You can perform various actions with fractions, for example, adding fractions. Addition of fractions can be divided into several types. Each type of addition of fractions has its own rules and algorithm of actions. Let's take a closer look at each type of addition.
Adding fractions with the same denominators.
For example, let's see how to add fractions with a common denominator.
The hikers went on a hike from point A to point E. On the first day, they walked from point A to B, or \(\frac(1)(5)\) all the way. On the second day they went from point B to D or \(\frac(2)(5)\) the whole way. How far did they travel from the beginning of the journey to point D?
To find the distance from point A to point D, add the fractions \(\frac(1)(5) + \frac(2)(5)\).
Adding fractions with the same denominators is that you need to add the numerators of these fractions, and the denominator will remain the same.
\(\frac(1)(5) + \frac(2)(5) = \frac(1 + 2)(5) = \frac(3)(5)\)
In literal form, the sum of fractions with the same denominators will look like this:
\(\bf \frac(a)(c) + \frac(b)(c) = \frac(a + b)(c)\)
Answer: the tourists traveled \(\frac(3)(5)\) all the way.
Adding fractions with different denominators.
Let's consider an example:
Add two fractions \(\frac(3)(4)\) and \(\frac(2)(7)\).
To add fractions with different denominators must first be found, and then use the rule for adding fractions with the same denominators.
For denominators 4 and 7, the common denominator is 28. The first fraction \(\frac(3)(4)\) must be multiplied by 7. The second fraction \(\frac(2)(7)\) must be multiplied by 4.
\(\frac(3)(4) + \frac(2)(7) = \frac(3 \times \color(red) (7) + 2 \times \color(red) (4))(4 \ times \color(red) (7)) = \frac(21 + 8)(28) = \frac(29)(28) = 1\frac(1)(28)\)
In literal form, we get the following formula:
\(\bf \frac(a)(b) + \frac(c)(d) = \frac(a \times d + c \times b)(b \times d)\)
Addition of mixed numbers or mixed fractions.
Addition occurs according to the law of addition.
For mixed fractions, add the integer parts to the integer parts and the fractional parts to the fractional parts.
If the fractional parts of mixed numbers have the same denominators, then add the numerators, and the denominator remains the same.
Add mixed numbers \(3\frac(6)(11)\) and \(1\frac(3)(11)\).
\(3\frac(6)(11) + 1\frac(3)(11) = (\color(red) (3) + \color(blue) (\frac(6)(11))) + ( \color(red) (1) + \color(blue) (\frac(3)(11))) = (\color(red) (3) + \color(red) (1)) + (\color( blue) (\frac(6)(11)) + \color(blue) (\frac(3)(11))) = \color(red)(4) + (\color(blue) (\frac(6 + 3)(11))) = \color(red)(4) + \color(blue) (\frac(9)(11)) = \color(red)(4) \color(blue) (\frac (9)(11))\)
If the fractional parts of mixed numbers have different denominators, then we find a common denominator.
Let's add mixed numbers \(7\frac(1)(8)\) and \(2\frac(1)(6)\).
The denominator is different, so you need to find a common denominator, it is equal to 24. Multiply the first fraction \(7\frac(1)(8)\) by an additional factor of 3, and the second fraction \(2\frac(1)(6)\) on 4.
\(7\frac(1)(8) + 2\frac(1)(6) = 7\frac(1 \times \color(red) (3))(8 \times \color(red) (3) ) = 2\frac(1 \times \color(red) (4))(6 \times \color(red) (4)) =7\frac(3)(24) + 2\frac(4)(24 ) = 9\frac(7)(24)\)
Related questions:
How to add fractions?
Answer: first you need to decide what type the expression belongs to: fractions have the same denominators, different denominators or mixed fractions. Depending on the type of expression, we proceed to the solution algorithm.
How to solve fractions with different denominators?
Answer: you need to find a common denominator, and then follow the rule of adding fractions with the same denominators.
How to solve mixed fractions?
Answer: Add integer parts to integer parts and fractional parts to fractional parts.
Example #1:
Can the sum of two result in a proper fraction? Wrong fraction? Give examples.
