T class type: ONZ (discovery of new knowledge - according to the technology of the activity method of teaching).

Basic goals:

1. Deduce methods of dividing a fraction by a natural number;
2. To form the ability to perform the division of a fraction by a natural number;
3. Repeat and consolidate the division of fractions;
4. Train the ability to reduce fractions, analyze and solve problems.

Equipment demo material:

1. Tasks for updating knowledge:

Compare expressions:

Reference:

2. Trial (individual) task.

1. Perform division:

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

References:

• When dividing a fraction by a natural number, you can multiply the denominator by this number, and 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.

During the classes

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

Purpose of the stage:

1. Organize the actualization of the requirements for the student on the part of educational activities (“must”);
2. Organize the activities of students to establish a thematic framework (“I can”);
3. To create conditions for the student to have 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 in math class. I hope it's mutual.

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

Right. What helps you divide fractions? (Rule, properties).

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

Well done! You did well in the last lesson. Would you like to discover new knowledge yourself today? (Yes).

Then - go! And the motto of the lesson is the statement “Mathematics cannot be learned by watching how your neighbor does it!”.

II. Actualization of knowledge and fixation of an individual difficulty in a trial action.

Purpose of the stage:

1. To organize the actualization of the studied methods of action, sufficient to build new knowledge. Fix these methods verbally (in speech) and symbolically (standard) and generalize them;
2. Organize the actualization of mental operations and cognitive processes sufficient to build new knowledge;
3. Motivate for a trial action and its independent implementation and justification;
4. Present an individual task for a trial action and analyze it in order to identify new educational content;
5. Organize the fixation of the educational goal and the topic of the lesson;
6. Organize the implementation of a trial action and fixing the difficulty;
7. 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 the tablet. (2)

How to write this number as a fraction?

How did you perform the division action? (Children pronounce the rule, the teacher hangs letters on the board)

2. Calculate and record only the results:

3. Add up your results and write down your 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 will try)

Try this.

4. Individual (trial) task.

Do the division: (example a only)

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

And now divide the fraction by a natural number in a simpler way, without performing the entire chain of calculations: (example b). I give you 3 seconds for this.

Who failed to complete the task in 3 seconds?

Who made 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 a new way)

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

III. Identification of the location and cause of the difficulty.

Purpose of the stage:

1. Organize the restoration of completed operations and fix (verbal and symbolic) place - step, operation, where the difficulty arose;
2. To organize the correlation of students' actions with the method (algorithm) used and the fixation in external speech of the cause of the difficulty - those specific knowledge, skills or abilities that are not enough 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 doing the whole chain of calculations)

What caused you difficulty? (Could not solve in a short time in a fast way)

What is the purpose of our lesson? (Find a quick way to divide a fraction by a natural number)

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

IV. Construction of the project of an exit from difficulty.

Purpose of the stage:

1. Clarification of the purpose of the project;
2. Choice of method (clarification);
3. Definition of means (algorithm);
4. Building a plan to achieve the goal.

Organization of the educational process at stage IV.

Let's go back to the test case. Did you say that you divided by the rule of dividing fractions? (Yes)

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

What step(s) do you think you can skip?

(The solution chain is open on the board:

Analyze and draw a conclusion. (Step 1)

If there is no answer, then we summarize through the questions:

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

Has the numerator changed? (Not)

So what step can be "omitted"? (Step 1)

Action plan:

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

V. Implementation of the constructed project.

Purpose of the stage:

1. Organize communicative interaction in order to implement the constructed project aimed at acquiring the missing knowledge;
2. Organize the fixation of the constructed method of action in speech and signs (with the help of a standard);
3. Organize the solution of the original problem and record the overcoming of the difficulty;
4. Organize a clarification of the general nature of the new knowledge.

Organization of the educational process at stage V.

Now run the test case in the new way quickly.

Are you able to complete the task quickly now? (Yes)

Explain how you did it? (Children speak)

This means that we have received 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, pronouncing the rule: when dividing a fraction by a natural number, you can multiply the denominator by this number, and leave the numerator the same.

(Everyone writes the formula in notebooks).

And now once again analyze the chain of solving the trial task, paying special attention to the answer. What did they 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 into the numerator of the new fraction, and leave the denominator the same)

Write this method in the form of a formula. (The student writes down the rule on the board. Everyone writes down the formula in notebooks.)

Let's go back to the first method. Can it be used if a:n? (Yes, this is the general way)

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

VI. Primary consolidation with pronunciation in external speech.

Purpose of the stage:

1. To organize the assimilation by children of a new method of action when solving typical 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) - perform at the blackboard, pronouncing the rule.
• No. 363 (d; f) - in pairs with a check on the sample.

