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Isolating an integer part from a fraction. What is a numerical fraction? The main property of a fraction

Human physiology [psychology]
28.12.2020

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, in their ability to “blow the mind”, surpass the rest of the algebra course.

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

A number fraction (or just a fraction) is a pair of integers written separated by a slash or a horizontal bar.

Fractions written through a horizontal line:

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

Fractions are usually written through a horizontal line - it’s easier to work with them this way, and they look better. The number written on top is called the numerator of the fraction, and the number written below is called the denominator.

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

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

If the denominator still has a zero, the fraction is called an indefinite fraction. Such a record is meaningless and cannot be used in calculations.

The main property of a fraction

Fractions a /b and c /d are said to be 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, since 1 4 = 2 2. Of course, there are many fractions that are not equal to each other. For example, 1/3 ≠ 5/4, since 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 very important property- remember it. Using the basic property of a fraction, you can simplify and shorten many expressions. In the future, it will constantly “pop up” in the form of various properties and theorems.

Improper fractions. Selecting a whole part

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

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

To isolate the whole part of an 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 (at most, 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 in the numerator of the new fraction;
  3. We rewrite the denominator without changes.

Well, is it difficult? At first glance, it may be difficult. But with a little practice, you will be able to do it almost orally. In the meantime, take a look at the examples:

Task. Select the whole part in the indicated fractions:

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

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

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

Going to an improper fraction

There is also a reverse operation, when we get rid of the whole part. This is called the improper fraction transition and is much more common because working with improper fractions is much easier.

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

  1. Multiply the whole part by the denominator. The result can be quite large numbers, but this should not bother us;
  2. Add the resulting number to the numerator of the original fraction. Write the result in the numerator of the improper fraction;
  3. Rewrite the denominator - again, without changes.

Here are specific examples:

Task. Convert to improper fraction:

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

Consider the case when the numerator or denominator of the 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 place minuses as fraction signs.

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

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

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

We deliberately do not consider the “plus on plus” case - with it, I think, everything is clear. Let's see how these rules work in practice:

Task. Take out the negatives of the four fractions written above.

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

Also, do not move minuses in fractions with the whole part highlighted. These fractions are first converted to improper fractions - and only then do calculations begin.

§ 1 Isolating the whole part from an improper fraction

In this lesson you will learn how to convert an improper fraction into a mixed number by highlighting the whole part, and also vice versa to obtain an improper fraction from a mixed number.

First, let's remember what a mixed number and an improper fraction are.

A mixed number is a special form of writing a number that contains an integer and a fractional part.

An improper fraction is a fraction whose numerator is greater than or equal to its denominator.

Let's consider the problem:

We will divide 8 candies among three children. How much will each person get?

To find out how many candies each child will receive, you need to

But it is not customary to write an improper fraction in the answer. It is first replaced either by an equal natural number (when the numerator is divisible by the denominator), or by the so-called separation of the whole part from the improper fraction (when the numerator is not divisible by the denominator).

Isolating an integer part from an improper fraction is replacing the fraction with an equal mixed number.

To separate the whole part from an improper fraction, you need to divide the numerator by the denominator with a remainder. In this case, the incomplete quotient will be the whole part, the remainder will be the numerator, and the divisor will be the denominator.

Let's return to the task.

So, we divide 8 by 3 with a remainder, we get 2 in the incomplete quotient and 2 in the remainder.

§ 2 Representation of a mixed number as an improper fraction

Let's do the following task:

Divide 49 by 13, we get 3 in the incomplete quotient (this will be the integer part) and the remainder 10 (we will write this in the numerator of the fractional part).

To perform various operations with mixed numbers, the skill of representing mixed numbers as improper fractions is useful. It's time to figure out how such a translation is carried out.

To represent a mixed number as an improper fraction, you need to multiply the denominator of the fraction by the whole part and add the numerator to the resulting product. As a result, we get a number that will be the numerator of the new fraction, and the denominator remains unchanged.

The first step is to multiply the whole part of 5 by the denominator 7, we get 35.

The second step is to add the numerator 4 to the resulting product 35, it will be 39.

Now let's write 39 in the numerator and leave 7 in the denominator.