\(\frac(2)(7) + \frac(3)(7) = \frac(2 + 3)(7) = \frac(5)(7)\)
The fraction \(\frac(5)(7)\) is a proper fraction, it is the result of the sum of two proper fractions \(\frac(2)(7)\) and \(\frac(3)(7)\).
\(\frac(2)(5) + \frac(8)(9) = \frac(2 \times 9 + 8 \times 5)(5 \times 9) =\frac(18 + 40)(45) = \frac(58)(45)\)
The fraction \(\frac(58)(45)\) is an improper fraction, it is the result of the sum of the proper fractions \(\frac(2)(5)\) and \(\frac(8)(9)\).
Answer: The answer is yes to both questions.
Example #2:
Add fractions: a) \(\frac(3)(11) + \frac(5)(11)\) b) \(\frac(1)(3) + \frac(2)(9)\).
a) \(\frac(3)(11) + \frac(5)(11) = \frac(3 + 5)(11) = \frac(8)(11)\)
b) \(\frac(1)(3) + \frac(2)(9) = \frac(1 \times \color(red) (3))(3 \times \color(red) (3)) + \frac(2)(9) = \frac(3)(9) + \frac(2)(9) = \frac(5)(9)\)
Example #3:
Write a mixed fraction as a sum natural number and a proper fraction: a) \(1\frac(9)(47)\) b) \(5\frac(1)(3)\)
a) \(1\frac(9)(47) = 1 + \frac(9)(47)\)
b) \(5\frac(1)(3) = 5 + \frac(1)(3)\)
Example #4:
Calculate the sum: a) \(8\frac(5)(7) + 2\frac(1)(7)\) b) \(2\frac(9)(13) + \frac(2)(13) \) c) \(7\frac(2)(5) + 3\frac(4)(15)\)
a) \(8\frac(5)(7) + 2\frac(1)(7) = (8 + 2) + (\frac(5)(7) + \frac(1)(7)) = 10 + \frac(6)(7) = 10\frac(6)(7)\)
b) \(2\frac(9)(13) + \frac(2)(13) = 2 + (\frac(9)(13) + \frac(2)(13)) = 2\frac(11 )(thirteen) \)
c) \(7\frac(2)(5) + 3\frac(4)(15) = 7\frac(2 \times 3)(5 \times 3) + 3\frac(4)(15) = 7\frac(6)(15) + 3\frac(4)(15) = (7 + 3)+(\frac(6)(15) + \frac(4)(15)) = 10 + \frac (10)(15) = 10\frac(10)(15) = 10\frac(2)(3)\)
Task #1:
At dinner they ate \(\frac(8)(11)\) of the cake, and in the evening at dinner they ate \(\frac(3)(11)\). Do you think the cake was completely eaten or not?
Solution:
The denominator of the fraction is 11, it indicates how many parts the cake was divided into. At lunch, we ate 8 pieces of cake out of 11. At dinner, we ate 3 pieces of cake out of 11. Let's add 8 + 3 = 11, we ate pieces of cake out of 11, that is, the whole cake.
\(\frac(8)(11) + \frac(3)(11) = \frac(11)(11) = 1\)
Answer: They ate the whole cake.
In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". Here's how it sounds:Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet been able to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches; none of them became a universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.
From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like time slowing down to a complete stop at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.
If we turn the logic we are used to, everything falls into place. Achilles runs with constant speed. Each subsequent segment of its path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of "infinity" in this situation, then it would be correct to say "Achilles will infinitely quickly overtake the tortoise."
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise crawls a hundred steps in the same direction. During the next time interval, equal to the first, Achilles will run another thousand steps, and the tortoise will crawl one hundred steps. Now Achilles is eight hundred paces ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein's statement about the insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells of a flying arrow:
A flying arrow is motionless, since at each moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time the flying arrow is at rest at different points in space, which, in fact, is movement. There is another point to be noted here. From one photograph of a car on the road, it is impossible to determine either the fact of its movement or the distance to it. To determine the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (of course, you still need additional data for calculations, trigonometry will help you). What I want to point out in particular is that two points in time and two points in space are two different things that should not be confused as they provide different opportunities for exploration.
