Do you want to feel like a sapper? Then this lesson is for you! Because now we will study fractions - these are such simple and harmless mathematical objects that surpass the rest of the algebra course in their ability to “take out the brain”.

The main danger of fractions is that they occur in real life. In this they differ, for example, from polynomials and logarithms, which can be passed and easily forgotten after the exam. Therefore, the material presented in this lesson, without exaggeration, can be called explosive.

A numeric fraction (or simply a fraction) is a pair of integers written through a slash or horizontal bar.

Fractions written through a horizontal bar:

The same fractions written with a slash:
5/7; 9/(−30); 64/11; (−1)/4; 12/1.

Usually fractions are written through a horizontal line - it's easier to work with them, and they look better. The number written on top is called the numerator of the fraction, and the number written on the bottom is called the denominator.

Any whole number can be represented as a fraction with a denominator of 1. For example, 12 = 12/1 is the fraction from the above example.

In general, you can put any whole number in the numerator and denominator of a fraction. The only restriction is that the denominator must be different from zero. Remember the good old rule: “You can’t divide by zero!”

If the denominator is still zero, the fraction is called indefinite. Such a record does not make sense and cannot participate in calculations.

Basic property of a fraction

Fractions a /b and c /d are called equal if ad = bc.

From this definition it follows that the same fraction can be written in different ways. For example, 1/2 = 2/4 because 1 4 = 2 2. Of course, there are many fractions that are not equal to each other. For example, 1/3 ≠ 5/4 because 1 4 ≠ 3 5.

A reasonable question arises: how to find all fractions equal to a given one? We give the answer in the form of a definition:

The main property of a fraction is that the numerator and denominator can be multiplied by the same number other than zero. This will result in a fraction equal to the given one.

This is a very important property - remember it. With the help of the basic property of a fraction, many expressions can be simplified and shortened. In the future, it will constantly “emerge” in the form of various properties and theorems.

Incorrect fractions. Selection of the whole part

If the numerator is less than the denominator, such a fraction is called proper. Otherwise (that is, when the numerator is greater than or at least equal to the denominator), the fraction is called an improper fraction, and an integer part can be distinguished in it.

The integer part is written as a large number in front of the fraction and looks like this (marked in red):

To highlight the whole part in improper fraction you need to follow three simple steps:

1. Find how many times the denominator fits in the numerator. In other words, find the maximum integer that, when multiplied by the denominator, will still be less than the numerator (in the extreme case, equal). This number will be the integer part, so we write it in front;
2. Multiply the denominator by the integer part found in the previous step, and subtract the result from the numerator. The resulting "stub" is called the remainder of the division, it will always be positive (in extreme cases, zero). We write it down in the numerator of the new fraction;
3. We rewrite the denominator unchanged.

Well, is it difficult? At first glance, it may be difficult. But it takes a little practice - and you will do it almost verbally. For now, take a look at the examples:

A task. Select the whole part in the given fractions:

In all examples, the integer part is highlighted in red, and the remainder of the division is in green.

Pay attention to the last fraction, where the remainder of the division turned out to be zero. It turns out that the numerator is completely divided by the denominator. This is quite logical, because 24: 6 \u003d 4 is a harsh fact from the multiplication table.

If everything is done correctly, the numerator of the new fraction will necessarily be less than the denominator, i.e. fraction becomes correct. I also note that it is better to highlight the whole part at the very end of the task, before writing the answer. Otherwise, you can significantly complicate the calculations.

Transition to improper fraction

There is also an inverse operation, when we get rid of the whole part. This is called the improper fraction transition and is much more common because improper fractions are much easier to work with.

The transition to an improper fraction is also done in three steps:

1. Multiply the integer part by the denominator. The result can be quite big numbers, but we should not be embarrassed;
2. Add the resulting number to the numerator of the original fraction. Write the result in the numerator of an improper fraction;
3. Rewrite the denominator - again, no change.

Here are specific examples:

A task. Convert to an improper fraction:

For clarity, the integer part is again highlighted in red, and the numerator of the original fraction is in green.

Consider the case when the numerator or denominator of a fraction contains a negative number. For example:

In principle, there is nothing criminal in this. However, working with such fractions can be inconvenient. Therefore, in mathematics it is customary to take out minuses as a fraction sign.

This is very easy to do if you remember the rules:

1. Plus times minus equals minus. Therefore, if there is a negative number in the numerator, and a positive number in the denominator (or vice versa), feel free to cross out the minus and put it in front of the whole fraction;
2. "Two negatives make an affirmative". When the minus is in both the numerator and the denominator, we simply cross them out - no additional action is required.

