Yurgel Olga Alexandrovna
1st grade (1-4)
Target:
- consolidate knowledge of the names of the components of addition and subtraction; to continue work on the formation of strong, conscious, automatic computing skills within 20;
- develop mathematical speech of students;
- cultivate accuracy when working in a notebook.
Equipment: an image of aliens, letters with examples, a ruler with drawings and examples for it.
During the classes:
I Org. moment.
II Oral account.
Today we have guests at our lesson. These are extraordinary guests. Do you want to guess who it is? To do this, you need to solve the examples on the cards with letters and put them in order under the corresponding numbers:
Children solve examples on cards (addition and subtraction within 20 with answers from 1 to 12, according to the table). Read the word that appears: aliens.
- Correctly! These are aliens. And here they are. (A picture of aliens is attached to the board.)
Landing took place. They do not yet know our language and speak to me mentally. This is called telepathy. They tell me they want to study the Earth and people. And they want to get to know you.
The first thing they want to explore is your quick wits. To do this, they are asked to represent numbers in the form of tens and units. And what are these numbers, let's try to mentally read. The aliens are sending us a signal. Well, who can guess the numbers?
Children call the numbers, if the number is two-digit, then they correctly read the thoughts. The number is represented as the sum of bit terms.
On the planet where our guests live, other icons are used instead of numbers. Look, they brought a ruler with them:
a) Compare the numbers: leaf and cherry; pear and asterisk; carrot and flag; sun and mushroom.
Inequalities are recorded using these icons.
b) Solve the examples:
Flower + 1
Carrot - 1
Triangle + 2
Pear - 2
Cherry - 2
Write examples on the board.
And now let's show how we can solve our earthly examples:
Children solve examples on counting fans.
III Work on the topic of the lesson.
And now attention, the aliens are mentally trying to help you better remember the components of addition. What are the names of the numbers that we add? (Addends.)
Let's repeat in chorus.
Children repeat at first quietly, then louder and louder.
What is the result of addition called? (Sum.)
Name the terms and sum:
Now consider this example:
Now feel your memory kick in again. Did you feel?
19 is minuend.
They repeat in chorus.
Why do you think this component is named so? (Because this number will be smaller when subtracted.)
4 is subtrahend. (chorus)
Why is it called so? (We subtract it.)
And what happened as a result is difference. (Chorus.)
IV Work on the textbook.
Examples #4(Children work in pairs.)
Find examples where the result should be a sum. Write down and solve any. Now explain to your neighbor where are the terms and where is the sum.
Find examples where the difference will be in the answer. Write down and solve any. Explain to the neighbor where is the reduced, where is the subtraction, and where is the difference.
With. 55 No. 4- orally.
V Work in notebooks.
No. 1 - problem solving
No. 6 - independently (put signs >,< или =)
VI Summary of the lesson.
And now, guys, the aliens are asking you to repeat what we did today in the lesson, what did we repeat?
They brought with them the A's that they give in schools on their planet.
(The teacher distributes prizes to those children who were the most active in the lesson.)
There are four basic arithmetic operations: addition, subtraction, multiplication and division. They are the basis of mathematics, with their help all other, more complex calculations are performed. Addition and subtraction are the simplest of them and are mutually opposite. But with the terms used in addition, we often encounter in life.
We are talking about the "combination of efforts" in the effort to jointly obtain the desired result, about the "terms success" etc. The names associated with subtraction remain within the bounds of mathematics, rarely appearing in everyday speech. Therefore, the words "subtracted", "reduced", "difference" are less common. The rule of finding each of these components can be applied only if the meaning of these names is understood.
Unlike many scientific terms having Greek, Latin or Arabic origin, in this case words with Russian roots are used. So it is not difficult to understand their meaning, which means it is easy to remember what is denoted by what term.
If you look closely at the name itself, it becomes noticeable that it is related to the words "different", "difference". From this it can be concluded that what is meant is the established difference between the quantities.