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

Purpose of the stage:

1. To organize the students' independent fulfillment of tasks for a new mode of action;
2. Organize self-test based on comparison with the standard;
3. Based on the results of independent work, organize a reflection on the assimilation of a new mode of action.

Organization of the educational process at stage VII.

Calculate in a new way:

• No. 363 (b; c)

Students check the standard, note the correctness of the performance. 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 check their work.

VIII. Inclusion in the system of knowledge and repetition.

Purpose of the stage:

1. Organize the identification of the boundaries of the application of new knowledge;
2. Organize the repetition of educational content necessary to ensure meaningful continuity.

Organization of the educational process at stage VIII.

Organize the fixation of unresolved difficulties in the lesson as a direction for future learning activities;

Organize discussion and recording of homework.

Organization of the educational process at stage IX.

1. Dialog:

Guys, what new knowledge did you discover today? (We learned to divide a fraction by a natural number in a simple way)

Formulate a general way. (They say)

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

What is the advantage of the new method?

Have we reached our goal of the lesson? (Yes)

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

Have you succeeded?

What were the difficulties?

2. Homework: clause 3.2.4.; No. 365 (l, n, o, p); No. 370.

3. Teacher: I am glad that today everyone was active, managed to find a way out of the difficulty. And most importantly, they were not neighbors when a new one was opened and consolidated. Thanks for the lesson kids!

A fraction is one or more parts of a whole, which is usually taken as a unit (1). As with natural numbers, you can perform all basic arithmetic operations with fractions (addition, subtraction, division, multiplication), for this you need to know the features of working with fractions and distinguish between their types. There are several types of fractions: decimal and ordinary, or simple. Each type of fractions has its own specifics, but once you have thoroughly figured out how to deal with them once, you will be able to solve any examples with fractions, since you will know the basic principles for performing arithmetic calculations with fractions. Let's look at examples of how to divide a fraction by an integer using different types of fractions.

How to divide a fraction by a natural number?
Ordinary or simple fractions are called fractions that are written as such a ratio of numbers in which the dividend (numerator) is indicated at the top of the fraction, and the divisor (denominator) of the fraction is indicated below. How to divide such a fraction by an integer? Let's look at an example! Let's say we need to divide 8/12 by 2.

To do this, we must perform a series of actions:
Thus, if we are faced with the task of dividing a fraction by an integer, the solution scheme will look something like this:

Similarly, you can divide any ordinary (simple) fraction by an integer.

How to divide a decimal by an integer?
A decimal fraction is a fraction that is obtained by dividing a unit into ten, a thousand, and so on parts. Arithmetic operations with decimal fractions are quite simple.

Consider an example of how to divide a fraction by an integer. Let's say we need to divide the decimal fraction 0.925 by the natural number 5.

Summing up, we will focus on two main points that are important when performing the operation of dividing decimal fractions by an integer:

• to divide a decimal fraction by a natural number, division into a column is used;
  
• a comma is placed in the private when the division of the integer part of the dividend is completed.
  

By applying these simple rules, you can always easily divide any decimal or fraction by an integer.

With fractions, you can perform all actions, including division. This article shows the division of ordinary fractions. Definitions will be given, examples will be considered. Let us dwell on the division of fractions by natural numbers and vice versa. The division of an ordinary fraction by a mixed number will be considered.

Division of ordinary fractions

Division is the inverse of multiplication. When dividing, the unknown factor is at the known product and another factor, where its given meaning is preserved with ordinary fractions.

If it is necessary to divide the ordinary fraction a b by c d, then to determine such a number, you need to multiply by the divisor c d, this will eventually give the dividend a b. Let's get a number and write it a b · d c , where d c is the reciprocal of c d number. Equalities can be written using the properties of multiplication, namely: a b d c c d = a b d c c d = a b 1 = a b , where the expression a b d c is the quotient of dividing a b by c d .

From here we obtain and formulate the rule for dividing ordinary fractions:

Definition 1

To divide an ordinary fraction a b by c d, it is necessary to multiply the dividend by the reciprocal of the divisor.

Let's write the rule as an expression: a b: c d = a b d c

The rules of division are reduced to multiplication. To stick to it, you need to be well versed in performing multiplication of ordinary fractions.

Let's move on to the division of ordinary fractions.

Example 1

Perform division 9 7 by 5 3 . Write the result as a fraction.

Solution

The number 5 3 is the reciprocal of 3 5 . You must use the rule for dividing ordinary fractions. We write this expression as follows: 9 7: 5 3 \u003d 9 7 3 5 \u003d 9 3 7 5 \u003d 27 35.

Answer: 9 7: 5 3 = 27 35 .

When reducing fractions, you should highlight the whole part if the numerator is greater than the denominator.

