How to multiply a fraction by a natural number. IX. Reflection on learning activities in the classroom. Multiplying a common fraction by a common fraction

§ 87. Addition of fractions.

Adding fractions has many similarities to adding whole numbers. Addition of fractions is an action consisting in the fact that several given numbers (terms) are combined into one number (sum), containing all the units and fractions of the units of the terms.

We will consider three cases sequentially:

1. Addition of fractions with like denominators.
2. Addition of fractions with different denominators.
3. Addition of mixed numbers.

1. Addition of fractions with like denominators.

Consider an example: 1/5 + 2/5.

Let's take segment AB (Fig. 17), take it as one and divide it into 5 equal parts, then part AC of this segment will be equal to 1/5 of segment AB, and part of the same segment CD will be equal to 2/5 AB.

From the drawing it is clear that if we take the segment AD, it will be equal to 3/5 AB; but the segment AD is precisely the sum of the segments AC and CD. So we can write:

1 / 5 + 2 / 5 = 3 / 5

Considering these terms and the resulting sum, we see that the numerator of the sum was obtained by adding the numerators of the terms, and the denominator remained unchanged.

From this we get the following rule: To add fractions with the same denominators, you need to add their numerators and leave the same denominator.

Let's look at an example:

2. Addition of fractions with different denominators.

Let's add the fractions: 3 / 4 + 3 / 8 First they need to be reduced to the lowest common denominator:

The intermediate link 6/8 + 3/8 could not be written; we've written it here for clarity.

Thus, to add fractions with different denominators, you must first reduce them to the lowest common denominator, add their numerators and label the common denominator.

Let's consider an example (we will write additional factors above the corresponding fractions):

3. Addition of mixed numbers.

Let's add the numbers: 2 3/8 + 3 5/6.

Let’s first bring the fractional parts of our numbers to a common denominator and rewrite them again:

Now we add the integer and fractional parts sequentially:

§ 88. Subtraction of fractions.

Subtracting fractions is defined in the same way as subtracting whole numbers. This is an action with the help of which, given the sum of two terms and one of them, another term is found. Let us consider three cases in succession:

1. Subtracting fractions with like denominators.
2. Subtracting fractions with different denominators.
3. Subtraction of mixed numbers.

1. Subtracting fractions with like denominators.

Let's look at an example:

13 / 15 - 4 / 15

Let's take the segment AB (Fig. 18), take it as a unit and divide it into 15 equal parts; then part AC of this segment will represent 1/15 of AB, and part AD of the same segment will correspond to 13/15 AB. Let us set aside another segment ED equal to 4/15 AB.

We need to subtract the fraction 4/15 from 13/15. In the drawing, this means that segment ED must be subtracted from segment AD. As a result, segment AE will remain, which is 9/15 of segment AB. So we can write:

The example we made shows that the numerator of the difference was obtained by subtracting the numerators, but the denominator remained the same.

Therefore, to subtract fractions with like denominators, you need to subtract the numerator of the subtrahend from the numerator of the minuend and leave the same denominator.

2. Subtracting fractions with different denominators.

Example. 3/4 - 5/8

First, let's reduce these fractions to the lowest common denominator:

The intermediate 6 / 8 - 5 / 8 is written here for clarity, but can be skipped later.

Thus, in order to subtract a fraction from a fraction, you must first reduce them to the lowest common denominator, then subtract the numerator of the minuend from the numerator of the minuend and sign the common denominator under their difference.

Let's look at an example:

3. Subtraction of mixed numbers.

Example. 10 3/4 - 7 2/3.

Let us reduce the fractional parts of the minuend and subtrahend to the lowest common denominator:

We subtracted a whole from a whole and a fraction from a fraction. But there are cases when the fractional part of the subtrahend is greater than the fractional part of the minuend. In such cases, you need to take one unit from the whole part of the minuend, split it into those parts in which the fractional part is expressed, and add it to the fractional part of the minuend. And then the subtraction will be performed in the same way as in the previous example:

§ 89. Multiplication of fractions.

When studying fraction multiplication we will consider next questions:

1. Multiplying a fraction by a whole number.
2. Finding the fraction of a given number.
3. Multiplying a whole number by a fraction.
4. Multiplying a fraction by a fraction.
5. Multiplication of mixed numbers.
6. The concept of interest.
7. Finding the percentage of a given number. Let's consider them sequentially.

1. Multiplying a fraction by a whole number.

Multiplying a fraction by a whole number has the same meaning as multiplying a whole number by an integer. To multiply a fraction (multiplicand) by an integer (factor) means to create a sum of identical terms, in which each term is equal to the multiplicand, and the number of terms is equal to the multiplier.

This means that if you need to multiply 1/9 by 7, then it can be done like this:

We easily obtained the result, since the action was reduced to adding fractions with the same denominators. Hence,

Consideration of this action shows that multiplying a fraction by a whole number is equivalent to increasing this fraction as many times as there are units in the whole number. And since increasing a fraction is achieved either by increasing its numerator

or by reducing its denominator , then we can either multiply the numerator by an integer or divide the denominator by it, if such division is possible.

From here we get the rule:

To multiply a fraction by a whole number, you multiply the numerator by that whole number and leave the denominator the same, or, if possible, divide the denominator by that number, leaving the numerator unchanged.

When multiplying, abbreviations are possible, for example:

2. Finding the fraction of a given number. There are many problems in which you have to find, or calculate, part of a given number. The difference between these problems and others is that they give the number of some objects or units of measurement and you need to find a part of this number, which is also indicated here by a certain fraction. To facilitate understanding, we will first give examples of such problems, and then introduce a method for solving them.

Task 1. I had 60 rubles; I spent 1/3 of this money on buying books. How much did the books cost?

Task 2. The train must travel a distance between cities A and B equal to 300 km. He has already covered 2/3 of this distance. How many kilometers is this?

Task 3. There are 400 houses in the village, 3/4 of them are brick, the rest are wooden. How many brick houses are there in total?

These are some of the many problems we encounter to find a part of a given number. They are usually called problems to find the fraction of a given number.

Solution to problem 1. From 60 rub. I spent 1/3 on books; This means that to find the cost of books you need to divide the number 60 by 3:

Solving problem 2. The point of the problem is that you need to find 2/3 of 300 km. Let's first calculate 1/3 of 300; this is achieved by dividing 300 km by 3:

300: 3 = 100 (that's 1/3 of 300).

To find two-thirds of 300, you need to double the resulting quotient, i.e., multiply by 2:

100 x 2 = 200 (that's 2/3 of 300).

Solving problem 3. Here you need to determine the number of brick houses that make up 3/4 of 400. Let’s first find 1/4 of 400,

400: 4 = 100 (that's 1/4 of 400).

To calculate three quarters of 400, the resulting quotient must be tripled, i.e. multiplied by 3:

100 x 3 = 300 (that's 3/4 of 400).

Based on the solution to these problems, we can derive the following rule:

To find the value of a fraction from a given number, you need to divide this number by the denominator of the fraction and multiply the resulting quotient by its numerator.

3. Multiplying a whole number by a fraction.

Earlier (§ 26) it was established that the multiplication of integers should be understood as the addition of identical terms (5 x 4 = 5+5 +5+5 = 20). In this paragraph (point 1) it was established that multiplying a fraction by an integer means finding the sum of identical terms equal to this fraction.