This concept in mathematics means:
- the difference between two numbers;
- it is a measure of how much one quantity is greater or less than another;
- this is the result obtained when subtracting - such a definition is offered by the school curriculum.
Note! If the quantities are equal to each other, then there is no difference between them. So their difference is zero.
What is minuend and subtrahend
As the name suggests, less is what is done less. And you can make the quantity smaller by subtracting a part from it. Thus, a diminished number is a number from which a part is taken away.
Subtracted, respectively, is the number that is subtracted from it.
| Minuend | Subtrahend | Difference | ||
| 18 | — | 11 | = | 7 |
| 14 | — | 5 | = | 9 |
| 26 | — | 22 | = | 4 |
Useful video: reduced, subtracted, difference
Rules for finding an unknown element
Having understood the terms, it is easy to establish by which rule each of the elements of subtraction is located.
Since the difference is the result of this arithmetic operation, it is found using this operation, no other rules are required here. But they are there in case the other term of the mathematical expression is unknown.
How to find the minuend
This term, as it was found out, refers to the amount from which the part was subtracted. But if one was subtracted, and the other remained in the end, therefore, the number consists of these two parts. It turns out that you can find the unknown reduced by adding two known elements.
So, in this case, to find the unknown, you should add the subtrahend and the difference:
Likewise in all such cases:
| ? | – | 5 | = | 9 |
| 9 | + | 5 | = | 14 |
It can be seen from the example that a certain value was taken away from 18, and 7 remained. To find this value, it is necessary to subtract 7 from 18.
| 26 | – | ? | = | 4 |
| 26 | – | 4 | = | 22 |
Thus, knowing the exact meaning of the names, one can easily guess by what rule each unknown element should be searched.
Useful video: how to find an unknown minuend
Conclusion
The four basic arithmetic operations are the basis on which all mathematical calculations are based, from the simplest to the most complex. Of course, in our time, when people tend to entrust technology to everything down to the thought process, it is more common and faster to make calculations using a calculator. But any skill increases the independence of a person - from technical means, from others. It is not necessary to make mathematics your specialty, but to have at least minimal knowledge and skills means to have additional support for your own confidence.
The concept of subtraction is best understood with an example. You decide to drink tea with sweets. There were 10 candies in the vase. You ate 3 candies. How many candies are left in the vase? If we subtract 3 from 10, then 7 sweets will remain in the vase. Let's write the problem mathematically:
Let's take a closer look at the entry:
10 is the number from which we subtract or which we reduce, therefore it is called reduced.
3 is the number we are subtracting. Therefore it is called deductible.
7 is the result of subtraction or is also called difference. The difference shows how much the first number (10) more than a second number (3) or how much the second number (3) is less than the first number (10).
If you are in doubt whether you have found the difference correctly, you need to do verification. Add the second number to the difference: 7+3=10
When subtracting l, the minuend cannot be less than the subtrahend.
We draw a conclusion from what has been said. Subtraction- this is an action with the help of which the second term is found by the sum and one of the terms.
In literal form, this expression will look like this:
a -b=c
a - reduced,
b - subtracted,
c is the difference.
Properties of subtracting a sum from a number.
13 — (3 + 4)=13 — 7=6
13 — 3 — 4 = 10 — 4=6
The example can be solved in two ways. The first way is to find the sum of numbers (3 + 4), and then subtract from total number(13). The second way is to subtract the first term (3) from the total number (13), and then subtract the second term (4) from the resulting difference.
In literal form, the property for subtracting the sum from a number will look like this:
a - (b + c) = a - b - c
The property of subtracting a number from a sum.
(7 + 3) — 2 = 10 — 2 = 8
7 + (3 — 2) = 7 + 1 = 8
(7 — 2) + 3 = 5 + 3 = 8
To subtract a number from the sum, you can subtract this number from one term, and then add the second term to the result of the difference. Under the condition, the term will be greater than the subtracted number.