Example 2

Divide 8 15: 24 65 . Write the answer as a fraction.

Solution

The solution is to switch from division to multiplication. We write it in this form: 8 15: 24 65 = 2 2 2 5 13 3 5 2 2 2 3 = 13 3 3 = 13 9

It is necessary to make a reduction, and this is done as follows: 8 65 15 24 \u003d 2 2 2 5 13 3 5 2 2 2 3 \u003d 13 3 3 \u003d 13 9

We select the integer part and get 13 9 = 1 4 9 .

Answer: 8 15: 24 65 = 1 4 9 .

Division of an extraordinary fraction by a natural number

We use the rule of dividing a fraction by a natural number: to divide a b by a natural number n, you need to multiply only the denominator by n. From here we get the expression: a b: n = a b · n .

The division rule is a consequence of the multiplication rule. Therefore, representing a natural number as a fraction will give an equality of this type: a b: n \u003d a b: n 1 \u003d a b 1 n \u003d a b n.

Consider this division of a fraction by a number.

Example 3

Divide the fraction 1645 by the number 12.

Solution

Apply the rule for dividing a fraction by a number. We get an expression like 16 45: 12 = 16 45 12 .

Let's reduce the fraction. We get 16 45 12 = 2 2 2 2 (3 3 5) (2 2 3) = 2 2 3 3 3 5 = 4 135 .

Answer: 16 45: 12 = 4 135 .

Division of a natural number by a common fraction

The division rule is similar about the rule of dividing a natural number by an ordinary fraction: to divide a natural number n by an ordinary a b , it is necessary to multiply the number n by the reciprocal of the fraction a b .

Based on the rule, we have n: a b \u003d n b a, and thanks to the rule of multiplying a natural number by an ordinary fraction, we get our expression in the form n: a b \u003d n b a. It is necessary to consider this division with an example.

Example 4

Divide 25 by 15 28 .

Solution

We need to move from division to multiplication. We write in the form of an expression 25: 15 28 = 25 28 15 = 25 28 15 . Let's reduce the fraction and get the result in the form of a fraction 46 2 3 .

Answer: 25: 15 28 = 46 2 3 .

Division of a common fraction by a mixed number

When dividing an ordinary fraction by a mixed number, you can easily shine to dividing ordinary fractions. You need to convert a mixed number to an improper fraction.

Example 5

Divide the fraction 35 16 by 3 1 8 .

Solution

Since 3 1 8 is a mixed number, let's represent it as an improper fraction. Then we get 3 1 8 = 3 8 + 1 8 = 25 8 . Now let's divide the fractions. We get 35 16: 3 1 8 = 35 16: 25 8 = 35 16 8 25 = 35 8 16 25 = 5 7 2 2 2 2 2 2 2 (5 5) = 7 10

Answer: 35 16: 3 1 8 = 7 10 .

Dividing a mixed number is done in the same way as ordinary numbers.

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

Last time we learned how to add and subtract fractions (see the lesson "Addition and subtraction of fractions"). The most difficult moment in those actions was bringing fractions to a common denominator.

Now it's time to deal with multiplication and division. The good news is that these operations are even easier than addition and subtraction. To begin with, consider the simplest case, when there are two positive fractions without a distinguished integer part.

To multiply two fractions, you need to multiply their numerators and denominators separately. The first number will be the numerator of the new fraction, and the second will be the denominator.

To divide two fractions, you need to multiply the first fraction by the "inverted" second.

Designation:

From the definition it follows that the division of fractions is reduced to multiplication. To flip a fraction, just swap the numerator and denominator. Therefore, the entire lesson we will consider mainly multiplication.

As a result of multiplication, a reduced fraction can arise (and often does arise) - of course, it must be reduced. If, after all the reductions, the fraction turned out to be incorrect, the whole part should be distinguished in it. But what exactly will not happen with multiplication is reduction to a common denominator: no crosswise methods, maximum factors and least common multiples.

By definition we have:

Multiplication of fractions with an integer part and negative fractions

If there is an integer part in the fractions, they must be converted to improper ones - and only then multiplied according to the schemes outlined above.

If there is a minus in the numerator of a fraction, in the denominator or in front of it, it can be taken out of the limits of multiplication or removed altogether according to the following rules:

1. Plus times minus gives minus;
2. Two negatives make an affirmative.

Until now, these rules have only been encountered when adding and subtracting negative fractions, when it was required to get rid of the whole part. For a product, they can be generalized in order to “burn” several minuses at once:

1. We cross out the minuses in pairs until they completely disappear. In an extreme case, one minus can survive - the one that did not find a match;
2. If there are no minuses left, the operation is completed - you can start multiplying. If the last minus is not crossed out, since it did not find a pair, we take it out of the limits of multiplication. You get a negative fraction.