In both cases, multiplication consisted of finding the sum of identical terms.

Now we move on to multiplying a whole number by a fraction. Here we will encounter, for example, multiplication: 9 2 / 3. It is clear that the previous definition of multiplication does not apply to this case. This is evident from the fact that we cannot replace such multiplication by adding equal numbers.

Because of this, we will have to give a new definition of multiplication, i.e., in other words, answer the question of what should be understood by multiplication by a fraction, how this action should be understood.

The meaning of multiplying a whole number by a fraction is clear from the following definition: multiplying an integer (multiplicand) by a fraction (multiplicand) means finding this fraction of the multiplicand.

Namely, multiplying 9 by 2/3 means finding 2/3 of nine units. In the previous paragraph, such problems were solved; so it’s easy to figure out that we’ll end up with 6.

But now there is an interesting and important question: why are they like this at first glance? various actions How is finding the sum of equal numbers and finding the fraction of a number called by the same word “multiplication” in arithmetic?

This happens because the previous action (repeating the number with terms several times) and the new action (finding the fraction of the number) give the answer to homogeneous questions. This means that we proceed here from the considerations that homogeneous questions or tasks are solved by the same action.

To understand this, consider the following problem: “1 m of cloth costs 50 rubles. How much will 4 m of such cloth cost?

This problem is solved by multiplying the number of rubles (50) by the number of meters (4), i.e. 50 x 4 = 200 (rubles).

Let’s take the same problem, but in it the amount of cloth will be expressed as a fraction: “1 m of cloth costs 50 rubles. How much will 3/4 m of such cloth cost?”

This problem also needs to be solved by multiplying the number of rubles (50) by the number of meters (3/4).

You can change the numbers in it several more times, without changing the meaning of the problem, for example, take 9/10 m or 2 3/10 m, etc.

Since these problems have the same content and differ only in numbers, we call the actions used in solving them the same word - multiplication.

How do you multiply a whole number by a fraction?

Let's take the numbers encountered in the last problem:

According to the definition, we must find 3/4 of 50. Let's first find 1/4 of 50, and then 3/4.

1/4 of 50 is 50/4;

3/4 of the number 50 is .

Hence.

Let's consider another example: 12 5 / 8 =?

1/8 of the number 12 is 12/8,

5/8 of the number 12 is .

Hence,

From here we get the rule:

To multiply a whole number by a fraction, you need to multiply the whole number by the numerator of the fraction and make this product the numerator, and sign the denominator of this fraction as the denominator.

Let's write this rule using letters:

To make this rule completely clear, it should be remembered that a fraction can be considered as a quotient. Therefore, it is useful to compare the found rule with the rule for multiplying a number by a quotient, which was set out in § 38

It is important to remember that before performing multiplication, you should do (if possible) reductions, For example:

4. Multiplying a fraction by a fraction. Multiplying a fraction by a fraction has the same meaning as multiplying a whole number by a fraction, i.e., when multiplying a fraction by a fraction, you need to find the fraction that is in the factor from the first fraction (the multiplicand).

Namely, multiplying 3/4 by 1/2 (half) means finding half of 3/4.

How do you multiply a fraction by a fraction?

Let's take an example: 3/4 multiplied by 5/7. This means you need to find 5/7 of 3/4. Let's first find 1/7 of 3/4, and then 5/7

1/7 of the number 3/4 will be expressed as follows:

5/7 numbers 3/4 will be expressed as follows:

Thus,

Another example: 5/8 multiplied by 4/9.

1/9 of 5/8 is ,

4/9 of the number 5/8 is .

Thus,

From these examples the following rule can be deduced:

To multiply a fraction by a fraction, you need to multiply the numerator by the numerator, and the denominator by the denominator, and make the first product the numerator, and the second product the denominator of the product.

This is the rule in general view can be written like this:

When multiplying, it is necessary to make (if possible) reductions. Let's look at examples:

5. Multiplication of mixed numbers. Since mixed numbers can easily be replaced by improper fractions, this circumstance is usually used when multiplying mixed numbers. This means that in cases where the multiplicand, or the multiplier, or both factors are expressed as mixed numbers, they are replaced by improper fractions. Let's multiply, for example, mixed numbers: 2 1/2 and 3 1/5. Let's turn each of them into an improper fraction and then multiply the resulting fractions according to the rule for multiplying a fraction by a fraction:

Rule. To multiply mixed numbers, you must first convert them into improper fractions and then multiply them according to the rule for multiplying fractions by fractions.

Note. If one of the factors is an integer, then the multiplication can be performed based on the distribution law as follows:

6. The concept of interest. When solving problems and performing various practical calculations, we use all kinds of fractions. But it must be borne in mind that many quantities allow not just any, but natural divisions for them. For example, you can take one hundredth (1/100) of a ruble, it will be a kopeck, two hundredths is 2 kopecks, three hundredths is 3 kopecks. You can take 1/10 of a ruble, it will be "10 kopecks, or a ten-kopeck piece. You can take a quarter of a ruble, i.e. 25 kopecks, half a ruble, i.e. 50 kopecks (fifty kopecks). But they practically don’t take it, for example , 2/7 of a ruble because the ruble is not divided into sevenths.

The unit of weight, i.e. the kilogram, primarily allows for decimal divisions, for example 1/10 kg, or 100 g. And such fractions of a kilogram as 1/6, 1/11, 1/13 are not common.

In general, our (metric) measures are decimal and allow decimal divisions.

However, it should be noted that it is extremely useful and convenient in a wide variety of cases to use the same (uniform) method of subdividing quantities. Many years of experience have shown that such a well-justified division is the “hundredth” division. Let us consider several examples relating to the most diverse areas of human practice.

1. The price of books has decreased by 12/100 of the previous price.

Example. The previous price of the book was 10 rubles. It decreased by 1 ruble. 20 kopecks

2. Savings banks pay depositors 2/100 of the amount deposited for savings during the year.

Example. 500 rubles are deposited in the cash register, the income from this amount for the year is 10 rubles.

3. The number of graduates from one school was 5/100 of the total number of students.

EXAMPLE There were only 1,200 students at the school, of which 60 graduated.

The hundredth part of a number is called a percentage.

The word "percentage" is borrowed from Latin language and its root "cent" means one hundred. Together with the preposition (pro centum), this word means “for a hundred.” The meaning of such an expression follows from the fact that initially in ancient Rome interest was the money that the debtor paid to the lender “for every hundred.” The word “cent” is heard in such familiar words: centner (one hundred kilograms), centimeter (say centimeter).

For example, instead of saying that over the past month the plant produced 1/100 of all products produced by it was defective, we will say this: over the past month the plant produced one percent of defects. Instead of saying: the plant produced 4/100 more products than the established plan, we will say: the plant exceeded the plan by 4 percent.

The above examples can be expressed differently:

1. The price of books has decreased by 12 percent of the previous price.

2. Savings banks pay depositors 2 percent per year on the amount deposited in savings.

3. The number of graduates from one school was 5 percent of all school students.

To shorten the letter, it is customary to write the % symbol instead of the word “percentage”.

However, you need to remember that in calculations the % sign is usually not written; it can be written in the problem statement and in the final result. When performing calculations, you need to write a fraction with a denominator of 100 instead of a whole number with this symbol.