In literal form, the property for subtracting a number from a sum will look like this:
(7 + 3) — 2 = 7 + (3 — 2)
(a +b) —c=a + (b - c), provided b > c
(7 + 3) — 2=(7 — 2) + 3
(a + b) - c \u003d (a - c) + b, provided a > c
Subtraction property with zero.
10 — 0 = 10
a - 0 = a
If you subtract zero from the number then it will be the same number.
10 — 10 = 0
a -a = 0
If you subtract the same number from a number then it will be zero.
Related questions:
In the example 35 - 22 = 13, name the minuend, the subtrahend and the difference.
Answer: 35 - reduced, 22 - subtracted, 13 - difference.
If the numbers are the same, what is their difference?
Answer: zero.
Do a subtraction check 24 - 16 = 8?
Answer: 16 + 8 = 24
Subtraction table for natural numbers from 1 to 10.
Examples for tasks on the topic "Subtraction of natural numbers."
Example #1:
Insert the missing number: a) 20 - ... = 20 b) 14 - ... + 5 = 14
Answer: a) 0 b) 5
Example #2:
Is it possible to subtract: a) 0 - 3 b) 56 - 12 c) 3 - 0 d) 576 - 576 e) 8732 - 8734
Answer: a) no b) 56 - 12 = 44 c) 3 - 0 = 3 d) 576 - 576 = 0 e) no
Example #3:
Read the expression: 20 - 8
Answer: “Subtract eight from twenty” or “Subtract eight from twenty.” Pronounce words correctly
Long way to develop skills solving equations starts with solving the very first and relatively simple equations. By such equations we mean equations, on the left side of which is the sum, difference, product or quotient of two numbers, one of which is unknown, and on the right side there is a number. That is, these equations contain an unknown term, minuend, subtrahend, multiplier, dividend, or divisor. The solution of such equations will be discussed in this article.
Here we will give the rules that allow us to find an unknown term, multiplier, etc. Moreover, we will immediately consider the application of these rules in practice, solving characteristic equations.
Page navigation.
So, we substitute the number 5 instead of x into the original equation 3 + x = 8, we get 3 + 5 = 8 - this equality is correct, therefore, we correctly found the unknown term. If during the check we received an incorrect numerical equality, then this would indicate to us that we solved the equation incorrectly. The main reasons for this may be either the application of the wrong rule, or computational errors.
How to find the unknown minuend, subtrahend?
The connection between addition and subtraction of numbers, which we already mentioned in the previous paragraph, allows us to obtain a rule for finding an unknown minuend through a known subtrahend and difference, as well as a rule for finding an unknown subtrahend through a known minuend and difference. We will formulate them in turn, and immediately give the solution of the corresponding equations.
To find the unknown minuend, you need to add the subtrahend to the difference.
For example, consider the equation x−2=5 . It contains an unknown minuend. The above rule tells us that in order to find it, we must add the known subtrahend 2 to the known difference 5, we have 5+2=7. Thus, the required minuend is equal to seven.
If you omit the explanations, then the solution is written as follows:
x−2=5 ,
x=5+2 ,
x=7 .
For self-control, we will perform a check. We substitute the found reduced into the original equation, and we obtain the numerical equality 7−2=5. It is correct, therefore, we can be sure that we have correctly determined the value of the unknown minuend.
You can move on to finding the unknown subtrahend. It is found by adding according to the following rule: to find the unknown subtrahend, it is necessary to subtract the difference from the minuend.
We solve an equation of the form 9−x=4 using the written rule. In this equation, the unknown is the subtrahend. To find it, we need to subtract the known difference 4 from the known reduced 9 , we have 9−4=5 . Thus, the required subtrahend is equal to five.
Let's bring short version solutions to this equation:
9−x=4 ,
x=9−4 ,
x=5 .
It remains only to check the correctness of the found subtrahend. Let's make a check, for which we substitute the found value 5 instead of x into the original equation, and we get the numerical equality 9−5=4. It is correct, therefore the value of the subtrahend that we found is correct.