Wednesday, July 4, 2018
Very well the differences between set and multiset are described in Wikipedia. We look.
As you can see, "the set cannot have two identical elements", but if there are identical elements in the set, such a set is called a "multiset". Reasonable beings will never understand such logic of absurdity. This is the level of talking parrots and trained monkeys, in which the mind is absent from the word "completely." Mathematicians act as ordinary trainers, preaching their absurd ideas to us.
Once upon a time, the engineers who built the bridge were in a boat under the bridge during the tests of the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.
No matter how mathematicians hide behind the phrase "mind me, I'm in the house", or rather "mathematics studies abstract concepts", there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash desk, paying salaries. Here a mathematician comes to us for his money. We count the entire amount to him and lay it out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his "mathematical salary set". We explain the mathematics that he will receive the rest of the bills only when he proves that the set without identical elements is not equal to the set with identical elements. This is where the fun begins.
First of all, the deputies' logic will work: "you can apply it to others, but not to me!" Further, assurances will begin that there are different banknote numbers on banknotes of the same denomination, which means that they cannot be considered identical elements. Well, we count the salary in coins - there are no numbers on the coins. Here the mathematician will begin to convulsively recall physics: different coins there is a different amount of dirt, the crystal structure and the arrangement of atoms of each coin is unique...
And now I have the most interest Ask: where is the boundary beyond which elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science here is not even close.
Look here. We select football stadiums with the same field area. The area of the fields is the same, which means we have a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How right? And here the mathematician-shaman-shuller takes out a trump ace from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I will show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."
Sunday, March 18, 2018
The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but they are shamans for that, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Do you need proof? Open Wikipedia and try to find the "Sum of Digits of a Number" page. She doesn't exist. There is no formula in mathematics by which you can find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics, the task sounds like this: "Find the sum of graphic symbols representing any number." Mathematicians cannot solve this problem, but shamans can do it elementarily.
Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let's say we have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.
1. Write down the number on a piece of paper. What have we done? We have converted the number to a number graphic symbol. This is not a mathematical operation.
2. We cut one received picture into several pictures containing separate numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic characters to numbers. This is not a mathematical operation.
4. Add up the resulting numbers. Now that's mathematics.
The sum of the digits of the number 12345 is 15. These are the "cutting and sewing courses" from shamans used by mathematicians. But that's not all.
From the point of view of mathematics, it does not matter in which number system we write the number. So, in different systems reckoning, the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. With a large number of 12345, I don’t want to fool my head, consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We will not consider each step under a microscope, we have already done that. Let's look at the result.
As you can see, in different number systems, the sum of the digits of the same number is different. This result has nothing to do with mathematics. It's like finding the area of a rectangle in meters and centimeters would give you completely different results.
Zero in all number systems looks the same and has no sum of digits. This is another argument in favor of the fact that . A question for mathematicians: how is it denoted in mathematics that which is not a number? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists, no. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement of numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical action does not depend on the value of the number, the unit of measure used, and on who performs this action.
Ouch! Isn't this the women's restroom?
- Young woman! This is a laboratory for studying the indefinite holiness of souls upon ascension to heaven! Nimbus on top and arrow up. What other toilet?
Female... A halo on top and an arrow down is male.
If you have such a work of design art flashing before your eyes several times a day,
Then it is not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort on myself to see minus four degrees in a pooping person (one picture) (composition of several pictures: minus sign, number four, degrees designation). And I do not consider this girl a fool who does not know physics. She just has an arc stereotype of perception of graphic images. And mathematicians teach us this all the time. Here's an example.
1A is not "minus four degrees" or "one a". This is a "pooping man" or the number "twenty-six" in hexadecimal system reckoning. Those people who constantly work in this number system automatically perceive the number and letter as one graphic symbol.
And now, as you can understand from the title of the article, we will talk about addition.
Without the operation of addition, it is difficult to imagine our modern life, because addition is used almost everywhere. For example, you need to calculate the total price of all products in the basket or the number of fruits on the table. Addition is literally everywhere you look. Therefore, it is a basic operation and it must be mastered perfectly. Let's get started.
a+b=c
The simplest examples are on apples. Vasya had 3 apples, and Petya had 2 apples. If Petya gives Vasya 2 apples, how many will Vasya have? The answer is obvious, right? There will be 5 of them.
a- Vasya initially had apples.
b- apples from Petya initially.
c- Vasya has apples after the transfer.