You need to be able to replace an integer with the indicated icon with a fraction with a denominator of 100:

Conversely, you need to get used to writing an integer with the indicated symbol instead of a fraction with a denominator of 100:

7. Finding the percentage of a given number.

Task 1. The school received 200 cubic meters. m of firewood, with birch firewood accounting for 30%. How much birch firewood was there?

The meaning of this problem is that birch firewood made up only part of the firewood that was delivered to the school, and this part is expressed in the fraction 30/100. This means that we have a task to find a fraction of a number. To solve it, we must multiply 200 by 30/100 (problems of finding the fraction of a number are solved by multiplying the number by the fraction.).

This means that 30% of 200 equals 60.

The fraction 30/100 encountered in this problem can be reduced by 10. It would be possible to do this reduction from the very beginning; the solution to the problem would not have changed.

Task 2. There were 300 children of various ages in the camp. Children 11 years old made up 21%, children 12 years old made up 61% and finally 13 year old children made up 18%. How many children of each age were there in the camp?

In this problem you need to perform three calculations, i.e. sequentially find the number of children 11 years old, then 12 years old and finally 13 years old.

This means that here you will need to find the fraction of the number three times. Let's do it:

1) How many 11-year-old children were there?

2) How many 12-year-old children were there?

3) How many 13-year-old children were there?

After solving the problem, it is useful to add the numbers found; their sum should be 300:

63 + 183 + 54 = 300

It should also be noted that the sum of the percentages given in the problem statement is 100:

21% + 61% + 18% = 100%

This suggests that total number children in the camp were taken as 100%.

3 a d a h a 3. The worker received 1,200 rubles per month. Of this, he spent 65% on food, 6% on apartments and heating, 4% on gas, electricity and radio, 10% on cultural needs and 15% saved. How much money was spent on the needs indicated in the problem?

To solve this problem you need to find the fraction of 1,200 5 times. Let's do this.

1) How much money was spent on food? The problem says that this expense is 65% of total earnings, i.e. 65/100 of the number 1,200. Let’s do the calculation:

2) How much money did you pay for an apartment with heating? Reasoning similarly to the previous one, we arrive at the following calculation:

3) How much money did you pay for gas, electricity and radio?

4) How much money was spent on cultural needs?

5) How much money did the worker save?

To check, it is useful to add up the numbers found in these 5 questions. The amount should be 1,200 rubles. All earnings are taken as 100%, which is easy to check by adding up the percentage numbers given in the problem statement.

We solved three problems. Despite the fact that these problems dealt with different things (delivery of firewood for the school, the number of children of different ages, the worker's expenses), they were solved in the same way. This happened because in all problems it was necessary to find several percent of given numbers.

§ 90. Division of fractions.

As we study division of fractions, we will consider the following questions:

1. Divide an integer by an integer.
2. Dividing a fraction by a whole number
3. Dividing a whole number by a fraction.
4. Dividing a fraction by a fraction.
5. Division of mixed numbers.
6. Finding a number from its given fraction.
7. Finding a number by its percentage.

Let's consider them sequentially.

1. Divide an integer by an integer.

As was indicated in the department of integers, division is the action that consists in the fact that, given the product of two factors (dividend) and one of these factors (divisor), another factor is found.

We looked at dividing an integer by an integer in the section on integers. We encountered two cases of division there: division without a remainder, or “entirely” (150: 10 = 15), and division with a remainder (100: 9 = 11 and 1 remainder). We can therefore say that in the field of integers, exact division is not always possible, because the dividend is not always the product of the divisor by the integer. After introducing multiplication by a fraction, we can consider any case of division of integers as possible (only division by zero is excluded).

For example, dividing 7 by 12 means finding a number whose product by 12 would be equal to 7. Such a number is the fraction 7 / 12 because 7 / 12 12 = 7. Another example: 14: 25 = 14 / 25, because 14 / 25 25 = 14.

Thus, to divide a whole number by a whole number, you need to create a fraction whose numerator is equal to the dividend and the denominator is equal to the divisor.

2. Dividing a fraction by a whole number.

Divide the fraction 6 / 7 by 3. According to the definition of division given above, we have here the product (6 / 7) and one of the factors (3); it is required to find a second factor that, when multiplied by 3, would give the given product 6/7. Obviously, it should be three times smaller than this product. This means that the task set before us was to reduce the fraction 6/7 by 3 times.

We already know that reducing a fraction can be done either by decreasing its numerator or by increasing its denominator. Therefore you can write:

In this case, the numerator 6 is divisible by 3, so the numerator should be reduced by 3 times.

Let's take another example: 5 / 8 divided by 2. Here the numerator 5 is not divisible by 2, which means that the denominator will have to be multiplied by this number:

Based on this, a rule can be made: To divide a fraction by a whole number, you need to divide the numerator of the fraction by that whole number.(if possible), leaving the same denominator, or multiply the denominator of the fraction by this number, leaving the same numerator.

3. Dividing a whole number by a fraction.

Let it be necessary to divide 5 by 1/2, i.e., find a number that, after multiplying by 1/2, will give the product 5. Obviously, this number must be greater than 5, since 1/2 is a proper fraction, and when multiplying a number the product of a proper fraction must be less than the product being multiplied. To make this clearer, let's write our actions as follows: 5: 1 / 2 = X , which means x 1 / 2 = 5.

We must find such a number X , which, if multiplied by 1/2, would give 5. Since multiplying a certain number by 1/2 means finding 1/2 of this number, then, therefore, 1/2 of the unknown number X is equal to 5, and the whole number X twice as much, i.e. 5 2 = 10.

So 5: 1 / 2 = 5 2 = 10

Let's check:

Let's look at another example. Let's say you want to divide 6 by 2/3. Let's first try to find the desired result using the drawing (Fig. 19).

Fig.19

Let us draw a segment AB equal to 6 units, and divide each unit into 3 equal parts. In each unit, three thirds (3/3) of the entire segment AB is 6 times larger, i.e. e. 18/3. Using small brackets, we connect the 18 resulting segments of 2; There will be only 9 segments. This means that the fraction 2/3 is contained in 6 units 9 times, or, in other words, the fraction 2/3 is 9 times less than 6 whole units. Hence,

How to get this result without a drawing using calculations alone? Let's reason like this: we need to divide 6 by 2/3, i.e. we need to answer the question how many times 2/3 is contained in 6. Let's find out first: how many times 1/3 is contained in 6? In a whole unit there are 3 thirds, and in 6 units there are 6 times more, i.e. 18 thirds; to find this number we must multiply 6 by 3. This means that 1/3 is contained in b units 18 times, and 2/3 is contained in b units not 18 times, but half as many times, i.e. 18: 2 = 9. Therefore , when dividing 6 by 2/3 we did the following:

From here we get the rule for dividing a whole number by a fraction. To divide a whole number by a fraction, you need to multiply this whole number by the denominator of the given fraction and, making this product the numerator, divide it by the numerator of the given fraction.

Let's write the rule using letters:

To make this rule completely clear, it should be remembered that a fraction can be considered as a quotient. Therefore, it is useful to compare the rule found with the rule for dividing a number by a quotient, which was set out in § 38. Please note that the same formula was obtained there.