Thus, in this lesson you learned how to convert an improper fraction into a mixed number; to do this, you need to divide the numerator by the denominator with a remainder. Then the incomplete quotient will be the integer part, the remainder will be the numerator, and the divisor will be the denominator of the fractional part of the mixed number.

You also learned about representing a mixed number as an improper fraction. In order to represent a mixed number as an improper fraction, you need to multiply the denominator of the fractional part of the mixed number by the whole part and add the numerator to the resulting product.

List of used literature:

  1. Mathematics 5th grade. Vilenkin N.Ya., Zhokhov V.I. and others. 31st ed., erased. - M: 2013.
  2. Didactic materials in mathematics 5th grade. Author - Popov M.A. - year 2013
  3. We calculate without errors. Work with self-test in mathematics grades 5-6. Author - Minaeva S.S. - year 2014
  4. Didactic materials for mathematics grade 5. Authors: Dorofeev G.V., Kuznetsova L.V. - 2010
  5. Control and independent work in mathematics 5th grade. Authors - Popov M.A. - year 2012
  6. Mathematics. 5th grade: educational. for general education students. institutions / I. I. Zubareva, A. G. Mordkovich. - 9th ed., erased. - M.: Mnemosyne, 2009

Mathematics lesson in 4th grade topic: Isolating the whole part from an improper fraction Lesson topic: Isolating the whole part from an improper fraction. Didactic goal: to create conditions for the formation of new educational information. Goals and objectives of the lesson: 1. Form the concept of a mixed number. 2. Develop the ability to isolate the whole part from an improper fraction. 3. Develop computing skills. 4. Develop the ability to analyze and solve word problems to find a part of a number and a number from its part. 5. Develop logical thinking students. Planned learning outcomes, formation of UUD: Subject: expand the concept of number, develop skills in converting improper fractions into mixed numbers and apply the acquired knowledge and skills when performing various tasks. Meta-subject: develop the ability to see a mathematical problem in the context of a problem situation in other disciplines, in the surrounding life. Cognitive UUD: develop ideas about number; ability to work with a textbook, additional sources of information (analyze, extract the necessary information); the ability to make generalizations, conclusions, and establish cause-and-effect relationships. Communicative learning activities: cultivate respect for each other, develop the ability to enter into educational dialogue with the teacher, with classmates, observing the norms 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 stages of work, control your actions, detect and correct errors, critically evaluate the results of your work and the work of everyone based on existing criteria, develop the ability to mobilize strength and energy, to overcome obstacles. Personal learning achievements: to form learning motivation, initiative, develop skills of competent oral and written mathematical speech, and the ability to self-assess one’s actions. Resources: multimedia projector, presentation. Lesson type: learning new material. Stage of the lesson Teacher's activity Student's activity Organizational moment Greeting, checking readiness for the lesson, organizing children's attention. . Get involved in the business rhythm of the lesson. Methods, techniques, forms used Verbal Formed UUD Be able to express your 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 class we will continue working on fractions. Guys, during the lesson you should discover new knowledge, but, as you know, every new knowledge is related to what we have already learned. Therefore, we will start with repetition. Oral arithmetic Updating knowledge and skills Practical answers are written down in a column, we check the answers on the slides. pronounce in class Be able to sequence actions (Regulatory UUD). Be able to transform information from one form to another (Cognitive UUD). Be able to express your thoughts orally and in writing (Communicative UUD). Blitz survey: What rules did you use when: 1. Finding the sum of fractions. 2. Find the difference of fractions. 3. Find the number by part. 4. Find the part by number. They tell the rules. Participate in a conversation with the teacher. Be able to express your thoughts orally (Communicative UUD). Be able to navigate your knowledge system: 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. Statement of the problem Verbal Be able to formulate your thoughts orally (Communicative UUD). Be able to navigate. . your knowledge system: distinguish the new from the already known with the help (Cognitive teachers of UUD). Children express their options for solutions. 4. “Formulation of the problem and purpose of the lesson. Select a whole part from this fraction. What do you offer? What do you think is the goal of the lesson? The purpose of the lesson and topic are formulated by the students. Goal: Learn to isolate the whole part from an improper fraction Verbal, practical Be able to acquire new knowledge: find answers to questions using a textbook, your life experience and information received at (Cognitive UUD lesson). Be able to express 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 in a work lesson to be spoken according to Ability to collectively draw up a plan (Regulatory UUD). Know the sequence of actions (Regulatory UUD). Be able to express your thoughts orally and in writing; listen and understand the speech of others (Communicative UUD) Be able to sequence actions (Regulatory UUD). Be able to carry out work according to the proposed plan (Regulatory UUD). talk through the lesson on Acquiring new knowledge and methods of assimilation 5. Discovering something new: Explanation on the board. Write the fraction 16/5 as a quotient. What rule was used to isolate a whole part from an improper fraction? To isolate a whole part from an improper fraction, you need to: divide the numerator by the denominator with the remainder; the resulting incomplete quotient is recorded 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 to self-assess on the criterion of success in educational activities (Personal UUD). based on the whole part of the fraction; write the remainder into the numerator of the fraction; write the divisor into 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 – 1 example with explanation at the board. The rest with comments. No. 4 (a, b, c) – independently. Peer review. m is an integer, n and b are parts. In a fraction, the integer is always the numerator. The guys say the rule: to find a whole you need to multiply 6. Formulation of new knowledge. Let’s confirm our statement with a rule in the textbook. 7. Primary consolidation 8. Physical education lesson 9. Repetition of what has been learned Writing on the board: m/n = b Highlight where in the fraction the whole and the parts? How to find the whole? Applying the rule, we solve the equation. parts P. 28, task 10. What additional questions can be asked? P. 27, No. 8 – at the blackboard (a, b, c) – 3 students decide. The rest solve in pairs (d).Check Analysis of the problem. Self-recording of the 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? Separate the whole part from an improper fraction. Verbally visual What conclusion did you come to? It is necessary to isolate the whole part from an improper fraction; divide its numerator by the denominator, the quotient will be the whole part, the remainder will be the numerator, and the divisor will be the denominator of the fraction. Now let’s test ourselves how you learned this. Do it yourself. (mutual check). Information about homework Reflection 11. Homework: P. 26, No. 4 (d, e, f), learn the rule on p. 26 and p. 28 No. 11 If you think that you understand the topic of today’s lesson, then color the leaflet with a green pencil. what not If you think you have learned enough material in yellow. If you think that you did not understand the topic of today's lesson in red. Self-assessment Be able to assess the correctness of an action at the level of adequate retrospective assessment. (Regulatory UUD). based on the Ability to self-assessment criterion on the success of educational activities (Personal UUD).