Substitute in the formula: 2 + 3 = 5 ;
Types of additions
Add up online [there will be a simulator for addition]
Adding numbers
Adding numbers is very easy even for schoolchildren and some preschoolers. Addition is the sum of 2 or more numbers. For example, 2 + 3 = 5, and graphically this can be represented as follows:
A large number is divided into parts, let's take the number 1234, and in it: 4-ones, 3-tens, 2-hundreds, 1-thousands. So, if we add 4 to 7, then 4+7=10+1, that is, 1 ten and 1 unit. If adding numbers in one place (units, for example) you have a number greater than 10, but less than 20, then you add one to ten, and leave the rest in place of units.
Another example: 8 + 9, we get 10 + 7, which means we add 1 to the tens, and write 7 instead of the units, we get 17.
Next example: let's say 16+5. Here in the number 16 it has 1 ten and 6 units. We add 5 more units to them. Remember that 1 ten is ten ones. So, up to 20, 16s lack 4 units. We get 20+1. Result: 21.
In the same way, operations are performed with hundreds and thousands:
For example, 61+47. One hundred = ten tens. Let's represent the terms as 60+1 and 40+7. We get 60 + 40 and 1 + 7, since 6 + 4 \u003d 10, then 60 + 40 \u003d 100, so we get a hundred, and 1 + 7 \u003d 8. Result: 100+8=108.
Speeding up verbal counting
Addition of fractions
Imagine a circle of pizza. Pizza is one whole, and cutting in half we get something less than one, right? Half unit. How to write it down?
½, so we denote half of one whole pizza, and if we divide the pizza into 4 equal parts, then each of them will be denoted ¼. Etc…
How to add fractions?
It's simple. Let's add ¼ c ¼ th. When adding, it is important that the denominator (4) of one fraction coincides with the denominator of the second. (1) is called the numerator.
The fraction 2/4 can be reduced to the form ½.
Why? What is a fraction? ½ \u003d 1: 2, and if you divide 2 by 4, then this is the same as dividing 1 by 2. Therefore, the fraction 2/4 \u003d 1/2.
Adding fractions with different denominators
If you come across such fractions ½ + ¼, then you need to reduce to a common denominator. Among these denominators, the largest is 4. Since 2 can be doubled and get 4, we get the fraction 2/4 from the fraction ½. When multiplying the numerator, the denominator is also multiplied. We get 2/4 + 1/4 = 3/4.
Adding denominators
Perhaps you meant the addition of fractions, then their denominators are reduced to a common one and again the numerators are added, the denominators only increase.
Addition of numerators
Addition of mixed numbers
What is a mixed number? It is an integer with a fractional part. That is, if the numerator is less than the denominator, then the fraction is less than one, and if the numerator is greater than the denominator, then the fraction is greater than one. A mixed number is a fraction that is greater than one and has its integer part highlighted:
Addition properties
Displacement: a + b = b + a. From a change in the places of the terms, the sum does not change.
Associative: a + b + c = a + (b + c). The sum does not change if any group of adjacent terms is replaced by their sum.
a + 0 = 0 + a = a.
Adding zero to a number does not change that number.
Addition of limits
Adding limits is not difficult. Here a simple formula is enough, which says that if the limit of the sum of functions tends to the number a, then this is equivalent to the sum of these functions, the limit of each of which tends to the number a.
addition lesson
Addition is an arithmetic operation during which two numbers are added, and their result will be a new one - the third.
The addition formula is expressed as follows: a+b=c.
You can find examples and tasks below.
At adding fractions it should be remembered that:
So, let's add up. Make sure the denominators are the same. Then we add the numerators (1+1)/4, so we get 2/4. When adding fractions, only the numerators are added!
If the sum of fractions came across, for example, 1/3 and 1/2, then you will have to multiply not one fraction, but both to bring to a common denominator. The easiest way to do this is to multiply the first fraction by the denominator of the second, and the second fraction by the denominator of the first, we get: 2/6 and 3/6. We add (2+3)/6 and get 5/6.