When dividing, abbreviations are possible, for example:

4. Dividing a fraction by a fraction.

Let's say we need to divide 3/4 by 3/8. What will the number that results from division mean? It will answer the question how many times the fraction 3/8 is contained in the fraction 3/4. To understand this issue, let's make a drawing (Fig. 20).

Let's take a segment AB, take it as one, divide it into 4 equal parts and mark 3 such parts. Segment AC will be equal to 3/4 of segment AB. Let us now divide each of the four original segments in half, then the segment AB will be divided into 8 equal parts and each such part will be equal to 1/8 of the segment AB. Let us connect 3 such segments with arcs, then each of the segments AD and DC will be equal to 3/8 of the segment AB. The drawing shows that a segment equal to 3/8 is contained in a segment equal to 3/4 exactly 2 times; This means that the result of division can be written as follows:

3 / 4: 3 / 8 = 2

Let's look at another example. Let's say we need to divide 15/16 by 3/32:

We can reason like this: we need to find a number that, after multiplying by 3/32, will give a product equal to 15/16. Let's write the calculations like this:

15 / 16: 3 / 32 = X

3 / 32 X = 15 / 16

3/32 unknown number X are 15/16

1/32 of an unknown number X is ,

32 / 32 numbers X make up .

Hence,

Thus, to divide a fraction by a fraction, you need to multiply the numerator of the first fraction by the denominator of the second, and multiply the denominator of the first fraction by the numerator of the second, and make the first product the numerator, and the second the denominator.

Let's write the rule using letters:

When dividing, abbreviations are possible, for example:

5. Division of mixed numbers.

When dividing mixed numbers, they must first be converted to improper fractions and then divide the resulting fractions according to the rules for dividing fractional numbers. Let's look at an example:

Let's convert mixed numbers to improper fractions:

Now let's divide:

Thus, to divide mixed numbers, you need to convert them into improper fractions and then divide using the rule for dividing fractions.

6. Finding a number from its given fraction.

Among the various fraction problems, sometimes there are those in which the value of some fraction of an unknown number is given and you need to find this number. This type of problem will be the inverse of the problem of finding the fraction of a given number; there a number was given and it was required to find some fraction of this number, here a fraction of a number was given and it was required to find this number itself. This idea will become even clearer if we turn to solving this type of problem.

Task 1. On the first day, the glaziers glazed 50 windows, which is 1/3 of all the windows of the built house. How many windows are there in this house?

Solution. The problem says that 50 glazed windows make up 1/3 of all the windows of the house, which means there are 3 times more windows in total, i.e.

The house had 150 windows.

Task 2. The store sold 1,500 kg of flour, which is 3/8 of the total flour stock the store had. What was the store's initial supply of flour?

Solution. From the conditions of the problem it is clear that 1,500 kg of flour sold constitute 3/8 of the total stock; This means that 1/8 of this reserve will be 3 times less, i.e. to calculate it you need to reduce 1500 by 3 times:

1,500: 3 = 500 (this is 1/8 of the reserve).

Obviously, the entire supply will be 8 times larger. Hence,

500 8 = 4,000 (kg).

The initial stock of flour in the store was 4,000 kg.

From consideration of this problem, the following rule can be derived.

To find a number from a given value of its fraction, it is enough to divide this value by the numerator of the fraction and multiply the result by the denominator of the fraction.

We solved two problems on finding a number given its fraction. Such problems, as is especially clear from the last one, are solved by two actions: division (when one part is found) and multiplication (when the whole number is found).

However, after we have learned the division of fractions, the above problems can be solved with one action, namely: division by a fraction.

For example, the last task can be solved in one action like this:

In the future, we will solve problems of finding a number from its fraction with one action - division.

7. Finding a number by its percentage.

In these problems you will need to find a number knowing a few percent of that number.

Task 1. At the beginning of this year I received 60 rubles from the savings bank. income from the amount I put into savings a year ago. How much money have I put in the savings bank? (The cash desks give depositors a 2% return per year.)

The point of the problem is that I put a certain amount of money in a savings bank and stayed there for a year. After a year, I received 60 rubles from her. income, which is 2/100 of the money I deposited. How much money did I put in?

Consequently, knowing part of this money, expressed in two ways (in rubles and fractions), we must find the entire, as yet unknown, amount. This is an ordinary problem of finding a number given its fraction. The following problems are solved by division:

This means that 3,000 rubles were deposited in the savings bank.

Task 2. Fishermen fulfilled the monthly plan by 64% in two weeks, harvesting 512 tons of fish. What was their plan?

From the conditions of the problem it is known that the fishermen completed part of the plan. This part is equal to 512 tons, which is 64% of the plan. We don’t know how many tons of fish need to be prepared according to the plan. Finding this number will be the solution to the problem.

Such problems are solved by division:

This means that according to the plan, 800 tons of fish need to be prepared.

Task 3. The train went from Riga to Moscow. When he passed the 276th kilometer, one of the passengers asked a passing conductor how much of the journey they had already covered. To this the conductor replied: “We have already covered 30% of the entire journey.” What is the distance from Riga to Moscow?

From the problem conditions it is clear that 30% of the route from Riga to Moscow is 276 km. We need to find the entire distance between these cities, i.e., for this part, find the whole:

§ 91. Reciprocal numbers. Replacing division with multiplication.

Let's take the fraction 2/3 and replace the numerator in place of the denominator, we get 3/2. We got the inverse of this fraction.

In order to obtain a fraction that is the inverse of a given fraction, you need to put its numerator in place of the denominator, and the denominator in place of the numerator. In this way we can get the reciprocal of any fraction. For example:

3/4, reverse 4/3; 5/6, reverse 6/5

Two fractions that have the property that the numerator of the first is the denominator of the second, and the denominator of the first is the numerator of the second, are called mutually inverse.

Now let's think about what fraction will be the reciprocal of 1/2. Obviously, it will be 2 / 1, or just 2. By looking for the inverse fraction of the given one, we got an integer. And this case is not isolated; on the contrary, for all fractions with a numerator of 1 (one), the reciprocals will be integers, for example:

1/3, reverse 3; 1/5, reverse 5

Since in finding reciprocal fractions we also encountered integers, in what follows we will talk not about reciprocal fractions, but about reciprocal numbers.

Let's figure out how to write the inverse of an integer. For fractions, this can be solved simply: you need to put the denominator in place of the numerator. In the same way, you can get the inverse of an integer, since any integer can have a denominator of 1. This means that the inverse of 7 will be 1/7, because 7 = 7/1; for the number 10 the inverse will be 1/10, since 10 = 10/1

This idea can be expressed differently: the reciprocal of a given number is obtained by dividing one by a given number. This statement is true not only for whole numbers, but also for fractions. In fact, if we need to write the inverse of the fraction 5/9, then we can take 1 and divide it by 5/9, i.e.

Now let's point out one thing property reciprocal numbers, which will be useful to us: the product of reciprocal numbers is equal to one. Indeed:

Using this property, we can find reciprocal numbers in the following way. Let's say we need to find the inverse of 8.

Let's denote it by the letter X , then 8 X = 1, hence X = 1/8. Let's find another number that is the inverse of 7/12 and denote it by the letter X , then 7/12 X = 1, hence X = 1: 7 / 12 or X = 12 / 7 .

We introduced here the concept of reciprocal numbers in order to slightly supplement the information about dividing fractions.

When we divide the number 6 by 3/5, we do the following:

Pay special attention to the expression and compare it with the given one: .