Of course, these rules can also be applied in the opposite direction, i.e. you can add a minus under the fraction sign (most often - in the numerator).

We deliberately do not consider the case of “plus on plus” - with him, I think, everything is clear anyway. Let's take a look at how these rules work in practice:

A task. Take out the minuses of the four fractions written above.

Pay attention to the last fraction: it already has a minus sign in front of it. However, it is “burned” according to the rule “minus times minus gives plus”.

Also, do not move minuses in fractions with a highlighted integer part. These fractions are first converted to improper ones - and only then they begin to calculate.

Sections: Maths

Class: 4

Basic goals:

1. To form the ability to isolate the whole part from an improper fraction.
2. Revise the concepts of the numerator and denominator, correct and improper fractions, mixed numbers.
3. To update the ability to isolate the whole part from an improper fraction.

Mental operations necessary at the design stage: action by analogy, analysis, generalization.

Equipment:

Demo material:

1) Division formula with remainder.

Handout:

1) leaflets with the task (to stage 2)

2) Detailed sample for self-test (to step 6)

During the classes.

1 Self-determination to learning activities.

Goals:

1. Motivate students to learning activities by reinforcing the situation of success achieved in the previous lesson.
2. Determine the content of the lesson.

Organization of the educational process at stage 1.

For several lessons we have been working with some numbers. What numbers are we working with? (With fractional numbers).

What knowledge do we have about these numbers? (We know how to read, write, compare, solve problems).

I propose to continue our fruitful work. You are ready? (Yes).

Today we will continue to work with fractional numbers. I am sure that everything will work out perfectly for you and me. But first, let's repeat the material of the previous lessons.

2 Actualization of knowledge and fixation of difficulties in individual activities.

Goals:

1. Update the ability to find correct and improper fractions, mixed numbers, the definition of correct and improper fractions, mixed numbers.
2. Update the mental operations necessary and sufficient for the perception of new material.
3. Fix the situation when students cannot select the whole part from an improper fraction.

Organization of the educational process at stage 2.

What numbers did we learn in the previous lesson? (With mixed numbers).
What is a mixed number? (From the integer and fractional parts).

Fractions and mixed numbers are written on the board.

Into what groups can the presented numbers be divided?

Proper fractions ().

What fractions are right? (A fraction whose numerator is less than the denominator. A proper fraction is less than one).

Incorrect fractions. (…..)

What fractions are called improper? (A fraction in which the numerator is greater than the denominator or the numerator is equal to the denominator).

Which of the following improper fractions can be represented as a natural number?

()

What fraction can be represented as a mixed number? (an improper fraction where the numerator is greater than the denominator).

Define with number beam what is the mixed number of the fraction

Students have a sheet with a task (R-1), one student works at the blackboard, comments.

What is the smallest mixed number? ()

The greatest? ()

What arithmetic operation helped you? (Division. Division with remainder).

Prove it. (On the board: D-1).

12:7=1 (rest.5); 15:7=2 (rest.1); 25:7=3 (rest.4); 31:7=4 (rest.3)

Select the integer part of the fraction, write down the mixed number. Children work on the reverse side of the leaflet. Various answers are put on the board.

How did you act?

3 Identification of the causes of the difficulty and setting the goal of the activity.

Goals:

1. Organize communicative interaction to identify the distinctive properties of the task to select the whole part from an improper fraction.
2. Agree on the topic and purpose of the lesson.

Organization of the educational process at stage 3.

What task did you do? (It is necessary to select the whole part from the fraction).

How is this assignment different from the previous one? (The method that helped us to isolate the integer part from an improper fraction is not suitable for a fraction. It is inconvenient to show this fraction on a number line).

What do we see? (We got different answers).

Why? (We used different methods. We do not have an algorithm for extracting the integer part from an improper fraction).

What is the purpose of our lesson? (Build an algorithm and learn how to extract the integer part from an improper fraction).

Think and formulate the theme of our lesson. (“Separating the whole part from an improper fraction”).

Well done!

The name of the topic of the lesson is displayed on the board.

4 Building a project to get out of the difficulty.

Target:

1. Organize communicative interaction to build a new way of action to extract the whole part from an improper fraction.
2. Fix a new way in sign and verbal form and with the help of a standard.

Organization of the educational process at stage 4

In what way do you propose to find how many integer units are in a fractional number? (Numerator divided by denominator).

Which sign in the fraction notation told you how to act? (The line of a fraction is a division sign).

On the desk:

Let's write the fraction as a private: 65: 7.

What kind of division is this? (Division with remainder. On the board: D-1).