Given a fraction 7/4, we get that 7 is greater than 4, which means that 7/4 is greater than 1. How to select the whole part? (4+3)/4, then we get the sum of fractions 4/4 + 3/4, 4:4 + 3/4=1 + 3/4. Outcome: one whole, three fourths.
Addition 1 class
The first grade is the very beginning and the children still do not know how to count. Education should be conducted in the form of a game. Always in the first grade, addition begins with simple examples on apples, sweets, pears. This method is used for a reason, but because children love it when they play with them. And this is not the only reason. Children have seen apples, sweets and the like very often in their lives and have dealt with the transfer and quantity, so it will not be difficult to teach the addition of such things.
First-graders can come up with a huge number of addition tasks, for example:
Objective 1. In the morning, walking through the forest, the hedgehog found 4 mushrooms, and in the evening 2 more. How many mushrooms did the hedgehog have by the end of the day?
Objective 2. 2 birds flew across the sky from one city to another city, and an hour later 3 more birds joined them. How many birds are flying now?
Task 3. The ladder had a length of 2, and it seemed short to the owner, so he lengthened it by another 1. How long is the ladder now?
Task 4. Roma had 3 balls, and Sasha 4. If Roma gives Sasha all his balls, how many will Sasha have?
First-graders mostly solve problems in which the answer is a number from 1 to 10.
Addition 2 class
In the second grade, the tasks are more complex and will require more mental activity from the child.
Numerical assignments:
Single digits:
Double figures:
Text tasks
Misha is now 18 years old. How old will he be in 5 years? And after 16?
During the summer, Masha read 3 books. The first book had 23 pages, the second had 41 pages, and the third had 12 pages. How many pages did Masha read in total?
The tailor made 3 skirts. It took him 13 meters of fabric for each skirt. How much fabric did the tailor use in total?
The workers were repairing the road, which at the very beginning was 27 meters long. On the one hand, workers lengthened it by 18 meters, and on the other hand, by another 16 meters. What was the total length of the road after its repair?
On the first day, the tourists walked 17 km, and on the second day another 22. How many km did they walk in 2 days?
Pasha and grandma went to the store to buy vegetables. On the way back, Pasha carried a bag of potatoes, which weighed 5 kg, and grandmother carried cabbage and tomatoes, which weighed 12 kg each. How many kg of vegetables did Grandma and Pasha bring from the store in total?
On September 1, Tanya gave 2 bouquets to her favorite teachers. The first bouquet had 13 carnations, and the second had 4 more. How many carnations did Tanya give in total?
Vanya wants to get a copybook and a notebook for his birthday. How much money does dad need for a gift if the notebook costs 18 rubles, and Notebook 51 ruble?
Build 3-4 grade
The essence of addition in grades 3-4 is the addition of large numbers in a column.
How to fold into a column? Let's look at an example:
First of all, we write the numbers one under the other, and on the left between them we put a “+” sign, which means addition. Let's do it like this:
Now add the bottom number to the top number. The first ones add 1 and 8. 1+8=9.
3+7 and another ten from the previous column +1: 3+7+1. It turns out 11, we write down 1, and the ten is transferred again to the next column: 6 + 1 \u003d 7.
Now let's write an example in a line:
Total: 6748+381=7129
Addition 5 class
In the fifth grade, children begin to add fractions with the same and different denominators. I remember the rules:
1. Numerators are added, not denominators.
So, let's add up. Make sure the denominators are the same. Then we add the numerators (1+1)/4, so we get 2/4. When adding fractions, only the numerators are added!
2. To add, make sure the denominators are equal.
If the sum of fractions came across, for example, 1/3 and 1/2, then you will have to multiply not one fraction, but both to bring to a common denominator. The easiest way to do this is to multiply the first fraction by the denominator of the second, and the second fraction by the denominator of the first, we get: 2/6 and 3/6. We add (2+3)/6 and get 5/6.
3. Reducing a fraction is done by dividing the numerator and denominator by the same number.
The fraction 2/4 can be reduced to the form ½. Why? What is a fraction? ½ \u003d 1: 2, and if you divide 2 by 4, then this is the same as dividing 1 by 2. Therefore, the fraction 2/4 \u003d 1/2.