If we take the expression separately, without connection with the previous one, then it is impossible to solve the question of where it came from: from dividing 6 by 3/5 or from multiplying 6 by 5/3. In both cases the same thing happens. Therefore we can say that dividing one number by another can be replaced by multiplying the dividend by the inverse of the divisor.

The examples we give below fully confirm this conclusion.

) and denominator by denominator (we get the denominator of the product).

Formula for multiplying fractions:

For example:

Before you begin multiplying numerators and denominators, you need to check whether the fraction can be reduced. If you can reduce the fraction, it will be easier for you to make further calculations.

Dividing a common fraction by a fraction.

Dividing fractions involving natural numbers.

It's not as scary as it seems. As in the case of addition, we convert the integer into a fraction with one in the denominator. For example:

Multiplying mixed fractions.

Rules for multiplying fractions (mixed):

convert mixed fractions to improper fractions;
multiplying the numerators and denominators of fractions;
reduce the fraction;
If you get an improper fraction, then we convert the improper fraction into a mixed fraction.

Note! To multiply mixed fraction to another mixed fraction, you must first convert them to the form of improper fractions, and then multiply them according to the rule for multiplying ordinary fractions.

The second way to multiply a fraction by a natural number.

It may be more convenient to use the second method of multiplying a common fraction by a number.

Note! To multiply a fraction by natural number It is necessary to divide the denominator of the fraction by this number, and leave the numerator unchanged.

From the example given above, it is clear that this option is more convenient to use when the denominator of a fraction is divided without a remainder by a natural number.

Multistory fractions.

In high school, three-story (or more) fractions are often encountered. Example:

To bring such a fraction to its usual form, use division through 2 points:

Note! When dividing fractions, the order of division is very important. Be careful, it's easy to get confused here.

Note, For example:

When dividing one by any fraction, the result will be the same fraction, only inverted:

Practical tips for multiplying and dividing fractions:

1. The most important thing when working with fractional expressions is accuracy and attentiveness. Do all calculations carefully and accurately, concentratedly and clearly. It's better to write a few extra lines in your draft than to get lost in mental calculations.

2. In tasks with different types fractions - go to the form of ordinary fractions.

3. We reduce all fractions until it is no longer possible to reduce.

4. We transform multi-level fractional expressions into ordinary ones using division through 2 points.

5. Divide a unit by a fraction in your head, simply turning the fraction over.

Let's continue to study operations with ordinary fractions. Now in the spotlight multiplying common fractions. In this article we will give a rule for multiplying ordinary fractions and consider the application of this rule when solving examples. We will also focus on multiplying an ordinary fraction by a natural number. In conclusion, let's look at how to multiply three and more fractions.

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Multiplying a common fraction by a common fraction

Let's start with the wording rules for multiplying ordinary fractions: Multiplying a fraction by a fraction produces a fraction whose numerator is equal to the product of the numerators of the fractions being multiplied, and the denominator is equal to the product of the denominators.

That is, the formula corresponds to the multiplication of ordinary fractions a/b and c/d.

Let us give an example illustrating the rule for multiplying ordinary fractions. Consider a square with side 1 unit. , while its area is 1 unit 2. Divide this square into equal rectangles with sides of 1/4 units. and 1/8 units. , while the original square will consist of 4·8=32 rectangles, therefore, the area of each rectangle is 1/32 of the area of the original square, that is, it is equal to 1/32 units 2. Now let's paint over part of the original square. All our actions are reflected in the figure below.

The sides of the shaded rectangle are 5/8 units. and 3/4 units. , which means its area is equal to the product of the fractions 5/8 and 3/4, that is, units 2. But the shaded rectangle consists of 15 “small” rectangles, which means its area is 15/32 units 2. Hence, . Since 5·3=15 and 8·4=32, the last equality can be rewritten as , which confirms the formula for multiplying ordinary fractions of the form .

Note that using the stated multiplication rule, you can multiply both proper and improper fractions, and fractions with the same denominators, and fractions with different denominators.

Let's consider examples of multiplying ordinary fractions.

Multiply the common fraction 7/11 by common fraction 9/8 .

The product of the numerators of the multiplied fractions 7 and 9 is equal to 63, and the product of the denominators of 11 and 8 is equal to 88. Thus, multiplying the common fractions 7/11 and 9/8 gives the fraction 63/88.

Here is a short summary of the solution: .

We should not forget about reducing the resulting fraction if the multiplication results in a reducible fraction, and about separating the whole part from an improper fraction.

Multiply fractions 4/15 and 55/6.

Let's apply the rule for multiplying ordinary fractions: .

Obviously, the resulting fraction is reducible (the test of divisibility by 10 allows us to state that the numerator and denominator of the fraction 220/90 have common multiplier 10). Let's reduce the fraction 220/90: gcd(220, 90)=10 and . It remains to isolate the whole part from the resulting improper fraction: .

Note that the reduction of a fraction can be carried out before calculating the products of the numerators and the products of the denominators of the multiplied fractions, that is, when the fraction has the form . To do this, the numbers a, b, c and d are replaced by their expansions into prime factors, after which the same factors of the numerator and denominator are reduced.

For clarification, let's return to the previous example.

Calculate the product of fractions of the form .

According to the formula for multiplying ordinary fractions, we have .

Since 4=2·2, 55=5·11, 15=3·5 and 6=2·3, then . Now we reduce common prime factors: .

All that remains is to calculate the products in the numerator and denominator, and then isolate the whole part from the improper fraction: .

It should be noted that the multiplication of fractions is characterized by a commutative property, that is, the multiplied fractions can be swapped: .

Multiplying a common fraction by a natural number

Let's start with the wording rules for multiplying a common fraction by a natural number: Multiplying a fraction by a natural number produces a fraction whose numerator is equal to the product of the numerator of the fraction being multiplied by the natural number, and the denominator is equal to the denominator of the fraction being multiplied.

Using letters, the rule for multiplying a fraction a/b by a natural number n has the form .

The formula follows from the formula for multiplying two ordinary fractions of the form . Indeed, representing a natural number as a fraction with a denominator of 1, we get .

Let's look at examples of multiplying a fraction by a natural number.

Multiply the fraction 2/27 by 5.

Multiplying the numerator 2 by the number 5 gives 10, therefore, by virtue of the rule for multiplying a fraction by a natural number, the product of 2/27 by 5 is equal to the fraction 10/27.

It is convenient to write the whole solution like this: .

When multiplying a fraction by a natural number, the resulting fraction often has to be reduced, and if it is also incorrect, then represented as a mixed number.

Multiply the fraction 5/12 by the number 8.

According to the formula for multiplying a fraction by a natural number, we have . Obviously, the resulting fraction is reducible (the sign of divisibility by 2 indicates the common divisor 2 of the numerator and denominator). Let's reduce the fraction 40/12: since LCM(40, 12)=4, then . It remains to highlight the whole part: .

Here's the entire solution: .

Note that the reduction could be carried out by replacing the numbers in the numerator and denominator with their decompositions into prime factors. In this case, the solution would look like this: .

In conclusion of this point, we note that multiplying a fraction by a natural number has a commutative property, that is, the product of a fraction by a natural number is equal to the product of this natural number by the fraction: .