Find the result. (65: 7 = 9) (res. 2)

What does the quotient 9 and the remainder 2 mean in the resulting equality? (The quotient 9 means that 65 contains 9 times 7 and 2 remains).

What will the quotient 9 stand for in a mixed number? (9 is the integer part of the mixed number).

On the desk:

What will be the remainder 2 in a mixed number? (2 is the numerator of the fraction of the mixed number).

On the desk:

What about the denominator? (He remains, does not change).

On the desk:

What is the mixed number?

Did we complete the task? (Yes).

What mathematical action helped us? (Division with remainder. On the board: D-1).

The teacher returns to the answers on the sheets, summarizes, encourages with a word those who did it right. In group form, students deduce a new method in sign form on leaflets. The correct option is selected.

Write down, using the division formula with a remainder (D-1), what mixed number is the fraction equal to?

On the board: D-3

How to extract the whole part from an improper fraction?

To extract the whole part from an improper fraction, you need to divide its numerator by the denominator. The quotient will be the integer part, the remainder will be the numerator, and the denominator will not change.

Well done! Thank you!

Let's still check our opinion with the opinion of the textbook. Turn to page 26, Math 4 (part 2), read the rule first to yourself and then aloud.

We were right? (Yes).

Well done!

Fizminutka (at the choice of the teacher).

5 Primary consolidation in external speech.

Target:

Fix the method of extracting the integer part from an improper fraction in external speech.

Organization of the educational process at stage 5.

Let's repeat the algorithm for extracting the integer part from an improper fraction. D 2

We have compiled an algorithm for extracting the integer part from an improper fraction. What is the purpose of our future activities? (Practice).

No. 4 (a, b, c) p. 26 - with commentary according to the model.

No. 4 (d, e) p. 26 - in pairs.

6 Self-monitoring with self-test.

Target:

1. To organize the independent performance by students of the task of isolating the whole part from an improper fraction.
2. Train the ability for self-control and self-esteem.
3. Test your ability to isolate the whole part from an improper fraction.
4. Contribute to the creation of a situation of success.

Organization of the educational process at stage 6.

You managed to derive an algorithm for extracting the integer part from an improper fraction and practiced solving examples. I think now you can complete the task yourself.

Do it yourself:

No. 3 p. 26 - 1 option - 1 and 2 columns;

Option 2 - 3 and 4 columns;

Whoever wishes, can complete the task of another option.

The students complete the work, at the end of which they check themselves according to the model for self-examination. P-2 card is used.

Test yourself using the self-test template and record the result of the test using the “+” or “?” green pen.

Who made mistakes while doing the task? (…)

What is the reason? (…)

Who's got it right?

Well done!

You can organize work on correcting errors in groups or frontally. Students who have not made mistakes are appointed as consultants.

7 Inclusion in the knowledge system and repetition.

Target:

Train the ability to isolate the whole part from an improper fraction.

Organization of the educational process at stage 7.

Let's try to apply our knowledge when comparing a fraction and a mixed number.

Find an inequality in which you need to compare a proper fraction with an improper one.

What do we do?

Let's extract the integer part from the improper fraction.

Means?!

An improper fraction is larger than a proper one. We proved this by selecting the integer part.

Well done!

Finish the task, compare.

Let's check.

8 Reflection of learning activities in the classroom.

Goals:

1. Fix in speech the algorithm for extracting the integer part from an improper fraction.
2. Record the remaining difficulties and ways to overcome them.
3. Evaluate your own performance in class.
4. Coordinate homework.

Organization of the educational process at stage 8.

What did you learn in the lesson? (Separate the whole part from an improper fraction).

What algorithm have we built? (You can say the D-2 algorithm).

Who had difficulty? How will you act?

Who is happy today? Why?

I had a hard time in class.
I got the lesson, but I need practice.
- I understood the lesson well, but I need help.
- Well done, I understood the lesson perfectly.

Homework: come up with five improper fractions and highlight the whole part; No. 10, No. 11 p. 28 - optional; No. 15 p. 28 (a or b) - optional.

Well done! Thanks for the lesson!

How to extract the integer part from an improper fraction? To select an integer part from an improper fraction, you must: Divide the numerator by the denominator with the remainder; The incomplete quotient will be the whole part; The remainder (if any) gives the numerator, and the divisor gives the denominator of the fractional part. Do No. 1057, 1058, 1059, 1060. 1062, 1063. 1064. 7.