4. If the fraction is greater than one, then you can select the whole part.
Given a fraction 7/4, we get that 7 is greater than 4, which means that 7/4 is greater than 1. How to select the whole part? (4+3)/4, then we get the sum of fractions 4/4 + 3/4, 4:4 + 3/4=1 + 3/4. Outcome: one whole, three fourths.
Addition 6 class
Addition of the sixth grade is the addition of complex fractions and the addition of numbers with different signs, which you will learn about in our article Subtraction.
Addition presentation
Addition table
You can also use the addition table, if it is still difficult to calculate yourself.
To add two single digit numbers, just find one vertically and the other horizontally:
Sign up for the course "Speed up mental counting, NOT mental arithmetic" to learn how to quickly and correctly add, subtract, multiply, divide, square numbers and even take roots. In 30 days, you will learn how to use easy tricks to simplify arithmetic operations. Every lesson has new tricks understandable examples and helpful assignments.
Addition Examples
In the picture you can see examples for adding two-digit numbers, three two-digit numbers and examples in which you need to insert a number so that there is a correct answer:
Games for the development of mental counting
Special educational games developed with the participation of Russian scientists from Skolkovo will help improve oral counting skills in an interesting game form.
Game "Fast Addition"
The game "Quick Addition" develops thinking and memory. The main essence of the game is to choose numbers, the sum of which is equal to a given number. This game is given a matrix from one to sixteen. A given number is written above the matrix, you must select the numbers in the matrix so that the sum of these numbers is equal to the given number. If you answer correctly, you score points and continue playing.
Game "Fast addition reload"
The game "Fast Addition Reboot" develops thinking, memory and attention. The main essence of the game is to choose the correct terms, the sum of which will be equal to a given number. In this game, three numbers are given on the screen and the task is given, add the number, the screen indicates which number to add. You select the desired numbers from the three numbers and press them. If you answer correctly, then you score points and continue to play further.
Game "Quick Score"
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 "Visual Geometry"
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, they must be quickly counted, then they close. Four numbers are written below the table, you must select one correct number and click on it with the mouse. If you answer correctly, you score points and continue playing.
Piggy bank game
The game "Piggy bank" develops thinking and memory. The main essence of the game is to choose which piggy bank more money.In this game, four piggy banks are given, you need to calculate which piggy bank has more money and show this piggy bank with the mouse. If you answer correctly, then you score points and continue to play further.
Game "Mathematical matrices"
"Mathematical Matrices" great brain exercise for kids, which will help you develop his mental work, mental counting, quick search for the right components, attentiveness. The essence of the game is that the player has to find a pair from the proposed 16 numbers that will give a given number in total, for example, in the picture below, this number is “29”, and the desired pair is “5” and “24”.
Game "Mathematical Comparisons"
A wonderful game with which you can relax your body and tense your brain. The screenshot shows an example of this game, in which there will be a question related to the picture, and you will have to answer. Time is limited. How many times can you answer?
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Actions with fractions.
Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")
So, what are fractions, types of fractions, transformations - we remembered. Let's tackle the main question.
What can you do with fractions? Yes, everything is the same as with ordinary numbers. Add, subtract, multiply, divide.
All these actions with decimal operations with fractions are no different from operations with integers. Actually, this is what they are good for, decimal. The only thing is that you need to put the comma correctly.
Mixed numbers, as I said, are of little use for most actions. They still need to be converted to ordinary fractions.
And here are the actions with ordinary fractions will be smarter. And much more important! Let me remind you: all actions with fractional expressions with letters, sines, unknowns, and so on and so forth are no different from actions with ordinary fractions! Operations with ordinary fractions are the basis for all algebra. It is for this reason that we will analyze all this arithmetic in great detail here.
Addition and subtraction of fractions.