How to separate the whole part from an improper fraction? To isolate the whole part from an improper fraction, you must: Divide the numerator by the denominator with the remainder; An incomplete quotient will be a whole part; The remainder (if any) is given by the numerator, and the divisor is the denominator of the fraction. Complete numbers 1057, 1058, 1059, 1060. 1062, 1063. 1064. 7.

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Mixed numbers

“Mathematics lesson notes” - Follow the example. a) 4/7+2/7= (4+2)/7= 6/7 b, c, d (at the board) d) 7/9-2/9= (7-2)/9= 5/ 9 f, g, h (at the board). 12 kg of cucumbers were collected from 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 the subtraction rule. Learning new material:

“Comparing decimal fractions” - The purpose of the lesson. Compare numbers: Mental counting. 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. Rank decimals. Reinforcement lesson in 5th grade.

“Rules for rounding numbers” - 1.8. 48. Well done! 3. 3. Learn to apply the rounding rule using examples. Try to compare. Round whole numbers to the nearest ten. 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.

“Adding mixed numbers” - 25. Example 4. Find the value of the difference 3 4\9-1 5\6. 3 4\9=3 818; 1 5\6=1 15\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 notes in 6th grade

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

Remember!

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

Any improper fraction is always greater than a proper fraction.

How to select an entire part

An improper fraction can have a whole part. Let's look at how this can be done.

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

  1. divide the numerator by the denominator with the remainder;
  2. we write the resulting incomplete quotient into the whole part of the fraction;
  3. write the remainder into the numerator of the fraction;
  4. We write the divisor into the denominator of the fraction.
Example. Select the whole part from the 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 we can also do the opposite operation, 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 resulting amount from step 2 into the numerator of the fraction, and leave the denominator of the fractional part the same.

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