How to find the integer part of a proper fraction. Extraction from a fraction of an integer part online
Sections: Maths
Class: 4
Basic goals:
- To form the ability to isolate the whole part from an improper fraction.
- Revise the concepts of the numerator and denominator, correct and improper fractions, mixed numbers.
- 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:
- Motivate students to learning activities by reinforcing the situation of success achieved in the previous lesson.
- 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 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).
Determine with the help of a number ray what mixed number is a 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 for reverse side 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:
- Organize communicative interaction to identify the distinctive properties of the task to select the whole part from an improper fraction.
- 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 ways. 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:
- Organize communicative interaction to build a new way of action to extract the whole part from an improper fraction.
- Fix 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:
- To organize the independent performance by students of the task of isolating the whole part from an improper fraction.
- Train the ability for self-control and self-esteem.
- Test your ability to isolate the whole part from an improper fraction.
- 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:
- Fix in speech the algorithm for extracting the integer part from an improper fraction.
- Record the remaining difficulties and ways to overcome them.
- Evaluate your own performance in class.
- 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!
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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 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 isolate the whole part in an improper fraction, you need to follow three simple steps:
- 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;
- 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;
- 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:
- Multiply the integer part by the denominator. The result can be quite large numbers, but we should not be embarrassed;
- Add the resulting number to the numerator of the original fraction. Write the result in the numerator of an improper fraction;
- 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 is 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:
- 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;
- "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.
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:
- divide the numerator by the denominator with the remainder;
- the resulting incomplete quotient is written into the integer part of the fraction;
- the remainder is written in the numerator of the fraction;
- the divisor is written in the denominator of the 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:
- multiply its integer part by the denominator of the fractional part;
- add the numerator of the fractional part to the resulting product;
- 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.
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:
- divide the numerator by the denominator with the remainder;
- the resulting incomplete quotient is written into the integer part of the fraction;
- the remainder is written in the numerator of the fraction;
- the divisor is written in the denominator of the 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:
- multiply its integer part by the denominator of the fractional part;
- add the numerator of the fractional part to the resulting product;
- 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.