And before moving on to the next rule, we note that in the 6th grade, a rule for solving equations is considered, which allows you to transfer any term from one part of the equation to another with the opposite sign. So, all the rules considered above for finding an unknown term, reduced and subtracted, are fully consistent with it.
To find the unknown factor, you need to...
Let's take a look at the equations x 3=12 and 2 y=6 . In them, the unknown number is the factor on the left side, and the product and the second factor are known. To find the unknown factor, you can use the following rule: to find the unknown factor, you need to divide the product by the known factor.
This rule is based on the fact that we gave the division of numbers a meaning opposite to the meaning of multiplication. That is, there is a connection between multiplication and division: from the equality a b=c , in which a≠0 and b≠0, it follows that c:a=b and c:b=c , and vice versa.
For example, let's find the unknown factor of the equation x·3=12 . According to the rule, we need to divide famous work 12 by a known multiplier of 3 . Let's do : 12:3=4 . So the unknown factor is 4 .
Briefly, the solution of the equation is written as a sequence of equalities:
x 3=12 ,
x=12:3 ,
x=4 .
It is also desirable to check the result: we substitute the found value instead of the letter in the original equation, we get 4 3 \u003d 12 - the correct numerical equality, so we correctly found the value of the unknown factor.
And one more thing: acting according to the studied rule, we actually perform the division of both parts of the equation by a non-zero known multiplier. In grade 6, it will be said that both parts of the equation can be multiplied and divided by the same non-zero number, this does not affect the roots of the equation.
How to find the unknown dividend, divisor?
As part of our topic, it remains to figure out how to find the unknown dividend with a known divisor and quotient, as well as how to find an unknown divisor with a known dividend and quotient. The relationship between multiplication and division already mentioned in the previous paragraph allows you to answer these questions.
To find the unknown dividend, you need to multiply the quotient by the divisor.
Let's consider its application with an example. Solve the equation x:5=9 . To find the unknown divisible of this equation, it is necessary, according to the rule, to multiply the known quotient 9 by the known divisor 5, that is, we perform the multiplication of natural numbers: 9 5 \u003d 45. Thus, the desired dividend is 45.
Let's show a short notation of the solution:
x:5=9 ,
x=9 5 ,
x=45 .
The check confirms that the value of the unknown dividend is found correctly. Indeed, when substituting the number 45 into the original equation instead of the variable x, it turns into the correct numerical equality 45:5=9.
Note that the analyzed rule can be interpreted as the multiplication of both parts of the equation by a known divisor. Such a transformation does not affect the roots of the equation.
Let's move on to the rule for finding the unknown divisor: to find the unknown divisor, divide the dividend by the quotient.
Consider an example. Find the unknown divisor from equation 18:x=3 . To do this, we need to divide the known dividend 18 by the known quotient 3, we have 18:3=6. Thus, the required divisor is equal to six.
The solution can also be formulated as follows:
18:x=3 ,
x=18:3 ,
x=6 .
Let's check this result for reliability: 18:6=3 is the correct numerical equality, therefore, the root of the equation is found correctly.
It is clear that this rule can only be applied when the quotient is different from zero, so as not to encounter division by zero. When the quotient is zero, two cases are possible. If in this case the dividend is equal to zero, that is, the equation has the form 0:x=0 , then this equation satisfies any non-zero value of the divisor. In other words, the roots of such an equation are any numbers that are not equal to zero. If at zero the partial dividend is different from zero, then for any values of the divisor, the original equation does not turn into the correct numerical equality, that is, the equation has no roots. To illustrate, we present the equation 5:x=0 , it has no solutions.
Sharing Rules
Consistent application of the rules for finding the unknown term, minuend, subtrahend, multiplier, dividend and divisor allows solving equations with a single variable more than complex type. Let's deal with this with an example.
Consider the equation 3 x+1=7 . First, we can find the unknown term 3 x , for this we need to subtract the known term 1 from the sum 7, we get 3 x=7−1 and then 3 x=6 . Now it remains to find the unknown factor by dividing the product of 6 by the known factor 3 , we have x=6:3 , whence x=2 . So the root of the original equation is found.