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mixed numbers

"Summary of a lesson in mathematics" - Follow the model. a) 4/7+2/7= (4+2)/7= 6/7 b, c, d (at the board) e) 7/9-2/9= (7-2)/9= 5/ 9 f, f, h (at the board). 12 kg of cucumbers were harvested in the garden. 2/3 of all cucumbers were pickled. 6/7-3/7=(6-3)/7=3/7 2/11+5/11=(2+5)/22=7/22 9/10-8/10=(9-8 )/10=2/10. Show the fraction 2/8+3/8. Formulate a subtraction rule. Learning new material:

"Comparison of decimal fractions" - The purpose of the lesson. Compare numbers: Mental account. 9.85 and 6.97; 75.7 and 75.700; 0.427 and 0.809; 5.3 and 5.03; 81.21 and 81.201; 76.005 and 76.05; 3.25 and 3.502; Read the fractions: 41.1; 77.81; 21.005; 0.0203. 41.1; 77.81; 21.005; 0.0203. Equalize the number of decimal places. Lesson plan. Discharges decimal fractions. Consolidation lesson in 5th grade.

"Rules for rounding numbers" - 1.8. 48. Well done! 3. 3. Learn to apply the rounding rule with examples. Try to compare. Round whole numbers to tens. 1. Remember the rule for rounding numbers. Is it convenient to work with such a number? One hundred thousandths. 3. Write down the result. 5312. >. 2. Derive a rule for rounding decimal fractions to a given digit.

"Addition of mixed numbers" - 25. Example 4. Find the value of the difference 3 4\9-1 5\6. 3 4 \ 9 \u003d 3 818; 15\6=115\18. 3 4\9=3 8\18=3+8\18=2+1+8\18=2+8\18+18\18=2+ +26\18=2 26\18. Lesson abstract in grade 6