Everyone can add (subtract) fractions with the same denominators (I really hope!). Well, let me remind you that I’m completely forgetful: when adding (subtracting), the denominator does not change. The numerators are added (subtracted) to give the numerator of the result. Type:
In short, in general view:
What if the denominators are different? Then, using the main property of the fraction (here it came in handy again!), We make the denominators the same! For instance:
Here we had to make the fraction 4/10 from the fraction 2/5. Solely for the purpose of making the denominators the same. I note, just in case, that 2/5 and 4/10 are the same fraction! Only 2/5 is uncomfortable for us, and 4/10 is even nothing.
By the way, this is the essence of solving any tasks in mathematics. When we're out uncomfortable expressions do the same, but more convenient to solve.
Another example:
The situation is similar. Here we make 48 out of 16. By simple multiplication by 3. This is all clear. But here we come across something like:
How to be?! It's hard to make a nine out of a seven! But we are smart, we know the rules! Let's transform every fraction so that the denominators are the same. This is called "reduce to a common denominator":
How! How did I know about 63? Very simple! 63 is a number that is evenly divisible by 7 and 9 at the same time. Such a number can always be obtained by multiplying the denominators. If we multiply some number by 7, for example, then the result will certainly be divided by 7!
If you need to add (subtract) several fractions, there is no need to do it in pairs, step by step. You just need to find the denominator that is common to all fractions, and bring each fraction to this same denominator. For instance:
And what will be the common denominator? You can, of course, multiply 2, 4, 8, and 16. We get 1024. Nightmare. It is easier to estimate that the number 16 is perfectly divisible by 2, 4, and 8. Therefore, it is easy to get 16 from these numbers. This number will be the common denominator. Let's turn 1/2 into 8/16, 3/4 into 12/16, and so on.
By the way, if we take 1024 as a common denominator, everything will work out too, in the end everything will be reduced. Only not everyone will get to this end, because of the calculations ...
Solve the example yourself. Not a logarithm... It should be 29/16.
So, with the addition (subtraction) of fractions is clear, I hope? Of course, it is easier to work in a shortened version, with additional multipliers. But this pleasure is available to those who honestly worked in the lower grades ... And did not forget anything.
And now we will do the same actions, but not with fractions, but with fractional expressions. New rakes will be found here, yes ...
So, we need to add two fractional expressions:
We need to make the denominators the same. And only with the help multiplication! So the main property of the fraction says. Therefore, I cannot add one to x in the first fraction in the denominator. (But that would be nice!). But if you multiply the denominators, you see, everything will grow together! So we write down, the line of the fraction, leave an empty space on top, then add it, and write the product of the denominators below, so as not to forget:
And, of course, we don’t multiply anything on the right side, we don’t open brackets! And now, looking at the common denominator of the right side, we think: in order to get the denominator x (x + 1) in the first fraction, we need to multiply the numerator and denominator of this fraction by (x + 1). And in the second fraction - x. You get this:
Note! Parentheses are here! This is the rake that many step on. Not brackets, of course, but their absence. Parentheses appear because we multiply the whole numerator and the whole denominator! And not their individual pieces ...
In the numerator of the right side, we write the sum of the numerators, everything is as in numerical fractions, then we open the brackets in the numerator of the right side, i.e. multiply everything and give like. You don't need to open the brackets in the denominators, you don't need to multiply something! In general, in denominators (any) the product is always more pleasant! We get:
Here we got the answer. The process seems long and difficult, but it depends on practice. Solve examples, get used to it, everything will become simple. Those who have mastered the fractions in the allotted time, do all these operations with one hand, on the machine!
And one more note. Many famously deal with fractions, but hang on examples with whole numbers. Type: 2 + 1/2 + 3/4= ? Where to fasten a deuce? No need to fasten anywhere, you need to make a fraction out of a deuce. It's not easy, it's very simple! 2=2/1. Like this. Any whole number can be written as a fraction. The numerator is the number itself, the denominator is one. 7 is 7/1, 3 is 3/1 and so on. It's the same with letters. (a + b) \u003d (a + b) / 1, x \u003d x / 1, etc. And then we work with these fractions according to all the rules.
Well, on addition - subtraction of fractions, knowledge was refreshed. Transformations of fractions from one type to another - repeated. You can also check. Shall we settle a little?)
Calculate:
Answers (in disarray):
71/20; 3/5; 17/12; -5/4; 11/6
Multiplication / division of fractions - in the next lesson. There are also tasks for all actions with fractions.