Multiplying three or more fractions

The way we defined ordinary fractions and the operation of multiplication with them allows us to assert that all the properties of multiplying natural numbers also apply to multiplying fractions.

The commutative and associative properties of multiplication make it possible to unambiguously determine multiplying three or more fractions and natural numbers. In this case, everything happens by analogy with the multiplication of three or more natural numbers. In particular, fractions and natural numbers in a product can be rearranged for ease of calculation, and in the absence of parentheses indicating the order in which actions are performed, we can arrange the parentheses ourselves in any of the acceptable ways.

Let's look at examples of multiplying several fractions and natural numbers.

Multiply three common fractions 1/20, 12/5, 3/7 and 5/8.

Let's write down the product we need to calculate . By virtue of the rule for multiplying fractions, the written product is equal to a fraction whose numerator is equal to the product of the numerators of all fractions, and the denominator is equal to the product of the denominators: .

Before calculating the products in the numerator and denominator, it is advisable to replace all the factors with their decompositions into simple factors and perform a reduction (you can, of course, reduce a fraction after multiplication, but in many cases this requires a lot of computational effort): .

Multiply five numbers .

In this product, it is convenient to group the fraction 7/8 with the number 8, and the number 12 with the fraction 5/36, this will simplify the calculations, since with such a grouping the reduction is obvious. We have
.

Multiplying fractions

We will consider the multiplication of ordinary fractions in several possible options.

Multiplying a common fraction by a fraction

This is the simplest case in which you need to use the following rules for multiplying fractions.

To multiply fraction by fraction, necessary:

multiply the numerator of the first fraction by the numerator of the second fraction and write their product into the numerator of the new fraction;
multiply the denominator of the first fraction by the denominator of the second fraction and write their product into the denominator of the new fraction;

Before multiplying numerators and denominators, check to see if the fractions can be reduced. Reducing fractions in calculations will make your calculations much easier.

Multiplying a fraction by a natural number

To make a fraction multiply by a natural number You need to multiply the numerator of the fraction by this number, and leave the denominator of the fraction unchanged.

If the result of multiplication is an improper fraction, do not forget to turn it into a mixed number, that is, highlight the whole part.

Multiplying mixed numbers

To multiply mixed numbers, you must first turn them into improper fractions and then multiply according to the rule for multiplying ordinary fractions.

Another way to multiply a fraction by a natural number

Sometimes when making calculations it is more convenient to use another method of multiplying a common fraction by a number.

To multiply a fraction by a natural number, you need to divide the denominator of the fraction by this number, and leave the numerator the same.

As can be seen from the example, this version of the rule is more convenient to use if the denominator of the fraction is divisible by a natural number without a remainder.

Multiplying mixed numbers: rules, examples, solutions.

In this article we will look at multiplying mixed numbers. First, we will outline the rule for multiplying mixed numbers and consider the application of this rule when solving examples. Next we'll talk about multiplying a mixed number and a natural number. Finally, we will learn how to multiply a mixed number and a common fraction.

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Multiplying mixed numbers.

Multiplying mixed numbers can be reduced to multiplying ordinary fractions. To do this, it is enough to convert mixed numbers to improper fractions.

Let's write it down mixed number multiplication rule:

First, the mixed numbers being multiplied must be replaced by improper fractions;
Secondly, you need to use the rule for multiplying fractions by fractions.

Let's look at examples of applying this rule when multiplying a mixed number by a mixed number.

Perform multiplication of mixed numbers and .

First, let's represent the mixed numbers being multiplied as improper fractions: And . Now we can replace the multiplication of mixed numbers with the multiplication of ordinary fractions: . Applying the rule for multiplying fractions, we get . The resulting fraction is irreducible (see reducible and irreducible fractions), but it is improper (see proper and improper fractions), therefore, to obtain the final answer, it remains to isolate the whole part from the improper fraction: .

Let's write the entire solution in one line: .

To strengthen the skills of multiplying mixed numbers, consider solving another example.

Do the multiplication.

Funny numbers and are equal to the fractions 13/5 and 10/9, respectively. Then . At this stage, it’s time to remember about reducing a fraction: replace all the numbers in the fraction with their decompositions into prime factors, and perform a reduction of identical factors.

Multiplying a mixed number and a natural number

After replacing a mixed number with an improper fraction, multiplying a mixed number and a natural number leads to the multiplication of an ordinary fraction and a natural number.

Multiply a mixed number and the natural number 45.

A mixed number is equal to a fraction, then . Let's replace the numbers in the resulting fraction with their decompositions into prime factors, perform a reduction, and then select the whole part: .

Multiplication of a mixed number and a natural number is sometimes conveniently carried out using the distributive property of multiplication relative to addition. In this case, the product of a mixed number and a natural number is equal to the sum of the products of the integer part by the given natural number and the fractional part by the given natural number, that is, .

Calculate the product.

Let's replace the mixed number with the sum of the integer and fractional parts, after which we apply the distributive property of multiplication: .

Multiplying mixed numbers and fractions It is most convenient to reduce it to the multiplication of ordinary fractions by representing the mixed number being multiplied as an improper fraction.

Multiply the mixed number by the common fraction 4/15.

Replacing the mixed number with a fraction, we get .

Multiplying fractions

§ 140. Definitions. 1) Multiplying a fraction by an integer is defined in the same way as multiplying integers, namely: to multiply a number (multiplicand) by an integer (factor) means to compose a sum of identical terms, in which each term is equal to the multiplicand, and the number of terms is equal to the multiplier.

So multiplying by 5 means finding the sum:
2) Multiplying a number (multiplicand) by a fraction (factor) means finding this fraction of the multiplicand.

Thus, we will now call finding a fraction of a given number, which we considered before, multiplication by a fraction.

3) To multiply a number (multiplicand) by a mixed number (factor) means to multiply the multiplicand first by the whole number of the multiplier, then by the fraction of the multiplier, and add the results of these two multiplications together.

For example:

The number obtained after multiplication in all these cases is called work, i.e. the same as when multiplying integers.

From these definitions it is clear that the multiplication of fractional numbers is an action that is always possible and always unambiguous.

§ 141. The expediency of these definitions. To understand the advisability of introducing the last two definitions of multiplication into arithmetic, let’s take the following problem:

Task. A train, moving uniformly, covers 40 km per hour; how to find out how many kilometers this train will travel in a given number of hours?

If we remained with that one definition of multiplication, which is indicated in integer arithmetic (the addition of equal terms), then our problem would have three various solutions, namely:

If the given number of hours is an integer (for example, 5 hours), then to solve the problem you need to multiply 40 km by this number of hours.

If a given number of hours is expressed as a fraction (for example, an hour), then you will have to find the value of this fraction from 40 km.

Finally, if the given number of hours is mixed (for example, hours), then 40 km will need to be multiplied by the integer contained in the mixed number, and to the result add another fraction of 40 km, which is in the mixed number.

The definitions we have given allow us to give one general answer to all these possible cases:

you need to multiply 40 km by a given number of hours, whatever it may be.

Thus, if the problem is presented in general form as follows:

A train, moving uniformly, covers v km in an hour. How many kilometers will the train travel in t hours?

then, no matter what the numbers v and t are, we can give one answer: the desired number is expressed by the formula v · t.