Mathematics lesson in grade 4 topic: Extracting the integer part from an improper fraction Lesson topic: Extracting the integer part from an improper fraction. Didactic goal: to create conditions for the formation of new educational information. Aims and objectives of the lesson: 1. Form the concept of a mixed number. 2. To form the ability to isolate the whole part from an improper fraction. 3. Develop computing skills. 4. Develop the ability to analyze and solve text problems to find a part of a number and a number by its part. 5. Develop logical thinking students. Planned learning outcomes, the formation of UUD: Subject: to expand the concept of number, to form the ability to translate improper fractions into mixed numbers and apply the acquired knowledge and skills when performing various tasks. Metasubject: develop the ability to see math problem in the context of a problematic situation in other disciplines, in the surrounding life. Cognitive UUD: develop ideas about the number; the ability to work with a textbook, additional sources of information (analyze, extract the necessary information); the ability to make generalizations, conclusions, establish causal relationships. Communicative UUD: cultivate respect for each other, develop the ability to enter into an educational dialogue with the teacher, with classmates, observing the norms of speech behavior, the ability to ask questions, listen and answer questions from others, the ability to put forward a hypothesis. Regulatory UUD: determine the purpose of the task, learn to plan the stages of work, control your actions, detect and correct mistakes, critically evaluate the results of your work and the work of everyone, based on existing criteria, form the ability to mobilize forces and energy, to overcome obstacles. Personal UUD: form learning motivation , initiative, develop the skills of competent oral and written mathematical speech, the ability to self-evaluate their actions. Resources: multimedia projector, presentation. Type of lesson: learning new material. Stage of the lesson Teacher's activity Student's activity Organizational moment Greeting, checking readiness for the lesson, organizing the attention of children. . Included in the business rhythm of the lesson. Used methods, techniques, forms Verbal Formed UUD To be able to formulate their thoughts orally (Communicative UUD). The ability to listen and understand the speech of others (Communicative UUD). As you understand from what you read, today in the lesson we will continue to work on fractions. Guys, in the lesson you should discover new knowledge, but, as you know, every new knowledge is related to what we have already studied. So let's start with repetition. Oral count Actualization of knowledge and skills Practical Answers are written in a column, we check the answers on the slides. pronounce in the lesson Be able to follow the sequence of actions (Regulatory UUD). Be able to convert information from one form to another (Cognitive UUD). Be able to formulate your thoughts in oral and written form (Communicative UUD). Blitz poll: What rules did you use when: 1. Find the sum of fractions. 2. Find the difference between fractions. 3. Find the number by part. 4. Find a part by number. They tell the rules. Participate in a conversation with the teacher. Be able to formulate your thoughts orally (Communicative UUD). Be able to navigate in your knowledge system: to distinguish the new from the already known with the help of a teacher (Cognitive UUD). The ability to listen and understand the speech of others (Communicative UUD). Goal-setting and motivation 3. Problem statement Verbal Be able to formulate your thoughts orally (Communicative UUD). Know how to navigate. . own system of knowledge: to distinguish the new from the already known with the help (Cognitive teachers of UUD). Children express their options. 4. “Formulation of the problem and the purpose of the lesson Select an integer part from this fraction. What do you offer? What do you think is the purpose of the lesson? The purpose of the lesson and the topic are formulated by the students. Purpose: To learn how to isolate the whole part from an improper fraction Verbal, practical To be able to gain new knowledge: find answers to questions using a textbook, your life experience and information obtained in (Educational lesson UUD). Be able to formulate your thoughts orally; listen and understand speech (Communicative other UUD). So any improper fraction can be represented as a mixed number. The integer part is a natural number, and the fractional part is a proper fraction. . . Drawing up an algorithm. Verbally visually practical, reproductive analysis at the lesson to pronounce according to Be able to collectively drawn up a plan (Regulatory UUD). Know the sequence of actions (Regulatory UUD). Be able to formulate your thoughts orally and in writing; listen and understand the speech of others (Communicative UUD) Be able to follow the sequence of actions (Regulatory UUD). To be able to perform work according to the proposed plan (Regulatory UUD). pronounce the lesson on Assimilation of new knowledge and ways of assimilation 5. Discovery of the new: Explanation on the board. Write down the fraction 16/5 as a private What rule was used to select an integer part from an improper fraction To select an integer part from an improper fraction, you need to: divide the numerator by the denominator with the remainder; record the resulting incomplete quotient in Be able to make the necessary adjustments to the action after its completion based on its assessment and taking into account the nature of the errors made (Regulatory UUD). The ability for self-assessment on the criterion of success in educational activities (Personal UUD). the basis of the integer part of the fraction; write the remainder in the numerator of the fraction; put the divisor in the denominator of the fraction. 16:5=3(rest 1)) 3 - integer 1 - numerator 5 - denominator 16/5 = 3 1/5 Reading the rule in the textbook on p. 26, no. 3 - at the blackboard 1 example with explanation. The rest with comments. No. 4 (a, b, c) - independently. Mutual verification. m integer, n and b parts In a fraction, the integer is always the numerator. The guys say the rule to find the whole you need to multiply 6. Formulation of new knowledge. We will confirm our statement with a rule in the textbook. 7. Primary consolidation 8. Physical education 9. Repetition of what has been studied Writing on the board: m / n \u003d b Select where in the fraction is the whole and parts? How to find the whole? Applying the rule, we solve the equation. part C. 28, task 10. What additional questions can be asked? S. 27, No. 8 - at the blackboard (a, b, c) - 3 students decide. The rest solve in pairs (d) Verification Analysis of the problem. Self-recording solution. Answering questions, they analyze their work in the lesson Summing up the lesson Verbal, analysis 10. Lesson summary: What did you learn in the lesson? Extract the integer part from an improper fraction. Verbally visual What conclusion did you come to? In order to separate the integer part from an improper fraction, divide its numerator by the denominator, the quotient will be the integer part, the remainder the numerator, and the divisor the denominator of the fraction. And now let's check yourself how you learned this. Perform on their own. (mutual check). Information about homework Reflection 11. Homework: C. 26, No. 4 (d, e, f), learn the rule on p. 26 and p. 28 #11 If you think you have understood the topic of today's lesson, then color the piece of paper with a green pencil. what not If you think you have learned enough material in yellow. If you think you did not understand the topic of today's lesson in red. Self-assessment To be able to assess the correctness of the performance of an action at the level of an adequate retrospective assessment. (Regulatory UUD). based on the ability to self-assessment of the criterion for the success of educational activities (Personal UUD).

has a numerator greater than the denominator. Such fractions are called improper.

Remember!

An improper fraction has a numerator equal to or greater than the denominator. That's why improper fraction or equal to one or greater than one.

Any improper fraction is always greater than a proper one.

How to select whole part

An improper fraction can have an integer part. Let's see how this can be done.

To extract the whole part from an improper fraction, you need to:

1. divide the numerator by the denominator with the remainder;
2. the resulting incomplete quotient is written into the integer part of the fraction;
3. the remainder is written in the numerator of the fraction;
4. the divisor is written in the denominator of the fraction.

Example. Separate the integer part from an improper fraction

11

2

.

Remember!

The resulting number above, containing an integer and a fractional part, is called mixed number.

We got a mixed number from an improper fraction, but you can also perform the reverse action, that is represent a mixed number as an improper fraction.

To represent a mixed number as an improper fraction:

1. multiply its integer part by the denominator of the fractional part;
2. add the numerator of the fractional part to the resulting product;
3. write the amount received from paragraph 2 into the numerator of the fraction, and leave the denominator of the fractional part the same.

Example. Let's represent the mixed number as an improper fraction.