To consolidate the material, we present short solution one more equation (2 x−7): 3−5=2 .
(2 x−7):3−5=2 ,
(2 x−7):3=2+5 ,
(2 x−7):3=7 ,
2 x−7=7 3 ,
2x−7=21 ,
2x=21+7 ,
2x=28 ,
x=28:2 ,
x=14 .
Bibliography.
- Maths.. 4th grade. Proc. for general education institutions. At 2 o'clock, Part 1 / [M. I. Moro, M. A. Bantova, G. V. Beltyukova and others]. - 8th ed. - M.: Education, 2011. - 112 p.: ill. - (School of Russia). - ISBN 978-5-09-023769-7.
- Maths: studies. for 5 cells. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. - M.: Mnemosyne, 2007. - 280 p.: ill. ISBN 5-346-00699-0.
To find an unknown term, you need to …………………………………………………………….. The result of multiplying two or more factors is called…………………………………… ……… To find the dividend, you need ……………………………………………………………………………… The result of subtracting numbers is called …………………… …………………………………………… The result of adding two or more terms is called ……………………………………… To find an unknown factor, you need to…………… ……………………………………………. The result of dividing numbers is called……………………………………………………………………. To find the minuend you need……………………………………………………………………………………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………………………………………………… …. To find how much one number is more or less than another, you need to………………………….………………………………………………………………………… ……………………………………..To find how many times one number is greater or less than another, you need ……………………….……………………………… ……………………………………………………………………………………. In an expression without brackets, containing only addition and subtraction or multiplication and division, the actions are performed ………………… ……………………………………………………………. In expressions containing brackets, all actions are performed first ………………………..…………………………………………………………………………… ……………………………………………………………………………………………………………………………………… ………………….. The perimeter of a figure is ………………………………………………………………………………… The perimeter of a rectangle is ……… …………………………………………………………………………………………………………………………………………………………… ……………………………………. The semi-perimeter of a rectangle is ………………………………………………………………….. To find the side of a square, you need the value of its perimeter………………………… ……………… To find the area of a rectangle, you need …………………………………………………………… To find the width of a rectangle, you need its area………………… ………………………… To find the length of a rectangle, you need …………………………………………………………….
To find the unknown term, you need to subtract the other term from the sum.
The result of multiplying two or more factors is called a product.
To find the dividend, you need to multiply the divisor by the quotient.
The result of subtracting numbers is called the difference
The result of adding two or more terms is called the sum.
To find the unknown factor, you need to divide the product by another factor.
The result of dividing numbers is called quotient.
To find the minuend, add the difference to the minuend.
To find the divisor, divide the dividend by the quotient.
To find the subtrahend, subtract the difference from the minuend.
To find how much one number is greater or less than another, subtract the smaller number from the larger number.
……………………………………………………………………………………………………………..
To find how many times one number is greater or less than another, you need to more divide by less.
………………………………………………………………………………………………………………….
In an expression without
brackets containing only addition and subtraction or multiplication and division,
actions are performed in order.………………………………………………………………………………….
In expressions containing brackets, all actions in brackets are performed first.………………………..
……………………………………………………………………………………………………………………………………………………………………………………………………………………………………..
The perimeter of a figure is the sum of the lengths of all sides.
The perimeter of the rectangle is the sum of the two sides multiplied by 2. P \u003d 2 * (a + b)………………………………………………………………………
The perimeter of a square is equal to the length of the side times 4………………………………………………………………………………………………….
The semi-perimeter of a rectangle is the length of two sides…………………………………………………………………..
To find the side of a square, you need to divide the value of its perimeter by 4…………………………………………
To find the area of a rectangle, multiply the length value by the width value.
To find the width of a rectangle, divide its area by its length.………………………………………………
To find the length of a rectangle, divide its area by its width.…………………………………………………………….