Note. Finding some fraction of a given number, by our definition, means the same thing as multiplying a given number by this fraction; therefore, for example, finding 5% (i.e. five hundredths) of a given number means the same thing as multiplying a given number by or by ; finding 125% of a given number means the same as multiplying this number by or by, etc.

§ 142. A note about when a number increases and when it decreases from multiplication.

Multiplication by a proper fraction decreases the number, and multiplication by an improper fraction increases the number if this improper fraction more than one, and remains unchanged if it is equal to one.
Comment. When multiplying fractional numbers, as well as integers, the product is taken equal to zero if any of the factors is equal to zero, so .

§ 143. Derivation of multiplication rules.

1) Multiplying a fraction by a whole number. Let a fraction be multiplied by 5. This means increased by 5 times. To increase a fraction by 5 times, it is enough to increase its numerator or decrease its denominator by 5 times (§ 127).

That's why:
Rule 1. To multiply a fraction by a whole number, you need to multiply the numerator by this whole number, but leave the denominator the same; instead, you can also divide the denominator of the fraction by the given whole number (if possible), and leave the numerator the same.

Comment. The product of a fraction and its denominator is equal to its numerator.

So:
Rule 2. To multiply a whole number by a fraction, you need to multiply the whole number by the numerator of the fraction and make this product the numerator, and sign the denominator of this fraction as the denominator.
Rule 3. To multiply a fraction by a fraction, you need to multiply the numerator by the numerator and the denominator by the denominator, and make the first product the numerator, and the second the denominator of the product.

Comment. This rule can also be applied to multiplying a fraction by an integer and an integer by a fraction, if only we consider the integer as a fraction with a denominator of one. So:

Thus, the three rules now outlined are contained in one, which in general can be expressed as follows:
4) Multiplication of mixed numbers.

Rule 4th. To multiply mixed numbers, you need to convert them to improper fractions and then multiply according to the rules for multiplying fractions. For example:
§ 144. Reduction during multiplication. When multiplying fractions, if possible, it is necessary to make a preliminary reduction, as can be seen from the following examples:

Such a reduction can be done because the value of a fraction will not change if its numerator and denominator are reduced by the same number of times.

§ 145. Changing a product with changing factors. When the factors change, the product of fractional numbers will change in exactly the same way as the product of integers (§ 53), namely: if you increase (or decrease) any factor several times, then the product will increase (or decrease) by the same amount .

So, if in the example:
to multiply several fractions, you need to multiply their numerators with each other and the denominators with each other and make the first product the numerator, and the second the denominator of the product.

Comment. This rule can also be applied to such products in which some of the factors of the number are integers or mixed, if only we consider the integer as a fraction with a denominator of one, and we turn mixed numbers into improper fractions. For example:
§ 147. Basic properties of multiplication. Those properties of multiplication that we indicated for integers (§ 56, 57, 59) also apply to the multiplication of fractional numbers. Let us indicate these properties.

1) The product does not change when the factors are changed.

For example:

Indeed, according to the rule of the previous paragraph, the first product is equal to the fraction, and the second is equal to the fraction. But these fractions are the same, because their terms differ only in the order of the integer factors, and the product of integers does not change when the places of the factors are changed.

2) The product will not change if any group of factors is replaced by their product.

For example:

The results are the same.

From this property of multiplication the following conclusion can be drawn:

to multiply a number by a product, you can multiply this number by the first factor, multiply the resulting number by the second, etc.

For example:
3) Distributive law of multiplication (relative to addition). To multiply a sum by a number, you can multiply each term separately by that number and add the results.

This law was explained by us (§ 59) as applied to integers. It remains true without any changes for fractional numbers.

Let us show, in fact, that the equality

(a + b + c + .)m = am + bm + cm + .

(the distributive law of multiplication relative to addition) remains true even when the letters mean fractional numbers. Let's consider three cases.

1) Let us first assume that the factor m is an integer, for example m = 3 (a, b, c – any numbers). According to the definition of multiplication by an integer, we can write (limiting ourselves to three terms for simplicity):

(a + b + c) * 3 = (a + b + c) + (a + b + c) + (a + b + c).

Based on the associative law of addition, we can omit all the parentheses on the right side; By applying the commutative law of addition, and then again the associative law, we can obviously rewrite the right-hand side as follows:

(a + a + a) + (b + b + b) + (c + c + c).

(a + b + c) * 3 = a * 3 + b * 3 + c * 3.

This means that the distributive law is confirmed in this case.

Dividing a fraction by a natural number

Sections: Mathematics

T lesson type: ONZ (discovery of new knowledge - using the technology of the activity-based teaching method).

Deduce methods for dividing a fraction by a natural number;
Develop the ability to divide a fraction by a natural number;
Repeat and reinforce division of fractions;
Train the ability to reduce fractions, analyze and solve problems.

Equipment demonstration material:

1. Tasks for updating knowledge:

2. Trial (individual) task.

1. Perform division:

2. Perform division without performing the entire chain of calculations: .

When dividing a fraction by a natural number, you can multiply the denominator by that number, but leave the numerator the same.

If the numerator is divisible by a natural number, then when dividing a fraction by this number, you can divide the numerator by the number and leave the denominator the same.

I. Motivation (self-determination) for educational activities.

Organize the updating of requirements for the student in terms of educational activities (“must”);
Organize student activities to establish thematic frameworks (“I can”);
Create conditions for the student to develop an internal need for inclusion in educational activities (“I want”).

Organization of the educational process at stage I.

Hello! I'm glad to see you all at the math lesson. I hope it's mutual.

Guys, what new knowledge did you acquire in the last lesson? (Divide fractions).

Right. What helps you do division of fractions? (Rule, properties).

Where do we need this knowledge? (In examples, equations, problems).

Well done! You did well on the assignments in the last lesson. Do you want to discover new knowledge yourself today? (Yes).

Then - let's go! And the motto of the lesson will be the statement “You can’t learn mathematics by watching your neighbor do it!”

II. Updating knowledge and fixing individual difficulties in a trial action.

Organize the updating of learned methods of action sufficient to build new knowledge. Record these methods verbally (in speech) and symbolically (standard) and generalize them;
Organize updating mental operations and cognitive processes sufficient to construct new knowledge;
Motivate for a trial action and its independent implementation and justification;
Present individual task for a trial action and analyze it in order to identify new educational content;
Organize fixation of the educational goal and topic of the lesson;
Organize the implementation of a trial action and fix the difficulty;
Organize an analysis of the responses received and record individual difficulties in performing a trial action or justifying it.

Organization of the educational process at stage II.

Frontally, using tablets (individual boards).

1. Compare expressions:

(These expressions are equal)

What interesting things did you notice? (The numerator and denominator of the dividend, the numerator and denominator of the divisor in each expression increased by the same number of times. Thus, the dividends and divisors in the expressions are represented by fractions that are equal to each other).

Find the meaning of the expression and write it down on your tablet. (2)

How can I write this number as a fraction?

How did you perform the division action? (Children recite the rule, the teacher hangs it on the board letter designations)

2. Calculate and record the results only:

3. Add up the results and write down the answer. (2)

What is the name of the number obtained in task 3? (Natural)

Do you think you can divide a fraction by a natural number? (Yes, we'll try)

Try this.

4. Individual (trial) task.

Perform division: (example a only)

What rule did you use to divide? (According to the rule of dividing fractions by fractions)

Now divide the fraction by a natural number greater than in a simple way, without performing the entire chain of calculations: (example b). I'll give you 3 seconds for this.