A task. Find the value of the expression:

We translate all fractions into improper ones, and then we take out the minuses outside the limits of multiplication. What remains is multiplied according to the usual rules. We get:

Let me remind you once again that the minus that comes before a fraction with a highlighted integer part refers specifically to the entire fraction, and not just to its integer part (this applies to the last two examples).

Also pay attention to negative numbers: when multiplied, they are enclosed in brackets. This is done in order to separate the minuses from the multiplication signs and make the whole notation more accurate.

Reducing fractions on the fly

Multiplication is a very laborious operation. The numbers here are quite large, and to simplify the task, you can try to reduce the fraction even more before multiplication. Indeed, in essence, the numerators and denominators of fractions are ordinary factors, and, therefore, they can be reduced using the basic property of a fraction. Take a look at the examples:

A task. Find the value of the expression:

By definition we have:

In all examples, the numbers that have been reduced and what is left of them are marked in red.

Please note: in the first case, the multipliers were reduced completely. Units remained in their place, which, generally speaking, can be omitted. In the second example, it was not possible to achieve a complete reduction, but the total amount of calculations still decreased.

However, in no case do not use this technique when adding and subtracting fractions! Yes, sometimes there are similar numbers that you just want to reduce. Here, look:

You can't do that!

The error occurs due to the fact that when adding a fraction, the sum appears in the numerator of a fraction, and not the product of numbers. Therefore, it is impossible to apply the main property of a fraction, since this property deals specifically with the multiplication of numbers.

There is simply no other reason to reduce fractions, so the correct solution to the previous problem looks like this:

The right decision:

As you can see, the correct answer turned out to be not so beautiful. In general, be careful.

Multiplication and division of fractions.

Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")

This operation is much nicer than addition-subtraction! Because it's easier. I remind you: to multiply a fraction by a fraction, you need to multiply the numerators (this will be the numerator of the result) and the denominators (this will be the denominator). That is:

For example:

Everything is extremely simple. And please don't look for a common denominator! Don't need it here...

To divide a fraction by a fraction, you need to flip second(this is important!) fraction and multiply them, i.e.:

For example:

If multiplication or division with integers and fractions is caught, it's okay. As with addition, we make a fraction from a whole number with a unit in the denominator - and go! For example:

In high school, you often have to deal with three-story (or even four-story!) fractions. For example:

How to bring this fraction to a decent form? Yes, very easy! Use division through two points:

But don't forget about the division order! Unlike multiplication, this is very important here! Of course, we will not confuse 4:2 or 2:4. But in a three-story fraction it is easy to make a mistake. Please note, for example:

In the first case (expression on the left):

In the second (expression on the right):

Feel the difference? 4 and 1/9!

What is the order of division? Or brackets, or (as here) the length of horizontal dashes. Develop an eye. And if there are no brackets or dashes, like:

then divide-multiply in order, left to right!

And another very simple and important trick. In actions with degrees, it will come in handy for you! Let's divide the unit by any fraction, for example, by 13/15:

The shot has turned over! And it always happens. When dividing 1 by any fraction, the result is the same fraction, only inverted.

That's all the actions with fractions. The thing is quite simple, but gives more than enough errors. Take note of practical advice, and there will be fewer of them (mistakes)!

Practical Tips:

1. The most important thing when working with fractional expressions is accuracy and attentiveness! These are not common words, not good wishes! This is a severe need! Do all the calculations on the exam as a full-fledged task, with concentration and clarity. It is better to write two extra lines in a draft than to mess up when calculating in your head.

2. In examples with different types of fractions - go to ordinary fractions.

3. We reduce all fractions to the stop.

4. We reduce multi-level fractional expressions to ordinary ones using division through two points (we follow the order of division!).

5. We divide the unit into a fraction in our mind, simply by turning the fraction over.

Here are the tasks you need to complete. Answers are given after all tasks. Use the materials of this topic and practical advice. Estimate how many examples you could solve correctly. The first time! Without a calculator! And draw the right conclusions...

Remember the correct answer obtained from the second (especially the third) time - does not count! Such is the harsh life.

So, solve in exam mode ! This is preparation for the exam, by the way. We solve an example, we check, we solve the following. We decided everything - we checked again from the first to the last. But only after look at the answers.

Calculate:

Did you decide?

Looking for answers that match yours. I specifically wrote them down in a mess, away from the temptation, so to speak ... Here they are, the answers, written down with a semicolon.

0; 17/22; 3/4; 2/5; 1; 25.

And now we draw conclusions. If everything worked out - happy for you! Elementary calculations with fractions are not your problem! You can do more serious things. If not...

So you have one of two problems. Or both at once.) Lack of knowledge and (or) inattention. But this solvable Problems.

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.