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)
you can get acquainted with functions and derivatives.
Fractions are ordinary numbers, they can also be added and subtracted. But due to the fact that they have a denominator, more complex rules are required here than for integers.
Consider the simplest case, when there are two fractions with the same denominators. Then:
To add fractions with the same denominators, add their numerators and leave the denominator unchanged.
To subtract fractions with the same denominators, it is necessary to subtract the numerator of the second from the numerator of the first fraction, and again leave the denominator unchanged.
Within each expression, the denominators of the fractions are equal. By definition of addition and subtraction of fractions, we get:
As you can see, nothing complicated: just add or subtract the numerators - and that's it.
But even in such simple actions, people manage to make mistakes. Most often they forget that the denominator does not change. For example, when adding them, they also begin to add up, and this is fundamentally wrong.
Get rid of bad habit Adding the denominators is easy enough. Try to do the same when subtracting. As a result, the denominator will be zero, and the fraction (suddenly!) will lose its meaning.
Therefore, remember once and for all: when adding and subtracting, the denominator does not change!
Also, many people make mistakes when adding several negative fractions. There is confusion with the signs: where to put a minus, and where - a plus.
This problem is also very easy to solve. It is enough to remember that the minus before the fraction sign can always be transferred to the numerator - and vice versa. And of course, do not forget two simple rules:
- Plus times minus gives minus;
- Two negatives make an affirmative.
Let's analyze all this with specific examples:
Task. Find the value of the expression:
In the first case, everything is simple, and in the second, we will add minuses to the numerators of fractions:
What if the denominators are different
You cannot directly add fractions with different denominators. At least, this method is unknown to me. However, the original fractions can always be rewritten so that the denominators become the same.
There are many ways to convert fractions. Three of them are discussed in the lesson " Bringing fractions to a common denominator", so we will not dwell on them here. Let's take a look at some examples:
Task. Find the value of the expression:
In the first case, we bring the fractions to a common denominator using the "cross-wise" method. In the second, we will look for the LCM. Note that 6 = 2 3; 9 = 3 · 3. The last factors in these expansions are equal, and the first ones are coprime. Therefore, LCM(6; 9) = 2 3 3 = 18.
What if the fraction has an integer part
I can please you: different denominators of fractions are not the greatest evil. Much more errors occur when the integer part is highlighted in the fractional terms.
Of course, for such fractions there are own addition and subtraction algorithms, but they are rather complicated and require a long study. Better use a simple circuit below:
- Convert all fractions containing an integer part to improper. We get normal terms (even if with different denominators), which are calculated according to the rules discussed above;
- Actually, calculate the sum or difference of the resulting fractions. As a result, we will practically find the answer;
- If this is all that was required in the task, we perform the inverse transformation, i.e. we get rid of the improper fraction, highlighting the integer part in it.
The rules for switching to improper fractions and highlighting the integer part are described in detail in the lesson "What is a numerical fraction". If you don't remember, be sure to repeat. Examples:
Task. Find the value of the expression:
Everything is simple here. The denominators inside each expression are equal, so it remains to convert all fractions to improper ones and count. We have:
To simplify the calculations, I skipped some obvious steps in the last examples.
A small note to the last two examples, where fractions with a highlighted integer part are subtracted. The minus before the second fraction means that it is the whole fraction that is subtracted, and not just its whole part.
Reread this sentence again, look at the examples, and think about it. This is where beginners make a lot of mistakes. They love to give such tasks to control work. You will also meet them repeatedly in the tests for this lesson, which will be published shortly.
Summary: General Scheme of Computing
In conclusion, I will give a general algorithm that will help you find the sum or difference of two or more fractions:
- If an integer part is highlighted in one or more fractions, convert these fractions to improper ones;
- Bring all the fractions to a common denominator in any way convenient for you (unless, of course, the compilers of the problems did this);
- Add or subtract the resulting numbers according to the rules for adding and subtracting fractions with the same denominators;
- Reduce the result if possible. If the fraction turned out to be incorrect, select the whole part.
Remember that it is better to highlight the whole part at the very end of the task, just before writing the answer.
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