expression this order of actions is performed:
1. multiplication; 2. division;. 3. addition; 4. subtraction; 5. addition. Find the meaning of this expression.
3. Make up three expressions in which the following order of actions is performed:
1. multiplication; 2. addition; 3. subtraction
1. addition; 2. subtraction; 3. addition
1. multiplication; 2. division; 3. addition
Find the meaning of these expressions.