Who couldn't complete the task in 3 seconds?

Who did it? (There are no such)

Why? (We don't know the way)

What did you get? (Difficulty)

What do you think we will do in class? (Divide fractions by natural numbers)

That's right, open your notebooks and write down the topic of the lesson: “Dividing a fraction by a natural number.”

Why does this topic sound new when you already know how to divide fractions? (Need new way)

Right. Today we will establish a technique that simplifies the division of a fraction by a natural number.

III. Identifying the location and cause of the problem.

Organize the restoration of completed operations and record (verbal and symbolic) the place - step, operation - where the difficulty arose;
Organize the correlation of students’ actions with the method (algorithm) used and fixation in external speech of the cause of the difficulty - that specific knowledge, skills or abilities that are lacking to solve the initial problem of this type.

Organization of the educational process at stage III.

What task did you have to complete? (Divide a fraction by a natural number without going through the entire chain of calculations)

What caused you difficulty? (Couldn't decide for a short time fast way)

What goal do we set for ourselves in the lesson? (Find quick way dividing a fraction by a natural number)

What will help you? (Already known rule for dividing fractions)

IV. Building a project for getting out of a problem.

Clarification of the project goal;
Choice of method (clarification);
Determination of means (algorithm);
Building a plan to achieve the goal.

Organization of the educational process at stage IV.

Let's return to the test task. You said you divided according to the rule for dividing fractions? (Yes)

To do this, replace the natural number with a fraction? (Yes)

What step (or steps) do you think can be skipped?

(The solution chain is open on the board:

Analyze and draw a conclusion. (Step 1)

If there is no answer, then we lead you through questions:

Where did the natural divisor go? (Into the denominator)

Has the numerator changed? (No)

So which step can you “omit”? (Step 1)

Multiply the denominator of a fraction by a natural number.
We do not change the numerator.
We get a new fraction.

V. Implementation of the constructed project.

Organize communicative interaction in order to implement the constructed project aimed at acquiring the missing knowledge;
Organize the recording of the constructed method of action in speech and signs (using a standard);
Organize the solution to the initial problem and record how to overcome the difficulty;
Organize clarification of the general nature of new knowledge.

Organization of the educational process at stage V.

Now run the test case in a new way quickly.

Now you were able to complete the task quickly? (Yes)

Explain how you did this? (Children talk)

This means that we have gained new knowledge: the rule for dividing a fraction by a natural number.

Well done! Say it in pairs.

Then one student speaks to the class. We fix the rule-algorithm verbally and in the form of a standard on the board.

Now enter the letter designations and write down the formula for our rule.

The student writes on the board, saying the rule: when dividing a fraction by a natural number, you can multiply the denominator by this number, but leave the numerator the same.

(Everyone writes the formula in their notebooks).

Now analyze the chain of solving the test task again, paying special attention to the answer. What did you do? (The numerator of the fraction 15 was divided (reduced) by the number 3)

What is this number? (Natural, divisor)

So how else can you divide a fraction by a natural number? (Check: if the numerator of a fraction is divisible by this natural number, then you can divide the numerator by this number, write the result in the numerator of the new fraction, and leave the denominator the same)

Write this method down as a formula. (The student writes the rule on the board while pronouncing it. Everyone writes the formula in their notebooks.)

Let's return to the first method. You can use it if a:n? (Yes it general method)

And when is it convenient to use the second method? (When the numerator of a fraction is divided by a natural number without a remainder)

VI. Primary consolidation with pronunciation in external speech.

Organize children’s assimilation of a new method of action when solving standard problems with their pronunciation in external speech (frontally, in pairs or groups).

Organization of the educational process at stage VI.

Calculate in a new way:

No. 363 (a; d) - performed at the board, pronouncing the rule.
No. 363 (e; f) - in pairs with checking according to the sample.

VII. Independent work with self-test according to the standard.

Organize students’ independent completion of tasks for a new way of action;
Organize self-test based on comparison with the standard;
Based on the results of execution independent work organize reflection on the assimilation of a new way of action.

Organization of the educational process at stage VII.

Calculate in a new way:

Students check against the standard and mark the correctness of execution. The causes of errors are analyzed and errors are corrected.

The teacher asks those students who made mistakes, what is the reason?

At this stage, it is important that each student independently checks their work.

Before solving task 8), consider an example from the textbook:

IX. Reflection on learning activities in the classroom.

Organize recording of new content learned in the lesson;
Organize a reflective analysis of educational activities from the point of view of fulfilling the requirements known to students;
Organize students’ assessment of their own activities in the lesson;
Organize the recording of unresolved difficulties in the lesson as a direction for future educational activities;
Organize a discussion and recording of homework.

Organization of the educational process at stage IX.

Guys, what new knowledge have you discovered today? (Learned how to divide a fraction by a natural number in a simple way)

Formulate a general method. (They say)

In what way and in what cases can you use it? (They say)

What is the advantage of the new method?

Have we achieved our lesson goal? (Yes)

What knowledge did you use to achieve your goal? (They say)

Did everything work out for you?

What were the difficulties?

Page navigation.

Multiplying mixed numbers.

Multiplying mixed numbers can be reduced to multiplying ordinary fractions. To do this, it is enough to convert mixed numbers to improper fractions.

Let's write it down mixed number multiplication rule:

First, the mixed numbers being multiplied must be replaced by improper fractions;
Secondly, you need to use the rule for multiplying fractions by fractions.

Let's look at examples of applying this rule when multiplying a mixed number by a mixed number.

Perform multiplication of mixed numbers and .

Let's write the entire solution in one line: .

To strengthen the skills of multiplying mixed numbers, consider solving another example.

Do the multiplication.

Multiplying a mixed number and a natural number

After replacing a mixed number with an improper fraction, multiplying a mixed number and a natural number leads to the multiplication of an ordinary fraction and a natural number.

Multiply a mixed number and the natural number 45.

A mixed number is equal to a fraction, then . Let's replace the numbers in the resulting fraction with their decompositions into prime factors, perform a reduction, and then select the whole part: .

Calculate the product.

Let's replace the mixed number with the sum of the integer and fractional parts, after which we apply the distributive property of multiplication: .

Multiplying mixed numbers and fractions It is most convenient to reduce it to the multiplication of ordinary fractions by representing the mixed number being multiplied as an improper fraction.

Multiply the mixed number by the common fraction 4/15.

Replacing the mixed number with a fraction, we get .

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Multiplying fractions

So multiplying by 5 means finding the sum:
2) Multiplying a number (multiplicand) by a fraction (factor) means finding this fraction of the multiplicand.

Thus, we will now call finding a fraction of a given number, which we considered before, multiplication by a fraction.

For example:

The number obtained after multiplication in all these cases is called work, i.e. the same as when multiplying integers.

From these definitions it is clear that the multiplication of fractional numbers is an action that is always possible and always unambiguous.

§ 141. The expediency of these definitions. To understand the advisability of introducing the last two definitions of multiplication into arithmetic, let’s take the following problem:

Task. A train, moving uniformly, covers 40 km per hour; how to find out how many kilometers this train will travel in a given number of hours?

If we remained with the one definition of multiplication that is indicated in integer arithmetic (the addition of equal terms), then our problem would have three different solutions, namely: